Water resource systems planning and management : an introduction to methods, models, and applications

Loucks, Daniel P.

Beek, Eelco van

2017

Springer

This revised, updated textbook presents a systems approach to the planning, management, and operation of water resources infrastructure in the environment. Previously published in 2005 by UNESCO and Deltares (Delft Hydraulics at the time), this new edition, written again with contributions from Jery R. Stedinger, Jozef P. M. Dijkman, and Monique T. Villars, is aimed equally at students and professionals. It introduces readers to the concept of viewing issues involving water resources as a system of multiple interacting components and scales. It offers guidelines for initiating and carrying out water resource system planning and management projects. It introduces alternative optimization, simulation, and statistical methods useful for project identification, design, siting, operation and evaluation and for studying post-planning issues. The authors cover both basin-wide and urban water issues and present ways of identifying and evaluating alternatives for addressing multiple-purpose and multi-objective water quantity and quality management challenges. Reinforced with cases studies, exercises, and media supplements throughout, the text is ideal for upper-level undergraduate and graduate courses in water resource planning and management as well as for practicing planners and engineers in the field.

Environmental Technology

English

978-3-319-44234-1

978-3-319-44232-7

Daniel P. Loucks
Eelco van Beek

Water Resource
Systems Planning
and Management
An Introduction to Methods,
Models, and Applications

Water Resource Systems Planning
and Management

Daniel P. Loucks Eelco van Beek
•

Water Resource
Systems Planning
and Management
An Introduction to Methods, Models,
and Applications

With Contributions by
Jery R. Stedinger
Jos P.M. Dijkman
Monique T. Villars

123

Eelco van Beek
Deltares
Delft
The Netherlands

Daniel P. Loucks
Cornell University
Ithaca, NY
USA

ISBN 978-3-319-44232-7
DOI 10.1007/978-3-319-44234-1

ISBN 978-3-319-44234-1

(eBook)

Jointly published with Deltares and UNESCO-IHE
Library of Congress Control Number: 2016950366
© The Editor(s) (if applicable) and The Author(s) 2017. This book is published open access.
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Foreword

Water resources are special. In their natural states
they are beautiful. People like to live and vacation near rivers, lakes and coasts. Water is also
powerful. Water can erode rock, alter existing
landscapes and form new ones. Life on this
planet depends on water. Most of our economic
activities consume water. All the food we grow,
process and eat requires water. Much of our
waste is transported and assimilated by water.
The importance of water to our well-being is
beyond question. Our dependence on water will
last forever.
So, what is the problem? The answer is simply that water, although
plentiful, is not distributed as we might wish. There is often too much or too
little, or what exists is too polluted or too expensive. A further problem is that
the overall water situation is likely to further deteriorate as a result of global
changes. This is a result not only of climatic change but also of other global
change drivers such as population growth, land use changes, urbanization and
migration from rural to urban areas, all of which will pose challenges never
before seen. Water obviously connects all these areas and any change in these
drivers has an impact on it. Water has its own dynamics that are fairly nonlinear. For example, while population growth in the twentieth century
increased threefold—from 1.8 to 6 billion people—water withdrawal during
the same period increased sixfold! That is clearly unsustainable. Freshwater,
although a renewable resource, is finite and is very vulnerable. If one considers all the water on Earth, 97.5 % is located in the seas and oceans and what
is available in rivers, lakes and reservoirs for immediate human consumption
comprises no more than a mere 0.007 % of the total. This is indeed very
limited and on average is roughly equivalent to 42,000 km3 per year.
If one looks at the past 30 years only in terms of reduction in per capita
water availability in a year the picture is even more disturbing. While in 1975
availability stood at around 13,000 m3 per person per year, it has now
dropped to 6000 m3; meanwhile water quality has also severely deteriorated.
While this cannot be extrapolated in any meaningful manner, it nevertheless
indicates the seriousness of the situation. This will likely be further exacerbated by the expected impacts of climate change. Although as yet unproven
v

vi

to the required rigorous standards of scientific accuracy, increasing empirical
evidence indicates that the hydrological cycle is accelerating while the
amount of water at a given moment in time is remains the same. If this
acceleration hypothesis is true then it will cause an increase in the frequency
and magnitude of flooding. At the other end of the spectrum, the prevailing
laws of continuity mean that the severity and duration of drought will also
increase. These increased risks are likely to have serious regional implications. Early simulation studies suggest that wet areas will become even more
humid while dry areas will become increasingly arid. This will not occur
overnight; similarly, appropriate countermeasures will need time to establish
policies that integrate the technical and social issues in a way that takes
appropriate consideration of the cultural context.
Tremendous efforts and political will are needed to substantially reduce
the number of human beings who have no access to safe drinking water and
adequate sanitation facilities respectively. The substantial growth of human
populations—especially as half of humanity already lives in urban areas—
and the consequent expansion of agricultural and industrial activities with a
high water demand, have only served to increase problems of water availability, quality—and in many regions—waterborne disease. There is now an
increasing urgency in the UN system to protect water resources through
better management. Data on the scale of deforestation with subsequent land
use conversion, soil erosion, desertification, urban sprawl, loss of genetic
diversity, climate change and the precariousness of food production through
irrigation, all reveal the growing seriousness of the problem. We have been
forced to recognize that society’s activities can no longer continue unchecked
without causing serious damage to the very environment and ecosystems we
depend upon for our survival. This is especially critical in water scarce
regions, many of which are found in the developing world and are dependent
on water from aquifers that are not being recharged as fast as their water is
being withdrawn and consumed. Such practices are clearly not sustainable.
Proper water resources management requires consideration of both supply
and demand. The mismatch of supply and demand over time and space has
motivated the development of much of the water resources infrastructure that
is in place today. Some parts of the globe witness regular flooding as a result
of monsoons and torrential downpours, while other areas suffer from the
worsening of already chronic water shortages. These conditions are often
aggravated by the increasing discharge of pollutants resulting in a severe
decline in water quality.
The goal of sustainable water management is to promote water use in such
a way that society’s needs are both met to the extent possible now and in the
future. This involves protecting and conserving water resources that will be
needed for future generations. UNESCO’s International Hydrological Programme (IHP) addresses these short- and long-term goals by advancing our
understanding of the physical and social processes affecting the globe’s water
resources and integrating this knowledge into water resources management.
This book describes the kinds of problems water managers can and do face
and the types of models and methods one can use to define and evaluate
alternative development plans and management policies. The information

Foreword

Foreword

vii

derived from these models and methods can help inform stakeholders and
decision-makers alike in their search for sustainable solutions to water
management problems. The successful application of these tools requires
collaboration among natural and social scientists and those in the affected
regions, taking into account not only the water-related problems but also the
social, cultural and environmental values.
On behalf of UNESCO it gives me great pleasure to introduce this book. It
provides a thorough introduction to the many aspects and dimensions of
water resources management and presents practical approaches for analyzing
problems and identifying ways of developing and managing water resources
systems in a changing and uncertain world. Given the practical and academic
experience of the authors and the contributions they have made to our profession, I am confident that this book will become a valuable asset to those
involved in water resources planning and management. I wish to extend our
deepest thanks to Profs. Pete Loucks and Eelco van Beek for offering their
time, efforts and outstanding experience, which is summarized in this book
for the benefit of the growing community of water professionals.
András Szöllösi-Nagy
Past Deputy Assistant Director-General, UNESCO
and Past Secretary, International Hydrological Programme
Past Rector Magnificus, UNESCO-IHE, Delft, The Netherlands

Preface

Water resource systems planning and management issues are rarely simple.
Demands for reliable supplies of clean water to satisfy the energy, food, and
industrial demands of an increasing population and to maintain viable natural
ecosystems are growing. This is happening at the same time changes in our
climate are increasing the risks of having to deal with too little or too much
water in many river basins, watersheds, and urban areas. Societies are
becoming increasingly aware of the importance of water and its management
and use; their governing institutions are becoming increasingly involved in
water resources development and management decision-making processes.
To gain a better understanding of the complex interactions among all the
hydrologic, ecologic, economic, engineering and social components of water
resource systems, analyses based on systems perspectives are useful. While
analyses of such complex systems can be challenging, integrated systems
approaches are fundamental for identifying and evaluating options for
improving system performance and security for the benefit of all of us.
Just how well we are able to plan and manage our water availability,
quality, and variability is a major determinant of the survival of species, the
functioning and resilience of ecosystems, the strength of economies, and the
vitality of societies. To aid in the analysis of planning and managing options,
a variety of modelling approaches have been developed. This book introduces the science and art of developing and applying various modelling
approaches in support of water resources planning and management. Its main
emphasis is on the practice of developing and using models to address
specific water resources planning and management issues and problems.
Their purpose is to provide relevant, objective, timely and meaningful
information to those who are responsible for deciding how we develop,
manage, and use our water resources.
Readers of this book are not likely to learn the art of systems modelling
and analyses unless they actually do it. The modelling approaches, examples
and case studies contained in this book, together with the exercises offered at
the end of most chapters, we believe and hope, will facilitate the process of
becoming a skilled water resources systems modeler, analyst and planner.
This has been our profession, indeed our hobby and source of enjoyment, and
we can highly recommend it to others.
Water resource systems planning and management is a multidisciplinary
activity. The modelling and analysis of water resources systems involves

ix

x

inputs from the applicable natural and social sciences and from the people,
the stakeholders, who will be impacted. It is a challenge.
Although we have attempted to incorporate into each chapter current
approaches to water resources systems planning and analysis, this book does
not pretend to be a review of the state-of-the-art of water resources systems
analysis. Rather it is intended to introduce readers to the art of developing
and using models and modelling approaches applied to the planning and
managing of water resources systems. We have tried to organize our discussion in a way useful for teaching and self-study. The contents reflect our
belief that the most appropriate methods for planning and management are
often the simpler ones, chiefly because they are easier to understand and
explain, require less input data and time, and are easier to apply to specific
issues or problems. This does not imply that more sophisticated and complex
models are less useful. Sometimes their use is the only way one can provide
the needed information.
In this book, we attempt to give readers the knowledge to make appropriate choices regarding model complexity. These choices will depend in part
on factors such as the issues being addressed and the information needed, the
level of accuracy desired, the availability of data and their cost, and the time
required and available to carry out the analysis. While many analysts have
their favourite modelling approaches, the choice of a particular model and
solution method should be based on the knowledge of various modelling
approaches and their advantages and limitations. There is no one best
approach for analyzing all the issues one might face in this profession.
This book assumes readers have had some mathematical training in
algebra, calculus, geometry and the use of vectors and matrices. Readers will
also benefit from some background in probability and statistics and some
exposure to micro-economic theory and welfare economics. Some knowledge of hydrology, hydraulics and environmental engineering will also be
beneficial, but not absolutely essential. Readers wanting an overview of some
of natural processes that take place in watersheds, river basins, estuaries and
coastal zones can refer to the Appendices (available on the internet along
with the book itself). An introductory course in optimization and simulation
methods, typically provided in either an operations research or an economic
theory course, can also benefit the reader, but again it is not essential.
Chapter 1 introduces water resources systems planning and management
and reviews some examples of water resources systems projects in which
modelling has had a critical role. These projects also serve to identify some
of the current issues facing water managers in different parts of the world.
Chapter 2 introduces the general modelling approach and the role of models
in water resources planning and management activities. Chapter 3 begins the
discussion of optimization and simulation modelling and how they are
applied and used in practice. Chapter 4 focuses on the development and use
of various optimization methods for the preliminary definition of infrastructure design and operating policies. These preliminary results define
alternatives that usually need to be further analyzed and improved using
simulation methods. The advantages and limitations of different

Preface

Preface

xi

optimization/simulation approaches are illustrated using some simple water
allocation, reservoir operation and water quality management problems.
Chapter 5 extends this discussion of optimization to problems characterized by more qualitative objectives and/or constraints. In addition, it introduces some of the more recently developed methods of statistical modelling,
including artificial neural networks and evolutionary search methods
including genetic algorithms and genetic programming. This chapter expects
interested readers desiring more detail will refer to other books and papers,
many of which are solely devoted to just these topics. Chapters 6 through 8
are devoted to probabilistic models, uncertainty and sensitivity analyses.
These methods are useful not only for identifying more realistic, reliable, and
robust infrastructure designs and operating policies for the given hydrological variability and uncertain parameter values and objectives but also for
estimating some of the major uncertainties associated with model predictions.
Such probabilistic and stochastic models can also help identify just what
model input data are needed and how accurate those data need be with
respect to their influence on the decisions being considered.
Water resources planning and management today inevitably involve
multiple goals or objectives, many of which may be conflicting. It is difficult,
if not impossible, to please all stakeholders all the time. Models containing
multiple objectives can be used to identify the tradeoffs among conflicting
objectives. This is the information useful to decision-makers who must
decide what to do given these tradeoffs among conflicting performance criteria that stakeholders care about. Chapter 9 on multi-objective modelling
identifies various types of economic, environmental and physical objectives,
and some commonly used ways of including multiple objectives in optimization and simulation models.
Chapter 10 is devoted to various approaches for modelling water quality
in surface water bodies. Chapter 11 focuses on modelling approaches for
multiple purpose water quantity planning and management in river basins.
Chapter 12 zooms into urban areas and presents some ways of analyzing
urban water systems. Finally, Chap. 13 describes how projects involving the
analyses of water resource systems can be planned and executed.
Following these thirteen chapters are four appendices. They are not
contained in the book but are available on the internet where this book can be
downloaded. They contain descriptions of (A) natural hydrological and
ecological processes in river basins, estuaries and coastal zones, (B) monitoring and adaptive management, (C) drought management, and (D) flood
management.
For university teachers, the contents of this book represent more than can
normally be covered in a single quarter or semester course. A first course
might include Chaps. 1 through 5, and possibly Chaps. 9 and 10 or 11 or 12
or 13 depending on the background and interest of the participants in the
class. A second course could include Chaps. 6 through 8 and/or any combination of Chaps. 10 through 12, as desired. Exercises are offered at the end
of each chapter, and instructors using this text in their academic courses can
contact the authors for the solutions of those exercises if desired.

xii

Preface

Many have helped us prepare this book. Jery Stedinger contributed to
Chaps. 6, 7 and 8, Nicki Villars helped substantially with Chap. 10, and Jozef
Dijkman contributed a major portion related to flood management. Tjitte
Nauta, Laura Basco Carrera and Thijs Stoffelen contributed to Chap. 13.
Others who offered advice and who helped review earlier chapter drafts
include Vladan Babovic, Martin Baptist, Henk van den Boogaard, Herman
Breusers, Harm Duel, Herman Gerritsen, Peter Gijsbers, Jos van Gils, Simon
Groot, Karel Heynert, Joost Icke, Hans Los, Marcel Marchand, Tony Minns,
Erik Mosselman, Arthur Mynett, Roland Price, Erik Ruijgh, Johannes Smits,
Mindert de Vries and Micha Werner. Engelbert Vennix and Hans van Bergem created most of the figures and tables in this book. We again thank
Deltares and all these individuals and others who provided assistance and
support on various aspects during the entire time in 2005 and when this
second edition was being prepared.
We have also benefited from the comments of Profs. Jan-Tai Kuo at
National Taiwan University in Taipei, Jay Lund at the University of
California at Davis, Daene McKinney of the University of Texas in Austin,
Peter Rogers at Harvard University in Cambridge, MA, Tineke Ruijgh-van
der Ploeg at TU-Delft, Robert Traver at Villanova University in Philadelphia,
and Jinwen Wang at Huazhong University of Science and Technology in
Wuhan, all of whom have used earlier drafts of this book in their classes.
Finally we acknowledge with thanks the critical support of Andras Szöllösi-Nagy, recently retired as rector of UNESCO-IHE in Delft, NL, and the
publishing staff at UNESCO for publishing the first edition of this book and
making its electronic version free and unrestricted. We are especially grateful
to Michael Luby of Springer for all his assistance and guidance, and to
Deltares and UNESCO-IHE in Delft, NL, for sharing the cost of publishing
this second edition. We have written this book for an international audience,
and hence we are especially grateful and pleased to have this connection to
and support from Deltares and UNESCO-IHE.
Most importantly, we wish to acknowledge and thank all our teachers,
students and colleagues throughout the world who have taught us all we
know and added to the quality of our professional and personal lives. We
have tried our best to make this book error free, but inevitably somewhere
there will be flaws. For that, we apologize and take responsibility for any
errors of fact, judgment or science that may be contained in this book. We
will be most grateful if you let us know of any or have other suggestions for
improving this book.
Ithaca, NY, USA
Delft, The Netherlands
July 2016

Daniel P. Loucks
Eelco van Beek

Contents

1

Water Resources Planning and Management:
An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Planning and Management Issues: Some Case
Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Kurds Seek Land, Turks Want Water. . . . . .
1.2.2 Sharing the Water of the Jordan River
Basin: Is There a Way? . . . . . . . . . . . . . . .
1.2.3 Mending the “Mighty and Muddy” Missouri.
1.2.4 The Endangered Salmon . . . . . . . . . . . . . .
1.2.5 Wetland Preservation: A Groundswell
of Support and Criticism . . . . . . . . . . . . . .
1.2.6 Lake Source Cooling: Aid to Environment,
or Threat to Lake?. . . . . . . . . . . . . . . . . . .
1.2.7 Managing Water in the Florida Everglades . .
1.2.8 Restoration of Europe’s Rivers and Seas . . .
1.2.9 Flood Management on the Senegal River . . .
1.2.10 Nile Basin Countries Striving to Share
Its Benefits . . . . . . . . . . . . . . . . . . . . . . . .
1.2.11 Shrinking Glaciers at Top of the World . . . .
1.2.12 China, a Thirsty Nation . . . . . . . . . . . . . . .
1.2.13 Managing Sediment in China’s
Yellow River . . . . . . . . . . . . . . . . . . . . . .
1.2.14 Damming the Mekong (S.E. Asia),
the Amazon, and the Congo . . . . . . . . . . . .
1.3
So, Why Plan, Why Manage?. . . . . . . . . . . . . . . . .
1.3.1 Too Little Water . . . . . . . . . . . . . . . . . . . .
1.3.2 Too Much Water. . . . . . . . . . . . . . . . . . . .
1.3.3 Too Polluted. . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Too Expensive . . . . . . . . . . . . . . . . . . . . .
1.3.5 Ecosystem Too Degraded . . . . . . . . . . . . . .
1.3.6 Other Planning and Management Issues . . . .
1.4
System Planning Scales . . . . . . . . . . . . . . . . . . . . .
1.4.1 Spatial Scales for Planning
and Management. . . . . . . . . . . . . . . . . . . .
1.4.2 Temporal Scales for Planning
and Management. . . . . . . . . . . . . . . . . . . .

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Contents

1.5

2

3

Planning and Management Approaches . . . . . . . . .
1.5.1
Top-Down Planning and Management . . . .
1.5.2
Bottom-Up Planning and Management . . . .
1.5.3
Integrated Water Resources Management . .
1.5.4
Water Security and the Sustainable
Development Goals (SDGs) . . . . . . . . . . .
1.5.5
Planning and Management Aspects . . . . . .
1.6
Planning and Management Characteristics . . . . . . .
1.6.1
Integrated Policies and Development Plans .
1.6.2
Sustainability . . . . . . . . . . . . . . . . . . . . .
1.7
Meeting the Planning and Management
Challenges—A Summary . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Water Resource Systems Modeling: Its Role
in Planning and Management . . . . . . . . . . . . . . . . . . . .
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Modeling Water Resource Systems . . . . . . . . . . . .
2.2.1 An Example Modeling Approach . . . . . . .
2.2.2 Characteristics of Problems to be Modeled .
2.3
Challenges Involving Modeling. . . . . . . . . . . . . . .
2.3.1 Challenges of Planners and Managers . . . .
2.3.2 Challenges of Modelers . . . . . . . . . . . . . .
2.3.3 Challenges of Applying Models in Practice
2.3.4 Evaluating Modeling Success . . . . . . . . . .
2.4
Developments in Modeling. . . . . . . . . . . . . . . . . .
2.4.1 Technology . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Interactive Model-Building Environments . .
2.4.4 Open Modeling Systems . . . . . . . . . . . . .
2.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Models for Identifying and Evaluating Alternatives
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Model Components . . . . . . . . . . . . .
3.2
Plan Formulation and Selection. . . . . . . . . . .
3.2.1 Plan Formulation. . . . . . . . . . . . . . .
3.2.2 Plan Selection . . . . . . . . . . . . . . . . .
3.3
Conceptual Model Development . . . . . . . . . .
3.4
Simulation and Optimization . . . . . . . . . . . .
3.4.1 Simulating a Simple Water Resources
System. . . . . . . . . . . . . . . . . . . . . .
3.4.2
Defining What to Simulate . . . . . . . .
3.4.3
Simulation Versus Optimization . . . .
3.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4

5

An Introduction to Optimization Models and Methods .
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Comparing Time Streams of Economic Benefits
and Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Interest Rates . . . . . . . . . . . . . . . . . . . .
4.2.2 Equivalent Present Value . . . . . . . . . . . .
4.2.3 Equivalent Annual Value . . . . . . . . . . . .
4.3
Nonlinear Optimization Models and Solution
Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Solution Using Calculus. . . . . . . . . . . . .
4.3.2 Solution Using Hill Climbing . . . . . . . . .
4.3.3 Solution Using Lagrange Multipliers . . . .
4.4
Dynamic Programming . . . . . . . . . . . . . . . . . . .
4.4.1 Dynamic Programming Networks
and Recursive Equations . . . . . . . . . . . .
4.4.2 Backward-Moving Solution Procedure . . .
4.4.3 Forward-Moving Solution Procedure . . . .
4.4.4 Numerical Solutions . . . . . . . . . . . . . . .
4.4.5 Dimensionality . . . . . . . . . . . . . . . . . . .
4.4.6 Principle of Optimality. . . . . . . . . . . . . .
4.4.7 Additional Applications . . . . . . . . . . . . .
4.4.8 General Comments on Dynamic
Programming . . . . . . . . . . . . . . . . . . . .
4.5
Linear Programming . . . . . . . . . . . . . . . . . . . . .
4.5.1 Reservoir Storage Capacity-Yield Models.
4.5.2 A Water Quality Management Problem . .
4.5.3 A Groundwater Supply Example . . . . . . .
4.5.4 A Review of Linearization Methods . . . .
4.6
A Brief Review . . . . . . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data-Fitting, Evolutionary, and Qualitative Modeling.
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Artificial Neural Networks . . . . . . . . . . . . . . . .
5.2.1 The Approach. . . . . . . . . . . . . . . . . . .
5.2.2 An Example . . . . . . . . . . . . . . . . . . . .
5.3
Evolutionary Algorithms . . . . . . . . . . . . . . . . .
5.3.1 Genetic Algorithms . . . . . . . . . . . . . . .
5.3.2 Example Iterations. . . . . . . . . . . . . . . .
5.3.3 Differential Evolution . . . . . . . . . . . . .
5.3.4 Covariance Matrix Adaptation Evolution
Strategy . . . . . . . . . . . . . . . . . . . . . . .
5.4
Genetic Programming . . . . . . . . . . . . . . . . . . .
5.5
Qualitative Functions and Modeling . . . . . . . . .
5.5.1 Linguistic Functions . . . . . . . . . . . . . .
5.5.2 Membership Functions . . . . . . . . . . . . .
5.5.3 Illustrations of Qualitative Modeling . . .

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xvi

Contents

5.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6

7

An Introduction to Probability, Statistics,
and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Probability Concepts and Methods . . . . . . . . . . .
6.2.1 Random Variables and Distributions . . . .
6.2.2 Expected Values . . . . . . . . . . . . . . . . . .
6.2.3 Quantiles, Moments, and Their Estimators
6.2.4 L-Moments and Their Estimators . . . . . .
6.3
Distributions of Random Events . . . . . . . . . . . . .
6.3.1 Parameter Estimation . . . . . . . . . . . . . . .
6.3.2 Model Adequacy. . . . . . . . . . . . . . . . . .
6.3.3 Normal and Lognormal Distributions . . . .
6.3.4 Gamma Distributions . . . . . . . . . . . . . . .
6.3.5 Log-Pearson Type 3 Distribution. . . . . . .
6.3.6 Gumbel and GEV Distributions. . . . . . . .
6.3.7 L-Moment Diagrams . . . . . . . . . . . . . . .
6.4
Analysis of Censored Data . . . . . . . . . . . . . . . . .
6.5
Regionalization and Index-Flood Method . . . . . . .
6.6
Partial Duration Series . . . . . . . . . . . . . . . . . . . .
6.7
Stochastic Processes and Time Series . . . . . . . . .
6.7.1 Describing Stochastic Processes . . . . . . .
6.7.2 Markov Processes and Markov Chains . . .
6.7.3 Properties of Time Series Statistics . . . . .
6.8
Synthetic Streamflow Generation . . . . . . . . . . . .
6.8.1 Introduction . . . . . . . . . . . . . . . . . . . . .
6.8.2 Streamflow Generation Models . . . . . . . .
6.8.3 A Simple Autoregressive Model . . . . . . .
6.8.4 Reproducing the Marginal Distribution. . .
6.8.5 Multivariate Models. . . . . . . . . . . . . . . .
6.8.6 Multiseason, Multisite Models . . . . . . . .
6.9
Stochastic Simulation . . . . . . . . . . . . . . . . . . . .
6.9.1 Generating Random Variables. . . . . . . . .
6.9.2 River Basin Simulation . . . . . . . . . . . . .
6.9.3 The Simulation Model . . . . . . . . . . . . . .
6.9.4 Simulation of the Basin . . . . . . . . . . . . .
6.9.5 Interpreting Simulation Output . . . . . . . .
6.10
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modeling Uncertainty . . . . . . . . . . . . . . . . . . . .
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . .
7.2
Generating Values from Known Probability
Distributions. . . . . . . . . . . . . . . . . . . . . .
7.3
Monte Carlo Simulation. . . . . . . . . . . . . .

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xvii

7.4
7.5
7.6

Chance Constrained Models . . . . . . . . . . . . .
Markov Processes and Transition Probabilities
Stochastic Optimization . . . . . . . . . . . . . . . .
7.6.1 Probabilities of Decisions . . . . . . . . .
7.6.2 A Numerical Example . . . . . . . . . . .
7.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8

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308
311
316
317
327
327
328

System Sensitivity and Uncertainty Analysis . . . . . . . .
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Issues, Concerns, and Terminology . . . . . . . . . . .
8.3
Variability and Uncertainty in Model Output . . . .
8.3.1 Natural Variability. . . . . . . . . . . . . . . . .
8.3.2 Knowledge Uncertainty . . . . . . . . . . . . .
8.3.3 Decision Uncertainty . . . . . . . . . . . . . . .
8.4
Sensitivity and Uncertainty Analyses . . . . . . . . . .
8.4.1 Uncertainty Analyses . . . . . . . . . . . . . . .
8.4.2 Sensitivity Analyses. . . . . . . . . . . . . . . .
8.5
Performance Indicator Uncertainties. . . . . . . . . . .
8.5.1 Performance Measure Target Uncertainty .
8.5.2 Distinguishing Differences Between
Performance Indicator Distributions . . . . .
8.6
Communicating Model Output Uncertainty. . . . . .
8.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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370
371
373

Performance Criteria . . . . . . . . . . . . . . . . . . . . . . .
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Informed Decision-Making . . . . . . . . . . . . . . .
9.3
Performance Criteria and General Alternatives .
9.3.1 Constraints on Decisions . . . . . . . . . .
9.3.2 Tradeoffs Among Performance Criteria
9.4
Quantifying Performance Criteria . . . . . . . . . .
9.4.1 Economic Criteria . . . . . . . . . . . . . . .
9.4.2 Environmental Criteria . . . . . . . . . . . .
9.4.3 Ecological Criteria. . . . . . . . . . . . . . .
9.4.4 Social Criteria. . . . . . . . . . . . . . . . . .
9.5
Multicriteria Analyses . . . . . . . . . . . . . . . . . .
9.5.1 Dominance . . . . . . . . . . . . . . . . . . . .
9.5.2 The Weighting Method . . . . . . . . . . .
9.5.3 The Constraint Method . . . . . . . . . . .
9.5.4 Satisficing . . . . . . . . . . . . . . . . . . . .
9.5.5 Lexicography . . . . . . . . . . . . . . . . . .
9.5.6 Indifference Analysis . . . . . . . . . . . . .
9.5.7 Goal Attainment . . . . . . . . . . . . . . . .

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xviii

Contents

9.5.8 Goal Programming . . . . . . . . . . . . .
9.5.9 Interactive Methods . . . . . . . . . . . . .
9.5.10 Plan Simulation and Evaluation. . . . .
9.6
Statistical Summaries of Performance Criteria .
9.6.1 Reliability . . . . . . . . . . . . . . . . . . .
9.6.2 Resilience. . . . . . . . . . . . . . . . . . . .
9.6.3 Vulnerability. . . . . . . . . . . . . . . . . .
9.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10

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409
409
409
410
411
411

Water Quality Modeling and Prediction . . . . . . . . .
10.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
10.2
Establishing Ambient Water Quality Standards .
10.2.1 Water Use Criteria. . . . . . . . . . . . . . .
10.3
Water Quality Model Use . . . . . . . . . . . . . . .
10.3.1 Model Selection Criteria. . . . . . . . . . .
10.3.2 Model Chains . . . . . . . . . . . . . . . . . .
10.3.3 Model Data . . . . . . . . . . . . . . . . . . .
10.4
Models of Water Quality Processes . . . . . . . . .
10.4.1 Mass Balance Principles. . . . . . . . . . .
10.4.2 Steady-State Models . . . . . . . . . . . . .
10.4.3 Design Streamflows for Setting
and Evaluating Quality Standards . . . .
10.4.4 Temperature . . . . . . . . . . . . . . . . . . .
10.4.5 Sources and Sinks . . . . . . . . . . . . . . .
10.4.6 First-Order Constituents . . . . . . . . . . .
10.4.7 Dissolved Oxygen . . . . . . . . . . . . . . .
10.4.8 Nutrients and Eutrophication. . . . . . . .
10.4.9 Toxic Chemicals . . . . . . . . . . . . . . . .
10.4.10 Sediments. . . . . . . . . . . . . . . . . . . . .
10.4.11 Processes in Lakes and Reservoirs. . . .
10.5
Simulation Methods . . . . . . . . . . . . . . . . . . .
10.5.1 Numerical Accuracy . . . . . . . . . . . . .
10.5.2 Traditional Approach . . . . . . . . . . . . .
10.5.3 Backtracking Approach . . . . . . . . . . .
10.5.4 Model Uncertainty. . . . . . . . . . . . . . .
10.6
Conclusions—Implementing a Water Quality
Management Policy . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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River
11.1
11.2
11.3
11.4

Basin Modeling . . . . . .
Introduction . . . . . . . .
Model Time Periods . .
Streamflow Estimation.
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11.5

Lakes and Reservoirs. . . . . . . . . . . . . . . . . . . . .
11.5.1 Estimating Active Storage Capacity . . . . .
11.5.2 Reservoir Storage-Yield Functions. . . . . .
11.5.3 Evaporation Losses . . . . . . . . . . . . . . . .
11.5.4 Over- and Within-Year Reservoir Storage
and Yields . . . . . . . . . . . . . . . . . . . . . .
11.6
Drought and Flood Risk Reduction . . . . . . . . . . .
11.6.1 Drought Planning and Management . . . . .
11.6.2 Flood Protection and Damage Reduction .
11.7
Hydroelectric Power Production . . . . . . . . . . . . .
11.8
Withdrawals and Diversions . . . . . . . . . . . . . . . .
11.9
Lake-Based Recreation . . . . . . . . . . . . . . . . . . .
11.10 Model Synthesis . . . . . . . . . . . . . . . . . . . . . . . .
11.11 Project Scheduling . . . . . . . . . . . . . . . . . . . . . .
11.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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473
474
476
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479
489
489
490
502
504
505
506
511
515
515
516

Water Systems . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
Water Treatment . . . . . . . . . . . . . . . . . . . . . . .
Water Distribution. . . . . . . . . . . . . . . . . . . . . .
12.3.1 Open-Channel Networks . . . . . . . . . . .
12.3.2 Pressure Pipe Networks . . . . . . . . . . . .
12.3.3 Water Quality . . . . . . . . . . . . . . . . . . .
12.4
Wastewater Collection . . . . . . . . . . . . . . . . . . .
12.4.1 Sewer Networks . . . . . . . . . . . . . . . . .
12.5
Wastewater Treatment . . . . . . . . . . . . . . . . . . .
12.6
Urban Drainage Systems . . . . . . . . . . . . . . . . .
12.6.1 Rainfall . . . . . . . . . . . . . . . . . . . . . . .
12.6.2 Runoff . . . . . . . . . . . . . . . . . . . . . . . .
12.6.3 Surface Pollutant Loading and Washoff .
12.6.4 Water Quality Impacts . . . . . . . . . . . . .
12.6.5 Green Urban Infrastructure . . . . . . . . . .
12.7
Urban Water System Modeling . . . . . . . . . . . . .
12.7.1 Optimization. . . . . . . . . . . . . . . . . . . .
12.7.2 Simulation . . . . . . . . . . . . . . . . . . . . .
12.8
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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527
527
529
531
533
533
535
536
536
537
539
539
541
549
553
558
558
558
560
561
561
564

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12

Urban
12.1
12.2
12.3

13

Project Planning: Putting It All Together . . . . . . . . . . .
13.1
Water Management Challenges . . . . . . . . . . . . . . .
13.2
Water Resources System Components, Functions,
and Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.1 Components . . . . . . . . . . . . . . . . . . . . . .
13.2.2 Functions . . . . . . . . . . . . . . . . . . . . . . . .
13.2.3 Goals, Strategies, Decisions, and Scenarios.
13.2.4 Systems Approaches to WRS Planning
and Decision Making. . . . . . . . . . . . . . . .

. . . 567
. . . 567
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568
568
569
570

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xx

Contents

13.3

Conceptual Description of WRS . . . . . . . . . . . .
13.3.1 Characteristics of the Natural Resources
System. . . . . . . . . . . . . . . . . . . . . . . .
13.3.2 Characteristics of the Socioeconomic
System. . . . . . . . . . . . . . . . . . . . . . . .
13.3.3 Characteristics of the Administrative
and Institutional System . . . . . . . . . . . .
13.4
Framework for Analysis and Implementation . . .
13.4.1 Step I—Inception Phase . . . . . . . . . . . .
13.4.2 Step II—Situation Analysis. . . . . . . . . .
13.4.3 Step III—Strategy Building . . . . . . . . .
13.4.4 Steps IV and V—Action Planning
and Implementation . . . . . . . . . . . . . . .
13.5
Making It Work . . . . . . . . . . . . . . . . . . . . . . .
13.5.1 Stakeholder Engagement . . . . . . . . . . .
13.5.2 Using Models in a Planning Process . . .
13.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 572
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575
576
579
584
592

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596
600
600
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609
614

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

617

1

Water Resources Planning
and Management: An Overview

1.1

Introduction

Water resource systems have benefited both
people and their economies for many centuries.
The services provided by such systems are
multiple. Yet in many regions of the world they
are not able to meet even basic drinking water
and sanitation needs. Nor can many of these
water resource systems support and maintain
resilient biodiverse ecosystems. Typical causes
include inappropriate, inadequate and/or degraded infrastructure, excessive withdrawals of river
flows, pollution from industrial and agricultural
activities, eutrophication resulting from nutrient
loadings, salinization from irrigation return
flows, infestations of exotic plant and animals,
excessive fish harvesting, flood plain and habitat
alteration from development activities, and
changes in water and sediment flow regimes. The
inability of water resource systems to meet the
diverse needs for water often reflect failures in
planning, management, and decision-making—
and at levels broader than water. Planning,
developing, and managing water resources to
ensure adequate, inexpensive, and sustainable
supplies and qualities of water for both humans
and natural ecosystems can only succeed if we
recognize and address the causal socioeconomic
factors, such as inadequate education, corruption,
population pressures, and poverty.
Over the centuries, surface and ground waters
have been a source of water supply for agricultural,
municipal, and industrial consumers. Rivers have
provided hydroelectric energy and inexpensive

ways of transporting bulk cargo. They have provided people water-based recreational opportunities and have been a source of water for wildlife
and their habitats. They have also served as a
means of transporting and transforming waste
products that are discharged into them. The
quantity and quality regimes of streams and rivers
have been a major factor in governing the type,
health, and biodiversity of riparian and aquatic
ecosystems. Floodplains have provided fertile
lands for agricultural crop production and relatively flat lands for the siting of roads and railways
and commercial and industrial complexes. In
addition to the economic benefits that can be
derived from rivers and their floodplains, the aesthetic beauty of most natural rivers has made lands
adjacent to them attractive sites for residential and
recreational development. Rivers and their floodplains have generated, and, if managed properly,
can continue to generate, substantial cultural,
economic, environmental, and social benefits for
their inhabitants.
Human activities undertaken to increase the
benefits obtained from rivers and their floodplains may also increase the potential for costs
and damages such as when the river is experiencing periods of droughts, floods, and heavy
pollution. These costs and damages are physical,
economic, environmental, and social. They result
because of a mismatch between what humans
expect or demand, and what nature offers or
supplies. Human activities tend to be based on
the “usual or normal” range of river flow conditions. Rare or “extreme” flow conditions

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_1

1

2

1 Water Resources Planning and Management: An Overview

outside these normal ranges will continue to
occur, and possibly with increasing frequency as
climate change experts suggest. River-dependent
human activities that cannot adjust to these
extreme flow conditions will incur losses.
The planning of human activities involving
rivers and their floodplains must consider certain
hydrologic facts. One of these facts is that surface water flows and aquifer storage volumes
vary over space and time. They are also finite.
There are limits to the amounts of water that can
be withdrawn from them. There are also limits to
the amounts of pollutants that can be discharged
into them. Once these limits are exceeded, the
concentrations of pollutants in these waters may
reduce or even eliminate the benefits that could
be obtained from other users of the resource.
Water resources professionals have learned
how to plan, design, build, and operate structures
that together with nonstructural measures
increase the benefits people can obtain from the
water resources contained in aquifers, lakes,
rivers, and estuaries. However, there is a limit to
the services one can expect from these resources.
Rivers, estuaries, and coastal zones under stress
from over development and overuse cannot reliably meet the expectations of those depending on
them. How can these resources best be managed
and used? How can this be accomplished in an
environment of uncertain and varying supplies
and uncertain and increasing demands, and consequently of increasing conflicts among individuals having different interests in their
management and use? The central purpose of
water resources planning, management, and
analysis activities is to address, and if possible
answer, these questions. These questions have
scientific, technical, political (institutional), and
social dimensions. Thus water resources planning processes and products are must.
River basin, estuarine, and coastal zone managers—those responsible for managing the
resources in those areas—are expected to manage
those resources effectively and efficiently, meeting the demands or expectations of all users, and
reconciling divergent needs. This is no small task,
especially as demands increase, as the variability

of hydrologic and hydraulic processes become
more pronounced, and as stakeholder expectations of system performance increase in complexity. The focus or goal is no longer simply to
maximize economic net benefits while making
sure the distribution of those benefits is equitable.
There are also environmental and ecological
goals to consider. Rarely are management questions one-dimensional, such as how can we provide, at acceptable costs, more high-quality water
to municipalities, industry, or to irrigation areas in
the basin. Now added to that question is how
would those withdrawals affect the downstream
hydrologic water quantity and quality regimes,
and in turn the riparian and aquatic ecosystems.
Problems and opportunities change over time.
Just as the goals of managing and using water
change over time, so do the processes of planning to meet these changing goals. Planning
processes evolve not only to meet new demands,
expectations, and objectives, but also in response
to new perceptions of how to plan and manage
more effectively.
This chapter reviews some of the issues
requiring water resources planning and management. It provides some context and motivation
for the following chapters that outline in more
detail our understanding of “how to plan” and
“how to manage” and how computer-based programs and models can assist those involved in
these activities. Additional information is available in many of the references listed at the end of
this chapter.

1.2

Planning and Management
Issues: Some Case Studies

Managing water resources certainly requires
knowledge of the relevant physical sciences and
technology. But at least as important, if not more
so, are the multiple institutional, social, or
political issues confronting water resources
planners and managers. The following brief
descriptions of some water resources planning
and management studies at various geographic
scales illustrate some of these issues.

1.2 Planning and Management Issues: Some Case Studies

1.2.1 Kurds Seek Land, Turks Want
Water
The Tigris and Euphrates Rivers (Fig. 1.1) created the “Fertile Crescent” where some of the
first civilizations emerged. Today their waters are
critical resources, politically as well as geographically. In one of the world’s largest public
works undertakings, Turkey’s Southeast Anatolia
Project includes 13 irrigation and hydropower
schemes, and the construction of 22 dams and 19
hydroelectric power plants on both the Tigris and
the Euphrates. Upon completion, it is expected to
provide up to 25% of the country’s electricity.
Its centerpiece, the Ataturk Dam (Fig. 1.2) on
the Euphrates River, is already completed. In the
lake formed behind the dam, sailing and swimming competitions are being held on a spot
where for centuries there was little more than
desert (Fig. 1.3).
When the multireservoir project is completed
it is expected to increase the amount of irrigated
land in Turkey by 40% and provide up to a

3

quarter of the country’s electric power needs.
Planners hope this can improve the standard of
living of six million of Turkey’s poorest people,
most of the Kurds, and thus undercut the appeal
of revolutionary separatism. It will also reduce
the amount of water Syria and Iraq believe they
need—water that Turkey fears might ultimately
be used in anti-Turkish causes.
The region of Turkey where Kurd’s predominate is more or less the same region covered by
the Southeast Anatolia Project, encompassing an
area about the size of Austria. Giving that region
autonomy by placing it under Kurdish self-rule
could weaken the central Government’s control
over the water resource that it recognizes as a
keystone of its future power.
In other ways also, Turkish leaders are using
their water as a tool of foreign as well as domestic
policy. Among their most ambitious projects
considered is a 50-mile undersea pipeline to carry
water from Turkey to the parched Turkish enclave
on northern Cyprus. The pipeline, if actually
built, will carry more water than northern Cyprus

Fig. 1.1 The Tigris and Euphrates Rivers in Turkey, northern Syria, and Iraq

4

1 Water Resources Planning and Management: An Overview

Fig. 1.2 Ataturk Dam on the Euphrates River in Turkey (DSI)

Fig. 1.3 Water sports on Ataturk Reservoir on the Euphrates River in Turkey (DSI)

1.2 Planning and Management Issues: Some Case Studies

can use. Foreign mediators, frustrated by their
inability to break the political deadlock on
Cyprus, are hoping that the excess water can be
sold to the ethnic Greek republic on the southern
part of the island as a way of promoting peace.
As everyone knows, the Middle East is currently (2016) witnessing considerable turmoil so
who knows the fate of any water resources project
in this region, including the one just described in
Turkey and the following example in Jordan. One
can only hope that the management and use of
this scarce resource will lead to more peaceful
resolutions of conflicts not only involving water
but of other political issues as well.

1.2.2 Sharing the Water of the
Jordan River Basin: Is
There a Way?
A growing population—approximately 12 million people—and intense economic development
in the Jordan River Basin (Fig. 1.4) are placing
heavy demands on its scarce freshwater resources. This largely arid region receives less than
250 mm of rainfall each year, yet total water use
for agricultural and economic activities has been
steadily increasing. This plus encroaching urban
development have degraded many sources of
high-quality water in the region.
The combined diversions by the riparian water
users have changed the river in its lower course
into little better than a sewage ditch. From the
1300 million cubic meters (mcm) of water that
flowed into the Dead Sea in the 1950s only a
small fraction remains at present. In normal years
the flow downstream from Lake Tiberias (also
called the Sea of Galilee or Lake Kinneret) is
some 60 million cubic meters (mcm)—about
10% of the natural discharge in this section. It
mostly consists of saline springs and sewage
water. These flows are then joined by what
remains of the Yarmouk, by some irrigation
return flows, and by winter runoff, adding up to
an annual total of from 200–300 mcm. Both in
quantity and quality this water is unsuitable for
irrigation and does not sufficiently supply natural
systems either. The salinity of the Jordan River

5

reaches up to 2000 parts per million (ppm) in the
lowest section, which renders it unfit for crop
irrigation. Only in flood years is fresh water
released into the lower Jordan Valley.
One result of this increased pressure on freshwater resources is the deterioration of the region’s
wetlands. These wetlands are important for water
purification and flood and erosion control. As
agricultural activities expand, wetlands are being
drained, and rivers, aquifers, lakes, and streams
are being polluted with runoff containing fertilizers and pesticides. Reversing these trends by
preserving natural ecosystems is essential to the
future availability of fresh water in the region.
To ensure that an adequate supply of fresh,
high-quality water is available for future generations, Israel, Jordan, and the Palestinian
Authority will have to work together to preserve
aquatic ecosystems (White et al. 1999). Without
these natural ecosystems, it will be difficult and
expensive to sustain high-quality water supplies.
The role of ecosystems in sustaining water supplies has largely been overlooked in the context
of the region’s water supplies. Vegetation controls storm water runoff and filters polluted water,
and it reduces erosion and the amount of sediment that makes its way into water supplies.
Streams assimilate wastewater, lakes store clean
water, and surface waters provide habitat for
many plants and animals.
The Jordan River Basin just like most river
basins should be evaluated and managed as a
whole system, to permit the comprehensive
assessment of the effects of water management
options on wetlands, lakes, the lower river, and
the Dead Sea coasts. Damage to ecosystems and
loss of animal and plant species should be
weighed against the potential benefits of developing land and creating new water resources. For
example, large river-management projects that
divert water to dry areas have promoted intensive
year-round farming and urban development, but
available river water is declining and becoming
increasingly polluted. Attempting to meet current
demands solely by withdrawing more ground
and surface water could result in widespread
environmental degradation and depletion of
freshwater resources.

6

1 Water Resources Planning and Management: An Overview

Fig. 1.4 The Jordan River
between Israel and Jordan

There are policies that if implemented could
help preserve the capacity of the Jordan River to
meet future demands. Most of the options relate
to improving the efficiency of water use—that is,
they involve conservation and better use of proven technologies. Also being considered are
policies that emphasize economic efficiency and
reduce overall water use. Charging higher rates
for water use in peak periods, and surcharges for
excessive use, would encourage conservation. In
addition, new sources of fresh water can be
obtained by capturing rainfall through rooftop
cisterns, catchment systems, and storage ponds.
However before such measures are required, one
should assess the impact on local aquifer
recharge, storage, and withdrawals.
Thus there are alternatives to a steady deterioration of the water resources of the Jordan Basin.
They will require coordination and cooperation

among all those living in the basin. Will this be
possible?

1.2.3 Mending the “Mighty
and Muddy” Missouri
Nearly two centuries after an epic expedition
through the Western US in search of a northwest
river passage to the Pacific Ocean, there is little
enchantment left to the Missouri River. Shown in
Figs. 1.5 and 1.6, it has been dammed, diked,
and dredged since the 1930s mainly to control
floods and float cargo barges. The river nicknamed the “Mighty Missouri” and the “Big
Muddy” by its explorers is today neither mighty
nor muddy. The conservation group American
Rivers perennially lists the Missouri among the
USA’s 10 most endangered rivers.

1.2 Planning and Management Issues: Some Case Studies

7

Fig. 1.5 Major river basins in the continental US

Fig. 1.6 The Missouri Basin’s Reservoirs (not to scale)
constructed for navigation and flood control

Its wilder upper reaches are losing their cottonwood trees to dam operations and cattle that
trample seedlings along the river’s banks. Its vast
middle contains multiple dams that hold back
floods, generate power, and provide pools for
boats and anglers.
Its lower one-third is a narrow canal sometimes called “The Ditch” that is deep enough for

commercial towboats. Some of the river’s banks
are armored with rock and concrete retaining
walls that protect half a million acres of farm
fields from flooding. Once those floods produced
and maintained marshlands and side streams—
habitats for a wide range of wildlife. Without
these habitats, many wild species are unable to
thrive, and in some cases even survive.
Changes to restore at least some of the Missouri
to a more natural state are being implemented.
These changes add protection of fish and wildlife
habitat to the list of objectives to be achieved by the
government agencies managing the Missouri. The
needs of wildlife are now as important as other
competing interests on the river including navigation and flood control. This is in reaction, in part, to
the booming $115 million-a-year outdoor recreation industry. Just how much more emphasis will
be given to these back-to-nature goals depends on
whether the Missouri River Basin Association, an
organization representing eight states and 28 Native
American tribes, can reach a compromise with the
traditional downstream uses of the river.

8

1 Water Resources Planning and Management: An Overview

1.2.4 The Endangered Salmon
Greater Seattle in the northwestern US state of
Washington may be best known around the
world for Microsoft, but residents know it for
something less flashy: its dwindling stock of wild
salmon. The Federal Government has placed
seven types of salmon and two types of trout on
its list of threatened or endangered species.
Saving the fish from extinction could slow land

development in one of the fastest growing
regions of the U.S.
Before the Columbia River and its tributaries in
NW US were blocked with dozens of dams, about
10–16 million salmon made the annual run back
up to their spawning grounds (Fig. 1.7). In 1996, a
little less than 1 million did. But the economy of
the NW depends on the dams and locks that have
been built in the Columbia that provide cheap
hydropower production and navigation.

Fig. 1.7 The Snake and Columbia River reservoirs identified by the Columbia and Snake Rivers Campaign for
modification or dismantling to permit salmon passage

1.2 Planning and Management Issues: Some Case Studies

For a long time, engineers tried to modify the
system so that fish passage would be possible. As
shown in Fig. 1.8b, this included even the use of
trucks to transport captured juvenile salmon
around dams for release downstream. (It is not
clear that the trucks will be there when the fish
return to spawn upstream of the dams.) These
measures have not worked all that well. Still too
many young fish enter the hydropower turbines
on their way down the river. Now, as the debate
over whether or not to remove some dams takes
Fig. 1.8 A salmon
swimming upstream
(a) and measures taken to
protect young juvenile
salmon pass by
hydropower dams on their
way downstream (b) (US
Fish and Wildlife Service
and US Army Corps of
Engineers, Pacific region)

9

place, fish are caught and trucked around the
turbines. The costs of keeping these salmon
alive, if not completely happy, are enormous.
Over a dozen national and regional environmental organizations have joined together to
bring back salmon and steelhead by modifying or
partially dismantling five federal dams on the
Columbia and Snake Rivers. Partial removal of
the four dams on the lower Snake River in
Washington State and lowering the reservoir
behind John Day dam on the Columbia bordering

10

1 Water Resources Planning and Management: An Overview

Oregon and Washington (see Fig. 1.8) should
help restore over 200 miles of vital river habitat.
Running the rivers more like rivers may return
salmon and steelhead to harvestable levels of the
1960s before the dams were built.
Dismantling part of the four Lower Snake
dams will leave most of each dam whole. Only
the dirt bank connecting the dam to the riverbank
will be removed. The concrete portion of the dam
will remain in place, allowing the river to flow
around it. The process is reversible and, the
Campaign argues, it will actually save taxpayers
money in planned dam maintenance, by eliminating subsidies to shipping industries and
agribusinesses, and by ending current salmon
recovery measures that are costly. Only partially
removing the four Lower Snake River dams and
modifying John Day dam will help restore rivers,
save salmon, and return balance to the Northwest’s major rivers.

1.2.5 Wetland Preservation:
A Groundswell of Support
and Criticism
The balmy beach community of Tiger Point
near Pensacola, Florida, bordering the Gulf of
Mexico, is booming with development. New
subdivisions, a Wal-Mart discount retail store
and a recreation center dot the landscape.
Most—if not all—of this neighborhood was
once a wetland that soaked up rain during
downpours. Now, water runs off the parking lots
and the roofs and into resident’s living rooms.
Some houses get flooded nearly every year.
A federal agency oversees wetland development. Critics say the agency is permitting in this
area one of the highest rates of wetland loss in the
nation. Obviously local developers wish they did
not have to deal with the agency at all. The tension in Tiger Point reflects the debate throughout
the US about whether the government is doing
enough—or too much—to protect the nation’s
environment, and in this case, its wetlands.

Environmentalists and some homeowners
value wetlands because they help reduce water
pollution and floods, as well as nurture a diverse
wildlife population. But many landowners and
developers see the open wetlands as prime territory for building houses and businesses, rather
than for breeding mosquitoes. They view existing federal wetland rules as onerous, illogical,
and expensive.
While some areas such as Tiger Point have
residents who want stricter laws to limit wetlands
development, others—such as the suburbs
around Seattle—have people who long for less
strict rules.
Federal regulators had tried to quell the controversy with a solution known as wetlands mitigation. Anyone who destroys a wetland is required
to build or expand another wetland somewhere
else. Landowners and developers also see mitigation as a way out of the torturous arguments over
wetlands. However, studies have shown many
artificial marshes do not perform as well as those
created by nature (NRC 2001). Many of the new,
artificial wetlands are what scientists call the “ring
around the pond” variety: open water surrounded
by cattails. Furthermore, the federal agency issuing permits for wetland replacement do not have
the resources to monitor them after they are
approved. Developers know this.

1.2.6 Lake Source Cooling:
Aid to Environment,
or Threat to Lake?
It seems to be an environmentalist’s dream: a
cost-effective system that can cool some
10 million square feet of high school and
university buildings simply by pumping cold
water from the depths of a nearby lake (Fig. 1.9).
No more chlorofluorocarbons, the refrigerants
that can destroy protective ozone in the atmosphere and at a cost substantially smaller than for
conventional air conditioners. The lake water is
returned to the lake, with a few added calories.

1.2 Planning and Management Issues: Some Case Studies

11

Fig. 1.9 The cold deep waters of Lake Cayuga are being used to cool the buildings of a local school and university
(Ithaca City Environmental Laboratory)

However, a group of local opponents insists
that Cornell University’s $55 million lake-sourcecooling plan that replaced its aging air conditioners is actually an environmental threat. They
believe it could foster algal blooms. Pointing to 5
years of studies, thousands of pages of data, and
more than a dozen permits from local and state
agencies, Cornell’s consultants say the system
could actually improve conditions in the lake. Yet
another benefit, they say, is that the system would
reduce Cornell’s contribution to global warming
by reducing the need to burn coal to generate
electricity.
For the most part, government officials agree.
But a small determined coalition of critics from
the local community argue over the expected

environmental impacts, and over the process that
took place in getting the required local, state, and
federal permits approved. This is in spite of the
fact that the planning process, that took over
5 years, requested and involved the participation
of all interested stakeholders (that would participate) from the very beginning. Even the local
Sierra Club chapter and biology professors at
other universities have endorsed the project.
However, in almost every project where the
environmental impacts are uncertain, there will
be debates among scientists as well as stakeholders. In addition, a significant segment of
society distrusts scientists anyway. “This is a
major societal problem,” wrote a professor and
expert in the dynamics of lakes. “A scientist says

12

1 Water Resources Planning and Management: An Overview

X and someone else says Y and you’re got chaos.
In reality, we are the problem. Every time we
flush our toilets, fertilize our lawns, gardens and
fields, or wash our cars we contribute to the
nutrient loading of the lake.”
The project has now been operating for over a
decade, and so far no adverse environmental
effects have been noticed at any of the many
monitoring sites.

1.2.7 Managing Water in the Florida
Everglades
The Florida Everglades (Fig. 1.10) is the largest
single wetland in the continental United States.
In the mid-1800s it covered a little over nine
million acres, but since that time the historical
Everglades has been drained and half of the area
devoted to agriculture and urban development.
The remaining wetland areas have been altered
by human disturbances both around and within
them. Water has been diverted for human uses,
flows have been lowered to protect against
floods, nutrient supplies to the wetlands from
runoff from agricultural fields and urban areas
have increased, and invasions of nonnative or
otherwise uncommon plants and animals have
out-competed native species. Populations of
wading birds (including some endangered species) have declined by 85–90% in the last
half-century, and many species of South Florida’s mammals, birds, reptiles, amphibians, and
plants are either threatened or endangered.
The present management system of canals,
pumps, and levees (Fig. 1.11) will not be able to
provide adequate water supplies to agricultural
and urban areas, or sufficient flood protection,
let alone support the natural (but damaged)
ecosystems in the remaining wetlands. The system is not sustainable. Problems in the greater
Everglades ecosystem relate to both water quality
and quantity, including the spatial and temporal

distribution of water depths, flows, and flooding
durations—called hydroperiods. Issues arise
because of variations from the natural/historical
hydrologic regime, degraded water quality, and
the sprawl from fast-growing urban areas.
To meet the needs of the burgeoning population and increasing agricultural demands for
water, and to begin the restoration of Everglades’
aquatic ecosystem to a more natural regime, an
ambitious plan has been developed by the U.S.
Army Corps of Engineers and its local sponsor,
the South Florida Water Management District.
The proposed Corps plan is estimated to cost over
$8 billion. The plan and its Environmental
Impact Statement (EIS) have received input from
many government agencies and nongovernmental
organizations, as well as from the public at large.
The plan to restore the Everglades is ambitious and comprehensive, involving change of
the current hydrologic regime in the remnant
Everglades to one that resembles a more natural
one, reestablishment of marshes and wetlands,
implementation of agricultural best management
practices, enhancements for wildlife and recreation, and provisions for water supply and flood
control.
Planning for and implementing the restoration
effort requires application of state-of-the-art large
systems analysis concepts, hydrological and
hydroecological data and models incorporated
within decision support systems, integration of
social sciences, and monitoring for planning and
evaluation of performance in an adaptive management context. These large, complex challenges of the greater Everglades restoration effort
demand the most advanced, interdisciplinary,
and scientifically sound analysis capabilities that
are available. They also require the political will
to make compromises and to put up with the
lawsuits by anyone possibly disadvantaged by
some restoration measure.
Who pays for all this? The taxpayers of
Florida and the taxpayers of the U.S.

1.2 Planning and Management Issues: Some Case Studies
Fig. 1.10 Scenes of the
Everglades in southern
Florida (South Florida
Water Management
District)

13

14

1 Water Resources Planning and Management: An Overview

Fig. 1.11 Pump station on a drainage canal in southern Florida (South Florida Water Management District)

1.2.8 Restoration of Europe’s Rivers
and Seas
1.2.8.1 North and Baltic Seas
The North and Baltic Seas (shown in Fig. 1.12)
are the most densely navigated seas in the world.
Besides shipping, military, and recreational uses,
an offshore oil industry and telephone cables
cover the seabed. The seas are rich and productive with resources that include not only fish but
also crucial minerals (in addition to oil) such as
gas, sand, and gravel. These resources and
activities play major roles in the economies of the
surrounding countries.
Being so intensively used and surrounded by
advanced industrialized countries, pollution
problems are serious. The main pollution sources
include various wastewater outfalls, dumping by
ships (of dredged materials, sewage sludge, and
chemical wastes) and operational discharges
from offshore installations. Deposition of atmospheric pollutants is an additional major source
of pollution.

Those parts of the seas at greatest risk from
pollution are where the sediments come to rest,
where the water replacement is slowest and
where nutrient concentrations and biological
productivity are highest. A number of warning
signals have occurred.
Algal populations have changed in number
and species. There have been algal blooms,
caused by excessive nutrient discharge from land
and atmospheric sources. Species changes show
a tendency toward more short-lived species of
the opportunistic type and a reduction, sometimes to the point of disappearance, of some
mammals and fish species and the sea grass
community. Decreases of ray, mackerel, sand eel,
and echinoderms due to eutrophication have
resulted in reduced plaice, cod, haddock and dab,
mollusk and scoter.
The impact of fishing activities is also considerable. Sea mammals, sea birds, and Baltic
fish species have been particularly affected by the
widespread release of toxins and pollutants
accumulate in the sediments and in the food web.

1.2 Planning and Management Issues: Some Case Studies

Fig. 1.12 Europe’s major rivers and seas

15

16

1 Water Resources Planning and Management: An Overview

Some animals, such as the gray seal and the sea
eagle, are threatened with extinction.
Particular concern has been expressed about
the Wadden Sea that serves as a nursery for many
North Sea species. Toxic PCB contamination, for
example, almost caused the disappearance of
seals in the 1970s. Also, the 1988 massive seal
mortality in the North and Wadden Seas,
although caused by a viral disease, is still thought
by many to have a link with marine pollution.
Although the North Sea needs radical and
lengthy treatment it is probably not a terminal
case. Actions are being taken by bordering
countries to reduce the discharge of wastes into
the sea. A major factor leading to agreements to
reduce discharges of wastewaters has been the
verification of predictive pollutant circulation
models of the sea that identify the impacts of
discharges from various sites along the sea
boundary.

1.2.8.2 The Rhine
The map of Fig. 1.13 shows the areas of the nine
countries that are part of river Rhine basin. In the
Dutch area of the Rhine basin, water is partly
routed northward through the IJssel and westward through the highly interconnected river
systems of the Rhine, Meuse, and Waal.
About 55 million people live in the Rhine
River basin and about 20 million of those people
drink the river water.
In the mid 1970s, some called the Rhine the
most romantic sewer in Europe. In November
1986, a chemical spill degraded much of the
upper Rhine’s aquatic ecosystem. This damaging
event was reported worldwide. The Rhine was
again world news in the first 2 months of 1995,
when its water level reached a height that occurs
on average once in a century. In the Netherlands,
some 200,000 people, 1,400,000 pigs and cows,
and 1,000,000 chickens had to be evacuated.
During the last 2 months of the same year there
was hardly enough water in the Rhine for navigation. It is fair to say these events have focused
increased attention on what needs to be done to
“restore” and protect the Rhine.
To address just how to restore the Rhine, it is
useful to look at what has been happening to the

river during the past 150 years. The Rhine was
originally a natural watercourse. It is the only
river connecting the Alps with the North Sea. To
achieve greater economic benefits from the river,
it was engineered for navigation, hydropower,
water supply, and flood protection. Flood plains
now “protected” from floods, provided increased
land areas suitable for development. The main
stream of the Rhine is now considerably shorter
and narrower and deeper than it was originally.
From an economic development point of
view, the engineering works implemented in the
river and its basin worked. The Rhine basin is
now one of the most industrialized regions in the
world. The basin is characterized by intensive
industrial and agricultural activities. Some 20%
of the world’s chemical industry is located in the
Rhine River basin. The River is reportedly the
busiest shipping waterway in the world, containing long canals with regulated water levels.
These canals connect the Rhine and its tributaries
with the rivers of almost all the surrounding river
basins including the Danube River. This provides
water transport to and from the North and Black
Seas.
From an environmental and ecological viewpoint, and from the viewpoint of flood control as
well, the economic development that has taken
place over the past two centuries has not worked
perfectly. The concerns growing from the recent
toxic spill and floods as from a generally
increasing interest by the inhabitants of the basin
in environmental and ecosystem restoration and
the preservation of natural beauty, has resulted in
basin-wide efforts to rehabilitate the basin to a
more “living” sustainable entity.
A Rhine Action Programme was created to
revive the ecosystem. The goal of that program is
the revival of the main stream as the backbone of
the ecosystem, particularly for migratory fish,
and the protection, maintenance, and the revival
of ecologically important areas along the Rhine.
The plan, implemented in the 1990s, was given
the name “Salmon 2000”. The return of salmon
to the Rhine is seen as a symbol of ecological
revival. A healthy salmon population will need to
swim throughout the river length. This will pose
a challenge, as no one pretends that the

1.2 Planning and Management Issues: Some Case Studies

Fig. 1.13 The Rhine River Basin of Western Europe and its extension in The Netherlands

17

18

1 Water Resources Planning and Management: An Overview

engineering works that provide navigation and
hydropower benefits, but which also inhibit fish
passage, are no longer needed or desired.

1.2.8.3 The Danube
The Danube River (shown in Fig. 1.14) is in the
heartland of Central Europe. Its basin includes to
a larger extent the territories of 15 countries. It
additionally receives runoff from small catchments located in four other countries. About
90 million people live in the basin. This river
encompasses perhaps more political, economic,
and social variations than arguably any other
river basin in Europe.
The river discharges into the Black Sea. The
Danube delta and the banks of the Black Sea
have been designated a Biosphere Reserve by
UNESCO. Over half of the Delta has been
declared a “wet zone of international significance.” Throughout its length the Danube River
provides a vital resource for drainage, communications, transport, power generation, fishing,

Germany

CzechRepublic
Morava
Slovak Republic

Munich

Inn

recreation, and tourism. It is considered to be an
ecosystem with irreplaceable environmental
values.
More than 40 dams and large barrages plus
over 500 smaller reservoirs have been constructed on the main Danube River and its
tributaries. Flood control dikes confine most of
the length of the main stem of the Danube River
and the major tributaries. Over the last 50 years
natural alluvial flood plain areas have declined
from about 26,000 km2 to about 6000 km2.
There are also significant reaches with river
training works and river diversion structures.
These structures trap nutrients and sediment in
the reservoirs. This causes changes in downstream flow and sediment transport regimes that
reduce the ecosystems’ habitats both longitudinally and transversely, and decrease the efficiency of natural purification processes. Thus
while these engineered facilities provide important opportunities for the control and use of the
river’s resources, they also illustrate the

Austria
Mura
Drova
Slovenia

Ukraine
Moldava

Vienna
Budapest

Tisza

Hungary
Zagreb

Croatia

Oltul
Sava
Belgrade

Bosnia and
Herzegovina

Bucarest

Serbia

Morava
Montenegro
Kosovo

Fig. 1.14 The Danube River in Central Europe

Romania

Danube
Bulgaria

1.2 Planning and Management Issues: Some Case Studies

difficulties of balancing these important economic activities with environmentally sound and
sustainable management.
The environmental quality of the Danube River
is also under intense pressure from a diverse range
of human activities, including point source and
nonpoint source agricultural, industrial, and
municipal wastes. Because of the poor water
quality (sometimes affecting human health) the
riparian countries of the Danube river basin have
been participating in environmental management
activities on regional, national, and local levels for
several decades. All Danube countries signed a
formal Convention on Cooperation for the Protection and Sustainable Use of the Danube River in
June 1994. The countries have agreed to take “…
all appropriate legal, administrative and technical
measures to improve the current environmental
and water quality conditions of the Danube River
and of the waters in its catchment area and to
prevent and reduce as far as possible adverse
impacts and changes occurring or likely to be
caused.”

1.2.9 Flood Management
on the Senegal River
As on many rivers in the tropical developing
world, dam constructions on the Senegal (and
conventional dam management strategies) can
change not only the riverine environment but also

Fig. 1.15 Senegal River
and its Manantali Reservoir
more than 1000 km
upstream in Mali

19

the social interactions and economic productivity
of farmers, fishers, and herders whose livelihoods
depend on the annual flooding of valley bottomlands. Although much of the Senegal River flows
through a low rainfall area, the naturally occurring annual flooding supported a rich and biologically diverse ecosystem. Living in a
sustainable relationship with their environment,
small-land holders farmed sandy uplands during
the brief rainy season, and then cultivated the clay
plains as floodwaters receded to the main channel
of the river. Livestock also benefited from the
succession of rain-fed pastures on the uplands and
flood-recession pastures on the plains. Fish were
abundant. As many as 30,000 tons were caught
yearly. Since the early 1970s, small irrigated rice
schemes added a fifth element to the production
array: rain-fed farming, recession farming, herding, fishing, and irrigation.
Completion of the Diama salt intrusion barrage
near the mouth of the river between Senegal and
Mauritania and Manantali High Dam more than
1000 km upstream in Mali (Fig. 1.15), and the
termination of the annual flood have had adverse
effects on the environment. Rather than insulating
the people from the ravages of drought, the dam
release policy can accelerate desertification and
intensify food insecurity. Furthermore, anticipation of donor investments in huge irrigation
schemes has, in this particular case, lead to the
expulsion of non-Arabic-speaking black Mauritanians from their floodplain lands.

20

1 Water Resources Planning and Management: An Overview

This is a common impact of dam construction:
increased hardships of generally politically powerless people in order that urban and industrial
sectors may enjoy electricity at reduced costs.
Studies in the Senegal Valley by anthropologists, hydrologists, agronomists, and others suggest that it may be entirely economically feasible
to create a controlled annual “artificial flood,”
assuring satisfaction of both urban, industrial,
and rural demands for the river’s water and
supporting groundwater recharge, reforestation,
and biodiversity.
Because of these studies, the government of
Senegal ended its opposition to an artificial flood,
and its development plans for the region are now
predicated on its permanence. However, due to
the common belief that releasing large quantities
of water to create an artificial flood is incompatible with maximum hydropower production, the
other members of the three-country consortium
managing the dams—Mali and Mauritania—have
resisted accepting this policy.

1.2.10 Nile Basin Countries Striving
to Share Its Benefits
The Nile River (Fig. 1.16) is one of the major
rivers of the world, serving millions and giving
birth to entire civilizations. It is one of the world’s
longest rivers, traversing about 6695 km from the
farthest source of its headwaters in Rwanda and
Burundi through Lake Victoria, to its delta in
Egypt on the Mediterranean Sea. Its basin
includes 11 African countries (Burundi, DR
Congo, Egypt, Eritrea, Ethiopia, Kenya, Rwanda,
South Sudan, The Sudan, and Tanzania) and
extends for more than 3 million square kilometers
which represents about 10% of Africa’s land mass
area. The basin includes the Sudd wetland system
in South Sudan.
Nile Basin countries are today home to more
than 437 million people and of these, 54%
(238 million) live within the basin and expect
benefits from the management and use of the
shared Nile Basin water resources.

Notwithstanding the basin’s natural and
environmental endowments and rich cultural
history, its people face considerable challenges
including persistent poverty with millions living
on less than a dollar a day; extreme weather
events associated with climate variability and
change such as floods and droughts; low access
to water and sanitation services; deteriorating
water quality; and very low access rate to modern
energy with most countries below 20% access
level. The region also has a history of tensions
and instability both between states and internal to
states.
Cooperative management and development
could bring a vast range of benefits including
increased hydropower and food production; better access to water for domestic use; improved
management of watersheds and reduced environmental degradation; reduced pollution and
more control over damage from floods and
droughts. Recognizing this the Nile Basin Initiative was created as a regional intergovernmental partnership that seeks to develop the
River Nile in a cooperative manner, share substantial socioeconomic benefits, and promote
regional peace and security. The partnership
includes 10 Member States namely Burundi, DR
Congo, Egypt, Ethiopia, Kenya, Rwanda, South
Sudan, The Sudan, Tanzania, and Uganda. Eritrea participates as an observer. NBI was conceived as a transitional institution until a
permanent institution can be created.
The partnership is guided by a Shared
Vision: “To achieve sustainable socio-economic
development through equitable utilization of,
and benefit from, the common Nile Basin Water
resources.” The shared belief is that countries
can achieve better outcomes for all the peoples
of the Basin through cooperation rather than
competition. It is supported by a “Shared Vision
Planning Model” built by experts from all the
basin countries. The model is designed to run
different scenarios and assess the basin-wide
impacts of different management policies and
assumptions that any country may wish to
perform.

21
Lebanon
Syria

Mediterranean Sea

Iraq
Israel/
Palestine
Jordan

Libya

Cairo
Saudi Arabia

Egypt
High Aswan Dam
Africa

Ni

le

Lake Nasser

Red Sea

Merowe
Atb

Eritrea

ara

Chad

Khartoum Jebel Aulia

Khashm El Girba
Tekeze

Sennar
Rumela/Burdana

Sudan

Roseires

Lake Tana

Renaissance

Central
African
Republic

Arab

hr

Ba

Machar
swamp

Malakal

Blue

Nile

So
ba
t
Jonglei
Canal

El
Ghazal

Bahr El

Addis Ababa

Ghazal
swamp

Ethiopia

Sudd
swamp

White
Nile

Uganda
Democratic
Republic of Congo

existing dam
dam under
construction
0

1 00

2 00

300

km

Lake
Kyoga

Lake
Albert

Kampala
Lake
Edward

Rwanda

Owen Falls

Lake
Victoria

Burundi
Tanzania

Fig. 1.16 The Nile River Basin

Kenya

Indian
Ocean

E030127n

1.2 Planning and Management Issues: Some Case Studies

22

1 Water Resources Planning and Management: An Overview

1.2.11 Shrinking Glaciers at Top
of the World
As shown in Fig. 1.17, Tibet lies north of India,
Nepal, Bhutan, and Myanmar, west of China, and
south of East Turkistan. The highest and largest
plateau on Earth, it stretches some 1500 miles
(2400 km) from east to west, and 900 miles
(1448 km) north to south, an area equivalent in
size to the United States region east of the Mississippi River. The Himalayas form much of its
southern boundary, and Tibet’s average altitude is
so high—11,000 feet (3350 km) above sea
level—that visitors often need weeks to acclimate.
The Tibetan Plateau serves as the headwaters
for many of Asia’s largest rivers, including the
Yellow, Yangtze, Mekong, Brahmaputra, Salween, and Sutlej, among others. A substantial
portion of the world’s population lives in the
watersheds of the rivers whose sources lie on the
Tibetan Plateau.
Recent studies—including several by the
Chinese Academy of Sciences—have documented a host of serious environmental challenges involving the quantity and quality of
Tibet’s freshwater reserves, most of them caused
by industrial activities. Deforestation has led to
large-scale erosion and siltation. Mining, manufacturing, and other human and industrial activities are producing record levels of air and water
pollution in Tibet, as well as elsewhere in China

Fig. 1.17 China, India,
and Southeast Asia,
highlighting the Tibetan
Plateau

(Wong 2013). Together, these factors portend
future water scarcity that could add to the
region’s political volatility.
Most important is that the region’s glaciers are
receding at one of the fastest rates anywhere in
the world, and in some regions of Tibet by three
3 m per year (IPPC 2007). The quickening
melting and evaporation is raising serious concerns in scientific and diplomatic communities,
in and outside China, about Tibet’s historic
capacity to store more freshwater than anyplace
on earth, except the North and South Poles.
Tibet’s water resources, they say, have become
an increasingly crucial strategic political and
cultural element that the Chinese are intent on
managing and controlling.

1.2.12 China, a Thirsty Nation
Why does China care about the freshwater in
Tibet? With more than a quarter of its land
classified as desert, China is one of the planet’s
most arid regions. Beijing is besieged each spring
by raging dust storms born in Inner Mongolia
where hundreds of square miles of grasslands are
turning to desert each year. In other parts of the
nation, say diplomats and economic development
specialists, Chinese rivers are either too polluted
or too filled with silt to provide all of China’s
people with adequate supplies of freshwater.

1.2 Planning and Management Issues: Some Case Studies

Chinese authorities have long had their eyes
on Tibet’s water resources. They have proposed
building dams for hydropower and spending
billions of dollars to build a system of canals to
tap water from the Himalayan snowmelt and
glaciers and transport it hundreds of miles north
and east to the country’s farm and industrial
regions.
But how long that frozen reservoir will last is
in doubt. In attempting to solve its own water
crisis, China could potentially create widespread
water shortages among its neighbors.
While the political issues involving Tibet are
complex, there is no denying that water plays a
role in China’s interest in the region. The water
of Tibet may prove to be one of its most
important resources in the long run—for China,
and for much of southern Asia. Figuring out how
to sustainably manage that water will be a key to
reducing political conflicts and tensions in the
region.

1.2.13 Managing Sediment in China’s
Yellow River
The scarcity of water is not the only issue China
has to address. So is sediment, especially in the
Yellow River (Fig. 1.18). The Yellow River basin
is the cradle of Chinese civilization, with agricultural societies appearing on the banks of the
river more than 7000 years ago. The Yellow River
originates in the Qinghai–Tibetan plateau and
discharges into the Bohai Gulf in the Yellow sea.
The basin is traditionally divided into the upper,
middle, and lower reaches, which can be described as three down-sloping steps: the Tibetan
Plateau, the Loess Plateau, and the alluvial plain.
Key management issues are many, but the most
visible one is sediment (Figs. 1.19 and 1.20).
The high sediment load of the Yellow River is
a curse if the sediment deposits on the bed of the
channel and reduces its capacity, thereby
increasing the risk of flooding. Also, rapid
deposition of sediment in reservoirs situated
along the river is a problem as it reduces their
effectiveness for flood control and water storage.

23

Another major management issue is the
ecosystem health of the river. The relative scarcity of water creates a tension between allocating
water for the benefit of river health, and for direct
social and economic benefit. Irrigation uses 80%
of the water consumed from the river, with the
rest supplying industry, and drinking water for
cities along the river and outside of the basin
(Tianjin, Cangzhou and Qingdao). During the
1980s and 1990s the lower river dried up nearly
every year, resulting in lost cereal production,
suspension of some industries, and insufficient
water supplies for more than 100,000 residents,
who had to queue daily for drinking water. As
well as costing around RmB40 billion in lost
production, there was a serious decline in the
ecological health of the river.
The diversity of habitat types and extensive
areas of wetlands within the Ramsar-listed Yellow River Delta support at least 265 bird species.
The birds, fish, and macroinvertebrates in the
delta rely on healthy and diverse vegetation
communities, which in turn depend upon on
annual freshwater flooding and the associated
high sediment loads. Degradation of the
ecosystem of the Delta has been documented,
especially from the late-1990s, due to increased
human activities and a significant decrease in the
flow of freshwater to the Delta wetlands. This has
led to saltwater intrusion and increased soil
salinity. Restoration activities involving the
artificial delivery of freshwater to the wetlands
began in 2002.

1.2.14 Damming the Mekong
(S.E. Asia), the Amazon,
and the Congo
The world’s most biodiverse river basins—the
Amazon, Congo, and Mekong—are attracting
hydropower developers. While hydropower projects address energy needs and offer the potential
of a higher standard of living, they also can
impact the river’s biodiversity, especially fisheries. The Amazon, Congo, and Mekong basins
hold roughly one-third of the world’s freshwater

24

1 Water Resources Planning and Management: An Overview

Fig. 1.18 The Yellow River Basin in China

fish species, most of which are not found elsewhere. Currently more than 450 additional dams
are planned for these three rivers (see Figs. 1.22
and 1.23) (Winemiller et al. 2016). Many of the
sites most appropriate for hydropower production
also are the habitats of many fish species. Given
recent escalation of hydropower development in
these basins, planning is needed to reduce biodiversity loss, as well as other adverse environmental, social, and economic impacts while
meeting the energy needs of the basins.
The Mekong River (Fig. 1.21) flows some
4200 km through Southeast Asia to the South
China Sea through Tibet, Myanmar (Burma),
Vietnam, Laos, Thailand, and Cambodia. Its
“development” has been restricted over the past

several decades due to regional conflicts, indeed
conflicts that have altered the history of the world.
Now that these conflicts are not resulting in military battles (at this writing), investment capital is
becoming available to develop the Mekong’s
resources for improved fishing, irrigation, flood
control, hydroelectric power, tourism, recreation,
and navigation. The potential benefits are substantial, but so are the environmental, ecological,
and social risks (Orr et al. 2012).
The economic value of hydroelectric power
currently generated from the Mekong brings in
welcome income however the environmental
impacts are harder to quantify. Today some
60 million people (12 million households) live in
the Lower Mekong Basin, and 80% rely directly

1.2 Planning and Management Issues: Some Case Studies

25

Fig. 1.19 Sediment flows in China’s Yellow River. http://yellowriver-china.blogspot.com/2011/09/book-review-onflood-discharge-and.html

on the river system for their food and livelihoods.
Most of these households would be affected by
alterations to fish availability since fish is their
main source of dietary protein. The food security
impacts on these people due to the existing and
proposed dam building and operation in Cambodia, Laos, Thailand, and Vietnam remain relatively unexplored. Dam builders have often
failed to recognize, or wish to ignore, the crucial
role of inland fisheries in meeting food security
needs.
During some months of the year the lack of
rainfall causes the Mekong to fall dramatically.
Salt water may penetrate as much as 500 km
inland. In other months the flow can be up to
30 times the low flows, causing the water in the
river to back up into wetlands and flood some
12,000 km2 of forests and paddy fields in the

Vietnamese delta region alone. The ecology of a
major lake, Tonle Sap, in Cambodia depends on
these backed up waters.
While flooding imposes risks on the inhabitants of the Mekong flood plain, there are also
distinct advantages. High waters deposit
nutrient-rich silts on the low-lying farmlands,
thus sparing the farmers from having to transport
and spread fertilizers on their fields. Also, shallow lakes and submerged lands provide spawning habitats for about 90% of the fish in the
Mekong basin. Fish yield totals over half a million tons annually.
What will happen to the social fabric and to the
natural environment if the schemes to build big
dams (see Fig. 1.22a) across the mainstream of
the Mekong are implemented? Depending on
their design, location, and operation, they could

26

1 Water Resources Planning and Management: An Overview

Fig. 1.20 Dams can be
designed and operated to
remove some of the
sediment that is trapped in
the upstream reservoir

disrupt the current fertility cycles and the habitats
and habits of the fish in the river resulting from
the natural flow and sediment regimes. Increased
erosion downstream from major reservoirs is also
a threat. Add to these possible adverse impacts
the need to evacuate and resettle thousands of
people displaced by the lake behind the dams.
How will they be resettled? And how long will it
take them to adjust to new farming conditions?
And will there even be a Delta? Together with sea
level rise and a blockage of Mekong’s sediment to

the Delta, its survival as a geologic feature, and as
a major source of food, is in doubt.
There have been suggestions that a proposed
dam in Laos could cause deforestation in a
wilderness area of some 3000 km2. Much of the
wildlife, including elephants, big cats, and other
rare animals, would have to be protected if they
are not to become endangered. Malaria-carrying
mosquitoes, liver fluke, and other disease bearers
might find ideal breeding grounds in the mud
flats of the shallow reservoir. These are among

1.2 Planning and Management Issues: Some Case Studies

27

Fig. 1.21 The Lower Mekong River Basin including Tonle Sap Lake in Cambodia and the Mekong Delta in Vietnam

28

1 Water Resources Planning and Management: An Overview

human populations and expanding deforestation
associated with new roads. Scheduled for completion in 2016, Brazil’s Belo Monte hydropower
complex was designed with installed capacity of
11,233 MW, ranking it the world’s third largest.
But it could also set a record for biodiversity loss
owing to selection of a site that is the sole habitat
for many species. The Congo has far fewer dams
than the Amazon or Mekong, yet most power
generated within the basin is from hydropower.
Inga Falls, a 14.5-km stretch of the lower Congo
that drops 96 m to near sea level, has greater
hydropower potential than anywhere else. The
Inga I and II dams, constructed in the 1970s and
1980s, currently yield 40% of the 2132-MW
installed capacity. Planned additional dams (Inga
III and Grand Inga) would harness as much as
83% of the Congo’s annual discharge, with most
of the energy to be exported. Grand Inga would
divert water and substantially reduce flow for at
least 20 km downstream from the falls. Again,
many trade-offs involved with dam building, and
all calling for comprehensive systems planning
and analyses to identify them.

Fig. 1.22 Lancang/Mekong River where reservoirs are
being planned on the river itself (a) and on many of its
tributaries (b). a http://khmerization.blogspot.com/2013/
10/wwf-expresses-alarm-over-laos-decision.html, 6/10/
13, and b reprinted from Wild and Loucks 2014, with
permission. © 2014. American Geophysical Union

the types of issues that need to be considered
now that increased development seems likely.
Similar issues face those who are planning
similar hydropower dam developments in the other
two most biodiverse river basins in the world—the
Amazon and the Congo (Fig. 1.23). Clarifying the
trade-offs between energy (economic), environmental, and social goals can inform governments
and funding institutions as they make their dam
siting, design, and operating decisions.
Hydropower accounts for more than
two-thirds of Brazil’s energy supply, and over
300 new Amazon dams have been proposed.
Impacts of these dams would extend beyond
direct effects on rivers to include relocation of

1.3

So, Why Plan, Why Manage?

Water resources planning and management
activities are usually motivated, as they were in
each of the previous section’s case examples, by
the realization that there are problems to solve
and/or opportunities to obtain increased benefits
by changing the management and use of water
and related land resources. These benefits can be
measured in many different ways. The best way
to do it is often not obvious. Whatever way is
proposed may provoke conflict. Hence there is
the need for careful study and research, as well as
full stakeholder involvement, in the search for
the best compromise plan or management policy.
Reducing the frequency and/or severity of the
adverse consequences of droughts, floods, and
excessive pollution are common goals of many
planning and management exercises. Other reasons include the identification and evaluation of
alternative measures that may increase the
available water supplies, hydropower, improve

1.3 So, Why Plan, Why Manage?

Fig. 1.22 (continued)

29

30

1 Water Resources Planning and Management: An Overview

Fig. 1.23 Fish diversity and dam locations in the
Amazon and Congo basins. In addition to basin-wide
biodiversity summaries (upper left), each basin can be
divided into ecoregions (white boundaries). Approximate

number of species (black numbers) and the total species
richness (shades of green) found in ecoregions differ
widely (Winemiller et al. 2016)

recreation and/or navigation, and enhance water
quality and aquatic ecosystems. Quantitative
system performance criteria can help one judge
the relative net benefits, however measured, of
alternative plans and management policies.
System performance criteria of interest have
evolved over time. They have ranged from being
primarily focused on safe drinking water just a
century ago to multipurpose economic development a half-century ago to goals that now include
environmental and ecosystem restoration and
protection, aesthetic and recreational experiences, and more recently, sustainability (ASCE
1998; GTT 2014).
Some of the multiple purposes served by a
river can be conflicting. A reservoir used solely
for hydropower, or water supply, is better able to
meet its objectives when it is full of water. On the
other hand, a reservoir used solely for downstream flood control is best left empty so it can
store more of the flood flows when they occur.
A single reservoir serving all three purposes
introduces conflicts over how much water to
store in it and discharge from it, i.e., how it
should be operated. In basins where diversion

demands exceed the available supplies, conflicts
will exist over water allocations. Finding the best
way to manage, if not resolve, these conflicts are
reasons for planning.

1.3.1 Too Little Water
Issues involving inadequate supplies to meet
demands can result from too little rain or snow.
They can also result from patterns of land and
water use. They can result from growing urbanization, the growing needs to meet instream flow
requirements, and conflicts over private property
and public rights regarding water allocations.
Other issues can involve transbasin water transfers and markets, objectives of economic efficiency versus the desire to keep nonefficient
activities viable, and demand management measures, including incentives for water reuse and
water reuse financing.
Measures to reduce the demand for water in
times of supply scarcity should be identified and
agreed upon before everyone must cope with an
actual water scarcity. The institutional authority

1.3 So, Why Plan, Why Manage?

to implement drought measures when their designated “triggers”—such as storage volumes in
reservoirs—have been met should be established
before they are needed. Such management measures may include increased groundwater
abstractions to supplement low-surface water
flows and storage volumes. Conjunctive use of
ground and surface waters can be sustainable as
long as the groundwater aquifers are recharged
during conditions of high flow and surface storage volumes. Many aquifers are subject to withdrawals exceeding recharge, and hence continued
withdrawals from them cannot be sustained.

1.3.2 Too Much Water
Damage due to flooding is a direct result of
floodplain development that is incompatible with
floods. This is a risk many take, and indeed on
average it may result in positive private net
benefits, especially when public agencies subsidize these private risk takers who incur losses in
times of flooding. In many river basins of
developed regions, annual expected flood damages are increasing over time, in spite of
increased expenditures in flood damage reduction measures. This is in part due to increased
economic development taking place on river
flood plains, not only of increased frequencies
and magnitudes of floods.
The increased economic value of developments
on floodplains often justifies increased development and increased expenditures on flood damage
reduction measures. Flood protection works
decrease the risks of flood damage, creating an
even larger incentive for increased economic
development. Then when a flood exceeding the
capacity of existing flood protection works occurs,
and it will, even more damage results. This cycle of
increasing flood damages and costs of protection is
a natural result of increasing values of flood plain
development. Just what is the appropriate level of
risk? It may depend, as Fig. 1.24 illustrates, on the
level of flood insurance or subsidy provided when
flooding occurs.
Flood damages will decrease only if there are
restrictions placed on floodplain development.

31

Fig. 1.24 The lowest risk of flooding on a floodplain
does not always mean the best risk, and what risk is
acceptable may depend on the amount of insurance or
subsidy provided when flood damage occurs

Analyses carried out during planning can help
identify the appropriate level of development and
flood damage protection works based on the
beneficial as well as adverse economic, environmental, and ecological consequences of flood
plain development. People are increasingly recognizing the economic as well as environmental
and ecological benefits of allowing floodplains to
do what they were formed to do—store flood
waters when floods occur.
Industrial development and related port
development may result in the demand for deeper
and wider rivers to allow the operation of larger
draft cargo vessels in the river. River channel
improvement cannot be detached from functions
such as water supply and flood control. Widening
and deepening a river channel for shipping purposes may also decrease flood water levels.

1.3.3 Too Polluted
Wastewater discharges by industry and households can have considerable detrimental effects
on water quality and hence on public and
ecosystem health. Planning and management
activities should pay attention to these possible
negative consequences of industrial development
and the intensive use and subsequent runoff of
pesticides and fertilizers in urban as well as in
agricultural areas.
Issues regarding the environment and water
quality include:

32

1 Water Resources Planning and Management: An Overview

• Upstream versus downstream conflicts on
meeting water quality standards,
• Threats from aquatic nuisance species,
• Threats from the chemical, physical, and
biological water quality of the watershed’s
aquatic resources,
• Quality standards for recycled water,
• Nonpoint source pollution discharges including sediment from erosion, and
• Inadequate groundwater protection, compacts,
and concerned institutions.
We still know too little about the environmental and health impacts of many of the
wastewater constituents found in river waters. As
more is learned about, for example, the harmful
effects of heavy metals and dioxins, pharmaceutical products, and micropollutants and
nanoparticles in our water supplies, water quality
standards, plans and management policies should
be adjusted accordingly. The occurrence of major
fish kills and algae blooms also point to the need
to manage water quality as well as quantity.

1.3.4 Too Expensive
Too many of the world’s population do not have
adequate water to meet all of their drinking and
sanitation needs. Much of this is not due to the
lack of technical options available to provide
water to meet those needs. Rather those options
are deemed to be too expensive. Doing so is
judged to be beyond the ability of those living in
poverty to pay and recover the costs of implementing, maintaining, and operating the needed
infrastructure. Large national and international
aid grants devoted to reducing water stress—
demands for clean water exceeding usable supplies—in stressed communities have not been
sustainable in the long run where recipients have
been unable to pay for the upkeep of whatever
water resource systems are developed and provided. If financial aid is to be provided, to be

effective it has to address all the root causes of
such poverty, not only the need for clean water.

1.3.5 Ecosystem Too Degraded
Aquatic and riparian ecosystems may be subject
to a number of threats. The most important ones
include habitat loss due to river training and
reclamation of floodplains and wetlands for
urban and industrial development, poor water
quality due to discharges of pesticides, fertilizers
and wastewater effluents, and the infestation of
aquatic nuisance species.
Exotic aquatic nuisance species can be major
threats to the chemical, physical, and biological
water quality of a river’s aquatic resources and a
major interference with other uses. The destruction and/or loss of the biological integrity of
aquatic habitats caused by introduced exotic
species is considered by many ecologists to be
among the most important problems facing natural aquatic and terrestrial ecosystems. Biological integrity of natural ecosystems is controlled
by habitat quality, water flows or discharges,
water quality, and biological interactions
including those involving exotic species.
Once exotic species are established, they are
usually difficult to manage and nearly impossible
to eliminate. This creates a costly burden for
current and future generations. The invasion in
North America of nonindigenous aquatic nuisance species such as the sea lamprey, zebra
mussel, purple loosestrife, European green crab,
and various aquatic plant species, for example,
has had pronounced economic and ecological
consequences for all who use or otherwise benefit from aquatic ecosystems.
Environmental and ecological effectiveness as
well as economic efficiency should be a guiding
principle in evaluating alternative solutions to
problems caused by aquatic nuisance organisms.
Funds spent in prevention and early detection
and eradication of aquatic nuisance species may

1.3 So, Why Plan, Why Manage?

reduce the need to spend considerably more
funds on management and control once such
aquatic nuisance species are well established.

1.3.6 Other Planning
and Management Issues
1.3.6.1 Navigation
Dredging river beds is a common practice to
keep river channels open for larger draft cargo
ships. The use of jetties as a way to increase the
flow in the main channel and hence increase
bottom scour is a way to reduce the amount of
dredging that may be needed, but any modification of the width and depth of a river channel can
impact its flood carrying capacity. It can also
alter the periodic flooding of the floodplain that
in turn can have ecological impacts.
1.3.6.2 River Bank Erosion
Bank erosion can be a serious problem where
towns are located close to morphologically active
(eroding) rivers. Predictions of changes in river
courses due to bank erosion and bank accretion
are important inputs to land use planning in river
valleys and the choice of locations for bridges,
buildings, and hydraulic structures.
1.3.6.3 Reservoir Related Issues
Degradation of the riverbeds upstream of reservoirs may increase the risks of flooding in those
areas. Reservoir construction inevitably results in
loss of land and forces the evacuation of residents due to impoundment. Reservoirs can be
ecological barriers for migrating fish species such
as salmon. The water quality in the reservoir may
deteriorate and the inflowing sediment may settle
and accumulate, reducing the active (useful)
water storage capacity of the reservoir and
causing more erosion downstream. Other potential problems may include those stemming from
stratification, water-related diseases, algae
growth, and abrasion of hydropower turbines.
Environmental and morphological impacts
downstream of the dam are often due to a
changed river hydrograph and decreased

33

sediment load in the water released from the
reservoir. Lower sediment concentrations result
in higher risks of scouring of downstream riverbeds and consequently a lowering of their
elevations. Economic as well as social impacts
include the risk of a dam break. Environmental
impacts may result from sedimentation control
measures (e.g., sediment flushing as shown in
Fig. 1.19) and reduced oxygen content of the
outflowing water.

1.4

System Planning Scales

1.4.1 Spatial Scales for Planning
and Management
Watersheds or river basins are usually considered
logical regions for water resources planning and
management. This makes sense if the impacts of
decisions regarding water resources management
are contained within the watershed or basin. How
land and water are managed in one part of a river
basin can impact the land and water in other parts
of the basin. For example, the discharge of pollutants or the clearing of forests in the upstream
portion of the basin may degrade the quality and
increase the variability of the flows and sedimentation downstream. The construction of a
dam or weir in the downstream part of a river
may block vessels and fish from traveling up- or
downstream through the dam site. To maximize
the economic and social benefits obtained from
the entire basin, and to insure that these benefits
and accompanying costs are equitably distributed, planning and management on a basin
scale is often undertaken.
While basin boundaries make sense from a
hydrologic point of view, they may be inadequate for addressing particular water resources
problems that are caused by events taking place
outside the basin. What is desired is the highest
level of performance, however defined, of the
entire physical, social-economic, and administrative water resource system. To the extent that
the applicable problems, stakeholders, and
administrative boundaries extend outside the

34

1 Water Resources Planning and Management: An Overview

river basin, then the physically based “river
basin” focus of planning and management should
be expanded to include the entire applicable
“problem-shed.” Hence consider the term “river
basin” used in this book to mean problem-shed
when appropriate.

1.4.2 Temporal Scales for Planning
and Management
Planning is a continuing iterative process. Water
resources plans need to be periodically updated
and adapt to new information, new objectives,
and updated forecasts of future demands, costs,
and benefits. Current decisions should not preclude future generations from options they may
want to consider, but otherwise current decisions
should be responsive to current needs and
opportunities, and have the ability to be adaptable in the future to possible changes in those
needs and opportunities.
The number and duration of within-year time
periods explicitly considered in the planning
process will depend in part on the need to consider the variability of the supplies of and
demands for water resources and on the purposes
to be served by the water resources. Irrigation
planning and summer season water recreation
planning may require a greater number of
within-year periods during the summer growing
and recreation season than might be the case if
one were considering only municipal water
supply planning, for example. Assessing the
impacts of alternatives for conjunctive surface
and groundwater management, or for water
quantity and quality management, require attention to processes that typically take place on
different spatial and temporal scales.

1.5

Planning and Management
Approaches

There are two general approaches to planning and
management. One is from the top-down, often
called command and control. The other is from the
bottom-up, often called the grassroots approach.

Both approaches, working together, can lead to an
integrated plan and management policy.

1.5.1 Top-Down Planning
and Management
Over much of the past half-century water
resources professionals have been engaged in
preparing integrated, multipurpose “master”
development plans for many of the world’s river
basins. These plans typically consist of a series of
reports, complete with numerous appendices,
describing all aspects of water resources management and use. In these documents alternative
structural and nonstructural management options
are identified and evaluated. Based on these
evaluations, the preferred plan is recommended.
This master planning exercise has typically
been a top-down approach. Professionals have
dominated the top-down approach. Using this
approach there is typically little if any active
participation of interested stakeholders. The
approach assumes that one or more institutions
have the ability and authority to develop and
implement the plan, i.e., to oversee and manage
the coordinated development and operation of
the basin’s activities impacting the surface and
ground waters of the basin. In today’s environment where publics are calling for less government oversight, regulation and control, and
increasing participation in planning and management activities, strictly top-down approaches
are becoming less desirable or acceptable.

1.5.2 Bottom-Up Planning
and Management
Within the past several decades water resources
planning and management processes have
increasingly involved the active participation of
interested stakeholders—those potentially affected by the decision being considered. Plans are
being created from the bottom-up rather
than top-down through a process of consensus
building. Concerned citizens, nongovernmental organizations, as well as professionals in

1.5 Planning and Management Approaches

governmental agencies are increasingly working
together toward the creation of adaptive comprehensive water management programs, policies, and plans.
Experiences trying to implement plans
developed primarily by professionals without
significant citizen involvement have shown that
even if such plans are technically sound they
have little chance of success if they do not take
into consideration the concerns and objectives of
affected stakeholders. To gain their support,
concerned stakeholders must be included in the
decision-making process as early as possible.
They must become part of the decision-making
process, not merely spectators, or even advisors,
to it. This will help gain their cooperation and
commitment to the plans eventually adopted.
Participating stakeholders will consider the
resulting plans as their plans as much as someone
else’s. They will have a sense of ownership, and
as such will strive to make them work. Such
adopted plans, if they are to be successfully
implemented, must fit within existing legislative,
permitting, enforcement, and monitoring programs. Stakeholder participation improves the
chance that the system being managed will be
sustainable.
Successful planning and management
involves motivating all potential stakeholders
and sponsors to join and participate in the water
resources planning and management process. It
will involve building a consensus on goals and
objectives and on how to achieve them. Ideally
this should occur before addressing conflicting
issues so that all involved know each other and
are able to work together more effectively.
Agreements on goals and objectives and on the
organization (or group formed from multiple
organizations) that will lead and coordinate the
water resources planning and management process should be reached before stakeholders bring
their individual priorities or problems to the

35

table. Once the inevitable conflicts become
identified, the settling of administrative matters
does not get any easier.
Bottom-up planning must strive to achieve a
common or “shared” vision among all stakeholders. It must either comply with all applicable
laws and regulations, or propose changes to
them. It should strive to identify and evaluate
multiple alternatives and performance criteria—
including sustainability criteria, and yet keep the
process from producing a wish list of everything
each stakeholder wants. In other words, it must
identify trade-offs among conflicting goals or
measures of performance, and prioritizing
appropriate strategies. It must value and compare, somehow, the intangible and nonmonetary
impacts of environmental and ecosystem protection and restoration with other activities
whose benefits and costs can be expressed in
monetary units. In doing all this, planners should
use modern information technology, as available,
to improve both the process and product. This
technology, however, will not eliminate the need
to reach conclusions and make decisions on the
basis of incomplete and uncertain data and scientific knowledge.
These process issues emphasize the need to
make water resources planning and management
as efficient and effective as possible and remain
participatory. Many issues will arise in terms of
evaluating alternatives and establishing performance criteria (prioritizing issues and possible
actions), performing incremental cost analysis,
and valuing monetary and nonmonetary benefits.
Questions must be answered as to how much
data must be collected and with what precision,
and what types of modern information technology (e.g., geographic information systems (GIS),
remote sensing, Internet and mobile Internet
networks, decision support systems, etc.) can be
beneficially used both for analyses as well as
communication.

36

1 Water Resources Planning and Management: An Overview

1.5.3 Integrated Water Resources
Management
The concept of integrated water resources management (IWRM) has been developing over the
past several decades. IWRM is the response to
the growing pressure on our water resources
systems caused by growing populations and
socioeconomic developments. Water shortages
and deteriorating water quality have forced many
countries in the world to reconsider their development policies with respect to the management
of their water resources. As a result water
resources management (WRM) has been undergoing a change worldwide, moving from a
mainly supply-oriented, engineering-biased
approach toward a demand-oriented, multisectoral approach, often labeled integrated water
resources management.
The concept of IWRM moves away from
top-down “water master planning” that usually
focuses on water availability and development, and
toward “comprehensive water policy planning”
that addresses the interaction between different
subsectors (Fig. 1.25), seeks to establish priorities,
considers institutional requirements, and deals with
the building of management capacity.

Box 1.1 Definition of IWRM
IWRM is a process which promotes the
coordinated development and management
of water, land, and related resources, in
order to maximize the resultant economic
and social welfare in an equitable manner
without compromising the sustainability of
vital ecosystems.
(GWP 2000)

IWRM (Box 1.1) considers the use of the
resources in relation to social and economic
activities and functions. These determine the
need for laws and regulations pertaining to the
sustainable and beneficial use of the water
resources. Infrastructure together with regulatory
measures allows more effective use of the
resource including meeting ecosystem needs.

1.5.4 Water Security
and the Sustainable
Development Goals
(SDGs)
While IWRM focuses on the process to improve
water management (the how), the term “water
security” focuses on the output (the what). The
World Economic Forum has identified Water
Security as one of the biggest global economic
development issues. Water Security is defined by
UN-Water (2013) as
the capacity of a population to safeguard sustainable access to adequate quantities of acceptable
quality water for sustaining livelihoods, human
well-being, and socio-economic development, for
ensuring protection against water-borne pollution
and water-related disasters, and for preserving
ecosystems in a climate of peace and political
stability.

Fig. 1.25 Interactions among the natural, administrative,
and socioeconomic water resource subsectors and
between them and their environment

Attempts are being made to identify the many
dimensions of water security and to quantify
them (van Beek and Arriens 2014; ADB 2016).
In 2015 the UN adopted the Sustainable Development Goals 2015–2030 that specify specific

1.5 Planning and Management Approaches

targets for various goals such as the provision of
water for drinking and sanitation, water productivity in agriculture, industry and energy, environment, and reduction of floods and droughts. It
is expected that many countries will expect their
water managers to use the SDGs as objectives in
water resources planning. This means that our
planning and management proposals need to be
able to quantify the impacts of possible plans and
policies in terms of the SDG targets.

1.5.5 Planning and Management
Aspects
1.5.5.1 Technical
Technical aspects of planning include hydrologic
assessments. Hydrologic assessments identify
and characterize the properties of, and interactions among, the resources in the basin or region.
This includes the land, the rainfall, the runoff, the
stream and river flows, and the groundwater.
Existing watershed land use and land cover,
and future changes in this use and cover, result in
part from existing and future changes in regional
population and economy. Planning involves
predicting changes in land use/covers and economic activities at watershed and river basin
levels. These will influence the amount of runoff,
and the concentrations of sediment and other
quality constituents (organic wastes, nutrients,
pesticides, etc.) in the runoff resulting from any
given pattern of rainfall over the land area. These
predictions will help planners estimate the
quantities and qualities of flows throughout a
watershed or basin, associated with any land use
and water management policy. This in turn provides the basis for predicting the type and health
of terrestrial and aquatic ecosystems in the basin.
All of this may impact the economic development of the region, which is what, in part,
determines the future demands for changes in
land use and land cover.
Technical aspects also include the estimation
of the costs and benefits of any measures taken to
manage the basin’s water resources. These measures might include:

37

• Engineering structures for making better use
of scarce water.
• Canals and water-lifting devices.
• Dams and storage reservoirs that can retain
excess water from periods of high flow for
use during the periods of low flow. By storage of floodwater they may also reduce flood
damage below the reservoir.
• Open channels that may take the form of a
canal, flume, tunnel, or partly filled pipe.
• Pressure conduits.
• Diversion structures, ditches, pipes, checks,
flow dividers, and other engineering facilities
necessary for the effective operation of irrigation and drainage systems.
• Municipal and industrial water intakes,
including water purification plants and transmission facilities.
• Sewerage and industrial wastewater treatment
plants, including waste collection and ultimate disposal facilities.
• Hydroelectric power storage, run-of-river, or
pumped storage plants.
• River channel regulation works, bank stabilization, navigation dams and barrages, navigation locks, and other engineering facilities
for improving a river for navigation.
• Levees and floodwalls for confining flows
within predetermined channels.
Not only must the planning process identify
and evaluate alternative management strategies
involving structural and nonstructural measures
that will incur costs and bring benefits, but it
must also identify and evaluate alternative time
schedules for implementing those measures. The
planning of development over time involving
interdependent projects, uncertain future supplies
and demands as well as costs, benefits, and interest (discount) rates is part of all water
resources planning and management processes.
With increasing emphasis placed on ecosystem
preservation and enhancement, planning must
include ecologic impact assessments. The mix of
soil types and depths and land covers together with
the hydrological quantity and quality flow and
storage regimes in rivers, lakes, wetlands, and

38

1 Water Resources Planning and Management: An Overview

aquifers all impact the riparian and aquatic ecology of the basin. Water managers are being asked
to consider ways of improving or restoring
ecosystems by, for example, reducing the
• destruction and/or loss of the biological
integrity of aquatic habitats caused by introduced exotic species or changes in flow and
sediment patterns due to upstream reservoir
operation.
• decline in number and extent of wetlands and
the adverse impacts to wetlands of proposed
land and water development projects.
• conflicts between the needs of people for
water supply, recreational, energy, flood
control, and navigation infrastructure and the
needs of ecological communities, including
endangered species.
And indeed there are and will continue to be
conflicts among alternative objectives and purposes of water management. Planners and managers must identify the trade-offs among
environmental, ecologic, economic, and social
impacts, however measured, and the management
alternatives that balance these often-conflicting
interests.

1.5.5.2 Financial and Economic
The overriding financial component of any planning process is to make sure that the recommended plans and projects will be able to pay for
themselves. Revenues are needed to recover
construction costs, if any, and to maintain, repair,
and operate any infrastructure designed to manage the basin’s water resources. This may require
cost-recovery policies that involve pricing the
outputs of projects. Recognizing water as an
economic good does not always mean that full
costs should be charged. Poor people have the
right to safe water and how this is to be achieved
should be taken into account. Yet beneficiaries
should be expected to pay at least something for
the added benefits they get. Planning must identify equitable cost and risk-sharing policies and
improved approaches to risk/cost management.

Financial viability is often viewed as a constraint that must be satisfied. It is not viewed as
an objective whose maximization could result in
a reduction in economic efficiency, equity, or
other nonmonetary objectives. In many developing countries a distinction is made between the
recovery of investment costs and the recovery of
O&M costs. Recovery of O&M costs is a minimum condition for a sustainable project. Without
that, it is likely that the performance of the project will deteriorate over time.
Many past failures in water resources management are attributable to the fact that water—its
quantity, reliability, quality, pressure, location—
has been and still is viewed as a free good. Prices
paid for irrigation and drinking water are in many
countries well below the full cost of the infrastructure and personnel needed to provide that
water, which comprises the capital charges
involved, the operation and maintenance (O&M)
costs, the opportunity cost, economic and environmental externalities (see GWP 2000). Charging for water at less than full cost means that the
government, society, and/or environment “subsidizes” water use and leads to an inefficient use of
the resource.

1.5.5.3 Institutional and Governance
The first condition for the successful implementation of plans and policies is to have an enabling
environment. There must exist national, provincial, and local policies, legislation and institutions that make it possible for the desired
decisions to be taken and implemented. The role
of the government is crucial. The reasons for
governmental involvement are manifold:
• Water is a resource beyond property rights: it
cannot be “owned” by private persons. Water
rights can be given to persons or companies,
but only the rights to use the water and not to
own it. Conflicts between users automatically
turn up at the table of the final owner of the
resource—the government.
• Water is a resource that often requires large
investments to develop, treat, store, distribute,

1.5 Planning and Management Approaches

and use, and then to collect, treat, and dispose
or reuse. Examples are multipurpose reservoirs and the construction of dykes along
coasts and rivers. The required investments
are large and typically can only be made by
governments or state-owned companies.
• Water is a medium that can easily transfer
external effects. The use of water by one
activity often has negative effects on other
water using activities (externalities). The
obvious example is the discharge of wastewater into a river may save the discharger
money but it may have negative effects on
downstream users requiring cleaner water.
Only the government can address many of
these issues and hence “good governance” is
necessary for good water management. An
insufficient institutional setting and the lack of a
sound economic base are the main causes of
water resources development project failure, not
technical inadequacy of design and construction.
This is also the reason why at present much
attention is given to institutional developments
and governance in both developed and developing regions and countries.
In Europe, various types of water agencies are
operational (e.g., the Agence de l’Eau in France
and the water companies in England), each
having advantages and disadvantages. The Water
Framework Directive of the European Union
requires that water management be carried out at
the scale of a river basin, particularly when this
involves transboundary management. It is very
likely that this will result in a shift in responsibilities of the institutions involved and the
establishment of new institutions. In other parts
of the world experiments are being carried out
with various types of river basin organizations,
combining local, regional, and sometimes
national governments.

1.5.5.4 Models for Impact Prediction
and Evaluation
Planning processes have undergone a significant
transformation over the past five decades, mainly
due to the continuing development of improved
computational technology. Planning today is

39

heavily dependent on the use of computer-based
impact prediction models. Such models are used
to assist in the identification and evaluation of
alternative ways of meeting various planning and
management objectives. They provide an efficient way of using spatial and temporal data in an
effort to predict the interaction and impacts, over
space and time, of various river basin components under alternative designs and operating
policies.
Many of the systems analysis approaches and
models discussed in the following chapters of
this book have been, and continue to be, central
to the planning and management process. Their
usefulness is directly dependent on the quality of
the data and models being used. Models can
assist planning and management at different
levels of detail. Some models are used for preliminary screening of alternative plans and policies, and as such do not require major data
collection efforts. Screening models can also be
used to estimate how significant certain data and
assumptions are to the decisions being considered, and hence can help guide additional data
collection activities. At the other end of the
planning and management spectrum, much more
detailed models can be used for engineering design. These more complex models are more data
demanding, and typically require higher levels of
expertise for their proper use.
The integration of modeling technology into
the social and political components of the planning and management processes in a way that
enhances those processes continues to be the
main challenge of those who develop planning
and management models. Efforts to build and
apply interactive generic modeling programs or
“shells” into which interested stakeholders can
“draw in” their system, enter their data and operating rules at the level of detail desired, simulate it, and discover the effect of alternative
assumptions and operating rules, has in many
cases helped to create a common or shared
understanding among these stakeholders. Getting
stakeholders involved in developing and experimenting with their own interactive data-driven
models has been an effective way of building a
consensus—a shared vision.

40

1 Water Resources Planning and Management: An Overview

1.5.5.5 Models for Shared Vision
or Consensus Building
Participatory planning involves conflict management. Each stakeholder or interest group has
its objectives, interests, and agendas. Some of
these may be in conflict. The planning and
management process is one of negotiation and
compromise. This takes time but from it can
come decisions that have the best chance of
being considered the right decisions by most
participants. Models can assist in this process of
reaching a common understanding and agreement among different stakeholders. This has a
greater chance of happening if the stakeholders
themselves are involved in the modeling process.
Involving stakeholders in collaborative model
building accomplishes a number of things. It
gives them a feeling of ownership. They will
have a much better understanding of just what
their model can do and what it cannot do. If they
are involved in model building, they will know
the assumptions built into their model.
Being involved in a modeling exercise is a
way to understand better the impacts of various
assumptions one must make when developing
and running models. While there may be no
agreement on the best of various assumptions to
make, stakeholders can learn which of those
assumptions matter and which do not. In addition, the involvement of stakeholders in the
process of model development will create discussions that will lead toward a better understanding of everyone’s interests and concerns.
Though such model building exercises, it is just
possible those involved will reach not only a
better understanding of everyone’s concerns, but
also a common or “shared” vision of at least how
their system (as represented by their model, of
course) works.
1.5.5.6 Models for Adaptive
Management
Recent emphasis has shifted from structural
engineering solutions to more nonstructural alternatives, especially for environmental and
ecosystem restoration. Part of this shift reflects
the desire to keep more options open for future
generations. It reflects the desire to be adaptive to

new information and to respond to surprises—
impacts not forecasted. As we learn more about
how river basins, estuaries, and coastal zones
work, and how humans can better manage those
resources, we do not want to regret what we have
done in the past that may preclude this
adaptation.
In some situations, it may be desirable to
create a “rolling” plan—one based on the results
of an optimization or simulation model of a
particular water resource system that can be
updated at any time. This permits responses to
resource management and regulatory questions
when they are asked, not just at times when new
planning and management exercises take place.
While this appears to be desirable, will planning
and management organizations have the financing and support to maintain and update the
modeling software used to estimate various
impacts, collect and analyze new data, and
maintain the expertise, all of which are necessary
for continuous planning (rolling plans)?

1.6

Planning and Management
Characteristics

1.6.1 Integrated Policies
and Development Plans
Clearly, a portion of any water resources planning and management study report should contain a discussion of the particular site-specific
water resource management issues and options.
Another part of the report might include a prioritized list of strategies for addressing existing
problems and available development or management opportunities in the basin.
Recent emphasis has shifted from structural
engineering solutions to more nonstructural alternatives, especially for environmental and
ecosystem restoration. Part of this shift reflects
the desire to keep more options open for future
generations. It reflects the desire to be adaptive to
new information and to respond to surprises—
impacts not forecasted. As we learn more about
how river basins, estuaries, and coastal zones
work, and how humans can better manage their

1.6 Planning and Management Characteristics

water resources, we do not want to be regretting
what we have done in the past that may preclude
this adaptation.
Consideration also needs to be given to
improving the quality of the water resources
planning and management review process and
focusing on outcomes themselves rather than
output measures. One of the outcomes should be
an increased understanding of some of the relationships between various human activities and
the hydrology and ecology of the basin, estuary,
or coastal zone. Models developed for predicting
the economic as well as ecologic interactions and
impacts due to changes in land and water management and use could be used to address
questions such as:
• What are the hydrologic, ecologic, and economic consequences of clustering or dispersing human land uses such as urban and
commercial developments and large residential areas? Similarly, what are the consequences of concentrated versus dispersed
patterns of reserve lands, stream buffers, and
forestland?
• What are the costs and ecological benefits of a
conservation strategy based on near-stream
measures (e.g., riparian buffers) versus
near-source (e.g., upland/site edge) measures?
What is the relative cost of forgone upland
development versus forgone valley or riparian
development? Do costs strongly limit the use
of stream buffer zones as mitigating for
agriculture,
residential,
and
urban
developments?
• Should large intensive developments be best
located in upland or valley areas? Does the
answer differ depending on economic, environmental, or aquatic ecosystem perspectives? From the same perspectives, is the
most efficient and desirable landscape highly
fragmented or highly zoned with centers of
economic activity?
• To what extent can riparian conservation and
enhancement mitigate upland human land use
effects? How do the costs of upland controls

41

•

•

•
•

•

•

•

•

compare with the costs of riparian mitigation
measures?
What are the economic and environmental
quality trade-offs associated with different
areas of different classes of land use such as
commercial/urban, residential, agriculture,
and forest?
Can adverse effects on hydrology, aquatic
ecology, and water quality of urban areas be
better mitigated with upstream or downstream
management approaches? Can land controls
like stream buffers be used at reasonable cost
within urban areas, and if so, how effective
are they?
Is there a threshold size for residential/
commercial areas that yield marked ecological effects?
What are the ecological states at the landscape scale that once attained become irreversible with reasonable mitigation measures?
For example, once stream segments in an
urban setting become highly altered by direct
and indirect effects (e.g., channel bank protection and straightening and urban runoff),
can they be restored with feasible changes in
urban land use or mitigation measures?
Mitigating flood risk by minimizing floodplain developments coincides with conservation of aquatic life in streams. What are
the economic costs of this type of risk
avoidance?
What are the economic limitations and ecologic benefits of having light residential zones
between waterways and commercial, urban,
or agriculture lands?
What are the economic development decisions that are irreversible on the landscape?
For example, once land is used for commercial development, it is normally too costly to
return it to agricultural land. This would
identify limits on planning and management
for conservation and development.
What are the associated ecological and economic impacts of the trend in residential,
commercial and forests lands replacing agricultural lands?

42

1 Water Resources Planning and Management: An Overview

The answers to these and similar questions
may well differ in different regions. However, if
we can address them on a regional scale, i.e., in
multiple river basins, we just might begin to
understand and predict better the interactions
among economy, environment ecology, and
people as a function of how we manage and use
its land and water. This in turn may help us better
manage and use our land and water resources for
the betterment of all—now and on into the future.

1.6.2 Sustainability
Sustainable water resource systems are those
designed and managed to best serve people living
in the future as well as those of us living today.
The actions that we as a society take now to
satisfy our own needs and desires should not
only depend on what those actions will do for us
but also on how they will affect our descendants.
This consideration of the long-term impacts on
future generations of actions taken now is the
essence of sustainable development. While the
word “sustainability” can mean different things
to different people, it always includes a consideration of the welfare of those living in the
future. While the debate over a more precise
definition of sustainability will continue, and
questions over just what it is that should be
sustained may remain unanswered, this should
not delay progress toward achieving water
resource systems that we judge best serves those
of us living today as well as our children and
their children living in the future.
The concept of environmental and ecological
sustainability has largely resulted from a growing
concern about the long-run health of our planet.
There is increasing evidence that our present
resource use and management activities and
actions, even at local levels, can significantly
affect the welfare of those living within much
larger regions in the future. Water resource
management problems at a river basin level are
rarely purely technical and of interest only to
those living within the individual river basins
where those problems exist. They are

increasingly related to broader societal structures,
demands, and goals.
What would future generations like us to do
for them? We do not know, but we can guess. As
uncertain as these guesses will be, we should
take them into account as we act to satisfy our
own immediate needs, demands, and desires.
There may be trade-offs between what we wish
to do for ourselves in our current generation
versus what we think future generations might
wish us to do for them. These trade-offs, if any,
between what present and future generations
would like should be considered. Once identified,
or at least estimated, just what decisions to make
should be debated and decided in the political
arena. There is no scientific theory to help us
identify which trade-offs, if any, are optimal.
The inclusion of sustainability criteria along
with the more common economic, environmental, ecological, and social criteria used to evaluate
alternative water resources development and
management strategies may identify a need to
change how we commonly develop and use our
water resources. We need to consider the impacts
of change itself. Change over time is certain; just
what it will be is uncertain. These changes will
impact the physical, biological, and social
dimensions of water resource systems. An
essential aspect in the planning, design and
management of sustainable systems is the anticipation of change. This includes change due to
geomorphologic processes, to aging of infrastructure, to shifts in demands or desires of a
changing society, and even due to increased
variability of water supplies, possibly because of
a changing climate. Change is an essential feature of sustainable water resources development
and management.
Sustainable water resource systems are those
designed and operated in ways that make them
more adaptive, robust, and resilient to an uncertain and changing future. Sustainable water
resource systems must be capable of effectively
functioning under conditions of changing supplies, management objectives, and demands.
Sustainable systems, like any others, may fail,
but when they fail they must be capable of

1.6 Planning and Management Characteristics

recovering and operating properly without undue
costs.
In the face of certain changes, but with
uncertain impacts, an evolving and adaptive
strategy for water resources development, management, and use is a necessary condition of
sustainable development. Conversely, inflexibility in the face of new information and new objectives and new social and political
environments is an indication of reduced system
sustainability. Adaptive management is a process
of adjusting management actions and directions,
as appropriate, in light of new information on the
current and likely future condition of our total
environment and on our progress toward meeting
our goals and objectives. Water resources
development and management decisions can be
viewed as experiments, subject to modification—
but with goals clearly in mind. Adaptive management recognizes the limitations of current
knowledge and experience and that we learn by
experimenting. It helps us move toward meeting
our changing goals over time in the face of this
incomplete knowledge and uncertainty. It accepts
the fact that there is a continual need to review
and revise management approaches because of
the changing as well as uncertain nature of our
socioeconomic and natural environments.
Changing the social and institutional components of water resource systems are often the
most challenging because they involve changing
the way individuals think and act. Any process
involving change will require that we change our
institutions—the rules under which we as a
society function. Individuals are primarily
responsible for, and adaptive to, changing political and social situations. Sustainability requires
that public and private institutions also change
over time in ways that are responsive to the needs
of individuals and society.
Given the uncertainty of what future generations will want, and the economic, environmental,
and ecological problems they will face, a guiding
principle for the achievement of sustainable water
resource systems is to provide options that allow
future generations to alter such systems. One of
the best ways to do this is to interfere as little as
possible with the proper functioning of natural

43

life cycles within river basins, estuaries, and
coastal zones. Throughout the water resource
system planning and management process, it is
important to identify all the beneficial and adverse
ecological, economic, environmental, and social
effects—especially the long-term effects—associated with any proposed planning and management project.

1.7

Meeting the Planning
and Management
Challenges—A Summary

Planning (the formulation of development and
management plans and policies) is an important
and often indispensable means to support and
improve operational management. Planning provides an opportunity to:
• assess the current state of the water resources
and the conflicts and priorities over their use,
formulate visions, set goals and targets, and
thus orient operational management,
• provide a framework for organizing policy
relevant research and public participation,
• increase the legitimacy, public acceptance of,
or even support for how the resources are to
be allocated or controlled, especially in times
of stress, and
• facilitate the interaction, discussion, and
coordination among managers and stakeholders, and generate a common point of
reference—a management plan or policy.
Many of the concerns and issues being
addressed by water resources planners and
managers today are similar to those faced by
planners and managers in the past. But some are
different. Most of the new ones are the result of
two trends: (1) a growing concern for the sustainability of natural ecosystems and (2) an
increased recognition for the need of the
bottom-up “grassroots” participatory approach to
planning, managing, and decision-making.
Today planners work for economic development and prosperity as they did in the past,
keeping in mind environmental impacts and

44

1 Water Resources Planning and Management: An Overview

goals as they have done in the past, but now
recognizing ecological impacts and values as
well. Water resources management may still be
focused on controlling and mitigating the adverse
impacts of floods and droughts and water pollution, on producing hydropower, on developing
irrigation, on controlling erosion and sediment,
and on promoting navigation, but only as these
and similar activities are compatible with healthy
ecosystems. Natural ecosystems generally benefit
from the variability of natural hydrologic
regimes. Other users prefer less variability. Much
of our engineering infrastructure is operated so as
to reduce hydrologic variability. Today water
resource systems are increasing, required to
provide rather than reduce hydrologic (and
accompanying sediment load) variability.
Reservoir operators, for example, can modify
their water release policies to increase this variability. Farmers and land use developers must
minimize rather than encourage land-disturbing
activities. Floodplains may need to get wet
occasionally. Rivers and streams may need to
meander and fish species requiring habitats along
the full length of rivers to complete their life
cycles must have access to those habitats. Clearly
these ecological objectives, added to all the other
economic and environmental ones, can only
compound the conflicts and issues with respect to
land and water management and use.
So, how can we manage all this conflict and
uncertainty? We know that water resources
planning and management should be founded on
sound science, efficient public program administration, and broad participation of stakeholders.
Yet obtaining each of these three conditions is a
difficult challenge. While the natural and social
sciences can help us predict the economic, environmental, and ecological impacts of alternative
decisions, those predictions are never certain. In
addition, these sciences offer no help in determining the best decision to make in the face of
multiple conflicting goals held by multiple
stakeholders—goals that have changed, and no
doubt will continue to change. Water resources
planning and management and decision-making
are not as easy as “we professionals can tell you
what to do. All you need is the will to do it.” Very

often it is not clear what should be done. Professionals administering the science, often from
public agencies, nongovernmental organizations,
or even from universities, are merely among all
the stakeholders having an interest in and contributing to the management of water.
Each governmental agency, consulting firm,
environmental interest group, and citizen typically has its own limitations, authorities, expertise and conflicts with other people, agencies and
organizations, all tending to detract from
achieving a fully integrated approach to water
resources planning and management. But just
because of this, the participation and contributions of all these stakeholders are needed. They
must come together in a partnership if indeed an
integrated approach to water resources planning
and management is to be achieved and sustained.
All views must be heard, considered, and acted
upon by all involved in the water resources
planning and management process.
Water resources planning and management is
not simply the application and implementation of
science. It is creating a social environment that
gets all of us who should be involved, from the
beginning, in a continuing planning process. This
process is one of
• educating ourselves about how our systems
work and function,
• identifying existing or potential options and
opportunities for enhancement and resource
development and use,
• resolving the inevitable problems and conflicts that will result over who gets what and
when and who pays who for what and when,
• making and implementing decisions, and
finally of
• monitoring the impacts of those decisions.
This process is repeated as surprises or new
opportunities or new knowledge dictates.
Successful water resources planning and
management requires the active participation of
all community institutions involved in economic
development and resource management. How
can this begin at the local stakeholder level? How
does anyone get others interested in preventing

1.7 Meeting the Planning and Management Challenges—A Summary

problems before those problems are apparent, or
especially before “unacceptable” solutions are
offered to deal with them? And how do you deal
with the inevitable group or groups of stakeholders who see it in their best interest not to
participate in the planning process, but to just
criticize it from the outside? Who is in a position
at the local level to provide that leadership and
needed financial support? In some regions, nongovernmental institutions have been instrumental
in initiating and coordinating this process at local
grassroot levels.
Water resources planning and management
processes should identify a vision that guides
development and operational activities in the
affected region. Planning and management processes should
• recognize and address the goals and expectations of the region’s stakeholders,
• identify and respond to the region’s
water-related problems,
• function effectively within the region’s
legal/institutional frameworks,
• accommodate both short- and long-term issues,
• generate a diverse menu of alternatives,
• integrate the biotic and abiotic parts of the
basin,
• take into account the allocation of water for
all needs, including those of natural systems,
• be stakeholder-driven,
• take a global perspective,
• be flexible and adaptable,
• drive regulatory processes, not be driven by them,
• be the basis for policy making,
• foster coordination among planning partners
and consistency among related plans,
• be accommodating of multiple objectives,
• be a synthesizer, recognize and deal with
conflicts, and
• produce recommendations that can be
implemented.
All too often integrated planning processes are
hampered by the separation of planning, management and implementing authorities, turf-protection
attitudes, shortsighted focusing of efforts, lack of

45

objectivity on the part of planners, and inadequate
funding. These deficiencies need addressing if
integrated holistic planning and management is to
be more than just something to write about.
Effective water resources planning and management is a challenge today, and will be an
increasing challenge into the foreseeable future.
This book introduces some of the tools that are
being used to meet these challenges. We consider
it only a first step toward becoming an accomplished planner or manager.

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1 Water Resources Planning and Management: An Overview

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Exercises
1:1 How would you define “Integrated Water
Resources Management” and what distinguishes it from “Sustainable Water
Resources Management”?
1:2 Can you identify some common water
management issues that are found in many
parts of the world?
1:3 Comment on the common practice of governments giving aid to those in drought or
flood areas without any incentives to alter
land use management practices in anticipation of the next drought or flood.
1:4 What tools and information are available for
developing integrated water resources plans
and management policies?
1:5 What structural and nonstructural measures
can be taken to address water resources
issues?
1:6 Find the following statistics:
• Percent of all freshwater resources
worldwide available for drinking;

Exercises

49

provide additional economic, environmen• Number of people who die each year
tal, or social benefits.
from diseases associated with unsafe
1:9. Describe some water resource systems condrinking water;
sisting of various interdependent natural,
• Percent of total freshwater resources in
physical, and social components. What are
polar regions;
the inputs to the systems and what are their
• Per capita annual withdrawal of cubic
outputs? How did you decide what to include
meters of freshwater in various countries;
in the system and what not to include?
• Average world per capita annual with1:10. Sustainability is a concept applied to
drawal of cubic meters of freshwater;
renewable resource management. In your
• Tons of pollutants entering lakes and
words define what that means and how it
rivers daily in various regions;
can be used in a changing and uncertain
• Average number of gallons of water
environment both with respect to water
consumed by humans in a lifetime;
supplies and demands. Over what space and
• Average number of kilometers per day a
timescales is it applicable, and how can one
woman in a developing country must
decide whether or not some plan or manwalk to fetch fresh water.
agement policy will be sustainable? How
1:7 Identify and briefly describe the six greatest
does this concept relate to the adaptive
rivers in the world.
management concept?
1:8. Identify some of the major water resource
management issues in the region where you 1:11. Identify and discuss briefly some of the
major issues and challenges facing water
live. What management alternatives might
managers today.
effectively reduce some of the problems or

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2

Water Resource Systems Modeling:
Its Role in Planning and Management

Planning, designing, and managing water
resource systems today inevitably involve impact
prediction. Impact prediction can be aided by the
use of models. While acknowledging the
increasingly important role of modeling in water
resource planning and management, we also
acknowledge the inherent limitation of models as
representations of any real system. Model structure, input data, objectives, and other assumptions
related to how the real system functions or will
behave under alternative infrastructure designs
and management policies or practices may be
controversial or uncertain. Future events are
always unknown and of course any assumptions
about them may affect model outputs, i.e., their
predictions. As useful as they may be, the results
of any quantitative analysis are always only a part
of the information that should be considered by
those involved in the overall planning and management decision-making process.

2.1

Introduction

Modeling provides a way, perhaps the principal
way, of predicting the behavior or performance of
proposed system infrastructure designs or management policies. The past 50 years have witnessed major advances in our abilities to model
the engineering, economic, ecologic, hydrologic,
and sometimes even the institutional or political
aspects of large complex multipurpose water
resource systems. Applications of models to real

systems have improved our understanding of
such systems, and hence have often contributed to
improved system design, management, and
operation. They have also taught us how limited
our modeling skills remain.
When design and management decisions are
made, they are based on what the decisionmakers assume will take place as a result of their
decisions. These predictions are based on qualitative information and beliefs in peoples’ heads,
as illustrated in Fig. 2.1, possibly informed by
quantitative information provided by mathematical or computer-based models as illustrated in
Fig. 2.2. Computer-based modeling is used to
enhance mental models. These quantitative
mathematical models are often considered
essential for carrying out environmental impact
assessments. Mathematical simulation and optimization models packaged within interactive
computer programs provide a common way for
planners and managers to predict the behavior of
any proposed water resources system design or
management policy before it is implemented.
Water resource systems are typically far more
complex than what analysts can model and
simulate. The reason is not primarily due to
computational limitations but rather it is because
we do not understand sufficiently the multiple
interdependent physical, biochemical, ecological,
social, legal, and political (human) processes that
govern the behavior of such water resource systems. People and their institutions impact the
performance of such systems, and the

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_2

51

52

2 Water Resource Systems Modeling: Its Role in Planning …

Fig. 2.1 Using mental models for prediction

performance of these systems impacts people.
System performance is affected by uncertainties
in things we can measure and processes we can
predict. They are also affected by the unpredictable actions of individuals and institutions as
they manage and use water in response to a
multitude of impacts they experience in their
physical and social environment. Some of these
impacts are water related. Others have nothing
directly to do with water.
The development and application of models,
i.e., the art, science, and practice of modeling, as
will be discussed in the following chapters,
should be preceded by a recognition of what can
and cannot be achieved from the use of models.
Models of real-world systems are always simplified representations of those systems. What
features of the actual system are represented in a

model, and what features are not, will depend in
part on what the modeler thinks is important with
respect to the issues being discussed or the
questions being asked. How well this is done will
depend on the skill of the modeler, the time and
money available, and, perhaps most importantly,
the modeler’s understanding of the real system
and decision-making process.
Developing models is an art. It requires
knowledge of the system being modeled, the
client’s objectives, goals, and information needs,
and some analytical and programming skills.
Models are always based on numerous assumptions or approximations, and some of these may
be at issue. Applying these approximations of
reality in ways that improve understandings and
eventually lead to a good decision clearly
requires not only modeling skills but also the

2.1 Introduction

ability to communicate and effectively work with
stakeholders and decision-makers.
Models produce information. They do not
make decisions or replace those individuals that
do. But they can inform them. Water resource
planners and managers must accept the fact that
decisions may not be influenced by the results of
their planning and management models. If model
results are not available when needed, they are
likely to be ignored when they become available.
If model results do not support the preferences of
decision-makers, they may also not be considered. To know, for example, that cloud seeding
may, on average, reduce the strength of hurricanes over a large region does not mean that such
cloud-seeding activities will or should be
undertaken. And it is unlikely everyone, even
so-called experts, will agree on any recommended course of action. Managers or operators
may know that not everyone may benefit from
what they would like to do, and those who lose
will likely scream louder than those who gain.
In addition, decision-makers may feel safer in
inaction than action (Shapiro 1990; Simon 1998).
There is a strong feeling in many cultures and
legal systems that fail to act (nonfeasance) is
considered more acceptable than acts that fail
(misfeasance or malfeasance). We all feel greater
responsibility for what we do than for what we
do not do. Yet our aversion to risks of failure
should not deter us from addressing sensitive
planning or policy issues in our models.
Modeling efforts should be driven by the need for
information and improved understanding. It is
that improved understanding (not improved
models per se) that may eventually lead to
improved system design, management, and/or
operation. Models used to aid water resource
planners and managers are not intended to be,
and rarely are (if ever), a replacement of their
judgment. This we have learned, if nothing else,
in our over 50 years of modeling experience.
This brief chapter serves to introduce this art of
modeling and its applications. The emphasis
throughout this book is on application. This
chapter is about modeling in practice more than in
theory. It is based on the considerable experience
and literature pertaining to how well, or how

53

poorly, professional practitioners and researchers
have done over the past five decades or more in
applying various modeling approaches or tools to
real problems with real clients (also see, for
example, Austin 1986; Brown et al. 2015; Cai
et al. 2013; Castelletti and Soncini-Sessa 2007;
Gass 1990; Kindler 1987, 1988; Loucks et al.
1985; Reynolds 1987; Rogers and Fiering 1986;
Russell and Baumann 2009; Watkins 2013).
In attempting to understand how modeling
can better support planners and managers, it may
be useful to examine just what planners and
managers of complex water resource systems do.
What planners or managers do governs to some
extent what they need to know. And what they
need to know governs to a large extent what
modelers or analysts should be trying to provide.
In this book the terms analysts or modelers,
planners, and managers can be the same person
or group of individuals. These terms are used to
distinguish the activities of individuals, not necessarily the individuals themselves.
First, we offer some general thoughts on the
major challenges facing water resource systems
planners and managers, the information they
need to meet these challenges, and the role analysts have in helping to provide this information.
Next, we review some criteria for evaluating the
success of any modeling activity designed to help
planners or managers solve real problems.
Finally, we argue why we think the practice of
modeling is in a continual state of transition, and
how current research and development in modeling as well as improvements in computing
technology are affecting that transition.

2.2

Modeling Water Resource
Systems

As will be discussed in greater detail in the following chapters of this book, there are many
types of models and modeling approaches that
have been developed and used to identify, study,
and evaluate alternative water resource designs,
management plans, and operating policies. But
before outlining these model types and modeling
approaches and how they can be used to best

2 Water Resource Systems Modeling: Its Role in Planning …

54
Fig. 2.2 Using
computer-based
mathematical models for
prediction

meet the needs of planners and managers, it may
be useful to describe a specific modeling example based on Borsuk et al. (2001). In this
example, a sequence of models are used to assess
how effective reductions in upstream nutrient
runoff may be in improving the habitat for fish
and shellfish in a downstream estuary.
This example is followed by a discussion of
the conditions needed that motivate the use of
models, whether solely mental (Fig. 2.1) or both
mental and mathematical (Fig. 2.2).

2.2.1

An Example Modeling
Approach

Consider for example the sequence or chain of
models illustrated in Fig. 2.3 required for the
prediction of fish and shellfish survival as a
function of nutrient loadings into an estuary. The
condition of the fish and shellfish are important
to the economy of the region and the income of
many stakeholders. One way to maintain healthy

stocks is to maintain sufficient levels of oxygen
in the estuary. The way to do this is to control
algae blooms. This in turn requires limiting the
nutrient loadings to the estuary that can cause
algae blooms and subsequent dissolved oxygen
deficits. The modeling challenge is to link
nutrient loading to fish and shellfish survival. In
other words, can some quantitative relationship
be defined relating the amount of nutrient loading
to the amount of fish and shellfish survival?
The negative effects of excessive nutrients
(e.g., nitrogen) in an estuary are shown in
Fig. 2.3. Nutrients stimulate the growth of algae.
Algae die and accumulate on the bottom where
bacteria consume them. Under calm wind conditions density stratification occurs. Oxygen is
depleted in the bottom water. As a consequence,
fish and shellfish may die or become weakened
and more vulnerable to disease.
A sequence of models, each providing input
data to the next model, can be defined to predict
shellfish and fish abundance in the estuary based
on upstream nutrient loadings. These models, for

2.2 Modeling Water Resource Systems

55

Fig. 2.3 The impacts of excessive nutrients in an estuary

each link shown in Fig. 2.4, can be a mix of
judgmental, mechanistic, and/or statistical ones.
Statistical models could range from simple
regressions to complex artificial neural networks.
Any type of model selected will have its
advantages and its limitations. Its appropriateness
may largely depend on the amount and precision
of the data available for model calibration and
verification.
The results of any modeling exercise should
be expressed in terms meaningful and of interest
to those that will be making decisions taking into
account those results. In this example ‘shell-fish
abundance’ and ‘number of fish-kills’ are meaningful indicators to stakeholders and can be
related to designated water body use.

2.2.2

Characteristics of Problems
to be Modeled

Problems motivating modeling and analyses
exhibit a number of common characteristics.
These are reviewed here because they provide
insight into whether a modeling study of a particular problem may be worthwhile. If the planners’ objectives are very unclear, if few
alternative courses of action exist, or if there is
little scientific understanding of the issues
involved, then mathematical modeling and other
more sophisticated methodologies are likely to
be of little use.
Successful applications of modeling are often
characterized by:

56

2 Water Resource Systems Modeling: Its Role in Planning …

Fig. 2.4 Cause and effect diagram for estuary eutrophication due to excessive nutrient loadings (after Borsuk et al.
2001)

• A systems focus or orientation. In such situations attention needs to be devoted to the
interdependencies and interactions of elements or components within the system as a
whole, as well as to the elements or components themselves.

• The use of interdisciplinary teams. In many
complex and nontraditional problems, it is not
at all clear from the start what mix of disciplinary viewpoints will turn out to be most
appropriate or acceptable. It is essential that
participants in such work—coming from

2.2 Modeling Water Resource Systems

different established disciplines—become
familiar with the techniques, vocabulary, and
concepts of the other disciplines involved.
Participation in interdisciplinary modeling
often requires a willingness to make mistakes
at the fringes of one’s technical competence
and to accept less than the latest advances in
one’s own discipline.
• The use of formal mathematics. Most analysts
prefer to use mathematical models to assist in
system description and the identification and
evaluation of efficient tradeoffs among
conflicting objectives, and to provide an
unambiguous record of the assumptions and
data used in the analysis.
Not all water resources planning and management problems are suitable candidates for
study using modeling methods. Modeling is most
likely to be appropriate when:
• The planning and management objectives are
reasonably well defined, and organizations
and individuals can be identified who can
benefit from obtaining and understanding the
model results.
• There are many alternative decisions that may
satisfy the stated objectives, and the best
decision is not obvious.
• The water resources system and the objectives
being analyzed are describable by reasonably
tractable mathematical representations.
• The information needed, such as the hydrological, economic, environmental, and ecological impacts resulting from any decision,
can be better estimated through the use of
models.
• The values of the model parameters are
estimable from readily obtainable data.

2.3

Challenges Involving Modeling

Modeling activities present challenges to those
who do it as well as those who sponsor it and
may potentially benefit from model results.

57

2.3.1

Challenges of Planners
and Managers

Planners and managers of water resource systems
are
responsible
for
solving
particular
water-related problems or meeting special water
resource needs. When they fail, they hear about
it. The public lets them know. (Example: the lead
contamination in the drinking water of Flint,
Michigan USA, after a switch in the water source
to reduce costs.) What makes their job particularly challenging is that stakeholders often have
different needs and expectations. Furthermore,
institutions where water resource planners and
managers work (or hire consultants to work for
them) are like most institutions these days. They
must do what they have been asked to do with
limited financial and human resources. Their
clients include all of us who use water, or at least
all of us who are impacted by the decisions they
make.
The overall objective of planners, managers,
and operators and their institutions is to provide a
service, such as reliable and inexpensive supplies
of water, assurance of water quality, production
of hydropower, protection from floods, provision
of commercial navigation and recreational
opportunities, preservation of wildlife and
enhancement of ecosystems, or some combination of these or other purposes. Furthermore they
are expected to do this at a cost no greater than
what people are willing to pay. Meeting these
goals, i.e., keeping everyone happy, is not always
easy, or even possible.
Simple technical measures or procedures are
rarely available that will ensure a successful
solution to any particular set of water resource
management problems. Furthermore, everyone
who has had any exposure to water resources
planning and management knows one cannot
design or operate a water resource system without making compromises. These compromises
often involve competing purposes (such as
hydropower and flood control) or competing
objectives (such as who benefits and who pays,
and how much and where and when). After
analysts, using their models of course, identify

58

2 Water Resource Systems Modeling: Its Role in Planning …

possible ways of achieving various goals and
objectives and provide estimates of associated
economical, environmental, ecological, and
social impacts, it is the decision-makers who
have the more difficult job. They must work with
and influence everyone who will be affected by
any decision.
Planning and managing involves not only
decision-making, but also developing among all
interested and influential individuals an understanding and consensus that legitimizes the
decisions and enhances their successful implementation. Planning and managing are processes
that take place in a social or political environment. These processes involve leadership and
communication among people and institutions.
Leadership and communication skills are learned
from experience working with people, not sitting
alone working with computers or models.
Moving an organization or institution into action
to achieve specific goals involves a number of
activities, including goal-setting, debating, coordinating, motivating, deciding, implementing, and
monitoring. Many of these must be done simultaneously and continuously, especially as conditions
(goals and objectives, water supplies, water
demands, financial budgets) change over time.
These activities create a number of challenges that
are relevant to modelers or analysts. Some include:
1. identifying creative ways of solving
problems.
2. finding out what each interest group wants to
know in order to reach an understanding of
the issues and a consensus on what to do.
3. developing and using models and presenting
their results so that everyone can reach a
common or shared understanding and agreement that is consistent with their individual
values.
4. making decisions and implementing them
given differences in opinions, social values,
and objectives.
In addressing these needs or challenges,
planners, and managers must consider the
relevant

• legal rules and regulations;
• history of previous decisions;
• preferences of important actors and interest
groups;
• probable reactions of those affected by any
decision;
• relative importance of various issues being
addressed; and finally;
• sciences, engineering, and economics—the
technical aspects of their work.
We mention these technical aspects last not to
suggest that they are the least important factor to
be considered. We do this to emphasize that they
are only among many factors and, probably in
the eyes of planners and managers, not the most
decisive or influential (Ahearne 1988; Carey
1988; Pool 1990; Thissen and Walker 2013;
Walker 1987).
So, does the scientific, technical, systematic
approach to modeling for planning and management really matter? We believe it can if it
addresses the issues of concern to their clients,
the planners, and managers. Analysts need to be
prepared to interact with the political or social
structure of the institutions they are attempting to
assist, as well as with the public and the press.
Analysts should also be prepared to have their
work ignored. Even if analysts are presenting
‘facts’ based on the current state of the sciences,
sometimes these sciences are not considered
relevant. Happily for scientists and engineers,
this is not always the case. The challenge of
modelers or analysts interested in having an
impact on the performance of water resource
systems is to become a part of the largely political planning and management process and to
contribute towards its improvement.

2.3.2

Challenges of Modelers

To engage in a successful water resource systems
study, the modeler must possess not only the
requisite mathematical and systems modeling
skills, but also an understanding of the environmental engineering, economic, political, cultural,

2.3 Challenges Involving Modeling

and social aspects of water resources planning
problems. Consider, for example, the study of a
large land development plan. The planner should
be able to predict how the proposed development
would affect the quantity and quality of the surface
and subsurface runoff and how this will impact the
quantity and quality of surface waters and ground
waters and their ecosystems. These impacts, in
turn, might affect the planned development itself,
or others downstream. To do this the analysts must
have an understanding of the biological, chemical,
and physical and even social processes that are
involved in water resources management.
A reasonable knowledge of economic theory,
law, regional planning, and political science can
be just as important as an understanding of
hydraulic, hydrogeologic, hydrologic, ecologic,
and environmental engineering disciplines. It is
obvious that the results of most water resources
management decisions have a direct impact on
people and their relationships. Hence, inputs
from those having knowledge of these disciplines
are useful during the comprehensive planning of
water resource systems.
Some of the early water resource systems
studies were often undertaken with a naive view
of the appropriate role and impact of models and
modelers in the policymaking process. Policymakers could foresee the need to make a decision. They would ask the systems group to study
the problem. These analysts would then model
the problem, identify feasible solutions and their
consequences, and recommend one or at most a
few alternative solutions. The policymakers, after
waiting patiently for these recommendations,
would then make a yes or no decision. Experience to date suggests the following:
1. A final solution to a water resources planning
problem rarely exists; plans and policies are
dynamic. They evolve over time as facilities
are added and modified to adapt to changes in
management objectives and in the demands
placed on the facilities.
2. For every major decision there are many
minor decisions, made by different agencies
or management organizations responsible for
different aspects of a system.

59

3. The times normally available to study particular water resources problems are shorter
than the times needed to do a thorough study,
or if there is sufficient time, the objectives of
the original study will likely have significantly shifted by the time the study is
completed.
This experience emphasizes some of the limitations and difficulties that any water resource
systems study may encounter, but more importantly, it underscores the need for constant
communication among the analysts, system
planners, managers and operators, and policymakers. The success or failure of many past
water resource studies is due largely to the efforts
expended or not expended in ensuring adequate,
timely and meaningful communication—communication among systems analysts, planners,
those responsible for system operation and design, and public officials responsible for major
decisions and setting general policies.
Decision-makers, who can benefit from the
information that can be derived from various
models and analyses, need it at particular times
and in a form useful and meaningful to them.
Once their window of opportunity for
decision-making has passed, such information,
no matter how well presented, is often useless.
At the beginning of any study, objectives are
usually poorly defined. As more is learned about
what can be achieved, stakeholders are better
able to identify what they want to achieve. Close
communication among analysts and all interested
stakeholders and decision-makers throughout the
modeling process is essential if systems studies
are to make their greatest contribution to the
planning process. Objectives as stated at the
beginning of a study often differ from the objectives as understood at the end of a study.
Furthermore, it is helpful if those who will use
models, and present the information derived from
models to those responsible for making decisions, are intimately involved with model development, solution, and analysis. Only then can
they appreciate the assumptions upon which any
particular model output is based, and hence
adequately evaluate the reliability of the results.

60

2 Water Resource Systems Modeling: Its Role in Planning …

Any water resource systems study that involves
only outside consultants, and minimal communication between consultants and planners within
a responsible management agency or involved
stakeholders, is not likely to have a significant
impact on the planning process. Models that are
useful tend to be those that are constantly being
modified and applied by those involved in plan
preparation, evaluation, and implementation.

2.3.3

Challenges of Applying Models
in Practice

The clients of modelers or analysts are typically
those who have problems to solve and who could
benefit from a better understanding of what
options they have and what impacts may result.
They want advice on what to do and why, what
will happen given what they do, and who will care
and how much. The aim of analysts is to provide
them with meaningful (understandable), useful,
accurate, and timely information. This information is to help them better understand their system,
its problems, and alternative ways to address
them. In short, the purpose of water resource
systems planning and management modeling is to
provide useful and timely information to those
involved in managing such systems.
Modeling is a process or procedure intended
to focus and force clearer thinking and to promote better decision-making. The approach
involves problem recognition, system definition,
and bounding; identification of various goals or
objectives; identification and evaluation of various alternatives; and very importantly, effective
communication of this information to those who
can benefit from it.
The focus of most books and articles on water
resource systems modeling is on modeling
methods. This book is no different. But what all
of us should also be interested in, and discuss
more than we do, is the use of these tools in the
processes of planning and management. If we
did, we could learn much from each other about

what tools are needed and how they can be better
applied in practice. We could extend the thoughts
of those who, in a more general way, addressed
these issues over four decades ago (Majoni and
Quade 1980; Tomlison 1980; Miser 1980; Stokey and Zeckhauser 1977).
There is always a gap between what
researchers in water resource systems modeling
produce and publish, and what the practitioner
finds useful and uses. Those involved in research
are naturally interested in developing new and
improved tools and methods for studying, identifying, and evaluating alternative water resource
system designs and management and operation
policies. If there were no gap between what is
being developed or advocated by researchers and
that which is actually used by practitioners, either
the research community would be very ineffective in developing new technology or the practitioners would be incredibly skilled in reading,
assimilating, evaluating, and adapting what is
worth adapting from this research to meet their
needs. Evaluation, testing, and inevitable modifications take time. Not all published research is
ready or suited for implementation. By definition
research is a work in progress.
How can modelers help reduce the time it
takes for new ideas and approaches to be adopted
and used in practice? Clearly, practitioners are
not likely to accept a new modeling approach or
even modeling itself unless it is obvious that it
will improve the performance of their work as
well as help them address problems they are
trying to solve. Will some new model or computer program make it easier for practitioners to
carry out their responsibilities? If it will, there is
a good chance that the model or computer program might be successfully used, eventually.
Successful use of the information derived from
models or programs is, after all, the ultimate test
of the value of those models or programs. Peer
review and publication is only one, and perhaps
not even a necessary, step towards that ultimate
test or measure of value of a particular model or
modeling approach.

2.3 Challenges Involving Modeling

2.3.4

Evaluating Modeling Success

There are a number of ways one can judge success (or failure) in applying models in practice.
Goeller (1988) suggested three measures as a
basis for judging success:
1. How the analysis was performed and presented (analysis success);
2. How it was used or implemented in the
planning and management processes (application success); and
3. How the information derived from the model
and its application affected the system design
or operation and the lives of those who used
the system (outcome success).
The extent to which the models and methods
and style of presentation are appropriate for the
problem being addressed, the resources and time
available for the study, and the institutional
environment of the client, are often hard to judge.
Publishing in peer-review journals and review
panels are two ways of judging. No model or
method is without its limitations. Two other
obvious indications are the feeling analysts have
about their own work and, very importantly, the
feeling the clients have about the analysts’ work.
Client satisfaction may not be an appropriate
indicator if, for example, they are unhappy only
because they are learning something they do not
want to accept. Producing results primarily to
reinforce a client’s prior position or opinions
might result in client satisfaction but, most would
agree, this is not an appropriate goal of modeling.
Application or implementation success
implies that the methods and/or results developed
in the study were seriously considered by those
involved in the planning and management process. One should not, it seems to us, judge success or failure based on whether or not any of the
model results, i.e., the computer ‘printout,’ were
directly implemented. What one hopes for is that
the information and understanding resulting from
model application helped define and focus the
problem and possible solutions, and helped
influence the debate among stakeholders and
decision-makers about what decisions to make or

61

actions to take. The extent to which this occurs is
the extent to which a modeling study will have
achieved application or implementation success.
Outcome success is based on what happened
to the problem situation once a decision (that was
largely influenced by the results of modeling)
was made and implemented. The extent to which
the information and understanding resulting from
modeling helped solve the problems or resolve
the issues, if it can be determined, is a measure of
the extent of outcome success.
It is clear that success based on any of the last
two of the three criteria will be strongly dependent on the success of the preceding criteria.
Modeling applications may be judged successful
based on the first two measures, but perhaps
because of unpredicted events, the problems
being addressed have become worse rather than
improved, or while those particular problems
were eliminated, their elimination caused one or
more even more severe problems. All of us can
think of examples where this has happened. The
previously mentioned lead contamination in the
drinking water of Flint, Michigan, resulting from
trying to reduce costs is one example. Any river
restoration project involving the removal of
engineering infrastructure is another example of
changing objectives or new knowledge following
previous decisions that no longer work very well.
Who knows—a broader systems study might
have helped planners, managers, and decisionmakers foresee such consequences, but one
cannot count on that. Hindsight is always clearer
than foresight. Much of what takes place in the
world is completely unpredictable. Given this, it
is not clear whether we should hold modelers or
analysts, or even planners or managers, completely responsible for any lack of ‘outcome
success’ if unforeseen events change society’s
goals, priorities, and understanding.
Problem situations and criteria for judging the
extent of success are likely to change over time.
By the time one can evaluate outcome success,
the system itself may have changed enough for
the outcome to be quite different than what was
predicted in the analysis. Monitoring the performance of any decision, whether or not based on a
successfully
analyzed
and
implemented

2 Water Resource Systems Modeling: Its Role in Planning …

62

modeling effort, is often neglected. But monitoring is very important if changes in system
design, management, and operation are to be
made to adapt to changing and unforeseen
conditions.
If the models, data, computer programs, documentation, and know-how are successfully
maintained, updated, and transferred to and used
by the client institutions, there is a good chance
that this methodology will be able to provide
useful information relevant to the changes that are
needed in system design, management, or operation. Until relatively recently, the successful
transfer of models and their supporting technology have involved a considerable commitment of
time and money for both the analysts as well as
the potential users of the tools and techniques. It
has been a slow process. Developments in interactive computer-based decision support systems
that provide a more easily understood human–
model–data–computer interface have substantially facilitated this technology transfer process.
These interactive interface developments have
had a major impact on the state of the practice in
using models in the processes of water resources
planning and management.

2.4
2.4.1

Developments in Modeling
Technology

The increasing developments in computer technology—from mobile devices to microcomputers
and workstations to supercomputers—and all
their software applications—have motivated the
concurrent development of an impressive set of
new models and accompanying software. This
software is aimed at facilitating model use and,
more importantly, interaction and communication between the analysts or modelers and their
clients. This new software includes
1. Interactive approaches to model operation
that put users more in control of their computers, models, and data;
2. Computer graphics that facilitate data input,
editing, display, and comprehension;

3. Geographic information systems that provide
improved spatial analysis and display
capabilities;
4. Expert systems that can help the user understand better how complex decision problems
might be solved and at the same time explain
to the users why one particular decision may
be better than another;
5. Cloud computing, electronic mail, and the
Internet that lets analysts, planners, and
managers communicate and share data and
information with others worldwide, and to
run models that are located and maintained at
distant sites;
6. Multimedia systems that permit the use of
sound and video animation in analyses, all
aimed at improved communication and
understanding.
These and other software developments are
giving planners and managers improved opportunities for increasing their understanding of their
water resource systems. Such developments in
technology should continue to aid all of us in
converting model output data to information, i.e.,
it should provide us with a clearer knowledge
and understanding of the alternatives, issues, and
impacts associated with potential solutions to
water resource systems problems. But once
again, this improved information and understanding will only be a part of what planners and
managers must consider.
Will all the potential benefits of new technology actually occur? Will analysts be able to
develop and apply these continual improvements
in new technology wisely? Will we avoid another
case of oversell or unfulfilled promises? Will we
avoid the temptation of generating fancy animated, full-color computer displays just because
we are easily able to, rather than being motivated
by the hope that such methods will add to
improved understanding of how to solve problems
more effectively? Will we provide the safeguards
needed to ensure the proper use and interpretation
of the information derived from increasingly
user-friendly computer programs? Will we keep a
problem-solving focus, and continue to work
towards increasing our understanding of how to

2.4 Developments in Modeling

improve the development and management of our
water resources whether or not our planning
models are incorporated into some sort of interactive computer-aided support system? We can,
but it will take discipline.
As modelers or researchers, we must discipline
ourselves to work more closely with our clients—
the planners, managers, and other specialists who
are responsible for the development and operation
of our water resource systems. We must study
their systems and their problems, and we must
identify their information needs. We must
develop better tools that they themselves can use
to model their water resource systems and obtain
an improved understanding—a shared vision—of
how their system functions and of their available
management options and associated impacts or
consequences. We must be willing to be multidisciplinary and capable of including all relevant
data in our analyses. We must appreciate and see
the perspectives of the agronomists, ecologists,
economists, engineers, hydrologists, lawyers, or
political and regional scientists—you name it—as
appropriate. Viewing a water resource system
from a single-discipline perspective is rarely
sufficient to meet today’s water resource systems
planning challenges.
Even if we have successfully incorporated all
relevant disciplines and data in our analyses, we
should have a healthy skepticism about our
resulting information. We must admit that this
information, especially concerning what might
happen in the future, is uncertain. If we are looking
into the future (whether using crystal balls or
mathematical models), we must admit that many
of our assumptions, e.g., parameter values, cannot
even be calibrated let alone verified. Our conclusions or estimates can be very sensitive to our
assumptions. One of our major challenges is to
communicate this uncertainty in understandable
ways to those who ask for our predictions.

2.4.2

Algorithms

Accompanying the improvements in the technology of computing that has had an enormous
impact on the capability of analysts to address

63

and study increasingly complex issues in water
resource systems planning and management,
improvements made in the mathematical and
computational algorithms have permitted the
modeling of more complex systems problems.
All our algorithms that have been applied to the
analysis of water resource systems, have their
strengths and limitations. We still lack the ‘perfect’ all-purpose algorithm. And it is not likely
that we will find one in the future. Probably the
major determinant of a particular algorithm or
software package chosen to address a particular
problem or development opportunity is that
which the analyst is most familiar with and
experienced in using.
Nevertheless, the menu of available algorithms that can be used for analyses is considerably larger today than what it was when the
seminal book on the design of water resource
systems (Maas et al. 1962) was published over
six decades ago. At that time mathematical programming (constrained optimization) software
applied to mainly deterministic linear and nonlinear problems dominated the interests of those
working toward improved models for preliminary screening of water resource systems prior to
more detailed simulation modeling. Simulations
were based on software and constrained by the
internal and magnetic tape memory capacity of
computers available at that time. Today our focus
is more on methods suited for enhancing stakeholder participation. Much of it based on the
results of research in artificial intelligence,
examples including evolutionary search methods
based on biological processes, multi-agent modeling, artificial neural networks, and data mining
methods.

2.4.3

Interactive Model-Building
Environments

Water resources planners and managers today
must consider the interests and goals of numerous stakeholders. The planning, managing, and
decision-making processes involve negotiation
and compromise among these numerous stakeholders, such as those shown in Fig. 2.5, who

64

2 Water Resource Systems Modeling: Its Role in Planning …

Fig. 2.5 Stakeholders
involved in river basin
planning and management,
each having different goals
and information needs

typically have different interests, objectives and
opinions about how their water resource system
should be managed. How do we model to meet
the information needs of all these different
stakeholders? How can we get them to believe in
and accept these models and their results? How
do we help them reach a common—shared—vision? How can we help create a shared vision
among all stakeholders of at least how their
system works and functions, if not how they
would like it to?
Today we know how to build some rather
impressive models of environmental systems. We
know how to incorporate within our models the
essential biology, chemistry and physics that
govern how the environmental system works. We
have also learned a little about how to include the
relevant economics, ecology, and engineering
into these models. Why do we do this? We do all

this modeling simply to be able to estimate, or
identify, and compare and evaluate the multiple
impacts resulting from different design and
management decisions we might make. Such
information, we assume, should be of value to
those responsible for choosing the ‘best’ decision.
If our goal is to help contribute to the solution
of, water resources problems, simply having
information from the world’s best models and
technology, as judged by our peers, is not a
guarantee of success. To be useful in the political
decision-making process, the information we
analysts generate with all our models and computer technology must be understandable, credible, and timely. It must be just what is needed
when it is needed. It must be not too little and not
too much.
The optimal format and level of detail and
precision of any information generated from

2.4 Developments in Modeling

models should depend on the needs and backgrounds of each individual involved in the
decision-making process. The value of such
information, even if the format and content are
optimal, will also depend on when it is available.
Information on an issue is only of value if it is
available during the time when the issue is being
considered—i.e., when there is an interest in that
issue and a decision concerning what to do about
it has not yet been made. That is the window of
opportunity when information can have an
impact. Information is of no value after the
decision is made unless of course that information results in opening up another window of
opportunity.
If there is truth in the expression “decision
makers don’t know what they want until they
know what they can get,” how do modelers know
what decision-makers will need before even they
do? How will modelers know what is the right
amount and detail of information? How will they
know especially if they are to have that information available, and in the proper form, before
or at, the time it is needed? Obviously modelers
cannot know this. However, over the past three
decades or so this challenge has been addressed
by developing and implementing decision support systems (DSSs) (Fedra 1992; Georgakakos
and Martin 1996; Loucks and da Costa 1991).
These interactive modeling and display technologies can, within limits, adapt to the level of
information needed and can give decision-makers
some control over data input, model operation,
and data output. But will each decision-maker,
each stakeholder, trust the model output? How
can they develop any confidence in the models
contained in a DSS? How can they modify those
models within a DSS to address issues the DSS
developer may not have considered? An answer
to these questions has been the idea of involving
the decision-makers themselves not only in
interactive model use, but in interactive model
building as well. This approach is commonly
termed collaborative modeling.
Figure 2.6 gives a general view of the components of many decision support systems. The
essential feature is the interactive interface that
permits easy and meaningful data entry and

65

display, and control of model (or computer)
operations. Depending on the particular issue at
hand, and more importantly the particular individuals and institutions involved, a decision
support system in the broadest sense can range
from minimal if any computer model use—
where the decision-makers provide all the data
and analyses, make the decision, and they or their
institutions implement those decisions—to decision support systems that are fully automated and
where no human involvement is present. The
latter are rare, but they do exist. The automatic
closing of the flood gates when there is a high
risk of flooding in Rotterdam harbor is an
example of this.
Involving stakeholders in model building
gives them a feeling of ownership. They will
have a much better understanding of just what
their model can do and what it cannot do. If they
are involved in model building, they will know
the assumptions built into their model. Being
involved in a joint modeling exercise is a way to
understand better the impacts of various
assumptions. While there may be no agreement
on the best of various assumptions to make,
stakeholders can learn which of those assumptions matter and which do not. In addition, just
the process of model development by numerous
stakeholders will create discussions that can lead
toward a better understanding of everyone’s
interests
and
concerns.
Though
such
model-building exercises, it is just possible those
involved will gain not only a better understanding of everyone’s concerns, but also a common
or ‘shared’ vision of at least how their water
resource system (as represented by their model,
of course) works. Experience in stakeholder
involvement in model building suggests such
model-building exercises can also help multiple
stakeholders reach a consensus on how their real
system should be developed and managed.
In the US, one of the major advocates of
shared vision or collaborative modeling is the
Institute for Water Resources of the US Army
Corps of Engineers. They have applied their
interactive general-purpose model-building platform in a number of exercises where conflicts
existed over the design and operation of water

66

2 Water Resource Systems Modeling: Its Role in Planning …

Fig. 2.6 Common components of many decision support systems

systems (Hamlet et al. 1996a, b, c; Palmer et al.
1995; Werick et al. 1996). Each of these
model-building ‘shared-vision’ exercises included numerous stakeholders together with experts
in the use of the software. Bill Werick of the
Corps writes:
Because experts and stakeholders can build these
models together, including elements that interest
each group, they become a trusted, consensus view
of how the water system works as a whole, and
how it affects stakeholders and the environment.
Without adding new bureaucracies or reassigning
decision making authority, the shared vision model
and the act of developing it create a connectedness
among problems solvers that resembles the natural
integration of the conditions they study.

Now the question is how to get all the
stakeholders, many who may not really want to
work together, involved in a model-building
exercise. This is our challenge! One step in that
direction is the development of improved technologies that will facilitate model development
and use by stakeholders having various backgrounds and interests. We need better tools for
building DSSs, not just better DSSs themselves.
We need to develop better modeling environments that people can use to make their own
models. Researchers need to be building the
model-building blocks, as opposed to the models
themselves. Researchers need to focus our
attention on improving those building blocks that

2.4 Developments in Modeling

can be used by others to build their own models.
Clearly if stakeholders are going to be involved
in model-building exercises, it will have to be an
activity that is enjoyable and require minimal
training and programming skills.
Traditional modeling experiences seem to
suggest that there are five steps in the modeling
process. First, the information the model is to
provide is identified. This includes measures of
system performance that are of interest to stakeholders. These system performance measures are
defined as functions of the behavior or state of
the system being modeled. Next this behavior
needs to be modeled so the state of the system
associated with any ‘external’ inputs can be
predicted. This requires modeling the physical,
chemical, biological, economic, ecological, and
social processes that take place, as applicable, in
the represented system. Third, these two parts are
put together along with a means of entering the
‘external’ inputs and obtaining in meaningful
ways the outputs. Next the model must be calibrated and verified or validated, to the extent it
can. Only now can the model be used to produce
the information desired.
This traditional modeling process is clearly
not going to work for those who are not especially trained or experienced or even interested in
these modeling activities. They need a
model-building environment where they can
easily create models that
• they understand,
• are compatible with available data,
• work and provide the level and amount of
information needed,
• are easily calibrated and verified when possible, and
• give them the interactive control over data
input, editing, model operation and output
display that they can understand and need in
order to make informed decisions.
The challenge in creating such model-building
environments is in making them sufficiently
useful and attractive so that multiple stakeholders
will want to use them. They will have to be

67

understandable. They will have to be relatively
easy and transparent, and even fun, to build.
They must be capable of simulating and producing different levels of detail with regard to
natural, engineering, economic, and ecological
processes that take place at different spatial and
temporal scales. And they must require no programming and debugging by the users. Just how
can this be done?
One approach is to develop interactive modeling ‘shells’ specifically suited to modeling
environmental problems. Modeling ‘shells’ are
data-driven programs that become models once
sufficient data have been entered into them.
There are a number of such generic modeling
shells for simulating water resource systems.
AQUATOOL, RIBASIM, MIKE-BASIN and
WEAP are representative of interactive riveraquifer simulation shells that require the system
to be represented by, and drawn in as, a network
of nodes and links (e.g., Fig 2.7 from WEAP).
Each node and link require data, and these data
depend on what that node and link represent, as
well as what the user wants to get from the output.
If what is of interest is the time series of quantities
of water flowing, or stored, within the system
resulting from reservoir operation and/or water
allocation policies, then water quality data need
not be entered, even though there is the capability
of modeling water quality. If water quality outputs are desired, then the user can choose the
desired various water quality constituents. Obviously, the more types of information desired or
the greater spatial or temporal resolution desired,
in the model output, the more input data required.
Interactive shells provide an interactive and
adaptive way to define models and their input
data. Once a model is defined, the shell provides
the interface for input data entry and editing,
model operation, and output data display.
To effectively use such shells, some training is
useful. This training pertains to the use of the
shell and what it can and cannot do. The developers of such shells have removed the need to
worry about data base management, solving
systems of equations, developing an interactive
interface, preserving mass balances and

2 Water Resource Systems Modeling: Its Role in Planning …

68

Fig. 2.7 The main interface of the WEAP program, which is typical of a variety of generic river basin models that are
able to simulate any river system drawn into the computer using a node-link network

continuity of flow, and the like. Any assumptions
built into the shell should be readily transparent
and acceptable by all before its use in any shared
vision exercises.

2.4.4

Open Modeling Systems

The next step in shared-vision modeling will be
to create a modeling environment that will enable
all stakeholders to include their own models in
the overall system description. Stakeholders tend
to believe their own models more than those
provided by governmental agencies or research
institutes. Their own models include the data
they trust, and are based on their own assumptions and views on how the system works. For

example, in transboundary water resources
issues, different countries may want to include
their own hydrodynamic models for the river
reaches in their country.
Various developments on open modeling
systems are taking place in Europe and the United
States, although most of them are still in a
research phase. The implementation of the Water
Framework Directive in Europe has stimulated
the development of OpenMI (European Open
Modelling Interface and Environment). OpenMI
will simplify the linking of water-related models
that will be used in the strategic planning required
by the Water Framework Directive (Gijsbers et al.
2002). An initiative in the United States aims to
establish a similar framework for Environmental
Models (Whelan and Nicholson 2002).

2.5 Conclusions

2.5

Conclusions

In our opinion the most important aspect of
model use today is communication. Unless water
resource planners and managers can articulate
well their needs for information, it will be difficult for modelers to generate such information. If
the modelers cannot communicate effectively
their modeling assumptions and results, or how
others can use their tools to obtain their own
results, little understanding will be gained from
such models. Both users and producers of modeling analyses must work together to improve
communication. This takes time, patience, and
the willingness to understand what each has to
say and what is really meant by what is said.
To expect everyone to communicate effectively and to fully understand one another may be
asking too much. As written in the Bible (Genesis; Chapter 11, Verses 1–9) there was a time
when everyone on the earth was together and
spoke one language. It seems these people decided to build a tower “whose top may reach into
the heaven.” Apparently this activity got the
attention of the Lord, who for some reason did
not like this tower building idea. So, according to
the Bible, the Lord came down to earth and
“confounded the peoples language so they could
not understand one another.” They could no
longer work together to build their tower.
Is it any wonder we have to work hard to
communicate more effectively with one another,
even in our single, but multidisciplinary, field of
water resources planning and management?
Let all of us modelers or analysts, planners, and
managers work together to build a new tower of
understanding. To do this we need to control our
jargon and take the time to listen, communicate,
and learn from each other and from all of our
experiences. Who knows, if we are successful,
we may even have another visit from the Lord.
Those who are involved in the development of
water resource systems modeling methodology
know that the use of these models cannot guarantee development of optimal plans for water
resources development and management. Given
the competing and changing objectives and

69

priorities of different interest groups, the concept
of an “optimal plan” is not very helpful or realistic. What modelers can do, however, is to
define and evaluate, in different levels of detail,
numerous alternatives that represent various
possible compromises among conflicting groups,
values, and management objectives. A rigorous
and objective analysis should help to identify the
possible tradeoffs among quantifiable objectives
so that further debate and analysis can be more
informed. The art of modeling is to identify those
issues and concerns that are important and significant and to structure the analysis to shed light
on these issues.
Although water resources planning and management processes are not restricted to mathematical modeling, such modeling is an important
part of those processes. Models can represent in a
fairly structured and ordered manner the important interdependencies and interactions among
the various control structures and users of a water
resource system. Models permit an evaluation of
the economic and physical consequences of
alternative engineering structures, of various
operating and allocating policies, and of different
assumptions regarding future supplies, demands,
technology, costs, and social and legal requirements. Although models cannot define the best
objectives or set of assumptions, they can help
identify the decisions that best meet any particular objective and assumptions.
We should not expect, therefore, to have the
precise results of any quantitative systems study
accepted and implemented. A measure of the
success of any systems study resides in the
answer to the following questions: Did the study
have a beneficial impact in the planning and
decision-making process? Did the results of such
studies lead to a more informed debate over the
proper choice of alternatives? Did it introduce
competitive alternatives that otherwise would not
have been considered?
There seems to be no end of challenging water
resource systems planning problems facing water
resources planners and managers. How one
models any specific water resource problem
depends on (a) the objectives of the analysis;

70

2 Water Resource Systems Modeling: Its Role in Planning …

(b) the data required to evaluate the projects;
(c) the time, data, money, and computational
facilities available for the analysis; and (d) the
modeler’s knowledge and skill. Model development is an art, requiring judgment in abstracting
from the real world the components that are
important to the decision to be made and that can
be illuminated by quantitative methods, and
judgment in expressing those components and
their interrelationships mathematically in the
form of a model. This art is to be introduced in
Chap. 3.

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Fifth Water Resources Operations Management
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Gijsbers, P. J. A., Moore, R. V., & Tindall, C. I. (2002).
Hamonil T: Towards OMI, an open modelling interface and environment to harmonise European developments in water related simulation software. The
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Goeller B. (1988). A framework for evaluating success in
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modeling in the ACT-ACF comprehensive study.
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Hamlet, A., et al. (1996c). Basic Stella II users manual for
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Mobile, Al.: USACE.
Kindler, J. (1987). Systems analysis approach to water
resources management: State of the art. In Symposium
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Analysis in Water Resources Management. Perugia,
Italy: WARREDOC.
Kindler, J. (1988). Systems analysis and water resources
planning. In Fourth IFAC International Symposium on
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Loucks, D. P., & da Costa, J. R. (Eds.). (1991). Decision
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References
Simon, H. A. (1998). Prediction and prescription in
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International Institute for Applied Systems Analysis.
Laxenburg, Austria: IIASA.
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analysis. New York, W.W: Norton.
Thissen, W. A. H., & Walker, W. E. (Eds.). (2013).
Public policy analysis: New developments (286 pp).
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doi:10.1007/978-1-4614-4602-6
(ISBN
978-1-4614-4601-9;
ISBN
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(eBook)).
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476.
Walker, W. E. (1987). Important hard problems in public
policy analysis (Report P-7282). Santa Monica, California: Rand Corporation.
Watkins, D. W. Jr., (Ed.). (2013). Water resources
systems analysis through case studies: Data and
models for decision making (168 pp). Reston, VA:
ASCE Press.
Werick, W. J., Whipple, W. Jr., & Lund, J. (1996).
ACTACF basinwide study. Mobile District, Mobile,
Al.: USACE.
Whelan, G., & Nicholson, T. J. (Eds.). (2002). Proceedings of the Environmental Software Systems Compability and Linkage Workshop (NRC report
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Washington.: Pacific Northwest National Laboratory.
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Raskin, P., Sieber, J. & Huber-Lee, A. (2001). Water
evaluation and planning system: User guide for
WEAP21. Boston, Mass: Tellus Institute.

Exercises
2:1
2:2
2:3
2:4
2:5
2:6
2:7

2:8

2:9

Additional References
(Further Reading)
Andreu, J., Capilla, J., & Sanchis, E. (1991). Aquatool.
A computer assisted support system for water
resources research management including conjunctive
use. In D. P. Loucks & J. R. da Costa (Eds.), Decision
support systems (pp. 333–355). Berlin: Springer.
Danish Hydraulic Institute (DHI). (1997). Mikebasin:
Operating manual and description. Hørsholm:
Denmark.
Delft Hydraulics. (2004). RIBASIM River basin simulation program operating manual and description. The
Netherlands, Delft.
Loucks, D. P. (1995, August). Developing and implementing decision support systems: A critique and a challenge. Water Resources Bulletin, Journal of American
Water Resources Association. 31(4), 571–82.
O’Callaghan, J. R. (Ed.). (1996). The interaction of
economic, ecology and hydrology. London: Chapman
& Hall.

2:10

What is a system?
What is systems analysis?
What is a mathematical model?
Why develop and use models?
What is a decision support system?
What is shared vision modeling and
planning?
What characteristics of water resources
planning or management problems make
them suitable for analysis using quantitative systems analysis techniques?
Identify some specific water resource
systems planning problems and for each
problem specify in words possible
objectives, the unknown decision variables whose values need to be determined, and the constraints or that must be
met by any solution of the problem.
From a review of the recent issues of various journals pertaining to water resources
and the appropriate areas of engineering,
economics, planning, and operations
research, identify those journals that contain articles on water resources systems
planning and analysis, and the topics or
problems currently being discussed.
Many water resource systems planning
problems involve considerations that are
very difficult if not impossible to quantify,
and hence they cannot easily be incorporated into any mathematical model for
defining and evaluating various alternative
solutions. Briefly discuss what value these
admittedly incomplete quantitative models may have in the planning process when
nonquantifiable aspects are also important.
Can you identify some planning problems
that have such intangible objectives?

72

2 Water Resource Systems Modeling: Its Role in Planning …

2:11 Define integrated water management and
what that entails as distinct from just
water management.
2:12 Water resource systems serve many
purposes and can satisfy many objectives. What is the difference between
purposes and objectives?
2:13 How would you characterize the steps of
a planning process aimed at solving a
particular problem?

2:14 Suppose you live in an area where the
only source of water (at a reasonable
cost) is from an aquifer that receives no
recharge. Briefly discuss how you might
develop a plan for its use over time.

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reproduce the material.

3

Models for Identifying
and Evaluating Alternatives

Water resources systems are characterized by
multiple interdependent components that produce
multiple economic, environmental, ecological,
and social impacts. Planners and managers
working to improve the performance of these
complex systems must identify and evaluate
alternative designs and operating policies, comparing their predicted performance with desired
goals or objectives. These alternatives are
defined by the values of numerous design,
management, and operating policy variables.
Constrained optimization together with simulation modeling is the primary way we have of
identifying the values of the unknown decision
variables that will best achieve specified goals
and objectives. This chapter introduces optimization and simulation modeling approaches
and describes what is involved in developing and
applying them to define and evaluate alternative
designs and operating policies.

3.1

Introduction

There are typically many different options
available to those planning and managing water
resource systems. It is not always clear what set
of particular design, management, and operating
policy decisions will result in the best overall
system performance. That is precisely why
modeling is done, to estimate the performance

associated with any set of decisions and
assumptions, and to predict just how well various
economic, environmental, ecosystem, and social
or political objectives or goals will be met.
One important criterion for plan identification
and evaluation is the economic benefit or cost a
plan would entail were it to be implemented.
Other criteria can include the extent to which any
plan meets environmental, ecological, and social
targets. Once planning or management performance measures (objectives) and various general
alternatives for achieving desired levels of these
performance measures have been identified,
models can be developed and used to help
identify specific alternative plans that best
achieve those objectives.
Some system performance objectives may be
in conflict, and in such cases models can help
identify the efficient tradeoffs among these
conflicting measures of system performance.
These tradeoffs indicate what combinations of
performance measure values can be obtained
from various system design and operating policy
variable values. If the objectives are the right
ones (that is, they are what the stakeholders
really care about), such quantitative tradeoff
information should be of value during the debate
over what decisions to make (Hipel et al. 2015).
Regional water resources development plans
designed to achieve various objectives typically
involve investments in land and infrastructure.

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_3

73

74

Achieving the desired economic, environmental,
ecological, and social objective values over time
and space may require investments in storage
facilities, pipes, canals, wells, pumps, treatment
plants, levees, and hydroelectric generating
facilities, or in fact the removal of some of them.
Many capital investments can result in irreversible economic and ecological impacts. Once
the forest in a valley is cleared and replaced by a
lake behind a dam, it is almost impossible to
restore the site to its original condition. In parts
of the world where river basin or coastal
restoration activities require the removal of
engineering structures, such as in the Florida
Everglades discussed in Chap. 1, engineers are
learning just how difficult and expensive that
effort can be.
The use of planning models is not going to
eliminate the possibility of making mistakes.
These models can, however, inform. They can
provide estimates of the different impacts associated with, say, a natural unregulated river system and a regulated river system. The former can
support a healthier ecosystem that provides a
host of flood protection and water quality
enhancement services. The latter can provide
more reliable and cheaper water supplies for
off-stream users and increased hydropower and
some protection from at least small floods for
those living on flood-prone lands. In short,
models can help stakeholders assess the future
consequences, the benefits and costs, and a
multitude of other impacts associated with
alternative plans or management policies.
This chapter introduces some mathematical
modeling approaches commonly used to study
and analyze water resources systems. The modeling approaches are illustrated by their application to some relatively simple water resources
planning and management problems. The purpose here is to introduce and compare some
commonly used modeling methods. This is not a
text on the state of the art of modeling. More
realistic and more complex problems usually
require much bigger and more complex models
than those introduced in this book, but these
bigger and more complex models are often based
on the principles and techniques presented here.

3

Models for Identifying and Evaluating Alternatives

The emphasis here is on the art of model development: just how one goes about constructing
a model that will provide information needed to
study and address particular problems, and various ways models might be solved. It is unlikely
anyone will ever use any of the specific models
developed in this or other chapters, simply
because they will not be solving the specific
example problems used to illustrate the different
approaches to model development and solution.
However, it is quite likely that water resources
managers and planners will use the modeling
approaches and solution methods presented in
this book to develop the models needed to analyze their own particular problems.
The water resource planning and management
problems and issues used here, or any others that
could have been used to illustrate model development, can be the core of more complex models
addressing more complex problems in practice.
Water resources planning and management today
is dominated by the use of optimization and
simulation models. While computer software is
becoming increasingly available for solving
various types of optimization and simulation
models, no software currently exists that will
build those models themselves. What to include
and what not to include and what parameter
values to assume in models of water resource
systems requires judgment, experience, and
knowledge of the particular problem(s) being
addressed, the system being modeled and the
decision-making environment. Understanding
the contents of, and performing the exercises
pertaining to, this chapter will be a first step
toward gaining some judgment and experience in
model development and solution.

3.1.1

Model Components

Mathematical models typically contain one or
more algebraic equations or inequalities. These
expressions include variables whose values are
assumed to be known and others that are
unknown and to be determined. Variables that
are assigned known values are usually called
parameters. Variables having unknown values

3.1 Introduction

that are to be determined by solving the model
are called decision variables. Models are developed for the primary purpose of identifying the
best values of the latter and for determining how
sensitive those derived values are to the assumed
parameter values.
Decision variables can include design and
operating policy variables of various water
resources system components. Design variables
can include the active and flood storage capacities of reservoirs, the power generating capacity
of hydropower plants, the pumping capacity of
pumping stations, the waste removal efficiencies
of wastewater treatment plants, the dimensions or
flow capacities of canals and pipes, the heights of
levees, the hectares of an irrigation area, the
targets for water supply allocations, and so on.
Operating variables can include releases of water
from reservoirs or the allocations of water to
various users over space and time. Unknown
decision variables can also include measures of
system performance, such as net economic benefits, concentrations of pollutants at specific sites
and times, ecological habitat suitability values or
deviations from particular ecological, economic,
or hydrological targets.
Models describe, in mathematical terms, the
system being analyzed and the conditions that the
system has to satisfy. These conditions are often
called constraints. Consider, for example, a
reservoir serving various water supply users
downstream. The conditions included in a model
of this reservoir would include the assumption
that water will flow in the direction of lower
heads (that is, downstream unless it is pumped
upstream), and the volume of water stored in a
reservoir cannot exceed its storage capacity. Both
the storage volume over time and the reservoir
capacity might be unknown and are to be determined. If the capacity is known or assumed, then
it is among the known model parameters.
Model parameter values, while assumed to be
known, can often be uncertain. The relationships
among various decision variables and assumed
known model parameters (i.e., the model itself)
may be uncertain. In these cases, the models can
be solved for a variety of assumed conditions and
parameter values. This provides an estimate of

75

just how important uncertain parameter values or
uncertain model structures are with respect to the
output of the model. This is called sensitivity
analysis. Sensitivity analyses will be discussed in
Chap. 8 in much more detail.
Solving a model means finding values of its
unknown decision variables. The values of these
decision variables can define a plan or policy.
They can also determine the costs and benefits or
the values of other measures of system performance associated with that particular management plan or policy. While the components of
optimization and simulation models can include
system performance indicators, model parameters and constraints, the process of model development and use also includes people. The
drawing shown in Fig. 3.1 (and in Chap. 2 as
well) illustrates some interested stakeholders
busy studying their river basin, in this case perhaps with the use of a physical simulation model.
(Further discussion of stakeholder involvement
in the planning and management process is in
Chap. 13).
Whether a mathematical model or physical
model is being used, one important consideration
is that if the modeling exercise is to be of any
value, it must provide the information desired
and in a form that the interested stakeholders and
decision-makers can understand.

3.2

Plan Formulation and Selection

Plan formulation can be thought of as assigning
particular values to each of the relevant decision
variables. Plan selection is the process of evaluating alternative plans and choosing the one that
best satisfies a particular objective or set of objectives. The processes of plan formulation and
selection involve modeling and communication
among all interested stakeholders, as the picture
in Fig. 3.1 suggests.
The planning and management issues being
discussed by the stakeholders in the basin pictured in Fig. 3.1 could well include surface and
ground water allocations, reservoir operation,
water quality management, and infrastructure
capacity expansion over time.

76

3

Models for Identifying and Evaluating Alternatives

Fig. 3.1 These stakeholders have an interest in how their
watershed or river basin is managed. Here they are using a
physical model to help them visualize and address

planning and management issues. Mathematical models
often replace physical models, especially for planning and
management studies

3.2.1

performance, such as costs and benefits. They
may also include environmental and social
measures not expressed in monetary units. (More
detail on performance criteria is contained in
Chap. 9).
To illustrate this plan formulation process,
consider the task of designing a tank that can
store a fixed volume, say V, of water. Once the
desired shape has been determined, the task is to
build a model that can determine the values of all
the design variables and the resulting cost. Different designs result in different sizes and
amounts of materials, and hence different costs.
Assume the purpose of the model is to define the
set of design variable values that results in the
minimum total cost, for a range of values of the
required volume, V.

Plan Formulation

Model building for defining alternative plans or
policies involves a number of steps. The first is to
clearly specify the issue or problem or decision
(s) to be made. What are the fundamental
objectives and possible alternatives? Such alternatives might require defining allocations of
water to various water users, the level of
wastewater treatment needed to maintain a
desired water quality in a receiving stream, the
capacities, and operating rules of multipurpose
reservoirs and hydropower plants, and the extent
and reliability of floodplain protection derived
from levees. Each of these decisions may affect
system performance criteria or objectives. Often
these objectives include economic measures of

3.2 Plan Formulation and Selection

The model of this problem must somehow
relate the unknown design variable values to the
cost of the tank. Assume, for example, a rectangular tank shape. The unknown design variables
are the tank length, L, width, W, and height,
H. These are the unknown decision variables. The
objective is to find the combination of L, W, and
H values that minimizes the total cost of providing a tank capacity of at least V units of water.
This volume V will be one of the model parameters. Its value is assumed known even though in
fact it may be unknown and dependent in part on
its cost. But for now assume V is known.
The cost of the tank will be the sum of the
costs of the base, the sides, and the top. These
costs will depend on the area of the base, sides,
and top. Assume that we know the average costs
per unit area of the base, sides, and top of the
tank. These average unit costs of the base, sides,
and top will probably differ. They can be denoted
as Cbase, Cside, and Ctop, respectively. These unit
costs together with the tank’s volume, V, are the
parameters of the model. If L, W, and H are
measured in meters, then the areas will be
expressed in units of square meters and the volume will be expressed in units of cubic meters.
The average unit costs will be expressed in
monetary units per square meter.
The final step of model building is to specify
all the relations among the model parameters and
decision variables. This includes defining the
objective (cost) function (in this case just one
unknown variable, Cost) and all the conditions
that must be satisfied while achieving that
objective. It is often helpful to first state these
relationships in words. The result is a word
model. Once that is written, mathematical notation can be defined and used to convert the word
model to a mathematical model.
The word model for this tank design problem
is to minimize total cost where:
• Total cost equals the sum of the costs of the
base, the sides, and the top.
• Cost of the sides is the cost-per-unit area of
the sides times the total side area.

77

• The total side area is twice the products of
length times height and width times height.
• Cost of the base is the cost-per-unit area of
the base times the total base area.
• Cost of the top is the cost-per-unit area of the
top times the total top area.
• The top and base area is the product of length
times width.
• The volume of the tank must at least equal the
required volume capacity.
• The volume of the tank is the product of the
length, width, and height of the tank.
Converting each of the above conditions to
mathematical expressions using the notation
defined above and inventing new notation when
needed results in:
• Total cost equals the sum of the costs of the
base, the sides, and the top.
Cost = sidecost + basecost + topcost
• Cost of the sides is the cost-per-unit area of
the sides times the total side area.
sidecost = Cside (sidearea)
• The total side area is twice the products of
length times height and width times height.
sidearea = 2(LH+WH)
• Cost of the base is the cost-per-unit area of
the base times the total base area.
basecost = Cbase (basearea)
• Cost of the top is the cost-per-unit area of the
top times the total top area.
topcost = Ctop (toparea)
• The top and base area is the product of length
times width.
toparea = basearea = LW
• The volume of the tank must at least equal the
required volume capacity.
tankvolume ≥ V
• The volume of the tank is the product of the
length, width, and height of the tank.
tankvolume = LWH
Combining some of the above conditions, a
mathematical optimization model can be written
as:

78

3

Models for Identifying and Evaluating Alternatives

a unit percent change in some parameter value
(what economists call elasticity)?
If indeed the decision variable values do not
Subject to:
change significantly with respect to a change in
Cost ¼ ðCbase þ Ctop ÞðLWÞ þ 2ðCside ÞðLH þ WHÞ an uncertain parameter value, there is no need to
ð3:2Þ devote more effort to reducing that uncertainty.
Any time and money available for further study
should be directed toward those parameters or
LWH  V
ð3:3Þ assumptions that substantially influence the
model’s decision variable values.
Equation 3.3 permits the tank’s volume to be
This capability of models to help identify
larger than that required. While this is allowed, it what data or assumptions are important and what
will cost more if the tank’s capacity is larger than are not can guide monitoring and data collection
V, and hence the least-cost solution of this model efforts. This is a beneficial attribute of modeling
will surely show that the product LWH will equal often overlooked.
the required volume V. In practice, however,
Continuing with the tank example, after
there may be practical, legal, and/or safety rea- determining, or estimating, all the values of the
sons why the decisions with respect to L, W, and model parameters and then solving the model to
H may result in a capacity that exceeds V.
obtain the cost-effective values of L, W and H,
In this model, the unknown decision variables we now have a design. It is just one of a number
include Cost, L, W, and H
of designs that could be proposed. Another deThe least-cost solution (using methods dis- sign might be for a cylindrical tank having a
cussed in the next chapter) is
radius and height as well as cost decision variables. For the same volume V and unit area costs,
W ¼ L ¼ ½2Cside V = ðCbase þ Ctop Þ1=3 ð3:4Þ we would find that the total cost is less, simply
because the areas of the base, side, and top are
and
less.
In the above discussion, the required volume
H ¼ V = ½2Cside V = ðCbase þ Ctop Þ2=3 ð3:5Þ capacity, V, has been assumed to be known. In
reality, it too may be a decision variable, and
or
what would be of greater value to
decision-makers is knowing the relationship
H ¼ V 1=3 ½ðCbase þ Ctop Þ = 2Cside 2=3
ð3:6Þ between various assumed values of V and their
respective minimum costs. Such a cost function
The modeling exercise should not end here. If can be defined by solving the model (defined by
there is any doubt about the value of any of the Eqs. 3.1, 3.2 and 3.3) for various values of V.
parameters, a sensitivity analyses can be perWhatever the final outcome of our modeling
formed on those uncertain parameters or efforts, there might be other considerations or
assumptions. In general, these assumptions could criteria that are not expressed or included in the
include the values of the cost parameters (e.g., model that might be important to those responthe costs-per-unit area) as well as the relation- sible for plan (tank design) selection.
ships expressed in the model (that is, the model
itself). How much does the total cost change with
respect to a change in any of the cost parameters 3.2.2 Plan Selection
or with the required volume V? How much does
any decision variable change with respect to There are various approaches to finding the
changes in those parameter values? What is the “best” plan or best set of decision variable values
percent change in a decision variable value given that satisfy an objective or goal. By trial and
Minimize Cost

ð3:1Þ

3.2 Plan Formulation and Selection

error, one could identify alternative plans, evaluate the performance of each plan, and select the
particular plan whose performance is judged
better than the others. This process could include
a systematic simulation of a range of possible
solutions in a search for the best. When there are
a large number of feasible alternatives—that is,
many decision variables and many possible values for each of them—it may no longer be
practical to identify and simulate all feasible
combinations of decision variable values, or even
a small percentage of them. It would simply take
too long. In this case it is often convenient to use
an optimization procedure.
Equations 3.1–3.3 represent an optimization
problem. There are an infinite number of feasible
tank designs, i.e., alternative values of L, W, and
H that satisfy the volume requirement. Our job is
to find the least-cost one. We can do this using a
mathematical optimization method. Mathematical optimization methods are designed to make
this search for the best solution (or better solutions) more efficient. Optimization methods are
used to identify those values of the decision
variables that satisfy specified objectives and
constraints
without
requiring
complete
enumeration.
While optimization models might help identify the decision variable values that will produce
the best plan directly, they are based on all the
assumptions incorporated in the model. Often
these assumptions are limiting. In these cases, the
solutions resulting from optimization models
should be analyzed in more detail, perhaps
through simulation methods, to improve the
values of the decision variables and to provide
more accurate estimates of the impacts associated
with those decision variable values. In these situations, optimization models are used for
screening out the clearly inferior solutions, not
for finding the very best one. Just how screening
can be performed using optimization models will
be discussed in the next chapter.
The values that the decision variables may
assume are rarely unrestricted. Usually various
functional relationships among these variables
must be satisfied. This is what is expressed in
constraint Eq. 3.3. For example, the tank has to

79

be able to contain a given amount of water. In a
water allocation problem, any water allocated to
and completely consumed by one user cannot
simultaneously or subsequently be allocated to
another user. Storage reservoirs cannot store
more water than their maximum storage capacities. Technological restrictions may limit the
capacities and sizes of pipes, generators, and
pumps to those commercially available. Water
quality concentrations should not exceed those
specified by water quality standards or regulations. There may be limited funds available to
spend on water resources development or
infrastructure projects. These are a few examples
of physical, legal, and financial conditions or
constraints that may restrict the ranges of decision variable values in the solution of a model.
Equations or inequalities can generally
express any physical, economic, legal, or social
restrictions on the values of the decision variables. Constraints can also simply define relationships among decision variables. For example,
Eq. 3.2 above defines a new decision variable
called Cost as a function of other decision variables and model parameters. In general, constraints describe in mathematical terms the
system being analyzed. They define the system
components and their interrelationships, and the
permissible ranges of values of the decision
variables, either directly or indirectly.
Typically, there exist many more decision
variables than constraints, and hence, if any
feasible solution exists, there may be many such
solutions that satisfy all the constraints. The
existence of many feasible alternatives is a
characteristic of most water resources systems
planning problems. Indeed it is a characteristic of
most engineering design and operation problems.
The particular feasible solution or plan that satisfies the objective function—that is, that maximizes or minimizes it—is called optimal. It is the
optimal solution of the mathematical model, but
it may not necessarily be considered optimal by
any decision-maker. What is optimal with respect
to a model may not be optimal with respect to
those involved in a planning or decision-making
process. To repeat what was written in Chap. 2,
models are used to provide information (useful

80

3

information, one hopes), to the decision-making
process. Model solutions are not replacements
for judgments of individuals involved in the
decision-making process.

3.3

Conceptual Model
Development

Prior to the selection or development of a quantitative model, it is often useful to develop a
conceptual one. Conceptual models are nonquantitative representations of a system. The
system components and their interactions are
defined often by diagrams similar to Fig. 3.2.
Figure 3.2 illustrates the form of a conceptual
model. This example conceptual model defines

Fig. 3.2 An outline of a conceptual model without its
detail (i.e., what exactly each component or box represents), showing the links representing interactions among

Models for Identifying and Evaluating Alternatives

the relationships between what land and water
managers can do and the eventual ecological
impacts of those actions. Once a conceptual
model has been quantified (expressed in mathematical terms), it becomes a mathematical model.
The model’s equations typically include variables whose values are unknown and can vary,
and parameters whose values are assumed
known.
The values of the model’s parameters need to
be determined. Model calibration involves finding the best values for these parameters. Calibration is based on comparisons of the model
results with observed data. Optimization methods
can sometimes be used to identify the values of
model parameters. This is called model calibration or identification. (Illustrations of the use of

components and between management decisions and
specific system impacts

3.3 Conceptual Model Development

optimization for estimating model parameter
values are presented in the following chapter.)
Sensitivity analysis may serve to identify the
impacts of uncertain parameter values and show
which parameter values substantially influence
the model’s results or solutions. Following calibration, the remaining uncertainties in the model
predictions may be quantified in an uncertainty
analysis as discussed in Chap. 8.
In addition to being calibrated, simulation
models should also be validated or verified. In
the validation or verification process, the model
results are compared with an independent set of
measured observations that were not used in
calibration. This comparison is made to determine whether or not the model describes the
system behavior sufficiently accurately.

3.4

Simulation and Optimization

The modeling approach to tank design discussed
in the previous section focused on the use of
optimization methods to identify the preferred
design variable values. Similar optimization
methods can be used to identify preferred design
variable values and operating policies for urban
stormwater runoff control or multiple reservoir
systems, given various assumptions regarding
parameter values and design and operating objectives. Once these preferred designs and operating policies have been identified, unless there is
reason to believe that a particular alternative is
really the best and needs no further analysis, each
of these preferred alternatives can be further
evaluated with the aid of more detailed and robust simulation models.
Simulation models address “what if” questions: What will likely happen over time at one or
more specific places if a particular design and/or
operating policy is implemented? Simulation
models are not limited by many of the assumptions incorporated into optimization models. For
example, the inputs to simulation models can
include a much longer time series of hydrological, economic, and environmental data such as
rainfall or streamflows, water supply demands,
pollutant loadings and so on, than would likely

81

be included in an optimization model. The
resulting outputs can better identify the variations
of multiple system performance indicator values:
that is, the multiple hydrological, ecological,
economic, environmental, and social impacts that
might be observed over time, given any particular system design and operating policy.
Simulating multiple sets of values defining the
designs and operating policies of a water
resources system can take a long time. Consider,
for example, 30 infrastructure capacity variables
whose values are to be determined. Even if only
two possible values are assumed for each of the
30 variables (such as to exist at some predetermined capacity or not), the number of combinations that could be simulated amounts to 230 or in
excess of 109. Simulating and comparing even
1% of these billion at a minute per simulation
amounts to over twenty years, continuously—
24 h per day. Most simulation models of water
resources systems contain many more variables,
each having a larger range of feasible values, and
are much more complex than this simple
30-binary-variable example. Mathematically, if
not in reality, there could be an infinite combination of feasible values for each of the decision
variables.
Simulation works well when there are only a
relatively few alternatives to be evaluated, not
when there are a large number of them. The trial
and error process of simulation can be time consuming. An important role of optimization methods is to reduce the number of alternatives for
simulation analyses. However, if only one method
of analysis is to be used to evaluate a complex
water resources system, simulation together with
human judgment concerning which alternatives to
simulate is often the method of choice.
Simulation can be based on either discrete
events or discrete time periods. Most simulation
models of water resources systems are designed
to simulate a sequence of events over a number
of discrete time periods. In each discrete time
period, the simulation model converts all the
initial conditions and inputs to outputs. The
duration of each period depends in part on the
particular system being simulated and the questions being addressed.

82

3.4.1

3

Simulating a Simple Water
Resources System

Consider the case of a potential reservoir releasing
water to downstream users (Fig. 3.3). A reservoir
and its operating policy can increase the benefits
each user receives over time by providing
increased flows during periods of otherwise low
flows relative to the user demands. Of interest is
whether or not the increased benefits the water
users obtain from an increased and more reliable
downstream flow conditions will offset the costs
of the reservoir.
Before this system can be simulated, one has
to define the active storage capacity of the reservoir and how much water is to be released
depending on the storage volume and time period.
In other words, one has to define the reservoir
operating policy. In addition, one must also define
the allocation policy: how much of the released
water to allocate to each user and to the river
downstream of the users.
There are literally an infinite number of possible design and operating policy variable values.
The next section will address the problem of
screening these alternatives to find those values
that are most worthy of further study using
simulation.
For this simple illustration assume the operating and allocation policies are as shown in
Fig. 3.4. Also for simplicity assume they apply
to each discrete time period. The reservoir

Models for Identifying and Evaluating Alternatives

operating policy, shown as a red line in upper
Fig. 3.4, attempts to meet a release target. If
insufficient water is available, all the water will
be released in the time period. If the inflow
exceeds the target flow and the reservoir is full, a
spill will occur.
This operating policy is sometimes called the
“standard” operating policy. It is not usually
followed in practice. Most operators, as indeed
specified by most reservoir operating policies,
will reduce releases in times of drought in an
attempt to save some water in the reservoir for
future releases in case of an extended period of
low inflows. This is called a hedging policy. Any
reservoir release policy, including a hedging
policy, can be defined within the blue portion of
the release policy plot shown in Fig. 3.4. The
dash–dot line in Fig. 3.4 is one such hedging
function. Once defined, any reservoir operating
policy can be simulated.
The simulation process for the three-user
system is shown in Fig. 3.5. It proceeds from
one time period to the next. The reservoir inflow,
obtained from a database, is added to the existing
storage volume, and a release is determined
based on the release policy (upper Fig. 3.4).
Once the release is known, the final storage
volume is computed and this becomes the initial
volume for the next simulation time period. The
reservoir release is then allocated to the three
downstream users and to the river downstream of
those users as defined by the allocation policy

Fig. 3.3 Conceptual model of a reservoir water allocation system to be simulated

3.4 Simulation and Optimization

83

Fig. 3.4 Reservoir operating policy defining the reservoir release to be made as a function of the current storage
volume and current inflow and the allocation policy for
the river flow downstream of the reservoir. The blue zone

in the reservoir release policy indicates the zone of
feasible releases. It is physically impossible to make
releases represented by points outside that blue zone

(lower Fig. 3.4). The resulting benefits can be
calculated and stored in an output database.
Additional data including storage volumes,
releases, and the allocations themselves can also
be stored in the output database, as desired. The

simulation process continues for the duration of
the simulation run. Then the output data can be
summarized for later comparison with other simulation results based on other reservoir capacities,
operation policies and/or allocation policies.

84

3

Models for Identifying and Evaluating Alternatives

Fig. 3.5 Flow diagram of the reservoir—user allocation system simulation process. The simulation terminates after
some predefined number of simulation time steps

It would not be too difficult to write a computer program to perform this simulation. In fact,
it can be done on a spreadsheet. However as easy
as that might be for anyone familiar with computer programming or spreadsheets, one cannot
expect it to be easy for many practicing water
resources planners and managers who are not
doing this type of work on a regular basis. Yet
they might wish to perform a simulation of their
particular system, and to do it in a way that
facilitates changes in many of its assumptions.
Computer programs capable of simulating a wide
variety of water resources systems are becoming
increasingly available. Simulation programs
together with their interfaces that facilitate the
input and editing of data and the display of
output data are typically called decision support
systems. Their input data define the components
of the water resources system and their configuration. Inputs also include hydrological data and
design and operating policy data. These generic
simulation programs are capable of simulating
surface and ground water flows, storage volumes

and qualities under a variety of system infrastructure designs and operating policies.

3.4.2

Defining What to Simulate

Before the simple system shown in Fig. 3.3 can
be simulated the design and operating policy of
the system, i.e., the information shown in
Fig. 3.4 needs to be defined. One way to do this
is to use optimization. Optimization is driven by
an objective function. Assume an overall measure of system performance has been decided
upon, and can be expressed as a function of the
decision variables. These decision variables
include all the information in Fig. 3.3, namely
the reservoir capacity and reservoir storage and
release and water user allocation decisions in
each time period. Of interest are the values of
these decision variables that achieve the highest
level of system performance. The use of an
optimization model will help in defining those
variable values.

3.4 Simulation and Optimization

85

Expressed in words, the optimization model is
to be developed and used to identify the decision
variable values that maximize system performance. Let B(K, S, R, A) represent the overall
system performance measure, as a function of the
reservoir capacity K, and all the initial storage
volumes, S(t), releases, R(t), and water allocations to users i, A(i,t), in each time period t for a
total of T time periods. Hence the objective is to
maximize B ðK; S; R; AÞ

ð3:7Þ

single-period optimization model containing only
the allocation benefits as the objective, B(A), and
constraint 3.10 for a single period, and various
values of the water available, R − Q, assuming
the benefits, B(A), do not change over time.
Maximize BðAð1Þ; Að2Þ; Að3ÞÞ

ð3:11Þ

Subject to:
Að1Þ þ Að2Þ þ Að3Þ  R  Q
for various values of R; given Q:

ð3:12Þ

while making sure that a mass balance of water is
maintained in the reservoir over time.
SðtÞ þ InflowðtÞRðtÞ ¼ Sðt þ 1Þ for each period t
ðand period T þ 1 ¼ 1Þ

ð3:8Þ
These mass balance equations define the
relationship between initial, S(t), and final, S
(t + 1) storage volume values in each period t,
and equate the final storage value in each period
to the initial value in the following period.
Finally, it assumes the entire simulation process
repeats itself after every T years.
The next set of constraints ensure that the
storage volumes, S(t), do not exceed the reservoir
storage capacity K and that the allocations, A(i,
t), to the three water users i do not exceed the
reservoir release, R(t), less the amount to remain
in the stream, Q(t).
SðtÞ  K

for each period t:

ð3:9Þ

Að1; tÞ þ Að2; tÞ þ Að3; tÞ  RðtÞQðtÞ
ð3:10Þ
for each period t:
This simple example ignores many of the
details one should consider when modeling
reservoirs and water users, and many of these
details will be discussed, and modeled, in subsequent chapters. But for now the model is sufficient to find values for each decision variable
shown in upper portion of Fig. 3.4. The allocation policies shown in the lower portion of
Fig. 3.4 can be obtained by solving a separate

3.4.3

Simulation Versus Optimization

Unlike simulation models, the solutions of optimization models are based on objective functions
that are to be maximized or minimized. The
objective function and constraints of an optimization model contain decision variables that are
unknown and parameters whose values are
assumed known. Constraints are expressed as
equations and inequalities. The tank model
(Eqs. 3.1, 3.2 and 3.3) is an example of an optimization model. So is the reservoir water allocation model, Eqs. 3.7–3.10 and the single-period
allocation model Eqs. 3.11 and 3.12.
The solution of an optimization model, if one
exists, contains the values of all of the unknown
decision variables. It is mathematically optimal
in that the values of the decision variables satisfy
all the constraints and maximize or minimize an
objective function. This “optimal” solution is of
course based on the assumed values of the model
parameters, the chosen objective function and the
structure of the model itself. At best these
assumptions can only approximate reality.
The assumptions made to permit model solution by optimization solution procedures (algorithms), may justify a more detailed and more
realistic simulation to check and improve on any
solution obtained from that optimization. While
the results from a simulation model may be more
realistic, both optimization and simulation models
are approximations of the real system being
modeled. The optimal solution of any model is
optimal only with respect to the particular model,

86

3

Models for Identifying and Evaluating Alternatives

Fig. 3.6 Distinguishing between simulation and optimization modeling. Simulation addresses “what if” questions; optimization can address “what should be”

questions. Both types of models are typically used in
water resources planning and management studies

not necessarily with respect to the real system. It is
important to realize this limited meaning of the
word “optimal,” a term commonly found in papers
published by water resources and other systems
analysts, planners, and engineers.
Figure 3.6 illustrates the broad differences
between simulation and optimization. Optimization models need explicit expressions of objectives. Simulation models do not. Simulation
simply addresses “what-if” scenarios—what may
happen if a particular scenario is assumed or if a
particular decision is made. Users of simulation
models must specify the values of design and
operating decision variables before a simulation
can be performed. Once these values of all
decision variables are defined, simulation can
provide more precise estimates of the impacts
that may result from those decisions.
While optimization will tell us what we should
do—what the best decision is—that solution is
often based on many limiting assumptions.
Because of this, we need to use optimization not
as a way to find the best solution, but to define a
relatively small number of good alternatives that
can later be tested, evaluated, and improved by
means of more detailed simulations. This process
of using optimization to reduce the large number

of plans and policies to a few that can then be
simulated and better evaluated is often called
preliminary screening.

3.5

Conclusions

This chapter has reviewed some basic types of
models and presented guidelines for their use.
Generic models for water resources system analyses are increasingly becoming available, saving
many organizations from having to develop their
own individual models. While many readers of
this book may get involved in writing their own
models, most of those involved in water resources
planning and management will be using existing
models and analyzing and presenting their results.
The information provided in this book is intended
to help those who wish to build their modeling
skills. Such skills will be useful to those involved
in water resource systems planning and management activities. Such skills may be useful even to
those who are expected to oversee or evaluate the
model results of others (say from various UN,
World Bank, or national aid agencies) who are
involved in analyzing particular water resource
systems in particular regions of the world.

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Winter Simulation Conference (pp. 115–21). Piscataway: N.J. Institute of Electrical and Electronics
Engineers.
Sargent, R. G. (1984b). Simulation model validation. In:
T. I. Ören, B. P. Zeigler, & M. S. Elzas (Eds.),
Simulation and model-based methodologies: An integrative view (pp. 537–55). Berlin: Springer (No. 10 in
the series: NATO ASI Series F: Computer and
Systems Sciences).
Scholten, H., Van Waveren, R. H., Groot, S., Van Geer, F.
C., Wösten, J. H. M., Koeze, R. D., et al. (2000). Good
Modelling Practice in Water Management, Proceedings Hydroinformatics 2000. Cedar Rapids, Iowa:
International Association for Hydraulic Research.
Singh, V. P. (Ed.). (1995). Computer models of watershed
hydrology. Littleton, Colorado: Water Resources
Publications.
Sun, L., & Liu, K. (2001). A method for interactive
articulation of information requirements for strategic
decision support. Information and Software Technology, 43(4), 247–263.
Van Waveren, R. H., Groot, S., Scholten, H., Van Geer,
F. C., Wösten, J. H. M., Koeze, R. D., et al. (1999).
Good modelling practice handbook (STOWA report
99–05). Lelystad, The Netherlands: Dutch Department
of Public Works, Institute for Inland Water Management and Waste Water Treatment (report 99.036).
Watkins, D. W. Jr., (Ed.). (2013). Water resources
systems analysis through case studies: Data and
models for decision making (168 pp). Reston, VA:
ASCE Press.

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Young, P. (1998). Data-based mechanistic modeling of
environmental, ecological, economic and engineering
systems. Environmental Modeling & Software, 13,
105–122.
Young, W. J., Lam, D. C. L., Ressel, V., & Wong, J. W.
(2000). Development of an environmental flows
decision support system. Environmental Modeling &
Software, 15, 257–265.
Zeigler, B. P. (1976). Theory of modelling and simulation.
Malabar, Florida: Robert E. Krieger.
Zeiler, M. (1999). Modeling our world: The ESRI guide to
geodatabase design. Redlands, California: ESRI.

3:6 Consider the following five alternatives
for
the
production
of
energy
(103 kwh/day) and irrigation supplies
(106 m3/month):

Exercises
3:1 Briefly outline why multiple disciplines
are needed to efficiently and effectively
manage water resources in major river
basins, or even in local watersheds.
3:2 Describe in a page or two what some of
the water management issues are in the
region where you live.
3:3 Define adaptive management, shared
vision modeling, and sustainability.
3:4 Distinguish what a manager does from
what an analyst (modeler) does.
3:5 Identify some typical or common water
resources planning or management problems that are suitable for analysis using
quantitative systems analysis techniques.

Models for Identifying and Evaluating Alternatives

Alternative

Energy production

Irrigation supply

A

22

20

B

10

35

C

20

32

D

12

21

E

6

25

Which alternative would be the best in your
opinion and why? Why might a decisionmaker select alternative E even realizing
other alternatives exist that can give more
hydropower energy and irrigation supply?
3:7 Define a model similar to Eqs. 3.1–3.3 for
finding the dimensions of a cylindrical
tank that minimizes the total cost of storing a specified volume of water. What are
the unknown decision variables? What are
the model parameters? Develop an iterative approach for solving this model.
3:8 Briefly distinguish between simulation and
optimization.
3:9 Consider a tank, a lake or reservoir or an
aquifer having inflows and outflows as
shown in the graph below.

Flows (m3/day)
Inflow

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time (days)

Exercises

89

(a) When was the inflow its maximum
and minimum values?
(b) When was the outflow its minimum
value?
(c) When was the storage volume its
maximum value?
(d) When was the storage volume its
minimum value?

(e) Write a mass balance equation for the
time series of storage volumes
assuming constant inflows and outflows during each time period.
3:10 Given the changing inflows and constant
outflow from a tank or reservoir, as shown
in the graph below, sketch a plot of the

Flows (m3/day)

100
Inflow

50

Ouƞlow

0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time (days)

300

RelaƟve Storage
(m3)

150

0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time (days)

90

3

storage volumes over the same period of
time, beginning at 150. Show how to
determine the value of the slope of the
storage volume plot at any time from the
inflow and outflow (= 50 m3/day) graph
below.
3:11 Describe, using words and a flow diagram,
how you might simulate the operation of a
storage reservoir over time. To simulate a
reservoir, what data do you need to have
or know?
3:12 Identify and discuss a water resources
planning situation that illustrates the need
for a combined optimization-simulation
study in order to identify the best plan and
its impacts.
3:13 Write a flow chart/computer simulation
program for computing the maximum
yield of water that can be obtained given
any value of active reservoir storage
capacity, K, using.

Year y

Flow Qy

Year y

1

5

9

3

2

7

10

6

3

8

11

8

4

4

12

9

5

3

13

3

6

3

14

4

7

2

15

9

8

1

Models for Identifying and Evaluating Alternatives

3:14

3:15

Flow Qy

3:16

3:17

Find the values of the storage capacity K
required for yields of 2, 3, 3.5, 4, 4.5, and 5.
How many different simulations of a water
resource system would be required to
ensure that there is at least a 95% chance
that the best solution obtained is within the
better 5% of all possible solutions that
could be obtained? What assumptions must
be made in order for your answer to be
valid? Can any statement be made comparing the value of the best solution
obtained from the all the simulations to the
value of the truly optimal solution?
Assume in a particular river basin 20
development projects are being proposed.
Assume each project has a fixed capacity
and operating policy and it is only a question of which of the 20 projects would
maximize the net benefits to the region.
Assuming 5 min of computer time is
required to simulate and evaluate each
combination of projects, show that it would
require 36 days of computer time even if
99% of the alternative combinations could
be discarded using “good judgment.” What
does this suggest about the use of simulation for regional interdependent multiproject water resources planning?
Assume you wish to determine the allocation of water Xj to three different users j,
who obtain benefits Rj(Xj). The total water
available is Q. Write a flow chart showing
how you can find the allocation to each user
that results in the highest total benefits.
Consider the allocation problem illustrated
below.

User 2

Gage site
User 1

User 3

Exercises

The allocation priority in each simulation period t is:
First 10 units of streamflow at the gage remain
in the stream.
Next 20 units go to User 3.
Next 60 units are equally shared by Users 1
and 2.
Next 10 units go to User 2.
Remainder goes downstream.

91

This will be a graph of the allocation as a
function of the flow at the allocation site.
(b) Simulate this allocation policy using any
river basin simulation model such as
RIBASIM, WEAP, Modsim, or other
selected model, including your own, for
any specified inflow series ranging from 0
to 130 units.

(a) Assume no incremental flow along the
stream and no return flow from users.
Define the allocation policy at each site.

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4

An Introduction to Optimization
Models and Methods

Water resource systems are characterized by
multiple interdependent components that together produce multiple economic, environmental,
ecological, and social impacts. As discussed in
the previous chapter, planners and managers
working toward improving the design and performance of these complex systems must identify
and evaluate alternative designs and operating
policies, comparing their predicted performance
with desired goals or objectives. Typically, this
identification and evaluation process is accomplished with the aid of optimization and simulation models. While optimization methods are
designed to provide preferred values of system
design and operating policy variables—values
that will lead to the highest levels of system
performance—they are often used to eliminate
the clearly inferior options. Using optimization
for a preliminary screening followed by more
detailed and accurate simulation is the primary
way we have, short of actually building physical
models, of estimating effective system designs
and operating policies. This chapter introduces
and illustrates the art of optimization model development and use in analyzing water resources
systems. The models and methods introduced in
this chapter are extended in subsequent chapters.

4.1

Introduction

This chapter introduces some optimization
modeling approaches for identifying ways of
satisfying specified goals or objectives. The

modeling approaches are illustrated by their
application to some relatively simple water
resources planning and management problems.
The purpose here is to introduce and compare
some commonly used optimization methods and
approaches. This is not a text on the state of the
art of optimization modeling. More realistic and
more complex problems usually require much
bigger and more complex models than those
developed and discussed in this chapter, but
these bigger and more complex models are often
based on the principles and techniques introduced here.
The emphasis here is on the art of model
development—just how one goes about constructing and solving optimization models that
will provide information useful for addressing
and perhaps even solving particular problems. It
is unlikely anyone will ever use any of the
specific models developed in this or other chapters simply because the specific examples used to
illustrate the approach to model development and
solution will not be the ones they face. However,
it is quite likely water resource managers and
planners will use these modeling approaches and
solution methods to analyze a variety of water
resource systems. The particular systems modeled and analyzed here, or any others that could
have been used, can be the core of more complex
models needed to analyze more complex problems in practice.
Water resources planning and management
today is dominated by the use of optimization
and simulation models. Computer software is

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_4

93

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4

becoming increasingly available for solving
various types of optimization and simulation
models. However, no software currently exists
that will build models of particular water
resource systems. What and what not to include
and assume in models requires judgment, experience, and knowledge of the particular problems
being addressed, the system being modeled and
the decision-making environment—including
what aspects can be changed and what cannot.
Understanding the contents of this and following
chapters and performing the suggested exercises
at the end of each chapter can only be a first step
toward gaining some judgment and experience in
model development.
Before proceeding to a more detailed discussion of optimization, a review of some methods
of dealing with time streams of economic
incomes or costs (engineering economics) may
be useful. Those familiar with this subject that is
typically covered in applied engineering economics courses can skip this next section.

4.2

Comparing Time Streams
of Economic Benefits and Costs

All of us make decisions that involve future
benefits and costs. The extent to which we value
future benefits or costs compared to present
benefits or costs is reflected by what is called a
discount rate. While economic criteria are only
one aspect of everything we consider when
making decisions, they are often among the
important ones. Economic evaluation methods
involving discount rates can be used to consider
and compare alternatives characterized by various benefits and costs that are expected to occur
now and in the future. This section offers a quick
and basic review of the use of discount rates that
enable comparisons of alternative time series of
benefits and costs. Many economic optimization
models incorporate discount rates in their economic objective functions.
Engineering economic methods typically
focus on the comparison of a discrete set of

An Introduction to Optimization Models and Methods

mutually exclusive alternatives (only one of
which can be selected) each characterized by a
time series of benefits and costs. Using various
methods involving the discount rate, the time
series of benefits and costs are converted to a
single net benefit that can be compared with
other such net benefits in order to identify the one
that is best. The values of the decision variables
(e.g., the design and operating policy variable
values) are known for each discrete alternative
being considered. For example, consider again
the tank design problem presented in the previous chapter. Alternative tank designs could be
identified, and then each could be evaluated, on
the basis of cost and perhaps other criteria as
well. The best would be called the optimal one, at
least with respect to the objective criteria used
and the discrete alternatives being considered.
The optimization methods introduced in the
following sections of this chapter extend those
engineering economics methods. Some methods
are discrete, some are continuous. Continuous
optimization methods, such as the model defined
by Eqs. 3.1–3.3 in Sect. 3.2 of the previous
chapter can identify the “best” tank design
directly without having to identify and compare
numerous discrete, mutually exclusive alternatives. Just how such models can be solved will be
discussed later in this chapter. For now, consider
the comparison of alternative discrete plans
p having different benefits and costs over time.
Let the net benefit generated at the end of time
period t by plan p be designated simply as Bp(t).
Each plan is characterized by the time stream
of net benefits it generates over its planning
period Tp.
fBp ð1Þ; Bp ð2Þ; Bp ð3Þ; . . .; Bp ðTp Þ

ð4:1Þ

Clearly, if in any time period t the benefits
exceed the costs, then Bp ðtÞ [ 0; and if the costs
exceed the benefits, Bp ðtÞ\0. This section
defines two ways of comparing different benefit,
cost or net-benefit time streams produced by
different plans perhaps having different planning
period durations Tp.

4.2 Comparing Time Streams of Economic Benefits and Costs

4.2.1

Interest Rates

Fundamental to the conversion of a time series of
incomes and costs to an equivalent single value, so
that it can be compared to other equivalent single
values of other time series, is the concept of the
time value of money. From time to time, individuals, private corporations, and governments need
to borrow money to do what they want to do. The
amount paid back to the lender has two components: (1) the amount borrowed and (2) an additional amount called interest. The interest amount
is the cost of borrowing money, of having the
money when it is loaned compared to when it is
paid back. In the private sector the interest rate, the
added fraction of the amount owed that equals the
interest, is often identified as the marginal rate of
return on capital. Those who have money, called
capital, can either use it themselves or they can
lend it to others, including banks, and receive
interest. Assuming people with capital invest their
money where it yields the largest amount of
interest, consistent with the risk they are willing to
take, most investors should be receiving at least the
prevailing interest rate as the return on their capital.
Any interest earned by an investor or paid by
a debtor depends on the size of the loan, the
duration of the loan, and the interest rate. The
interest rate includes a number of considerations.
One is the time value of money (a willingness to
pay something to obtain money now rather than
to obtain the same amount later). Another is the
risk of losing capital (not getting the full amount
of a loan or investment returned at some future
time). A third is the risk of reduced purchasing
capability (the expected inflation over time). The
greater the risks of losing capital or purchasing
power, the higher the interest rate compared to
the rate reflecting only the time value of money
in a secure and inflation-free environment.

4.2.2

Equivalent Present Value

To compare projects or plans involving different
time series of benefits and costs, it is often convenient to express these time series as a single
equivalent value. One way to do this is to convert

95

each amount in the time series to what it is worth
today, its present worth, that is, a single value at
the present time. This present worth will depend
on the prevailing interest rate in each future time
period. Assuming a value V0 is invested at the
beginning of a time period, e.g., a year, in a
project or a savings account earning interest at a
rate r per period, then at the end of the period the
value of that investment is (1 + r)V0.
If one invests an amount V0 at the beginning
of period t = 1 and at the end of that period
immediately reinvests the total amount (the
original investment plus interest earned), and
continues to do this for n consecutive periods at
the same period interest rate r, the value, Vn, of
that investment at the end of n periods would be
Vn ¼ V0 ð 1 þ r Þ n

ð4:2Þ

This results from V1 ¼ V0 =ð1 þ r Þ at the end of
period 1, V2 ¼ V1 =ð1 þ rÞ ¼ V0 ð1 þ rÞ2 at the end
of period 2, and so on until at the end of period n.
The initial amount V0 is said to be equivalent
to Vn at the end of n periods. Thus the present
worth or present value, V0, of an amount of
money Vn at the end of period n is
V0 ¼ Vn =ð 1 þ r Þ n

ð4:3Þ

Equation 4.3 is the basic compound interest
discounting relation needed to determine the
present value at the beginning of period 1 (or end
of period 0) of net benefits Vn that accrue at the
end of n time periods.
The total present value of the net benefits
generated by plan p, denoted V0p , is the sum of
the values of the net benefits Vp(t) accrued at the
end of each time period t times the discount
factor for that period t. Assuming the interest or
discount rate r in the discount factor applies for
the duration of the planning period, i.e., from
t = 1 to t = Tp.
V0p ¼

X

V p ðtÞ=ð1 þ r Þt

ð4:4Þ

t¼1;Tp

The present value of the net benefits achieved
by two or more plans having the same economic

96

4

An Introduction to Optimization Models and Methods

planning horizons Tp can be used as an economic
basis for plan selection. If the economic lives or
planning horizons of projects differ, then the
present value of the plans may not be an appropriate measure for comparison and plan selection.
A valid comparison of alternative plans using
present values is possible if all plans have the
same planning horizon or if funds remaining at
the end of the shorter planning horizon are
invested for the remaining time up until the longer
planning horizon at the same interest rate r.

This factor is often used to compute the
equivalent annual end-of-year cost of engineering structures that have a fixed initial construction cost C0 and annual end-of-year operation,
maintenance, and repair (OMR) costs. The
equivalent uniform end-of-year total annual cost,
TAC, equals the initial cost times the capital
recovery factor plus the equivalent annual
end-of-year uniform OMR costs.

4.2.3

For private investments requiring borrowed
capital, interest rates are usually established, and
hence fixed, at the time of borrowing. However,
benefits may be affected by changing interest
rates, which are not easily predicted. It is common practice in benefit–cost analyses to assume
constant interest rates over time, for lack of any
better assumption.
Interest rates available to private investors or
borrowers may not be the same rates that are used
for analyzing public investment decisions. In an
economic evaluation of public-sector investments, the same relationships are used even
though government agencies are not generally
free to loan or borrow funds on private money
markets. In the case of public-sector investments,
the interest rate to be used in an economic analysis is a matter of public policy; it is the rate at
which the government is willing to forego current
benefits to its citizens in order to provide benefits
to those living in future time periods. It can be
viewed as the government’s estimate of the time
value of public monies or the marginal rate of
return to be achieved by public investments.
These definitions and concepts of engineering
economics are applicable to many of the problems faced in water resources planning and
management. Each of the equations above is
applicable to discrete alternatives whose decision
variables (investments over time) are known. The
equations are used to identify the best alternative
from a set of mutually exclusive alternatives
whose decision variable values are known. More
detailed discussions of the application of

Equivalent Annual Value

If the lives of various plans differ, but the same
plans will be repeated on into the future, then one
need to only compare the equivalent constant
annual net benefits of each plan. Finding the
average or equivalent annual amount Vp is done
in two steps. First, one can compute the present
value, V0p , of the time stream of net benefits,
using Eq. 4.4. The equivalent constant annual
benefits, Vp, all discounted to the present must
equal the present value, V0p .
V0p ¼

X

V p =ð 1 þ r Þ t

t¼1;Tp

V p ¼ V0p =

X

or

1=ð1 þ r Þt

ð4:5Þ

t¼1;Tp

Using a little algebra the average annual
end-of-year benefits Vp of the project or plan p is

 

V p ¼ V0p rð1 þ rÞTp = ð1 þ rÞTp  1
ð4:6Þ
The capital recovery factor CRFn is the

 

expression rð1 þ rÞTp = ð1 þ rÞTp  1 in Eq. 4.6
that converts a fixed payment or present value V0p
at the beginning of the first time period into an
equivalent fixed periodic payment Vp at the end
of each time period. If the interest rate per period
is r and there are n periods involved, then the
capital recovery factor is
CRFn ¼ ½rð1 þ rÞn =½ð1 þ rÞn  1

ð4:7Þ

TAC ¼ CRFn C0 þ OMR

ð4:8Þ

4.2 Comparing Time Streams of Economic Benefits and Costs

engineering economics are contained in numerous texts on the subject. In the next section, we
introduce methods that can identify the best
alternative among those whose decision variable
values are not known. For example, engineering
economic methods can identify, for example, the
most cost-effective tank from among those whose
dimension values have been previously selected.
The optimization methods that follow can identify directly the values of the dimensions of most
cost-effective tank.

4.3

Nonlinear Optimization Models
and Solution Procedures

Constrained optimization involves finding the
values of decision variables given specified
relationships that have to be satisfied. Constrained optimization is also called mathematical
programming. Mathematical programming techniques include calculus-based Lagrange multipliers and various methods for solving linear and
nonlinear models including dynamic programming, quadratic programming, fractional programming, and geometric programming, to
mention a few. The applicability of each of these

97

as well as other constrained optimization procedures is highly dependent on the mathematical
structure of the model that in turn is dependent
on the system being analyzed. Individuals tend to
construct models in a way that will allow them to
use a particular optimization technique they think
is best. Thus, it pays to be familiar with various
types of optimization methods since no one
method is best for all optimization problems.
Each has its strengths and limitations. The
remainder of this chapter introduces and illustrates the application of some of the most common constrained optimization techniques used in
water resources planning and management.
Consider a river from which diversions are
made to three water-consuming firms that belong
to the same corporation, as illustrated in Fig. 4.1.
Each firm makes a product. Water is needed in
the process of making that product, and is the
critical resource. The three firms can be denoted
by the index j = 1, 2, and 3 and their water allocations by xj. Assume the problem is to determine the allocations xj of water to each of three
firms (j = 1, 2, 3) that maximize the total net
P
benefits,
j NBj ðxj Þ, obtained from all three
firms. The total amount of water available is
constrained or limited to a quantity of Q.

Fig. 4.1 Three water-using firms obtain water from a river. The amounts xj allocated to each firm j will depend on the
river flow Q

98

4

An Introduction to Optimization Models and Methods

Fig. 4.2 Concave net benefit functions for three water users, j, and their slopes at allocations xj

NB1 ðx1 Þ ¼ 6x1  x21

ð4:9Þ

NB2 ðx2 Þ ¼ 7x2  1:5x22

ð4:10Þ

units must remain in the river. Introducing this
constraint renders the previous solution infeasible. In this case we want to find the allocations
that maximize the total net benefits obtained from
all firms subject to having only 6 flow units
available for allocations. Using simple calculus
will not suffice.

NB3 ðx3 Þ ¼ 8x3  0:5x23

ð4:11Þ

4.3.2

Assume the net benefits, NBj(xj), derived from
water xj allocated to each firm j, are defined by

These are concave functions exhibiting
decreasing marginal net benefits with increasing
allocations. These functions look like hills, as
illustrated in Fig. 4.2.

4.3.1

Solution Using Calculus

Calculus can be used to find the allocations that
maximize each user’s net benefits, simply by
finding where the slope or derivative of the net
benefit function for each firm equals zero. The
derivative, dNB(x1)/dx1, of the net benefit function for Firm 1 is (6 − 2x1) and hence the allocation to Firm 1 that maximizes its net benefits
would be 6/2 or 3. The corresponding allocations
for Firms 2 and 3 are 2.33 and 8, respectively.
The total amount of water desired by all firms is
the sum of each firm’s desired allocation, or
13.33 flow units. However, suppose only 8 units
of flow are available for all three firms and 2

Solution Using Hill Climbing

One approach for finding, at least approximately,
the particular allocations that maximize the total
net benefit derived from all firms in this example
is an incremental steepest-hill-climbing method.
This method divides the total available flow
Q into increments and allocates each successive
increment so as to get the maximum additional
net benefit from that incremental amount of
water. This procedure works in this example
because each of the net benefit functions is concave; in other words, the marginal benefits
decrease as the allocation increases. This procedure is illustrated by the flow diagram in Fig. 4.3.
Table 4.1 lists the results of applying the
procedure shown in Fig. 4.3 to the problem when
(a) only 8 and (b) only 20 flow units are available. Here a minimum river flow of 2 is required
and is to be satisfied, when possible, before any
allocations are made to the firms.
The hill-climbing method illustrated in
Fig. 4.3 and Table 4.1 assigns each incremental

4.3 Nonlinear Optimization Models and Solution Procedures

99

Fig. 4.3 Steepest hill-climbing approach for finding allocation of a flow Qmax to the three firms, while meeting
minimum river flow requirements R

flow ΔQ to the use that yields the largest additional (marginal) net benefit. An allocation is
optimal for any total flow Q when the marginal
net benefits from each nonzero allocation are
equal, or as close to each other as possible given
the size of the increment ΔQ. In this example,
with a ΔQ of 1 and Qmax of 8, it just happens that
the marginal net benefits associated with each
allocation are all equal (to 4). The smaller the
ΔQ, the more precise will be the optimal allocations in each iteration, as shown in the lower
portion of Table 4.1, where ΔQ approaches 0.
Based on the allocations derived for various
values of available water Q, as shown in
Table 4.1, an allocation policy can be defined.
For this problem, the allocation policy that
maximizes total net benefits for any particular
value of Q is shown in Fig. 4.4.
This hill-climbing approach leads to optimal
allocations only if all of the net benefit functions
whose sum is being maximized are concave: that
is, the marginal net benefits decrease as the
allocation increases. Otherwise, only a local
optimum solution can be guaranteed. This is true

using any calculus-based optimization procedure
or algorithm.

4.3.3

Solution Using Lagrange
Multipliers

4.3.3.1 Approach
As an alternative to hill-climbing methods, consider a calculus-based method involving
Lagrange multipliers. To illustrate this approach,
a slightly more complex water-allocation example will be used. Assume that the benefit, Bj(xj),
each water-using firm receives is determined, in
part, by the quantity of product it produces and
the price per unit of the product that is charged.
As before, these products require water and water
is the limiting resource. The amount of product
produced, pj, by each firm j is dependent on the
amount of water, xj, allocated to it.
Let the function Pj(xj) represent the maximum
amount of product, pj, that can be produced by
firm j from an allocation of water xj. These are
called production functions. They are typically

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4

An Introduction to Optimization Models and Methods

Table 4.1 Hill-climbing iterations for finding allocations that maximize total net benefit given a flow of Qmax and a
required (minimum) streamflow of R = 2

Fig. 4.4 Water-allocation
policy that maximizes total
net benefits derived from
all three water-using firms

4.3 Nonlinear Optimization Models and Solution Procedures

concave: as xj increases the slope, dPj(xj)/dxj, of
the production function, Pj(xj), decreases. For
this example, assume the production functions
for the three water-using firms are
P1 ðx1 Þ ¼ 0:4ðx1 Þ0:9

ð4:12Þ

P2 ðx2 Þ ¼ 0:5ðx2 Þ0:8

ð4:13Þ

P3 ðx3 Þ ¼ 0:6ðx3 Þ0:7

ð4:14Þ

Next consider the cost of production. Assume
the associated cost of production can be expressed by the following convex functions:
C1 ¼ 3ðP1 ðx1 ÞÞ1:3

ð4:15Þ

C2 ¼ 5ðP2 ðx2 ÞÞ1:2

ð4:16Þ

C3 ¼ 6ðP3 ðx3 ÞÞ1:15

ð4:17Þ

Each firm produces a unique patented product,
and hence it can set and control the unit price of
its product. The lower the unit price, the greater
the demand and thus the more each firm can sell.
Each firm has determined the relationship
between the unit price and the amount that will

101

be demanded and sold. These are the demand
functions for that product. These unit price or
demand functions are shown in Fig. 4.5, where
the pj s are the amounts of each product produced. The vertical axis of each graph is the unit
price. To simplify the problem we are assuming
linear demand functions, but this assumption is
not a necessary condition.
The optimization problem is to find the water
allocations, the production levels, and the unit
prices that together maximize the total net benefit
obtained from all three firms. The water allocations plus the amount that must remain in the
river, R, cannot exceed the total amount of water
Q available.
Constructing and solving a model of this
problem for various values of Q, the total amount
of water available, will define the three allocation
policies as functions of Q. These policies can be
displayed as a graph, as in Fig. 4.4, showing the
three best allocations given any value of Q. This
of course assumes the firms can adjust to varying
allocations. In reality this may not be the case
(Chapter 9 examines this problem using more
realistic benefit functions that reflect the degree
to which firms can adapt to changing inputs over
time.)
The model:
Maximize Net benefit

ð4:18Þ

Fig. 4.5 Unit prices that will guarantee the sale of the specified amounts of products pj produced in each of the three
firms (linear functions are assumed in this example for simplicity)

102

4

Subject to
Definitional constraints:

Derivatives:

Net benefit ¼ Total return  Total cost
ð4:19Þ
Total return ¼ð12  p1 Þp1 þ ð20  1:5p2 Þp2
þ ð28  2:5p3 Þp3
ð4:20Þ
Total cost ¼ 3ðp1 Þ1:30 þ 5ðp2 Þ1:20 þ 6ðp3 Þ1:15
ð4:21Þ
Production functions defining the relationship
between water allocations xj and production pj
0:9

ð4:22Þ

p2 ¼ 0:5ðx2 Þ0:8

ð4:23Þ

p3 ¼ 0:6ðx3 Þ0:7

ð4:24Þ

p1 ¼ 0:4ðx1 Þ

An Introduction to Optimization Models and Methods

Water-allocation restriction
R þ x1 þ x2 þ x3 ¼ Q

ð4:25Þ

One can first solve this model for the values of
each pj that maximize the total net benefits,
assuming water is not a limiting constraint. This is
equivalent to finding each individual firm’s
maximum net benefits, assuming all the water that
is needed is available. Using calculus we can
equate the derivatives of the total net benefit
function with respect to each pj to 0 and solve
each of the resulting three independent equations:
Total Net benefit ¼ ½ð12  p1 Þp1 þ ð20  1:5p2 Þp2
h
þ ð28  2:5p3 Þp3   3ðp1 Þ1:30
i
þ 5ðp2 Þ1:20 þ 6ðp3 Þ1:15

ð4:26Þ

@ðNet benefit)=@p1 ¼ 0
¼ 12  2p1  1:3ð3Þp0:3
1
ð4:27Þ
@ðNet benefit)=@p2 ¼ 0
¼ 20  3p2  1:2ð5Þp0:2
2
ð4:28Þ
@ ðNet benefitÞ=@p3 ¼ 0
¼ 28  5p3  1:15ð6Þp0:15
3
ð4:29Þ
The result (rounded off) is p1 = 3.2, p2 = 4.0,
and p3 = 3.9 to be sold for unit prices of 8.77,
13.96, and 18.23, respectively, for a maximum
net revenue of 155.75. This would require water
allocations x1 = 10.2, x2 = 13.6, and x3 = 14.5,
totaling 38.3 flow units. Any amount of water
less than 38.3 will restrict the allocation to, and
hence the product production at, one or more of
the three firms.
If the total available amount of water is less
than that desired, constraint Eq. 4.25 can be
written as an equality, since all the water available, less any that must remain in the river, R,
will be allocated. If the available water supplies
are less than the desired 38.3 plus the required
streamflow R, then Eqs. 4.22–4.25 need to be
added. These can be rewritten as equalities since
they will be binding. Equation 4.25 in this case
can always be an equality since any excess water
will be allocated to the river, R.
To consider values of Q that are less than the
desired 38.3 units, constraints 4.22–4.25 can be
included in the objective function, Eq. 4.26, once
the right-hand side has been subtracted from the
left-hand side so that they equal 0. We set this
function equal to L.

4.3 Nonlinear Optimization Models and Solution Procedures

L ¼ ½ð12  p1 Þp1 þ ð20  1:5p2 Þp2 þ ð28  2:5p3Þp3 
h
i
 3ðp1 Þ1:30 þ 5ðp2 Þ1:20 þ 6ðp3 Þ1:15
h
i
h
i
 k1 p1  0:4ðx1 Þ0:9  k2 p2  0:5ðx2 Þ0:8
h
i
 k3 p3  0:6ðx3 Þ0:7  k4 ½R þ x1 þ x2 þ x3  Q

ð4:30Þ
Since each of the four constraint Eqs. 4.22–
4.25 included in Eq. 4.30 equals zero, each can
be multiplied by a variable λi without changing
the value of Eq. 4.30, or equivalently, Eq. 4.26.
These unknown variables λi are called the
Lagrange multipliers of constraints i. The value
of each multiplier, λi, is the marginal value of the
original objective function, Eq. 4.26, with
respect to a change in the value of the amount
produced, p, or in the case of constraint Eq. 4.25,
the amount of water available, Q. We will show
this shortly.
Differentiating Eq. 4.30 with respect to each
of the ten unknowns and setting the resulting
equations to 0 yields:
@L=@p1 ¼ 0 ¼ 12  2p1  1:3ð3Þp0:3
1  k1
ð4:31Þ
@L=@p2 ¼ 0 ¼ 20  3p2  1:2ð5Þp0:2
2  k2
ð4:32Þ

Table 4.2 Solutions to Eqs. 4.31–4.40

103

 k3
@L=@p3 ¼ 0 ¼ 28  5p3  1:15ð6Þp0:15
3
ð4:33Þ
 k4
@L=@x1 ¼ 0 ¼ k1 0:9ð0:4Þx0:1
1

ð4:34Þ

 k4
@L=@x2 ¼ 0 ¼ k2 0:8ð0:5Þx0:2
2

ð4:35Þ

 k4
@L=@x3 ¼ 0 ¼ k3 0:7ð0:6Þx0:3
3

ð4:36Þ

@L=@k1 ¼ 0 ¼ p1  0:4ðx1 Þ0:9

ð4:37Þ

@L=@k2 ¼ 0 ¼ p2  0:5ðx2 Þ0:8

ð4:38Þ

@L=@k3 ¼ 0 ¼ p3  0:6ðx3 Þ0:7

ð4:39Þ

@L=@k4 ¼ 0 ¼ R þ x1 þ x2 þ x3  Q

ð4:40Þ

These ten equations are the conditions necessary for a solution that maximizes Eq. 4.30, or
equivalently 4.26. They can be solved to obtain
the values of the ten unknown variables. The
solutions to these equations for various values of
Q, (found in this case using LINGO) are shown in
Table 4.2. (A free demo version of LINGO can be
obtained (downloaded) from www.LINDO.com.)

4.3.3.2

Meaning of Lagrange
Multiplier λ
In this example, Eq. 4.30 is the objective function that is to be maximized. It is maximized or

104

4

minimized by equating to zero each of its partial
derivatives with respect to each unknown variable. Equation 4.30 consists of the original net
benefit function plus each constraint i multiplied
by a weight or multiplier λi. This equation is
expressed in monetary units. The added constraints are expressed in other units: either the
quantity of product produced or the amount of
water available. Thus the units of the weights or
multipliers λi associated with these constraints
are expressed in monetary units per constraint
units. In this example, the multipliers λ1, λ2, and
λ3 represent the change in the total net benefit
value of the objective function (Eq. 4.26) per
unit change in the products p1, p2, and p3 produced. The multiplier λ4 represents the change in
the total net benefit per unit change in the water
available for allocation, Q − R.
Note in Table 4.2 that as the quantity of
available water increases, the marginal net benefits decrease. This is reflected in the values of
each of the multipliers, λi. In other words, the net
revenue derived from a quantity of product produced at each of the three firms, and from the
quantity of water available, is a concave function
of those quantities, as illustrated in Fig. 4.2.
To review the general Lagrange multiplier
approach and derive the definition of the multipliers, consider the general constrained optimization problem containing n decision variables
xj and m constraint equations i.
Maximize ðor minimize) FðXÞ

ð4:41Þ

subject to constraints
gi ðXÞ ¼ bi

i ¼ 1; 2; 3; . . .; m;

ð4:42Þ

where X is the vector of all xj. The Lagrange
function L(X, λ) is formed by combining
Eq. 4.42, each equaling zero, with the objective
function of Eq. 4.41.
LðX; kÞ ¼ F ðXÞ 

X

ki ð gi ð X Þ  bi Þ

ð4:43Þ

i

Solutions of the equations ∂L/∂xj = 0 for all
decision variables xj and ∂L/∂λi = 0 for all constraints gi are possible local optima.

An Introduction to Optimization Models and Methods

There is no guarantee that a global optimum
solution will be found using calculus-based
methods such as this one. Boundary conditions
need to be checked. Furthermore, since there is
no difference in the Lagrange multipliers procedure for finding a minimum or a maximum
solution, one needs to check whether in fact a
maximum or minimum is being obtained. In this
example, since each net benefit function is concave, a maximum will result.
The meaning of the values of the multipliers λi
at the optimum solution can be derived by
manipulation of ∂L/∂λi = 0. Taking the partial
derivative of the Lagrange function, Eq. 4.43,
with respect to an unknown variable xj and setting it to zero results in
@L=@xj ¼ 0 ¼ @F=@xj 

X

ki @ ðgi ðX ÞÞ=@xj

i

ð4:44Þ
Multiplying each term by ∂xj yields
@F ¼

X

ki @ ð gi ð X Þ Þ

ð4:45Þ

i

Dividing each term by ∂bk associated with a
particular constraint, say k, defines the meaning
of λk.
X
@F=@bk ¼
ki @ðgi ðXÞÞ=@bk ¼ kk ð4:46Þ
i

Equation 4.46 follows from the fact that
@ðgi ðXÞÞ=@bk equals 0 for constraints i ≠ k and
equals 1 for the constraint i = k. The latter is true
since bi = gi(X) and thus ∂(gi(X)) = ∂bi.
From Eq. 4.46, each multiplier λi is the marginal change in the original objective function F
(X) with respect to a change in the constant bi
associated with the constraint i. For nonlinear
problems, it is the slope of the objective function
plotted against the value of bi.
Readers can work out a similar proof if a slack
or surplus variable, Si, is included in inequality
constraints to make them equations. For a lessthan-or-equal constraint gi(X) ≤ bi a squared
slack variable S2i can be added to the left-hand
side to make it an equation gi ðXÞ þ S2i ¼ bi . For a

4.3 Nonlinear Optimization Models and Solution Procedures

greater-than-or-equal constraint gi(X) ≥ bi a
squared surplus variable S2i can be subtracted
from the left-hand side to make it an equation
gi ðXÞ  S2i ¼ bi . These slack or surplus variables
are squared to ensure they are nonnegative, and
also to make them appear in the differential
equations.
@L=@Si ¼ 0 ¼ 2Si ki ¼ Si ki

ð4:47Þ

Equation 4.47 shows that either the slack or
surplus variable, S, or the multiplier, λ, will
always be zero. If the value of the slack or surplus variable S is nonzero, the constraint is
redundant. The optimal solution will not be
affected by the constraint. Small changes in the
values, b, of redundant constraints will not
change the optimal value of the objective function F(X). Conversely, if the constraint is binding, the value of the slack or surplus variable
S will be zero. The multiplier λ can be nonzero if
the value of the function F(X) is sensitive to the
constraint value b.
The solution of the set of partial differential
Equations Eqs. 4.47 often involves a trial-anderror process, equating to zero a λ or a S for each
inequality constraint and solving the remaining
equations, if possible. This tedious procedure,
along with the need to check boundary solutions
when nonnegativity conditions are imposed,
detracts from the utility of classical Lagrange
multiplier methods for solving all but relatively
simple water resources planning problems.

4.4

Dynamic Programming

The water-allocation problems in the previous
section assumed a net-benefit function for each
water-using firm. In those examples, these functions were continuous and differentiable, a convenient attribute if methods based on calculus
(such as hill-climbing or Lagrange multipliers) are
to be used to find the best solution. In many
practical situations, these functions may not be so
continuous, or so conveniently concave for maximization or convex for minimization, making
calculus-based methods for their solution difficult.

105

A possible solution method for constrained
optimization problems containing continuous
and/or discontinuous functions of any shape is
called discrete dynamic programming. Each
decision variable value can assume one of a set
of discrete values. For continuous valued objective functions, the solution derived from discrete
dynamic programming may therefore be only an
approximation of the best one. For all practical
purposes this is not a significant limitation,
especially if the intervals between the discrete
values of the decision variables are not too large
and if simulation modeling is used to refine the
solutions identified using dynamic programming.
Dynamic programming is an approach that
divides the original optimization problem, with all
of its variables, into a set of smaller optimization
problems, each of which needs to be solved before
the overall optimum solution to the original
problem can be identified. The water supply allocation problem, for example, needs to be solved
for a range of water supplies available to each firm.
Once this is done the particular allocations that
maximize the total net benefit can be determined.

4.4.1

Dynamic Programming
Networks and Recursive
Equations

A network of nodes and links can represent each
discrete dynamic programming problem.
Dynamic programming methods find the best
way to get to, or go from, any node in that network. The nodes represent possible discrete
states of the system that can exist and the links
represent the decisions one could make to get
from one state (node) to another. Figure 4.6
illustrates a portion of such a network for the
three-firm allocation problem shown in Fig. 4.1.
In this case the total amount of water available,
Q − R, to all three firms is 10.
Thus, dynamic programming models involve
states, stages, and decisions. The relationships
among states, stages, and decisions are represented by networks, such as that shown in
Fig. 4.6. The states of the system are the nodes
and the values of the states are the numbers in the

106

4

An Introduction to Optimization Models and Methods

Fig. 4.6 A network
representing some of the
possible integer allocations
of water to three
water-consuming firms
j assuming 10 units of
water are available. The
circles or nodes represent
the discrete quantities of
water available to users not
yet allocated any water, and
the links represent feasible
allocation decisions xj to
the next firm j

nodes. Each node value in this example is the
quantity of water available to allocate to all
remaining firms, that is, to all connected links to
the right of the node. These state variable values
typically represent some existing condition either
before making, or after having made, a decision.
The stages of the system are the different components (e.g., firms) or time periods. Links
between (or connecting) initial and final states
represent decisions. The links in this example
represent possible allocation decisions for each
of the three different firms. Each stage is a separate firm. Each state is an amount of water that
remains to be allocated in the remaining stages.
Each link connects two nodes, the left node
value indicating the state of a system before a
decision is made, and the right node value indicating the state of a system after a decision is
made. In this case, the state of the system is the
amount of water available to allocate to the
remaining firms.
In the example shown in Fig. 4.6, the state
and decision variables are represented by integer
values—an admittedly fairly coarse discretization. The total amount of water available, in
addition to the amount that must remain in the

river, is 10. Note from the first row of Table 4.2
the exact allocation solution is x1 = 1.2,
x2 = 3.7, and x3 = 5.1. Normally, we would not
know this solution before solving for it using
dynamic programming, but since we do we can
reduce the complexity (number of nodes and
links) of the dynamic programming network so
that the repetitive process of finding the best
solution is clearer. Thus assume the range of x1
is limited to integer values from 0 to 2, the range
of x2 is from 3 to 5, and the range of x3 is from 4
to 6. These range limits are imposed here just to
reduce the size of the network. In this case, these
assumptions will not affect or constrain the
optimal integer solution. If we did not make
these assumptions the network would have, after
the first column of one node, three columns of
11 nodes, one representing each integer value
from 0 to 10. Finer (noninteger) discretizations
would involve even more nodes and connecting
links.
The links of Fig. 4.6 represent the water allocations. Note that the link allocations, the numbers on the links, cannot exceed the amount of
water available, that is, the number in the left
node of the link. The number in the right node is

4.4 Dynamic Programming

107

Fig. 4.7 Network as in
Fig. 4.6 representing
integer value allocations of
water to three
water-consuming firms.
The circles or nodes
represent the discrete
quantities of water
available, and the links
represent feasible allocation
decisions. The numbers on
the links indicate the net
benefits obtained from
these particular integer
allocation decisions

the quantity of water remaining after an allocation
has been made. The value in the right node, state
Sj+1, at the beginning of stage j + 1, is equal to the
value in the left node, Sj, less the amount of water,
xj, allocated to firm j as indicated on the link.
Hence, beginning with a quantity of water S1 that
can be allocated to all three firms, after allocating
x1 to Firm 1 what remains is S2:

(representing the allocation decisions we could
make) at each node (state) for each firm (stage).
Figure 4.7 shows the same network as in
Fig. 4.6; however the numbers on the links represent the net benefits obtained from the associated water allocations. For the three firms j = 1,
2, and 3, the net benefits, NBj(xj), associated with
allocations xj are

S1  x1 ¼ S2

NB1 ðx1 Þ ¼ maximumð12  p1 Þp1  3ðp1 Þ1:30

ð4:48Þ

where p1  0:4ðx1 Þ0:9

Allocating x2 to Firm 2, leaves S3.
S2  x2 ¼ S3

ð4:51Þ
ð4:49Þ

Finally, allocating x3 to Firm 3 leaves S4.
S3  x3 ¼ S4

NB2 ðx2 Þ ¼ maximumð20  1:5p2 Þp2  5ðp2 Þ1:20
where p2  0:5ðx2 Þ0:8
ð4:52Þ

ð4:50Þ

Figure 4.6 shows the different values of each
of these states, Sj, and decision variables xj
beginning with a quantity S1 = Q − R = 10. Our
task is to find the best path through the network,
beginning at the leftmost node having a state
value of 10. To do this we need to know the net
benefits we will get associated with all the links

NB3 ðx3 Þ ¼ maximumð28  2:5p3 Þp3  6ðp3 Þ1:15
where p3  0:6ðx3 Þ0:7
ð4:53Þ
The discrete dynamic programming algorithm
or procedure is a systematic way to find the best
path through this network, or any other suitable

108

4

network. What makes a network suitable for
dynamic programming is the fact that all the nodes
can be lined up in a sequence of vertical columns
and each link connects a node in one column to
another node in the next column of nodes. No link
passes over or through any other column(s) of
nodes. Links also do not connect nodes in the
same column. In addition, the contribution to the
overall objective value (in this case, the total net
benefits) associated with each discrete decision
(link) in any stage or for any firm is strictly a
function of the allocation of water to the firm. It is
not dependent on the allocation decisions associated with other stages (firms) in the network.
The main challenge in using discrete dynamic
programming to solve an optimization problem is
to structure the problem so that it fits this
dynamic programming network format. Perhaps
surprisingly, many water resources planning and
management problems do. But it takes practice to
become good at converting optimization problems to networks of states, stages, and decisions
suitable for solution by discrete dynamic programming algorithms.
In this problem the overall objective is to
Maximize

X

NBj ðxj Þ;

ð4:54Þ

An Introduction to Optimization Models and Methods

problem up into many optimization problems,
one for each possible discrete state (e.g., for each
node representing an amount of water available)
in each stage (e.g., for each firm). Given a particular state Sj and stage j—that is, a particular
node in the network—what decision (link) xj will
result in the maximum total net benefits, designated as Fj(Sj), given this state Sj for this and all
remaining stages or firms j, j + 1, j + 2 … ? This
question must be answered for each node in the
network before one can find the overall best set
of decisions for each stage: in other words, the
best allocations to each firm (represented by the
best path through the network) in this example.
Dynamic programming networks can be
solved in two ways—beginning at the most right
column of nodes or states and moving from right
to left, called the backward-moving (but forwardlooking) algorithm, or beginning at the leftmost
node and moving from left to right, called the
forward-moving (but backward-looking) algorithm. Both methods will find the best path
through the network. In some problems, however, only the backward-moving algorithm produces a useful solution. We will revisit this issue
when we get to reservoir operation where the
stages are time periods.

j

where NBj(xj) is the net benefit associated with an
allocation of xj to firm j. Equations 4.51–4.53
define these net benefit functions. As before, the
index j represents the particular firm, and each
firm is a stage for this problem. Note that the index
or subscript used in the objective function often
represents an object (like a water-using firm) at a
place in space or a time period. These places or
time periods are called the stages of a dynamic
programming problem. Our task is to find the best
path from one stage to the next: in other words, the
best allocation decisions for all three firms.
Dynamic programming can be viewed as a
multistage decision-making process. Instead of
deciding all three allocations in one single optimization procedure, like Lagrange multipliers,
the dynamic programming procedure divides the

4.4.2

Backward-Moving Solution
Procedure

Consider the network in Fig. 4.7. Again, the
nodes represent the discrete states—water available to allocate to all remaining users. The links
represent particular discrete allocation decisions.
The numbers on the links are the net benefits
obtained from those allocations. We want to
proceed through the node-link network from the
state of 10 at the beginning of the first stage to the
end of the network in such a way as to maximize
total net benefits. But without looking at all
combinations of successive allocations we cannot
do this beginning at a state of 10. However, we
can find the best solution if we assume we have
already made the first two allocations and are at

4.4 Dynamic Programming

109

any of the nodes or states at the beginning of the
final, third, stage with only one allocation decision remaining. Clearly at each node representing
the water available to allocate to the third firm, the
best decision is to pick the allocation (link) having the largest net benefits.
Denoting F3(S3) as the maximum net benefits
we can achieve from the remaining amount of
water S3, then for each discrete value of S3 we
can find the x3 that maximizes F3(S3). Those
shown in Fig. 4.7 include:

F3 ð5Þ ¼ MaximumfNB3 ðx3 Þg
x3  5
4  x3  6
¼ Maximumf27:9; 31:1g ¼ 31:1 when x3 ¼ 5

ð4:57Þ
F3 ð4Þ ¼ MaximumfNB3 ðx3 Þg
x3  4
4  x3  6
¼ Maximumf27:9g ¼ 27:9 when x3 ¼ 4

F3 ð7Þ ¼ MaximumfNB3 ðx3 Þg

ð4:58Þ

x3  7; the total flow available:
4  x3  6; the allowable range of allocations
= Maximumf27:9; 31:1; 33:7g ¼ 33:7 when x3 ¼ 6

ð4:55Þ
F3 ð6Þ ¼ MaximumfNB3 ðx3 Þg
x3  6
4  x3  6
¼ Maximumf27:9; 31:1; 33:7g ¼ 33:7 when x3 ¼ 6

ð4:56Þ

Fig. 4.8 Using the backward-moving dynamic programming method for finding the maximum remaining net
benefits, Fj(Sj), and optimal allocations (denoted by the
arrows on the links) for each state in Stage 3, then for
each state in Stage 2 and finally for the initial state in

These computations are shown on the network
in Fig. 4.8. Note that there are no benefits to be
obtained after the third allocation, so the decision
to be made for each node or state prior to allocating water to Firm 3 is simply that which
maximizes the net benefits derived from that last
(third) allocation. In Fig. 4.8 the links representing the decisions or allocations that result in
the largest net benefits are shown with arrows.

Stage 1 to obtain the allocation policy that maximizes
total net benefits, F1(10). The minimum flow to remain in
the river, R, is in addition to the ten units available for
allocation and is not shown in this network

110

4

Having computed the maximum net benefits,
F3(S3), associated with each initial state S3 for
Stage 3, we can now move backward (to the left) to
the discrete states S2 at the beginning of the second
stage. Again, these states represent the quantity of
water available to allocate to Firms 2 and 3. Denote
F2(S2) as the maximum total net benefits obtained
from the two remaining allocations x2 and x3 given
the quantity S2 water available. The best x2 depends
not only on the net benefits obtained from the
allocation x2 but also on the maximum net benefits
obtainable after that, namely the just-calculated
F3(S3) associated with the state S3 that results from
the initial state S2 and a decision x2. As defined in
Eq. 4.49, this final state S3 in Stage 2 obviously
equals S2 − x2. Hence for those nodes at the
beginning of Stage 2 shown in Fig. 4.8:
F2 ð10Þ ¼ MaximumfNB2 ðx2 Þ þ F3 ðS3 ¼ 10  x2 Þg
x2  10
3  x2  5
¼ Maximumf15:7 þ 33:7; 18:6
þ 33:7; 21:1 þ 31:1g ¼ 52:3 when x2 ¼ 4

ð4:59Þ
F2 ð9Þ ¼ MaximumfNB2 ðx2 Þ þ F3 ðS3 ¼ 9  x2 Þg
x2  9
3  x2  5
¼ Maximumf15:7 þ 33:7; 18:6
þ 31:1; 21:1 þ 27:9g ¼ 49:7
when x2 ¼ 4

ð4:60Þ
F2 ð8Þ ¼ MaximumfNB2 ðx2 Þ þ F3 ðS3 ¼ 8  x2 Þg
x2  8
3  x2  5 ðassume 4 instead of 5 since both
will not affect optimal solutionÞ
¼ Maximumf15:7 þ 31:1; 18:6
þ 27:9g ¼ 46:8 when x2 ¼ 3

ð4:61Þ
These maximum net benefit functions, F2(S2),
could be calculated for the remaining discrete
states from 7 to 0.

An Introduction to Optimization Models and Methods

Having computed the maximum net benefits
obtainable for each discrete state at the beginning
of Stage 2, that is, all the F2(S2) values, we can
move backward or left to the beginning of Stage
1. For this beginning stage there is only one state,
the state of 10 we are actually in before making
any allocations to any of the firms. In this case,
the maximum net benefits, F1(10), we can obtain
from given 10 units of water available, is
F1 ð10Þ ¼ MaximumfNB1 ðx1 Þ þ F2 ðS2 ¼ 10  x1 Þg
x1  10
0  x1  2
¼ Maximumf0 þ 52:3; 3:7
þ 49:7; 6:3 þ 46:8g ¼ 53:4 when x1 ¼ 1

ð4:62Þ
The value of F1(10) in Eq. 4.62 is the same as
the value of Eq. 4.54. This value is the maximum
net benefits obtainable from allocating the
available 10 units of water. From Eq. 4.62 we
know that we will get a maximum of 53.4 net
benefits if we allocate 1 unit of water to Firm 1.
This leaves 9 units of water to allocate to the two
remaining firms. This is our optimal state at the
beginning of Stage 2. Given a state of 9 at the
beginning of Stage 2, we see from Eq. 4.60 that
we should allocate 4 units of water to Firm 2.
This leaves 5 units of water for Firm 3. Given a
state of 5 at the beginning of Stage 3, Eq. 4.57
tells us we should allocate all 5 units to Firm 3.
All this is illustrated in Fig. 4.8.
Compare this discrete solution with the continuous one defined by Lagrange multipliers as
shown in Table 4.2. The exact solution, to the
nearest tenth, is 1.2, 3.7, and 5.1 for x1, x2, and
x3, respectively. The solution just derived from
discrete dynamic programming that assumed
only integer allocation values is 1, 4, and 5,
respectively.
To summarize, a dynamic programming
model was developed for the following problem:
Maximize Net benefit

ð4:63Þ

4.4 Dynamic Programming

111

Subject to
Net benefit ¼ Total return  Total cost
ð4:64Þ
Total return ¼ ð12  p1 Þp1 þ ð20  1:5p2 Þp2
þ ð28  2:5p3 Þp3
ð4:65Þ
Total cost ¼ 3ðp1 Þ1:30 þ 5ðp2 Þ1:20 þ 6ðp3 Þ1:15
ð4:66Þ
p1  0:4ðx1 Þ0:9

ð4:67Þ

p2  0:5ðx2 Þ0:8

ð4:68Þ

p3  0:6ðx3 Þ0:7

ð4:69Þ

x1 þ x2 þ x3  10

ð4:70Þ

The discrete dynamic programming version of
this problem required discrete states Sj representing the amount of water available to allocate
to firms j, j + 1, …. It required discrete allocations xj. Next it required the calculation of the
maximum net benefits, Fj(Sj), that could be
obtained from all firms j, beginning with Firm 3,
and proceeding backward as indicated in
Eqs. 4.71–4.73.
F3 ðS3 Þ ¼ maximumfNB3 ðx3 Þg over all x3  S3 ;
for all discrete S3 values between 0 and 10

ð4:71Þ
F2 ðS2 Þ ¼ maximumfNB2 ðx2 Þ þ F3 ðS3 Þg
over all x2  S2 and S3 ¼ S2  x2 ;

0  S2  10

The values of each NBj(xj) are obtained from
Eqs. 4.51 to 4.53.
To solve for F1(S1) and each optimal allocation xj we must first solve for all values of F3(S3).
Once these are known we can solve for all values
of F2(S2). Given these F2(S2) values, we can
solve for F1(S1). Equations 4.71 need to be
solved before Eqs. 4.72 can be solved, and
Eqs. 4.72 need to be solved before Eqs. 4.73 can
be solved. They need not be solved simultaneously, and they cannot be solved in reverse order.
These three equations are called recursive equations. They are defined for the backward-moving
dynamic programming solution procedure.
There is a correspondence between the nonlinear optimization model defined by Eqs. 4.63–
4.70 and the dynamic programming model
defined by the recursive Eqs. 4.71–4.73. Note
that F3(S3) in Eq. 4.71 is the same as
F3 ðS3 Þ ¼ Maximum NB3 ðx3 Þ

ð4:74Þ

Subject to
x 3  S3 ;

ð4:75Þ

where NB3(x3) is defined in Eq. 4.53.
Similarly, F2(S2) in Eq. 4.72 is the same as
F2 ðS2 Þ ¼ Maximum NB2 ðx2 Þ þ NB3 ðx3 Þ
ð4:76Þ
Subject to
x2 þ x3  S 2 ;

ð4:77Þ

where NB2(x2) and NB3(x3) are defined in
Eqs. 4.52 and 4.53.
Finally, F1(S1) in Eq. 4.73 is the same as
F1 ðS1 Þ ¼ Maximum NB1 ðx1 Þ þ NB2 ðx2 Þ þ NB3 ðx3 Þ

ð4:78Þ

ð4:72Þ
Subject to
F1 ðS1 Þ ¼ maximumfNB1 ðx1 Þ þ F2 ðS2 Þg
over all x1  S1 and S2 ¼ S1  x1 and S1 ¼ 10

ð4:73Þ

x1 þ x2 þ x3  S1 ¼ 10;

ð4:79Þ

where NB1(x1), NB2(x2), and NB3(x3) are defined
in Eqs. 4.51–4.53.

112

4

Alternatively, F3(S3) in Eq. 4.71 is the same as
F3 ðS3 Þ ¼ Maximumð28  2:5p3 Þp3  6ðp3 Þ1:15
ð4:80Þ

An Introduction to Optimization Models and Methods

successive states Sj and Sj+1 and the decision xj.
In the above example, these transition functions
are defined by Eqs. 4.48–4.50, or, in general
terms for all firms j, by
Sj þ 1 ¼ S j  x j

Subject to
p3  0:6ðx3 Þ0:7

ð4:81Þ

x 3  S3

ð4:82Þ

4.4.3

ð4:92Þ

Forward-Moving Solution
Procedure

Similarly, F2(S2) in Eq. 4.72 is the same as
F2 ðS2 Þ ¼ Maximumð20  1:5p2 Þp2
þ ð28  2:5p3 Þp3  5ðp2Þ1:20 6ðp3Þ1:15

ð4:83Þ
Subject to
p2  0:5ðx2 Þ0:8

ð4:84Þ

p3  0:6ðx3 Þ0:7

ð4:85Þ

x 2 þ x 3  S2

ð4:86Þ

Finally, F1(S1) in Eq. 4.73 is the same as
F1 ðS1 Þ ¼ Maximumð12  p1 Þp1
þ ð20  1:5p2 Þp2 þ ð28  2:5p3 Þp3
h
i
 3ðp1 Þ1:30 þ 5ðp2 Þ1:20 þ 6ðp3 Þ1:15
ð4:87Þ
Subject to

We have just described the backward-moving
dynamic programming algorithm. In that
approach at each node (state) in each stage we
calculated the best value of the objective function
that can be obtained from all further or remaining
decisions. Alternatively one can proceed forward, that is, from left to right, through a
dynamic programming network. For the
forward-moving algorithm at each node we need
to calculate the best value of the objective
function that could be obtained from all past
decisions leading to that node or state. In other
words, we need to find how best to get to each
state Sj+1 at the end of each stage j.
Returning to the allocation example, define
fj(Sj+1) as the maximum net benefits from the
allocation of water to firms 1, 2, …, j, given the
remaining water, state Sj+1. For this example, we
begin the forward-moving, but backward-looking,
process by selecting each of the ending states in the
first stage j = 1 and finding the best way to have
arrived at (or to have achieved) those ending states.
Since in this example there is only one way to get
to each of those states, as shown in Fig. 4.7 or
Fig. 4.8 the allocation decisions, x1, given a value
for S2 are obvious.

p1  0:4ðx1 Þ0:9

ð4:88Þ

p2  0:5ðx2 Þ0:8

ð4:89Þ

p3  0:6ðx3 Þ0:7

ð4:90Þ

f1 ðS2 Þ ¼ maximumfNB1 ðx1 Þg

x1 þ x2 þ x3  S1 ¼ 10

ð4:91Þ

x1 ¼ 10  S2

The transition function of dynamic programming defines the relationship between two

ð4:93Þ

Hence, f1(S2) is simply NB1(10 − S2). Once
the values for all f1(S2) are known for all discrete

4.4 Dynamic Programming

113

S2 between 0 and 10, move forward (to the right)
to the end of Stage 2 and find the best allocations
x2 to have made given each final state S3.
f2 ðS3 Þ ¼ maximumfNB2 ðx2 Þ þ f1 ðS2 Þg
0  x2  10  S3
ð4:94Þ
S2 ¼ S3 þ x 2
Once the values of all f2(S3) are known for all
discrete states S3 between 0 and 10, move forward to Stage 3 and find the best allocations x3 to
have made given each final state S4.
f3 ðS4 Þ ¼ maximumfNB3 ðx3 Þ þ f2 ðS3 Þg
for all discrete S4 between 0 and 10:
0  x3  10  S4
S3 ¼ S4 þ x 3
ð4:95Þ
Figure 4.9 illustrates a portion of the network
represented by Eqs. 4.93–4.95, and the fj(Sj+1)
values.
From Fig. 4.9, note the highest total net benefits are obtained by ending with 0 remaining
Fig. 4.9 Using the
forward-moving dynamic
programming method for
finding the maximum
accumulated net benefits,
fj(Sj + 1), and optimal
allocations (denoted by the
arrows on the links) that
should have been made to
reach each ending state,
beginning with the ending
states in Stage 1, then for
each ending state in Stage 2
and finally for the ending
states in Stage 3

water at the end of Stage 3. The arrow tells us
that if we are to get to that state optimally, we
should allocate 5 units of water to Firm 3. Thus
we must begin Stage 3, or end Stage 2, with
10 − 5 = 5 units of water. To get to this state at
the end of Stage 2 we should allocate 4 units of
water to Firm 2. The arrow also tells us we
should have had 9 units of water available at the
end of Stage 1. Given this state of 9 at the end of
Stage 1, the arrow tells us we should allocate 1
unit of water to Firm 1. This is the same allocation policy as obtained using the backwardmoving algorithm.

4.4.4

Numerical Solutions

The application of discrete dynamic programming to most practical problems will usually
require writing some software. There are no
general dynamic programming computer programs available that will solve all dynamic programming problems. Thus any user of dynamic
programming will need to write a computer
program to solve a particular problem unless they

114

do it by hand. Most computer programs written
for solving specific dynamic programming
problems create and store the solutions of the
recursive equations (e.g., Eqs. 4.93–4.95) in
tables. Each stage is a separate table, as shown in
Tables 4.3, 4.4, and 4.5 for this example
water-allocation problem. These tables apply to
only a part of the entire problem, namely that part
of the network shown in Figs. 4.8 and 4.9. The
backward solution procedure is used.
Table 4.3 contains the solutions of Eqs. 4.55–
4.58 for the third stage. Table 4.4 contains the
solutions of Eqs. 4.59–4.61 for the second stage.
Table 4.5 contains the solution of Eq. 4.62 for
the first stage.
From Table 4.5 we see that, given 10 units of
water available, we will obtain 53.4 net benefits

4

An Introduction to Optimization Models and Methods

and to get this we should allocate 1 unit to Firm
1. This leaves 9 units of water for the remaining
two allocations. From Table 4.4 we see that for a
state of 9 units of water available we should
allocate 4 units to Firm 2. This leaves 5 units.
From Table 4.3 for a state of 5 units of water
available we see we should allocate all 5 of them
to Firm 3.
Performing these calculations for various
discrete total amounts of water available, say
from 0 to 38 in this example, will define an
allocation policy (such as the one shown in
Fig. 4.5 for a different allocation problem) for
situations when the total amount of water is less
than that desired by all the firms. This policy can
then be simulated using alternative time series of
available amounts of water, such as streamflows,

Table 4.3 Computing the values of F3(S3) and optimal allocations x3 for all states S3 in Stage 3

Table 4.4 Computing the values of F2(S2) and optimal allocations x2 for all states S2 in Stage 2

4.4 Dynamic Programming

115

Table 4.5 Computing the values of F1(S1) and optimal allocations x1, for all states S1, in Stage 1

to obtain estimates of the time series (or statistical measures of those time series) of net benefits
obtained by each firm, assuming the allocation
policy is followed over time.

4.4.5

Dimensionality

One of the limitations of dynamic programming
is handling multiple state variables. In our
water-allocation example, we had only one state
variable: the total amount of water available. We
could have enlarged this problem to include other
types of resources the firms require to make their
products. Each of these state variables would
need to be discretized. If, for example, only
m discrete values of each state variable are considered, for n different state variables (e.g., types
of resources) there are mn different combinations
of state variable values to consider at each stage.
As the number of state variables increases, the
number of discrete combinations of state variable
values increases exponentially. This is called
dynamic programming’s “curse of dimensionality”. It has motivated many researchers to search
for ways of reducing the number of possible
discrete states required to find an optimal solution to large multistate-variable problems.

4.4.6

Principle of Optimality

The solution of dynamic programming models or
networks is based on a principal of optimality

(Bellman 1957). The backward-moving solution
algorithm is based on the principal that no matter
what the state and stage (i.e., the particular node
you are at), an optimal policy is one that proceeds forward from that node or state and stage
optimally. The forward-moving solution algorithm is based on the principal that no matter
what the state and stage (i.e., the particular node
you are at), an optimal policy is one that has
arrived at that node or state and stage in an
optimal manner.
This “principle of optimality” is a very simple
concept but requires the formulation of a set of
recursive equations at each stage. It also requires
that either in the last stage (j = J) for a
backward-moving algorithm, or in the first stage
(j = 1) for a forward-moving algorithm, the
future value functions, Fj+1(Sj+1), associated with
the ending state variable values, or past value
functions, f0(S1), associated with the beginning
state variable values, respectively, all equal some
known value. Usually that value is 0 but not
always. This condition is needed in order to
begin the process of solving each successive
recursive equation.

4.4.7

Additional Applications

Among the common dynamic programming
applications in water resources planning are
water allocations to multiple uses, infrastructure
capacity expansion, and reservoir operation.
The previous three-user water-allocation problem

116

4

(Fig. 4.1) illustrates the first type of application.
The other two applications are presented below.

4.4.7.1 Capacity Expansion
How much infrastructure should be built, when
and why? Consider a municipality that must plan
for the future expansion of its water supply system or some component of that system, such as a
reservoir, aqueduct, or treatment plant. The
capacity needed at the end of each future period
t has been estimated to be Dt. The cost, Ct(st, xt)
of adding capacity xt in each period t is a function
of that added capacity as well as of the existing
capacity st at the beginning of the period. The
planning problem is to find that time sequence of
capacity expansions that minimizes the present
value of total future costs while meeting the
predicted capacity demand requirements. This is
the usual capacity expansion problem.
This problem can be written as an optimization model: The objective is to minimize the
present value of the total cost of capacity
expansion.
X
Minimize
Ct ðst ; xt Þ;
ð4:96Þ
t

where Ct(st, xt) is the present value of the cost of
capacity expansion xt in period t given an initial
capacity of st.
The constraints of this model define the minimum required final capacity in each period t, or
equivalently the next period’s initial capacity, st+1,
as a function of the known existing capacity s1
and each expansion xt up through period t.
st þ 1 ¼ s1 þ

X

xs

An Introduction to Optimization Models and Methods

each future period t is no less than the capacity
required Dt at the end of that period.
st þ 1  D t

for t ¼ 1; 2; . . .; T

ð4:99Þ

There may also be constraints on the possible
expansions in each period defined by a set Ωt of
feasible capacity additions in each period t:
xt 2 Xt

ð4:100Þ

Figure 4.10 illustrates this type of capacity
expansion problem. The question is how much
capacity to add and when. It is a significant
problem for several reasons. One is that the cost
functions Ct(st, xt) typically exhibit fixed costs
and economies of scale, as illustrated in Fig. 4.11.
Each time any capacity is added there are fixed as
well as variable costs incurred. Fixed and variable
costs that show economies of scale (decreasing
average costs associated with increasing capacity
additions) motivate the addition of excess
capacity, capacity not needed immediately but
expected to be needed in the future to meet an
increased demand for additional capacity.
The problem is also important because any
estimates made today of future demands, costs
and interest rates are likely to be wrong. The
future is uncertain. Its uncertainties increase the
further the future. Capacity expansion planners
need to consider the future if their plans are to be
cost-effective and not myopic from assuming

for t ¼ 1; 2; . . .; T

s¼1;t

ð4:97Þ
Alternatively these equations may be expressed by a series of continuity relationships:
st þ 1 ¼ st þ xt

for t ¼ 1; 2; . . .; T

ð4:98Þ

In this problem, the constraints must also
ensure that the actual capacity st+1 at the end of

Fig. 4.10 A demand projection (solid blue line) and a
possible capacity expansion schedule (red line) for
meeting that projected demand over time

4.4 Dynamic Programming

Fig. 4.11 Typical cost function for additional capacity
given an existing capacity. The cost function shows the
fixed costs, C0, required if additional capacity is to be
added, and the economies of scale associated with the
concave portion of the cost function

there is no future. Just how far into the future do
they need to look? And what about the uncertainty in all future costs, demands, and interest
rate estimates? These questions will be addressed
after showing how the problem can be solved for
any fixed-planning horizon and estimates of
future demands, interest rates, and costs.
The constrained optimization model defined
by Eqs. 4.96–4.100 can be restructured as a
multistage decision-making process and solved
using either a forward or backward-moving discrete dynamic programming solution procedure.
The stages of the model will be the time periods
t. The states will be either the capacity st+1 at the
end of a stage or period t if a forward-moving
solution procedure is adopted, or the capacity st,
at the beginning of a stage or period t if a
backward-moving solution procedure is used.
A network of possible discrete capacity states
and decisions can be superimposed onto the
demand projection of Fig. 4.9, as shown in
Fig. 4.12. The solid blue circles in Fig. 4.12
represent possible discrete states, St, of the system, the amounts of additional capacity existing
at the end of each period t − 1 or equivalently at
the beginning of period t.
Consider first a forward-moving dynamic
programming algorithm. To implement this,
define ft(st+1) as the minimum cost of achieving a
capacity st+1, at the end of period t. Since at the
beginning of the first period t = 1, the accumulated least cost is 0, f0(s1) = 0.

117

Fig. 4.12 Network of discrete capacity expansion decisions (links) that meet the projected demand

Hence, for each final discrete state s2 in stage
t = 1 ranging from D1 to the maximum demand
DT, define
f1 ðs2 Þ ¼ minfC1 ðs1 ; x1 Þg in which the discrete x1
¼ s2 and s1 ¼ 0
ð4:101Þ
Moving to stage t = 2, for the final discrete
states s3 ranging from D2 to DT,
f2 ðs3 Þ ¼ minfC2 ðs2 ; x2 Þ þ f1 ðs2 Þg
over all discrete x2 between 0 ð4:102Þ
and s3  D1 and s2 ¼ s3  x2
Moving to stage t = 3, for the final discrete
states s4 ranging from D3 to DT,
f3 ðs4 Þ ¼minfC3 ðs3 ; x3 Þ þ f2 ðs3 Þg
over all discrete x3 between 0
and s4  D2 and s3 ¼ s4  x3
ð4:103Þ
In general for all stages t between the first and
last:
ft ðst þ 1 Þ ¼ minfCt ðst ; xt Þ þ ft1 ðst Þg
over all discrete xt between 0
and st þ 1  Dt1 and st ¼ st þ 1  xt

ð4:104Þ
For the last stage t = T and for the final discrete state sT+1 = DT,

118

4

fT ðsT þ 1 Þ ¼ minfCT ðsT ; xT Þ þ fT1 ðsT Þg
over all discrete xT
between 0 and DT  DT1
where sT ¼ sT þ 1  xT
ð4:105Þ
The value of fT(sT+1) is the minimum present
value of the total cost of meeting the demand for
T time periods. To identify the sequence of
capacity expansion decisions that results in this
minimum present value of the total cost requires
backtracking to collect the set of best decisions xt
for all stages t. A numerical example will illustrate this.
A numerical example
Consider the five-period capacity expansion
problem shown in Fig. 4.12. Figure 4.13 is the

Fig. 4.13 A discrete capacity expansion network showing the present value of the expansion costs associated
with each feasible expansion decision. Finding the best

An Introduction to Optimization Models and Methods

same network with the present value of the expansion costs on each link. The values of the
states, the existing capacities, represented by the
nodes, are shown on the left vertical axis. The
capacity expansion problem is solved on
Fig. 4.14 using the forward-moving algorithm.
From the forward-moving solution to the
dynamic programming problem shown in
Fig. 4.14, the present value of the cost of the
optimal capacity expansion schedule is 23 units
of money. Backtracking (moving left against the
arrows) from the farthest right node, this schedule adds 10 units of capacity in period t = 1, and
15 units of capacity in period t = 3.
Next consider the backward-moving algorithm applied to this capacity expansion problem.
The general recursive equation for a
backward-moving solution is

path through the network can be done using forward or
backward-moving discrete dynamic programming

4.4 Dynamic Programming

119

Fig. 4.14 A capacity-expansion example, showing the
results of a forward-moving dynamic programming
algorithm. The numbers next to the nodes are the

minimum cost to have reached that particular state at
the end of the particular time period t

Ft ðst Þ ¼ minimumfCt ðst ; xt Þ þ Ft þ 1 ðst þ 1 Þg
over all discrete xt from Dt  st to DT  st ;
for all discrete states st from Dt1 to DT

gives us a time series of future capacity expansion decisions for the next 5 time periods, what is
important to decision-makers is what additional
capacity to add in the current period, i.e., now,
not what capacity to add in future periods. Does
the uncertainty of future demands and costs and
the 5-period planning horizon affect this first
decision, x1? This is the question to ask. If the
answer is no, then one can place some confidence
in the value of x1. If the answer is yes, then more
study may be warranted to determine which
demand and cost scenario to assume, or, if
applicable, how far into the future to extend the
planning horizon.
Future capacity expansion decisions in time
periods 2, 3, and so on can be based on updated
information and analyses carried out closer to the

ð4:106Þ
where FT+1(DT) = 0 and as before each cost
function is the discounted cost.
Once again, as shown in Fig. 4.14, the minimum total present value cost is 23 if 10 units of
additional capacity are added in period t = 1 and
15 in period t = 3.
Now consider the question of the uncertainty
of future demands, Dt, discounted costs, Ct(st, xt),
as well as to the fact that the planning horizon
T is only 5 time periods. Of importance is just
how these uncertainties and finite planning
horizon affect our decisions. While the model

120

4

An Introduction to Optimization Models and Methods

Fig. 4.15 A capacity-expansion example, showing the
results of a backward-moving dynamic programming
algorithm. The numbers next to the nodes are the

minimum remaining cost to have the particular capacity
required at the end of the planning horizon given the
existing capacity of the state

time those decisions are to be made. At those
times, the forecast demands and economic cost
estimates can be updated and the planning horizon extended, as necessary, to a period that again
does not affect the immediate decision. Note that
in the example problem shown in Figs. 4.14 and
4.15, the use of 4 periods instead of 5 would have
resulted in the same first-period decision. There
is no need to extend the analysis to 6 or more
periods.
To summarize: What is important to
decision-makers is what additional capacity to
add now. While the current period’s capacity
addition should be based on the best estimates of
future costs, interest rates and demands, once a
solution is obtained for the capacity expansion
required for this and all future periods up to some

distant time horizon, one can then ignore all but
that first decision, x1: that is, what to add now.
Then just before the beginning of the second
period, the forecasting and analysis can be
redone with updated data to obtain an updated
solution for what if any capacity to add in period
2, and so on into the future. Thus, these
sequential decision making dynamic programming models can be designed to be used in a
sequential decision-making process.

4.4.7.2 Reservoir Operation
Reservoir operators need to know how much
water to release and when. Reservoirs designed to
meet demands for water supplies, recreation,
hydropower, the environment and/or flood control need to be operated in ways that meet those

4.4 Dynamic Programming

121

demands in a reliable and effective manner. Since
future inflows or storage volumes are uncertain,
the challenge, of course, is to determine the best
reservoir release or discharge for a variety of
possible inflows and storage conditions that could
exist or happen in each time period t in the future.
Reservoir release policies are often defined in
the form of what are called “rule curves.” Figure 4.17 illustrates a rule curve for a single
reservoir on the Columbia River in the northwestern United States. It combines components
of two basic types of release rules. In both of
these, the year is divided into various discrete
within-year time periods. There is a specified
release for each value of storage in each
within-year time period. Usually higher storage

zones are associated with higher reservoir releases. If the actual storage is relatively low, then
less water is usually released so as to hedge
against a continuing water shortage or drought.
Release rules may also specify the desired
storage level for the time of year. The operator is
to release water as necessary to achieve these
target storage levels. Maximum and minimum
release constraints might also be specified that
may affect how quickly the target storage levels
can be met. Some rule curves define multiple
target storage levels depending on hydrological
(e.g., snow pack) conditions in the upstream
watershed, or on the forecast climate conditions
as affected by ENSO cycles, solar geomagnetic
activity, ocean currents and the like.

Fig. 4.16 An example reservoir rule curve specifying
the storage targets and some of the release constraints,
given the particular current storage volume and time of
year. The release constraints also include the minimum

and maximum release rates and the maximum downstream channel rate of flow and depth changes that can
occur in each month

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Reservoir release rule curves for a year, such
as that shown in Fig. 4.16, define a policy that
does not vary from one year to the next. The
actual releases will vary, however, depending on
the inflows and storage volumes that actually
occur. The releases are often specified independently of future inflow forecasts. They are typically based only on existing storage volumes and
within-year periods—the two axes of Fig. 4.16.
Release rules are typically derived from trial
and error simulations. To begin these simulations

it is useful to have at least an approximate idea of
the expected impact of different alternative policies on various system performance measures or
objectives. Policy objectives could be the maximization of expected annual net benefits from
downstream releases, reservoir storage volumes,
hydroelectric energy and flood control, or the
minimization of deviations from particular
release, storage volume, hydroelectric energy or
flood flow targets or target ranges. Discrete
dynamic programming can be used to obtain

Fig. 4.17 Network representation of the four-season
reservoir release problem. Given any initial storage
volume St at the beginning of a season t, and an expected

inflow of Qt during season t, the links indicate the
possible release decisions corresponding to those in
Table 4.7

4.4 Dynamic Programming

initial estimates of reservoir-operating policies
that meet these and other objectives. The results
of discrete dynamic programming can be
expressed in the form shown in Fig. 4.17.

123

evaporation and seepage losses based on different
discrete combinations of initial and final storage
volumes for each within-year period t.
Rounding these losses to the nearest integer
value, Table 4.7 shows the net releases associA numerical example
ated with initial and final storage volumes. They
As a simple example, consider a reservoir having
are computed using Eq. 4.107. The information
an active storage capacity of 20 million cubic
in Table 4.7 allows us to draw a network repremeters, or for that matter any specified volume
senting each of the discrete storage volume states
units. The active storage volume in the reservoir
(the nodes), and each of the feasible releases (the
can vary between 0 and 20. To use discrete
links). This network for the four seasons t in the
dynamic programming, this range of possible
year is illustrated in Fig. 4.17.
storage volumes must be divided into a set of
This reservoir-operating problem is a multidiscrete values. These will be the discrete state
stage decision-making problem. As Fig. 4.17
variable values. In this example let the range of
illustrates, at the beginning of any season t, the
storage volumes be divided into intervals of 5
storage volume can be in any of the five discrete
storage volume units. Hence, the initial storage
states. Given the state, a release decision is to be
volume, St, can assume values of 0, 5, 10, 15, and
made. This release will depend on the state: the
20 for all periods t.
initial storage volume and the mean inflow, as
For each period t, let Qt be the mean inflow,
well as the losses that may be estimated based on
Lt(St, St+1) the evaporation and seepage losses
the initial and final storage volumes, as defined in
that depend on the initial and final storage volTable 4.6. The release will also depend on what
umes in the reservoir, and Rt the release or disis to be accomplished—that is, the objectives to
charge from the reservoir. Each variable is
be satisfied.
expressed as volume units for the period t.
For this example, assume there are various
Storage volume continuity requires that in
targets that water users would like to achieve.
each period t the initial active storage volume, St,
Downstream water users want reservoir operators
plus the inflow, Qt, less the losses, Lt(St, St+1),
to meet their flow targets. Individuals who use
and release, Rt, equals the final storage, or
the lake for recreation want the reservoir operaequivalently the initial storage, St+1, in the foltors to meet storage volume or storage level
lowing period t + 1. Hence
targets. Finally, individuals living on the downstream floodplain want the reservoir operators to
St þ Qt  Rt  Lt ðSt ; St þ 1 Þ ¼ St þ 1 for each period t:
provide storage capacity for flood protection.
ð4:107Þ Table 4.8 identifies these different targets that are
to be met, if possible, for the duration of each
To satisfy the requirement (imposed for conseason t.
venience in this example) that each storage volClearly, it will not be possible to meet all
ume variable be a discrete value over the range
these storage volume and release targets in all
from 0 to 20 in units of 5, the releases, Rt, must
four seasons, given inflows of 24, 12, 6, and 18,
be such that when Qt − Rt − Lt(St, St+1) is added
respectively. Hence, the objective in this example
to St the resulting value of St+1 is one of the five
will be to do the best one can: to minimize a
discrete numbers between 0 and 20.
weighted sum of squared deviations from each of
Assume four within-year periods t in each year
these targets. The weights reflect the relative
(kept small for this illustrative example). In these
importance of meeting each target in each season
four seasons assume the mean inflows, Qt, are 24,
t. Target deviations are squared to reflect the fact
12, 6, and 18, respectively. Table 4.6 defines the

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Table 4.6 Evaporation and seepage losses based on initial and final storage volumes for example reservoir-operating
problem

4.4 Dynamic Programming

125

Table 4.7 Discrete releases associated with initial and final storage volumes for example reservoir-operating problem

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Table 4.8 Storage volume and release targets for the example reservoir operation problem

that the marginal “losses” associated with deviations increase with increasing deviations. Small
deviations are not as serious as larger deviations,
and it is better to have numerous small deviations
rather than a few larger ones.
During the recreation season (periods 2 and 3),
deviations below or above the recreation storage
lake volume targets are damaging. During the
flood season (period 1), any storage volume in
excess of the flood control storage targets of 15
reduces the flood storage capacity. Deviations
below that flood control target are not penalized.
Flood control and recreation storage targets during each season t apply throughout the season,
thus they apply to the initial storage St as well as
to the final storage St+1 in appropriate periods t.
The objective is to minimize the sum of total
weighted squared deviations, TSDt, over all
seasons t from now on into the future:
Minimize

X
t

TSDt ;

ð4:108Þ

where
h
i
TSDt ¼ wst ðTS  St Þ2 þ ðTS  St þ 1 Þ2
h
i


þ wf t ðESt Þ2 þ ðESt þ 1 Þ2 þ wrt DR2t
ð4:109Þ

In the above equation, when t = 4, the last
period of the year, the following period t + 1 = 1,
the first period in the following year. Each ESt is
the storage volume in excess of the flood storage
target volume, TF. Each DRt is the difference
between the actual release, Rt, and the target
release, TRt, when the release is less than the target.
The excess storage, ESt, above the flood target
storage TF at the beginning of each season t can
be defined by the constraint:
St  TF þ ESt

for periods t ¼ 1 and 2:
ð4:110Þ

The deficit release, DRt, during period t can
be defined by the constraint:
Rt  TRt  DRt

for all periods t:

ð4:111Þ

The first component of the right side of
Eq. 4.109 defines the weighted squared deviations from a recreation storage target, TS, at the
beginning and end of season t. In this example
the recreation season is during periods 2 and 3.
The weights, wst, associated with the recreation
component of the objective are 1 in periods 2 and
3. In periods 1 and 4 the weights, wst, are 0.
The second component of Eq. 4.109 is for
flood control. It defines the weighted squared

4.4 Dynamic Programming

deviations associated with storage volumes in
excess of the flood control target volume, TF, at
the beginning and end of the flood season, period
t = 1. In this example, the weights, wft, are 1 for
period 1 and 0 for periods 2, 3, and 4. Note the
conflict between flood control and recreation at
the end of period 1 or equivalently at the
beginning of period 2.
Finally, the last component of Eq. 4.109
defines the weighted squared deficit deviations
from a release target, TRt, In this example all
release weights, wrt, equal 1.
Associated with each link in Fig. 4.17 is the
release, Rt, as defined in Table 4.7. Also associated with each link is the sum of weighted
squared deviations, TSDt, that result from the
particular initial and final storage volumes and the
storage volume and release targets identified in
Table 4.8. They are computed using Eq. 4.109,
with the releases defined in Table 4.7 and targets
defined in Table 4.8, for each feasible combination of initial and final storage volumes, St and
St+1, for each of the four seasons or periods in a
year. These computed weighted squared deviations for each link are shown in Table 4.9.
The goal in this example problem is to find the
path through a multiyear network—each year of
which is as shown in Fig. 4.17—that minimizes
the sum of the squared deviations associated with
each of the path’s links. Again, each link’s
weighted squared deviations are given in
Table 4.9. Of interest is the best path into the
future from any of the nodes or states (discrete
storage volumes) that the system could be in at
the beginning of any season t.
These paths can be found using the
backward-moving solution procedure of discrete
dynamic programming. This procedure begins at
any arbitrarily selected time period or season
when the reservoir presumably produces no further benefits to anyone (and it does not matter
when that time is—just pick any time) and proceeds backward, from right to left one stage (i.e.,
one time period) at a time, toward the present. At
each node (representing a discrete storage

127

volume St and inflow Qt), we can calculate the
release or final storage volume in that period that
minimizes the remaining sum of weighted
squared deviations for all remaining seasons.
Denote this minimum sum of weighted squared
deviations for all n remaining seasons t as
Ftn ðSt ; Qt Þ. This value is dependent on the state
(St, Qt), and stage, t, and the number n of
remaining seasons. It is not a function of the
decision Rt or St+1.
This minimum sum of weighted squared
deviations for all n remaining seasons t is equal
to
Ftn ðSt ; Qt Þ ¼ min

X

TSDt ðSt ; Rt ; St þ 1 Þ

t¼1;n

over all feasible values of Rt ;
ð4:112Þ
where
St þ 1 ¼ St þ Qt  Rt  Lt ðSt ; St þ 1 Þ

ð4:113Þ

and
St  K; the capacity of the reservoir ð4:114Þ
The policy we want to derive is called a
steady-state policy. Such a policy assumes the
reservoir will be operating for a relatively long
time with the same objectives and a repeatable
hydrologic time series of seasonal inputs. We can
find this steady-state policy by first assuming that
at some time all future benefits, losses or penal
ties, Ft ðSt ; Qt Þ, will be 0.
We can begin in that last season t of reservoir
operation and work backwards toward the present, moving left through the network one season
t at a time. We can continue for multiple years
until the annual policy begins repeating itself
each year. In other words, when the optimal Rt
associated with a particular state (St, Qt) is the
same in two or more successive years, and this
applies for all states (St, Qt) in each season t, a
steady-state policy has probably been obtained.

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Table 4.9 Total sum of squared deviations, TSDt, associated with initial and final storage volumes

These are calculated using Eqs. 4.109–4.111

4.4 Dynamic Programming

129

(A more definitive test of whether or not a
steady-state policy has been reached will be
discussed later.) A steady-state policy will occur
if the inflows, Qt, and objectives, TSDt (St, Rt,
St+1), remain the same for specific within-year
periods from year to year. This steady-state policy is independent of the assumption that the
operation will end at some point.
To find the steady-state operating policy for
this example problem, assume the operation ends
in some distant year at the end of season 4 (the
right-hand side nodes in Fig. 4.17). At the end of
this season the number of remaining seasons, n,
equals 0. The values of the remaining minimum

sums of weighted squared deviations, Ft ðSt ; Qt Þ
associated with each state (St, Qt), i.e., each node,
equal 0. Since for this problem there is no future.
Now we can begin the process of finding the best
releases Rt in each successive season t, moving
backward to the beginning of stage t = 4, then
stage t = 3, then to t = 2, and then to t = 1, and
then to t = 4 of the preceding year, and so on,
each move to the left increasing the number of
remaining seasons n by one.
At each stage, or season t, for each discrete
state (St, Qt) we can compute the release Rt or
equivalently the final storage volume St+1, that
minimizes
Fnt ðSt ; Qt Þ ¼ MinimumfTSDt ðSt ; Rt ; St þ 1 Þ
þ Ftn1
þ 1 ðSt þ 1 ; Qt þ 1 Þg for all 0  St  20

ð4:115Þ
The decision variable can be either the release,
Rt, or the final storage volume, St+1. If the decision variable is the release, then the constraints
on that release Rt are
Rt  St þ Qt  Lt ðSt ; St þ 1 Þ

ð4:116Þ

Rt  St þ Qt  Lt ðSt ; St þ 1 Þ  20 ðthe capacityÞ
ð4:117Þ
and
St þ 1 ¼ St þ Qt  Rt  Lt ðSt ; St þ 1 Þ

ð4:118Þ

If the decision variable is the final storage
volume, St+1, the constraints on that final storage
volume are
0  St þ 1  20

ð4:119Þ

St þ 1  St þ Qt  Lt ðSt ; St þ 1 Þ

ð4:120Þ

Rt ¼ St þ Qt  St þ 1  Lt ðSt ; St þ 1 Þ

ð4:121Þ

and

Note that if the decision variable is St+1 in
season t, this decision becomes the state variable
in season t + 1. In both cases, the storage volumes in each season are limited to discrete values
0, 5, 10, 15, and 20.
Tables 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16,
4.17, 4.18 and 4.19 show the values obtained
from solving the recursive equations for 10 successive seasons or stages (2.5 years). Each table
represents a stage or season t, beginning with
Table 4.10 at t = 4 and the number of remaining
seasons n = 1. The data in each table are
obtained from Tables 4.7 and 4.9. The last two
columns of each table represent the best release
and final storage volume decision(s) associated
with the state (initial storage volume and inflow).
Note that the policy defining the release or
final storage for each discrete initial storage
volume in season t = 3 in Table 4.12 is the same
as in Table 4.16, and similarly for season t = 4 in
Tables 4.13 and 4.17, and for season t = 1 in
Tables 4.14 and 4.18, and finally for season t = 2
in Tables 4.15 and 4.19. The policy differs over
each state, and over each different season, but not
from year to year for any specified state and
season. This indicates we have reached a
steady-state policy. If we kept on computing the
release and final storage policies for preceding
seasons, we would get the same policy as that
found for the same season in the following year.
The policy is dependent on the state—the initial
storage volume in this case—and on the season t,

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Table 4.10 Calculation of minimum squared deviations associated with various discrete storage states in season t = 4
with only n = 1 season remaining for reservoir operation

Table 4.11 Calculation of minimum squared deviations associated with various discrete storage states in season t = 3
with n = 2 seasons remaining for reservoir operation

4.4 Dynamic Programming

131

Table 4.12 Calculation of minimum squared deviations associated with various discrete storage states in season t = 2
with n = 3 seasons remaining for reservoir operation

Table 4.13 Calculation of minimum squared deviations associated with various discrete storage states in season t = 1
with n = 4 seasons remaining for reservoir operation

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Table 4.14 Calculation of minimum squared deviations associated with various discrete storage states in season t = 4
with n = 5 seasons remaining for reservoir operation

Table 4.15 Calculation of minimum squared deviations associated with various discrete storage states in season t = 3
with n = 6 seasons remaining for reservoir operation

4.4 Dynamic Programming

133

Table 4.16 Calculation of minimum squared deviations associated with various discrete storage states in season t = 2
with n = 7 seasons remaining for reservoir operation

Table 4.17 Calculation of minimum squared deviations associated with various discrete storage states in season t = 1
with n = 8 seasons remaining for reservoir operation

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Table 4.18 Calculation of minimum squared deviations associated with various discrete storage states in season t = 4
with n = 9 seasons remaining for reservoir operation

Table 4.19 Calculation of minimum squared deviations associated with various discrete storage states in season t = 3
with n = 10 seasons remaining for reservoir operation

4.4 Dynamic Programming

135

Table 4.20 The discrete steady-state reservoir-operating policy as computed for this example problem in Tables 4.16,
4.17, 4.18 and 4.19

but not on the year. This policy as defined in
Tables 4.16, 4.17, 4.18 and 4.19 is summarized
in Table 4.20.
This policy can be defined as a rule curve, as
shown in Fig. 4.18. It provides a first approximation of a reservoir release rule curve that one
can improve upon using simulation.
Table 4.20 and Fig. 4.18 define a policy that
can be implemented for any initial storage

Fig. 4.18 Reservoir rule
curve based on policy
defined in Table 4.20. Each
season is divided into
storage volume zones. The
releases associated with
each storage volume zone
are specified. Also shown
are the storage volumes that
would result if in each year
the actual inflows equaled
the inflows used to derive
this rule curve

volume condition at the beginning of any season
t. This can be simulated under different flow
patterns to determine just how well it satisfies the
overall objective of minimizing the weighted
sum of squared deviations from desired, but
conflicting, storage and release targets. There are
other performance criteria that may also be
evaluated using simulation, such as measures of
reliability, resilience, and vulnerability (Chap. 9).

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Table 4.21 A simulation of the derived operating policy in Table 4.20

The storage volumes and releases in each period twill repeat themselves each year, after the first year. The annual total squared
deviations, TSDt for the specific initial and final storage volumes and release conditions are obtained from Table 4.9

Assuming the inflows that were used to derive
this policy actually occurred each year, we can
simulate the derived sequential steady-state policy to find the storage volumes and releases that
would occur in each period, year after year, once
a repetitive steady-state condition were reached.
This is done in Table 4.21 for an arbitrary initial
storage volume of 20 in season t = 1. You can
try other initial conditions to verify that it
requires only 2 years at most to reach a repetitive
steady-state policy.
As shown in Table 4.21, if the inflows were
repetitive and the optimal policy was followed,
the initial storage volumes and releases would

begin to repeat themselves once a steady-state
condition has been reached. Once reached, the
storage volumes and releases will be the same
each year (since the inflows are the same). These
storage volumes are denoted as a blue line on the
rule curve shown in Fig. 4.18. The annual total
squared deviations will also be the same each
year. As seen in Table 4.21, this annual minimum weighted sum of squared deviations for this
example equals 186. This is what would be
observed if the inflows assumed for this analysis
repeated themselves.
Note from Tables 4.12, 4.13, 4.14, 4.15 and
4.16, 4.17, 4.18, 4.19 that once the steady-state

4.4 Dynamic Programming

137

sequential policy has been reached for any
specified storage volume, St, and season t, the
annual difference of the accumulated minimum
sum of squared deviations equals a constant,
namely the annual value of the objective function. In this case that constant is 186.
Ftn þ 1 ðSt ; Qt Þ  Ftn ðSt ; Qt Þ ¼ 186
for all St ; Qt and t:

ð4:122Þ

This condition indicates a steady-state policy
has been achieved.
This policy in Table 4.21 applies only for the
assumed inflows in each season. It does not
define what to do if the initial storage volumes or
inflows differ from those for which the policy is
defined. Initial storage volumes and inflows can
and will vary from those specified in the solution
of any deterministic model. One fact is certain:
no matter what inflows are assumed in any
model, the actual inflows will always differ.
Hence, a policy as defined in Table 4.20 and
Fig. 4.18 is much more useful than that in
Table 4.21. In Chap. 8 we will modify this
reservoir operation model to define releases or
final storage volumes as functions of not only
discrete storage volumes St but also of discrete
possible inflows Qt. However, the policy defined
by any relatively simple optimization model
policy should be simulated, evaluated, and further refined in an effort to identify the policy that
best meets the operating policy objectives.

4.4.8

General Comments
on Dynamic
Programming

Before ending this discussion of using dynamic
programming methods for analyzing water
resources planning, management and operating
policy problems, we should examine a major
assumption that has been made in each of the

applications presented. The first is that the net
benefits or costs or other objective values
resulting at each stage of the problem are
dependent only on the state and decision variable
values in each stage. They are independent of
decisions made at other stages. If the returns at
any stage are dependent on the decisions made at
other stages, then dynamic programming, with
some exceptions, becomes more difficult to
apply. Dynamic programming models can be
applied to design problems, such as the capacity
expansion problem or to operating problems,
such as the water-allocation and reservoir operation problems, but rarely to problems having
both unknown design and operating policy
decision variables at the same time. While there
are some tricks that may allow dynamic programming to be used to find the best solutions to
both design and operating problems encountered
in water resources planning, management and
operating policy studies, other optimization
methods, perhaps combined with dynamic programming where appropriate, are often more
useful.

4.5

Linear Programming

If the objective function and constraints of an
optimization model are all linear, many readily
available computer programs exist for finding its
optimal solution. Surprisingly many water
resource systems problems meet these conditions
of linearity. These linear optimization programs
are very powerful, and unlike many other optimization methods, they can be applied successfully to very large optimization problems
containing many variables and constraints. Many
water resources problems are too large to be
easily solved using nonlinear or dynamic programming methods. The number of variables and
constraints simply defining mass balances and
capacity limitations in numerous time periods

138

can become so big as to preclude the practical
use of most other optimization methods. Linear
programming procedures or algorithms for solving linear optimization models are often the most
efficient ways to find solutions to such problems.
Hence there is an incentive to convert large
optimization models to a linear form. Some ways
of doing this are discussed later in this chapter.
Because of the availability of computer programs that can solve linear programming problems, linear programming is arguably the most
popular and commonly applied optimization
algorithm in practical use today. It is used to
identify and evaluate alternative plans, designs
and management policies in agriculture, business, commerce, education, engineering, finance,
the civil and military branches of government,
and many other fields.
In spite of its power and popularity, for most
real-world water resources planning and management problems, linear programming, like the
other optimization methods already discussed in
this chapter, is best viewed as a preliminary
screening tool. Its value is more for reducing the
number of alternatives for further more detailed
simulations than for finding the best decision.
This is not just because approximation methods
may have been used to convert nonlinear functions to linear ones, but more likely because it is
difficult to incorporate all the complexity of the
system and all the objectives considered important to all stakeholders into a linear model.
Nevertheless, linear programming, like other
optimization methods, can provide initial designs
and operating policy information that simulation
models require before they can simulate those
designs and operating policies.
Equations 4.41 and 4.42 define the general
structure of any constrained optimization problem. If the objective function F(X) of the vector
X of decision variables xj is linear and if all the
constraints gi(X) in Eq. 4.42 are linear, then the
model becomes a linear programming model.

4

An Introduction to Optimization Models and Methods

The general structure of a linear programming
model is
Maximize or minimize

X

Pj xj

ð4:123Þ

j

Subject to
X

aij xj  or  bi

for i ¼ 1; 2; 3; . . .; m

j

ð4:124Þ
xj  0

for j ¼ 1; 2; 3; . . .; n:

ð4:125Þ

If any model fits this general form, where the
constraints can be any combination of equalities
(=) and inequalities (≥ or ≤), then a large variety
of linear programming computer programs can
be used to find the “optimal” values of all the
unknown decision variables xj. Variable nonnegativity is enforced within the solution algorithms of most commercial linear programming
programs, eliminating the need to have to specify
these conditions in any particular application.
Potential users of linear programming algorithms need to know how to construct linear
models and how to use the computer programs
that are available for solving them. They do not
have to understand all the mathematical details of
the solution procedure incorporated in the linear
programming codes. But users of linear programming computer programs should understand
what the solution procedure does and what the
computer program output means. To begin this
discussion of these topics, consider some simple
examples of linear programming models.

4.5.1

Reservoir Storage
Capacity-Yield Models

Linear programming can be used to define storage capacity-yield functions for a single or

4.5 Linear Programming

Fig. 4.19 Two storage-yield functions for a single
reservoir defining the maximum minimum dependable
release. These functions can be defined for varying levels
of yield reliability

multiple reservoirs. A storage capacity-yield
function defines the maximum constant “dependable” reservoir release or yield that will be
available, at a given level of reliability, during
each period of operation, as a function of the
active storage volume capacity. The yield from
any reservoir or group of reservoirs will depend
on the active storage capacity of each reservoir
and the water that flows into each reservoir, i.e.,
their inflows. Figure 4.19 illustrates two typical
storage-yield functions for a single reservoir.
To describe what a yield is and how it can be
increased, consider a sequence of 5 annual flows,
say 2, 4, 1, 5, and 3, at a site in an unregulated
stream. Based on this admittedly very limited
record of flows, the minimum (historically) “dependable” annual flow yield of the stream at that
site is 1, the minimum observed flow. Assuming
the flow record is representative of what future
flows might be, a discharge of 1 can be “guaranteed” in each period of record. (In reality, that
or any nonzero yield will have a reliability less
than 1, as will be considered in Chaps. 6 and 10.)
If a reservoir having an active storage capacity
of 1 is built, it could store 1 volume unit of flow
when the flow is greater than 2. It could then
release it along with the natural flow when the

139

natural flow is 1, increasing the minimum
dependable flow to 2 units in each year. Storing 2
units when the flow is 5, releasing 1 and the
natural flow when that natural flow is 2, and
storing 1 when the flow is 4, and then releasing
the stored 2 units along with the natural flow
when the natural flow is 1, will permit a yield of
3 in each time period with 2 units of active
capacity. This is the maximum annual yield that
is possible at this site, again based on these five
annual inflows and their sequence. The maximum annual yield cannot exceed the mean
annual flow, which in this example is 3. Hence,
the storage capacity-yield function equals 1 when
the active capacity is 0, 2 when the active
capacity is 1, and 3 when the active capacity is 2.
The annual yield remains at 3 for any active
storage capacity in excess of 2.
This storage-yield function is dependent not
only on the natural unregulated annual flows but
also on their sequence. For example if the
sequence of the same 5 annual flows were 5, 2, 1,
3, 4, the needed active storage capacity is 3
instead of 2 volume units as before to obtain a
dependable flow or yield of 3 volume units. In
spite of these limitations of storage
capacity-yield functions, historical records are
still typically used to derive them. (Ways of
augmenting the historical flow record are discussed in Chap. 6.)
There are many methods available for deriving storage-yield functions. One very versatile
method, especially for multiple reservoir systems, uses linear programming. Others are discussed in Chap. 10.
To illustrate a storage capacity-yield model,
consider a single reservoir that must provide at
least a minimum release or yield Y in each period
t. Assume a record of known (historical or synthetic) streamflows at the reservoir site is available. The problem is to find the maximum
constant yield Y obtainable from a given active
storage capacity. The objective is to

140

4

maximize Y

ð4:126Þ

This maximum yield is constrained by the
water available in each period, and by the
reservoir capacity. Two sets of constraints are
needed to define the relationships among the
inflows, the reservoir storage volumes, the
yields, any excess release, and the reservoir
capacity. The first set of continuity equations
equate the unknown final reservoir storage volume St+1 in period t to the unknown initial
reservoir storage volume St plus the known
inflow Qt, minus the unknown yield Y and excess
release, Rt, if any, in period t. (Losses are being
ignored in this example.)
St þ Qt  Y  Rt ¼ St þ 1 for each period t
¼ 1; 2; 3; . . .; T: T þ 1 ¼ 1
ð4:127Þ
If, as indicated in Eq. 4.127, one assumes that
period 1 follows the last period T, it is not necessary to specify the value of the initial storage
volume S1 and/or final storage volume ST+1.
They are set equal to each other and that variable
value remains unknown. The resulting
“steady-state” solution is based on the inflow
sequence that is assumed to repeat itself as well
as the available storage capacity, K.
The second set of required constraints ensures
that the reservoir storage volumes St at the

An Introduction to Optimization Models and Methods

beginning of each period t are no greater than the
active reservoir capacity K.
St  K

t ¼ l; 2; 3; . . .; T

ð4:128Þ

To derive a storage-yield function, the model
defined by Eqs. 4.126–4.128 must be solved for
various assumed values of capacity K. Only the
inflow values Qt and reservoir active storage
capacity K are assumed known. All other storage,
release and yield variables are unknown. Linear
programming will be able to find their optimal
values. Clearly, the upper bound on the yield
regardless of reservoir capacity will equal the mean
inflow (less any losses if they were included).
Alternatively, one can solve a number of linear programming models that minimize an
unknown storage capacity K needed to achieve
various specified yields Y. The resulting
storage-yield functions will be same. The minimum capacity needed to achieve a specified yield
will be the same as the maximum yield obtainable from the corresponding specified capacity
K. However, the specified yield Y cannot exceed
the mean inflow. If an assumed value of the yield
exceeds the mean inflow, there will be no feasible solution to the linear programming model.
Box 4.1 illustrates an example storage-yield
model and its solutions to find the storage-yield
function. For this problem, and others in this
chapter, the program LINGO (freely obtained
from www.lindo.com) is used.

4.5 Linear Programming

141

E020903a

Box 4.1. Example storage capacity-yield model and its solution from LINGO

! Reservoir Storage-Yield Model:
Define St as the initial active res. storage, period t,
Y as the reliable yield in each period t,
Rt as the excess release from the res., period t,
Qt as the known inflow volume to the res., period t
K as the reservoir active storage volume capacity.
;
Max = Y ; !Applies to Model 1. Must be omitted for Model 2;
Min = K ; !Applies to Model 2. Must be omitted for Model 1;
!
Subject to:
Mass balance constraints for each of 5 periods t.
;
S1 + Q1 - Y - R1 = S2;
S2 + Q2 - Y - R2 = S3;
S3 + Q3 - Y - R3 = S4;
S4 + Q4 - Y - R4 = S5;
S5 + Q5 - Y - R5 = S1; ! assumes a steady-state condition;
!
Capacity constraints on storage volumes.
;
S1 < K; S2 < K; S3 < K; S4 < K; S5 < K;
Data:
Q1 = 10; Q2 = 5; Q3 = 30; Q4 = 20; Q5 = 15;
!Note mean = 16;
K = ? ; ! Use for Model 1 only. Allows user to enter
any value of K during model run.;
Y = ? ; ! Use for Model 2 only. Allows user to enter
any value of Y during model run.
;
Enddata
:

0
5

5
10

0
5

10

12.5

10

0
5

0
0

0
5

7.5

0

2.5

0
5

5
0

0
0

25
15

15
10

10
5

10

0

0

15

0

2.5

15

15

10

10

0

15

15

0

0

0

5

0

18

16

17

11

0

14

18

0

0

0

0

0

Before moving to another application of linear
programming, consider how this storage-yield
problem, Eqs. 4.126–4.128, can be formulated as
a discrete dynamic programming model. The use of
discrete dynamic programming is clearly not the
most efficient way to define a storage-yield function but the problem of finding a storage-yield

function provides a good exercise in dynamic
programming. The dynamic programming network has the same form as shown in Fig. 4.19,
where each node is a discrete storage and inflow
state, and the links represent releases. Let Ftn ðSt Þ be
the maximum yield obtained given a storage
volume of St at the beginning of period t of a year

142

4

with n periods remaining of reservoir operation.
For initial conditions, assume all values of Ft0 ðSt Þ
for some final period t with no more periods n remaining equal a large number that exceeds the
mean annual inflow. Then for the set of feasible
discrete total releases Rt:



ð4:129Þ
Fnt ðSt Þ ¼ max min Rt ; Ftn1
þ 1 ðSt þ 1 Þ
This applies for all discrete storage volumes St
and for all within-year periods t and remaining
periods n. The constraints on the decision variables Rt are

An Introduction to Optimization Models and Methods

volumes will result in higher yields than will
lower ones. The highest yield will be that associated with the highest storage volumes and it
will equal the same value obtained from either of
the two linear programming models.

4.5.2

A Water Quality Management
Problem

The steady-state yields Ft(St) will depend on
the storage volumes St. High initial storage

Some linear programming modeling and solution techniques can be demonstrated using the
simple water quality management example
shown in Fig. 4.21. In addition, this example can
serve to illustrate how models can help identify
just what data are needed and how accurate they
must be for the decisions that are being
considered.
The stream shown in Fig. 4.20 receives
wastewater effluent from two point sources
located at sites 1 and 2. Without some wastewater treatment at these sites, the concentration of
some pollutant, Pj mg/l, at sites j = 2 and 3, will
continue to exceed the maximum desired concentration Pmax
. The problem is to find the level
j
of wastewater treatment (waste removed) at sites
i = 1 and 2 that will achieve the desired concentrations just upstream of site 2 and at site 3 at
a minimum total cost.

Fig. 4.20 A stream pollution problem that requires
finding the waste removal efficiencies (x1, x2) of wastewater treatment at sites 1 and 2 that meet the stream quality

standards at sites 2 and 3 at minimum total cost. W1 and
W2 are the amounts of pollutant prior to treatment at sites
1 and 2

Rt  S t þ Q t
Rt  St þ Qt  K;

and

ð4:130Þ

S t þ 1 ¼ St þ Q t  Rt
These recursive Eqs. 4.129 together with
constraint Eqs. 4.130 can be solved, beginning
with n = 1 and then for successive values of
seasons t and remaining periods n, until a
steady-state solution is obtained, that is, until
Ftn ðSt Þ ¼ Ftn1 ðSt Þ
for all values of St and periods t:

ð4:131Þ

4.5 Linear Programming

143

This is the classic water quality management
problem that is frequently found in the literature,
although least-cost solutions have rarely if ever
been applied in practice. There are valid reasons
for this that we will review later. Nevertheless,
this particular problem can serve to illustrate the
development of some linear models for determining data needs as well as for finding, in this
case, cost-effective treatment efficiencies. This
problem can also serve to illustrate graphically
the general mathematical procedures used for
solving linear programming problems.
The first step is to develop a model that predicts the pollutant concentrations in the stream as
a function of the pollutants discharged into it. To
do this we need some notation. Define Wj as the
mass of pollutant generated at site j (j = 1, 2)
each day. Without any treatment and assuming
no upstream pollution concentration, the discharge of W1 (in units of mass per unit time,
(M/T) at site j = 1 results in pollutant concentration of P1 in the stream at that site. This
concentration, (M/L3) equals the discharge W1
(M/T) divided by the streamflow Q1 (L3/T) at
that site. For example, assuming the concentration is expressed in units of mg/l and the flow is
in terms of m3/s, and mass of pollutant discharged is expressed as kg/day, and the flow
component of the wastewater discharge is negligible compared to the streamflow, the resulting
streamflow concentration P1 at site j = 1 is
W1/86.4 Q1:
P1 ðmg=lÞ ¼ Mass W1 discharged at site 1 ðkg=dayÞ=


streamflow Q1 at site 1 m3 =s =




kg=106 mg ð86; 400 s=dayÞ 103 L=m3
¼ W1 =86:4 Q1

ð4:132Þ
Each unit of a degradable pollutant mass in
the stream at site 1 in this example will decrease
as it travels downstream to site 2. Similarly each
unit of the pollutant mass in the stream at site 2

will decrease as it travels downstream to site 3.
The fraction αij of the mass at site i that reaches
site j is often assumed to be


aij ¼ exp ktij ;

ð4:133Þ

where k is a rate constant (1/time unit) that
depends on the pollutant and the temperature,
and tij is the time (number of time units) it takes a
particle of pollutant to flow from site i to site
j. The actual concentration at the downstream
end of a reach will depend on the streamflow at
that site as well as on the initial pollutant mass,
the time of travel and decay rate constant k.
In this example problem, the fraction of pollutant mass at site 1 that reaches site 3 is the
product of the transfer coefficients α12 and α23:
a13 ¼ a12 a23

ð4:134Þ

In general, for any site k between sites i and j:
aij ¼ aik akj

ð4:135Þ

Knowing the αij values for any pollutant and
the time of flow tij permits the determination of
the rate constant k for that pollutant and reach, or
contiguous series of reaches, from sites i to j,
using Eq. 4.133. If the value of k is 0, the pollutant is called a conservative pollutant; salt is an
example of this. Only increased dilution by less
saline water will reduce its concentration.
For the purposes of determining wastewater
treatment efficiencies or other capital investments
in infrastructure designed to control the pollutant
concentrations in the stream, some “design”
streamflow conditions have to be established.
Usually the design streamflow conditions are set
at low-flow values (e.g., the lowest monthly
average flow expected once in twenty years, or
the minimum 7-day average flow expected once
in ten years). Low design flows are based on the
assumption that pollutant concentrations will be
higher in low-flow conditions than in higher flow

144

4

conditions because of less dilution. While
low-flow conditions may not provide as much
dilution, they result in longer travel times, and
hence greater reductions in pollutant masses
between water quality monitoring sites. Hence
the pollutant concentrations may well be greater
at some downstream site when the flow conditions are higher than those of the design low-flow
value.
In any event, given particular design streamflow and temperature conditions, our first job is
to determine the values of these dimensionless
transfer coefficients αij. They will be independent
of the amount of waste discharged into the
stream as long as the stream stays aerobic. To
determine both α12 and α23 in this example
problem (Fig. 4.20) requires a number of pollutant concentration measurements at sites 1, 2 and
3 during design streamflow conditions. These
measurements of pollutant concentrations must
be made just downstream of the wastewater
effluent discharge at site 1, just upstream and
downstream of the wastewater effluent discharge
at site 2, and at site 3.
Assuming no change in streamflow and no
extra pollutant entering the reach that begins at
site 1 and ends just upstream of site 2, the mass
(kg/day) of pollutants just upstream of site 2 will
equal the mass at site 1, W1, times the transfer
coefficient α12:
Mass just upstream of site 2 ¼ W1 a12

ð4:136Þ

From this equation and 4.132 one can calculate the concentration of pollutants just upstream
of site 2.
The mass of additional pollutant discharged
into site 2 is W2. Hence the total mass just
downstream of site 2 is W1α12 + W2. At site 3 the
pollutant mass will equal the mass just downstream of site 2, times the transfer coefficient a23.
Given a streamflow of Q3 m3/s and pollutant
masses W1 and W2 kg/day, the pollutant concentration P3 expressed in mg/l will equal

An Introduction to Optimization Models and Methods

P3 ¼ ½W1 a12 þ W2  a23 =ð86:4Q3 Þ

ð4:137Þ

4.5.2.1 Model Calibration
Sample measurements are needed to estimate the
values of each reach’s pollutant transport coefficients aij. Assume five pairs of sample pollutant
concentration measurements have been taken in
the two stream reaches (extending from site 1 to
site 2, and from site 2 to site 3) during design
flow conditions. For this example, also assume
that the design streamflow just downstream of
site 1 and just upstream of site 2 are the same and
equal to 12 m3/s. The concentration samples
taken just downstream from site 1 and just
upstream of site 2 during this design flow condition can be used to solve for the transfer
coefficients α12 and α23 after adding error terms.
More than one sample is needed to allow for
measurement errors and other random effects
such as those from varying temperature, wind,
incomplete mixing or varying wasteload discharges within a day.
Denote the concentrations of each pair of
sample measurements s in the first reach (just
downstream of site 1 and just upstream of site 2)
as P1s and P2s and their combined error as Es.
Thus
P2s þ Es ¼ P1s a12 ðQ1 =Q2 Þ

ð4:138Þ

The problem is to find the best estimates of
the unknown α12. One way to do this is to define
“best” as those values of α12 and all Es that
minimize the sum of the absolute values of all the
error terms Es. This objective could be written
X
ð4:139Þ
Minimize
jEs j
s

The set of Eqs. 4.138 and 4.139 is an optimization model. The absolute value signs in
Eq. 4.139 can be removed by writing each error
term as the difference between two nonnegative
variables, PEs − NEs. Thus for each sample pair s:

4.5 Linear Programming

Es ¼ PEs  NEs

145

ð4:140Þ

If any Es is negative, PEs will be 0 and −NEs
will equal Es. The actual value of NEs is nonnegative. If Es is positive, it will equal PEs, and NEs
will be 0. The objective function, Eq. 4.139, that
minimizes the sum of absolute value of error terms,
can now be written as one that minimizes the sum
of the positive and negative components of Es:
Minimize

X

ðPEs þ NEs Þ

ð4:141Þ

s

Equations 4.139 and 4.140, together with
objective function 4.141 and a set of measurements, P1s and P2s, upstream and downstream of
the reach between sites 1 and 2 define a linear
programming model that can be solved to find
the transfer coefficient α12. An example illustrating the solution of this model for the stream
reach between site 1 and just upstream of site 2 is
presented in Box 4.2. (In this model the measured concentrations are denoted as SPjs rather
than Pjs. Again, the program LINGO (www.
lindo.com) is used to solve the model).
Box 4.3 contains the model and solution for
the reach beginning just downstream of site 2 to
site 3. In this reach the design streamflow is
12.5 m3/s due to the addition of wastewater flow
at site 2.
As shown in Boxes 4.2 and 4.3, the values of
the transfer coefficients are α12 = 0.25 and
α23 = 0.60. Thus from Eq. 4.134, α12
α23 = α13 = 0.15.

4.5.2.2 Management Model
Now that these parameter values αij are known, a
water quality management model can be developed. The water quality management problem,

illustrated in Fig. 4.20, involves finding the
fractions xi of waste removal at sites i = 1 and 2
that meet the stream quality standards at the end
of the two reaches at a minimum total cost.
The pollutant concentration, P2, just upstream
of site 2 that results from the pollutant concentration at site 1 equals the total mass of pollutant
at site 1 times the fraction α12 that remains at site
2, divided by the streamflow Q2 at site 2. The
total mass of pollutant at site 1 at the wastewater
discharge point is the sum of the mass just
upstream of the discharge site, P1Q1, plus the
mass discharged into the stream, W1(1 − x1), at
site 1. The parameter W1 is the total mass of
pollutant entering the treatment plant at site 1.
Similarly for site 2. The fraction of waste
removal, x1, at site 1 is to be determined. Hence
the concentration of pollutant just upstream of
site 2 is
P2 ¼ ½P1 Q1 þ W1 ð1  x1 Þa12 =Q2

ð4:142Þ

The terms P1 and Q1 are the pollutant concentration (M/L3) and streamflow (L3/T) just
upstream of the wastewater discharge outfall at
site 1. Their product is the mass of pollutant at
that site per unit time period (M/T).
The pollutant concentration, P3, at site 3 that
results from the pollutant concentration at site 2
equals the total mass of pollutant at site 2 times
the fraction a23. The total mass of pollutant at site
2 at the wastewater discharge point is the sum of
what is just upstream of the discharge site, P2Q2,
plus what is discharged into the stream,
W2(1 − x2). Hence the concentration of pollutant
at site 3 is
P3 ¼ ½P2 Q2 þ W2 ð1  x2 Þa23 =Q3

ð4:143Þ

146

4

An Introduction to Optimization Models and Methods

Box 4.2. Calibration of water quality model transfer coefficient parameter a12

Equations 4.142 and 4.143 will become the
predictive portion of the water quality management model. The remaining parts of the model
include the restrictions on the pollutant concentrations just upstream of site 2 and at site 3, and
limits on the range of values that each waste
removal efficiency, xi, can assume.
Pj  Pmax
j
0  xi  1:0

for j ¼ 2 and 3
for i ¼ 1 and 2:

ð4:144Þ
ð4:145Þ

Finally, the objective is to minimize the total
cost of meeting the stream quality standards Pmax
2
and Pmax
specified in Eqs. 4.144. Letting Ci(xi)
3

represent the cost function of wastewater treatment at sites i = 1 and 2, the objective can be
written:
Minimize

C1 ðx1 Þ þ C2 ðx2 Þ

ð4:146Þ

The complete optimization model consists of
Eqs. 4.142–4.146. There are four unknown
decision variables, x1; x2, P2, and P3.
Some of the constraints of this optimization
model can be combined to remove the two
unknown concentration values, P2 and P3.
Combining Eqs. 4.142 and 4.144, the concentration just upstream of site 2 must be no greater
than Pmax
2 :

4.5 Linear Programming

½P1 Q1 þ W1 ð1  x1 Þa12 =Q2  Pmax
2

147

ð4:147Þ

Combining Eqs. 4.143 and 4.144, and using
the fraction α13 (see Eq. 4.134) to predict the
contribution of the pollutant concentration at site
1 on the pollutant concentration at Site 3:
f½P1 Q1 þ W1 ð1  x1 Þa13 þ ½W2 ð1  x2 Þa23 g=Q3  Pmax
3

ð4:148Þ

Equation 4.148 assumes that each pollutant
discharged into the stream can be tracked
downstream, independent of the other pollutants
in the stream. Alternatively, Eq. 4.148 computes
the sum of all the pollutants found at site 2 and
then uses that total mass to compute the concentration at site 3. Both modeling approaches
give the same results if the parameter values and
cost functions are the same.

Box 4.3. Calibration of water quality model transfer coefficient parameter a23

148

4

An Introduction to Optimization Models and Methods

Table 4.22 Parameter values selected for the water quality management problem illustrated in Fig. 4.20

To illustrate the solution of either of these
models, assume the values of the parameters are
as listed in Table 4.22. Rewriting the water
quality management model defined by
Eqs. 4.145–4.148 and substituting the parameter
values in place of the parameters, and recalling
that kg/day = 86.4 (mg/l)(m3/s):
The water quality constraint at site 2,
Eq. 4.147, becomes
½ð32Þð10Þ þ 250; 000ð1  x1 Þ=86:40:25=12  20
that when simplified is
x1  0:78:

ð4:149Þ

The water quality constraint at site 3,
Eq. 4.148, becomes
f½ð32Þð10Þ þ 250; 000ð1  x1 Þ=86:40:15
þ ½80; 000ð1  x2 Þ=86:40:60g=13  20
that when simplified is

x1 þ 1:28x2  1:79:

ð4:150Þ

Restrictions on fractions of waste removal,
Eq. 4.145, must also be added to this model.
The feasible combinations of x1 and x2 can be
shown on a graph, as in Fig. 4.21. This graph is a
plot of each constraint, showing the boundaries
of the region of combinations of x1 and x2 that
satisfy all the constraints. This red shaded region
is called the feasible region.
To find the least-cost solution we need the cost
functions C1(x1) and C2(x2) in Eqs. 4.146. Suppose
these functions are not known. Can we determine
the least-cost solution without knowing these costs?
Models like the one just developed can be used to
determine just how accurate these cost functions (or
the values of any of the model parameters) need to
be for the decisions being considered.
While the actual cost functions are not known
in this example, their general form can be
assumed, as shown in Fig. 4.22. Since the
wasteloads produced at site 1 are substantially

4.5 Linear Programming

149

Fig. 4.21 Plot of the
constraints of water quality
management model
identifying those values of
the unknown (decision)
variables x1 and x2 that
satisfy all the constraints.
These feasible values are
contained in and on the
boundaries of the red
region

X2
1.5
equation 145

equation 150

equation 157

1.0

feasible region
0.5
equation 149

0.5

E020103o

0.0
0.0

1.0

1.5

2.0
X1

Fig. 4.22 General form of
total cost functions for
wastewater treatment
efficiencies at sites 1 and 2
in Fig. 4.20. The dashed
straight-line slopes c1 and
c2 are the average cost per
unit (%) removal for 80%
treatment. The actual
average costs clearly
depend on the values of the
waste removal efficiencies
x1 and x2, respectively

greater than those produced at site 2, and given
similar site, labor, and material cost conditions, it
seems reasonable to assume that the cost of
providing a specified level of treatment at site 1
would exceed (or certainly be no less than) the
cost of providing the same specified level of
treatment at Site 2. It would also seem the marginal costs at site 1 would be greater than, or at
least no less than, the marginal costs at site 2 for

any given treatment efficiency. The relative
positions of the cost functions shown in Fig. 4.23
are based on these assumptions.
Rewriting the cost function, Eq. 4.146, as a
linear function converts the model defined by
Eqs. 4.145–4.148 into a linear programming
model. For this example problem, the linear
programming model can be written as:

150

4

An Introduction to Optimization Models and Methods

Fig. 4.23 Plots of various
objective functions (dashed
lines) together with the
constraints of the water
quality management model

Minimize c1 x1 þ c2 x2

ð4:151Þ

Equation 4.151 is minimized subject to constraints 4.145, 4.149 and 4.150. The values of c1
and c2 depend on the values of x1 and x2 and both
pairs are unknown. Even if we knew the values of
x1 and x2 before solving the problem, in this
example the cost functions themselves (Fig. 4.22)
are unknown. Hence, we cannot determine the
values of the marginal costs c1 and c2. However,
we might be able to judge which marginal cost
will likely be greater than the other for any particular values of the decision variables x1 and x2.
In this example that is all we need to know.
First, assume c1 equals c2. Let c1 x1 + c2 x2
equal c and assume c/c1 = 1. Thus the cost
function is x1 + x2 = 1.0. This line can be plotted
onto the graph in Fig. 4.21, as shown by line “a”
in Fig. 4.23.
Line “a” in Fig. 4.23 represents equal values
for c1 and c2, and the total cost, c1 x1 + c2 x2,
equal to 1. Keeping the slope of this line constant
and moving it upward, representing increasing
total costs, to line “b”, where it covers the nearest
point in the feasible region, will identify the
least-cost combination of x1 and x2, again
assuming the marginal costs are equal. In this
case the solution is approximately 80% treatment
at both sites.

Note this particular least-cost solution also
applies for any value of c1 greater than c2 (for
example line “c” in Fig. 4.23). If the marginal
cost of 80% treatment at site 1 is no less than the
marginal cost of 80% treatment at site 2, then
c1 ≥ c2 and indeed the 80% treatment efficiencies
will meet the stream standards for the design
streamflow and wasteload conditions at a total
minimum cost. In fact, from Fig. 4.23 and
Eq. 4.150, it is clear that c2 has to exceed c1 by a
multiple of 1.28 before the least-cost solution
changes to another solution. For any other
assumption regarding c1 and c2, 80% treatment at
both sites will result in a least-cost solution to
meeting the water quality standards for those
design wasteload and streamflow conditions.
If c2 exceeds 1.28c1, as illustrated by line “d”,
then the least-cost solution would be x1 = 100%
and x2 = 62%. Clearly, in this example the
marginal cost, c1, of providing 100% wasteload
removal at site 1 will exceed the marginal cost,
c2, of 60% removal at site 2, and hence, that
combination of efficiencies would not be a
least-cost one. Thus we can be confident that the
least-cost solution is to remove 80% of the waste
produced at both waste-generating sites.
Note the least-cost wasteload removal efficiencies have been determined without knowing
the cost functions. Why spend money defining

4.5 Linear Programming

these functions more precisely? The answer:
costs need to be known for financial planning, if
not for economic analyses. No doubt the actual
costs of installing the least-cost treatment efficiencies of 80% will have to be determined for
issuing bonds or making other arrangements for
paying the costs. However, knowing the
least-cost removal efficiencies means we do not
have to spend money defining the entire cost
functions Ci(xi). Estimating the construction and
operating costs of achieving just one wastewater
removal efficiency at each site, namely 80%,
should be less expensive than defining the total
costs for a range of practical treatment plant
efficiencies that would be required to define the
total cost functions, such as shown in Fig. 4.22.
Admittedly this example is relatively simple.
It will not always be possible to determine the
“optimal” solutions to linear programming
problems, or other optimization problems, without knowing more about the objective function
than was assumed for this example. However,
this exercise illustrates the use of modeling for
purposes other than finding good or “optimal”
solutions. Models can help define the necessary
precision of the data needed to find those
solutions.
Modeling and data collection and analysis
should take place simultaneously. All too often
planning exercises are divided into two stages:
data collection and then analysis. Until one
knows what data one will need, and how accurate
those data must be, one need not spend money
and time collecting them. Conversely, model
development in the absence of any knowledge of
the availability and cost of obtaining data can
lead to data requirements that are costly, or even
impossible, to obtain, at least in the time available for decision-making. Data collection and
model development are activities that should be
performed simultaneously.
Because software is widely available to solve
linear programming programs, because these
software programs can solve very large problems
containing thousands of variables and constraints, and finally because there is less chance
of obtaining a local “nonoptimal” solution when

151

the problem is linear (at least in theory), there is
an incentive to use linear programming to solve
large optimization problems. Especially for large
optimization problems, linear programming is
often the only practical alternative for finding at
least an approximate optimal solution. Yet
models representing particular water resources
systems may not be linear. This motivates the use
of methods that can approximate nonlinear
functions with linear ones, or the use of other
search algorithms such as those discussed in
Chap. 5).
The following simple groundwater supply
problem illustrates the application of some linearization methods commonly applied to nonlinear separable functions—functions of only one
unknown variable.
These approximation methods typically
increase the number of variables and constraints
in a model. Some of these methods require integer
variables, or variables that can have values of
only 0 or 1. There is a practical limit on the
number of integer variables any linear programming software program can handle. Hence, for
large models there may be a need to perform
some preliminary screening designed to reduce
the number of alternatives that should be considered in more detail. This example can be used
to illustrate an approach to preliminary screening.

4.5.3

A Groundwater Supply
Example

Consider a water-using industry that plans to
obtain water from a groundwater aquifer. Two
wellfield sites have been identified. The first
question is how much will the water cost, and the
second, given any specified amount of water
delivered to the user, is how much should come
from each wellfield. This situation is illustrated
in Fig. 4.24.
Wells and pumps must be installed and operated to obtain water from these two wellfields.
The annual cost of wellfield development will
depend on the pumping capacity of the wellfield.
Assume that the annual costs associated with

152

Fig. 4.24 Schematic of a potential groundwater supply
system that can serve a water-using industry. The
unknown variables are the flows, QA and QB, from each
wellfield

various capacities QA and QB for Wellfields A and
B, respectively, are as shown in Fig. 4.25. These
are nonlinear functions that contain both fixed
and variable costs and hence are discontinuous.
The fixed costs result from the fact that some of
the components required for wellfield development come in discrete sizes. As indicated in the

4

An Introduction to Optimization Models and Methods

figure, the maximum flow capacity of Wellfields
A and B are 17 and 13, respectively.
In Fig. 4.25, the nonlinear functions on the
left have been approximated by piecewise linear
functions on the right. This is a first step in linearizing nonlinear separable functions. Increasing the number of linear segments can reduce the
difference between the piecewise linear approximation of the actual nonlinear function and the
function itself. At the same time it will increase
the number of variables and possibly constraints.
When approximating a nonlinear function by a
series of straight lines, model developers should
consider two factors. The first is just how accurate
need be the approximation of the actual function
for the decisions that will be made, and second is
just how accurate is the actual (in this case nonlinear) function in the first place. There is little
value in trying to eliminate relatively small errors
caused by the linearization of a function when the
function itself is highly uncertain. Most cost and
benefit functions, especially those associated with
future activities, are indeed uncertain.

Fig. 4.25 Annual cost functions associated with the Wellfields A and B as shown in Fig. 4.24. The actual functions are
shown on the left, and two sets of piecewise linear approximations are shown on the right

4.5 Linear Programming

153

4.5.3.1 A Simplified Model
Two sets of approximations are shown in
Fig. 4.26. Consider first the approximations
represented by the light blue dot-dash lines.
These single straight lines are very crude
approximations of each function. In this example
these straight-line cost functions are lower
bounds of the actual nonlinear costs. Hence, the
actual costs may be somewhat higher than those
identified in the solution of a model.
Using the blue dot-dash linear approximations
in Fig. 4.26, the linear programming model can
be written as follows:
Minimize

CostA þ CostB

ð4:152Þ

Subject to
CostA ¼ 8IA þ ½ð40  8Þ=17QA
linear approximation of CðQA Þ
ð4:153Þ
CostB ¼ 15IB þ ½ð26  15Þ=13QB
linear approximation of CðQB Þ
ð4:154Þ
IA ; IB are 0; 1 integer ðbinaryÞ variables
ð4:155Þ
QA  17IA limits QA to 17 and forces IA ¼ 1
if QA [ 0
ð4:156Þ

Fig. 4.26 Least-cost
wellfield use given total
demand Q based on model
defined by Eqs. 4.152 to
4.160

QB  13IB limits QB to 13 and forces IB ¼ 1
if QB [ 0
ð4:157Þ
QA þ QB ¼ Q mass balance

ð4:158Þ

Q; QA ; QB  0
non-negativity of all decision variables

ð4:159Þ
Q ¼ some specified amount from 0 to 30:
ð4:160Þ
The expressions within the square brackets,
[ ], in Eqs. 4.154 and 4.155 above represent the
slopes of the dot-dash linear approximations of
the cost functions. The integer 0, 1 variables are
required to include the fixed costs in the model.
Solving this linear model for various values of
the water demand Q provides some interesting
results. Again, they are based on the dot-dash
linear cost functions in Fig. 4.25. As Q increases
from 0 to just under 6.8, all the water will come
from the less expensive Wellfield A. For any
Q from 6.8 to 13, Wellfield B becomes less
expensive and all the water will come from it. For
any Q greater than the capacity of Wellfield B of 13
but no greater than the capacity of Wellfield A, 17,
all of it will come from Wellfield A. Because of the
fixed costs, it is cheaper to use one rather than both
wellfields. Beyond Q = 17, the maximum capacity of A, water needs to come from both wellfields.
Wellfield B will pump at its capacity, 13, and the
additional water will come from Wellfield A.

154

4

Figure 4.26 illustrates these solutions. One
can understand why in situations of increasing
demands for Q over time, capacity expansion
modeling might be useful. One would not close
down a wellfield once developed, just to achieve
what would have been a least-cost solution if the
existing wellfield had not been developed.

4.5.3.2 A More Detailed Model
A more accurate representation of these cost
functions may change these solutions for various
values of Q, although not significantly. However
consider the more accurate cost minimization
model that includes the red solid-line piecewise
linearizations shown in Fig. 4.26.
Minimize

CostA þ CostB

ð4:161Þ

Subject to
linear approximation of cost functions:
CostA ¼ f8IA1 þ ½ð20  8Þ=5QA1 g
þ f26IA2 þ ½ð30  26Þ=ð10  5ÞQA2 g
þ f35IA3 þ ½ð40  35Þ=ð17  10ÞQA3 g
ð4:162Þ
CostB ¼ f15IB1 þ ½ð18  15Þ=3QB1 g
þ f18IB2 þ ½ð20  18Þ=ð10  3ÞQB2
þ ½ð26  20Þ=ð13  10ÞQB3 g
ð4:163Þ
QA and QB defined.
QA ¼ QA1 þ ð5IA2 þ QA2 Þ þ ð10IA3 þ QA3 Þ
ð4:164Þ
QB ¼ QB1 þ ð3IB2 þ QB2 þ QB3 Þ
IAi and IBi are 0; 1 integer variables
for all segments i

ð4:165Þ
ð4:166Þ

QA1  5IA1
QA2  ð10  5ÞIA2 ;
QA3  ð17  10ÞIA3

limits QAi to width of

segment i and forces I Ai ¼ 1 if QAi [ 0
ð4:167Þ

An Introduction to Optimization Models and Methods

IA1 þ IA2 þ IA3  1

limits solution to at most

only one cost function segment i:
ð4:168Þ
QB1  3IB1 ;
QB2  ð10  3ÞIB2 ;
QB3  ð13  10ÞIB2 limits QBi to width of
segment i and forces IBi ¼ 1 if QBi [ 0:
ð4:169Þ
IB1 þ IB2  1
Q ¼ QA þ QB mass balance

ð4:170Þ
ð4:171Þ

Q; QA ; QB  0
non-negativityof all decision variables
ð4:172Þ
Q ¼ some specified amount from 0 to 30
ð4:173Þ
Constraint (4.170) limits the solution to at
most only the first segment or to the second and
third segments of the cost function for wellfield
B. Note that a 0, 1 integer variable for the fixed
cost of the third segment of this function is not
needed since its slope exceeds that of the second
segment. However the flow, QB3, in that segment
must be bounded using the integer 0, 1 variable,
IB2, associated with the second segment, as
shown in the third of Eqs. 4.169.
The solution to this model, shown in Fig. 4.27,
differs from the solution of the simpler model, but
only in the details. Wellfield A supplies all the
water for Q ≤ 4.3. For values of Q in excess of 4.3
up to 13 all the water comes from Wellfield B. For
values of Q in excess of 13 up to 14.8, the capacity
of Wellfield B remains at its maximum capacity of
13 and Wellfield A provides the additional amount
of needed capacity over 13. As Q increases from
14.9 to 17, the capacity of Wellfield B drops to 0
and the capacity of Wellfield A increases from 14.9
to 17. For values of Q between 17 and 18 Wellfield
B provides 13, its maximum capacity, and the
capacity of A increases from 4 to 5. For values of

4.5 Linear Programming

155

Fig. 4.27 Least-cost
wellfield use given total
demand Q based on
Eqs. 4.161 to 4.173

Q from 18.1 to 20, Wellfield B decreases to a
constant 10, and Wellfield A increases from 8.1 to
10. For values of Q from 20 to 23, Wellfield
A remains at 10 and Wellfield B increases from 10
to 13. For values of Q from 23 to 27, Wellfield
B again drops to a constant 10 and Wellfield A increases from 13 to 17. For values of Q in excess of
27, Wellfield A remains at its maximum capacity of
17, and Wellfield B increases from 10 to 13.
As in the previous example, this shows the
effect on the least-cost solution when one cost
function has relatively lower fixed and higher
variable costs compared with another cost function having relatively higher fixed and lower
variable costs.

4.5.3.3 An Extended Model
In this example, the simpler model (Eqs. 4.152–
4.160) and the more accurate model (Eqs. 4.161–
4.173) provided essentially the same allocations
of wellfield capacities associated with a specified
total capacity Q. If the problem contained a larger number of wellfields, the simpler (and
smaller) model might have been able to eliminate
some of these wellfields from further consideration. This would reduce the size of any new
model that approximates the cost functions of the
remaining wellfields more accurately.
The model just described, like the capacity
expansion model and water quality management
model, is another example of a cost-effective
model. The objective was to find the least-cost

way of providing a specified amount of water to
a water user. It does not address the problem of
planning for an increasing demand for Q over
time. Clearly it makes no sense to implement the
particular cost-effective solution for any value of
Q, as shown in Fig. 4.27, as the demand for
Q increases, as in this example, from 0 to 30.
This is the capacity expansion problem, the
solution of which will benefit from models that
take time into account and that are not static as
illustrated previously in this chapter.
Next, consider a cost–benefit analysis in
which the question is just how much water
should users use. To address this question we
assume the user has identified the annual benefits
associated with various amounts of water. The
annual benefit function, B(Q), and its piecewise
linear approximations, are shown in Fig. 4.28.
The straight, blue, dot-dash linear approximation of the benefit function shown in Fig. 4.28 is an
upper bound of the benefits. Incorporating it into a
model that uses the dot-dash linear lower bound
approximations of each cost function, as shown in
Fig. 4.25 will produce an optimistic solution. It is
unlikely that the value of Q that is based on more
accurate and thus less optimistic benefit and cost
functions will be any greater than the one identified by this simple optimistic model. Furthermore,
if any wellfield is not in the solution of this optimistic model, with some care we might be able to
eliminate that wellfield from further consideration
when developing a more accurate model.

156

4

An Introduction to Optimization Models and Methods

Fig. 4.28 Benefit function of the amount of water provided to the water user. Piecewise linear approximations of that
function of flow are shown on the right

Any component of a water resources system
that does not appear in the solution of a model
that includes optimistic approximations of performance measures that are to be maximized,
such as benefits, or that are to be minimized, such
as costs, are candidates for omission in any more
detailed model. This is an example of the process
of preliminary screening.
The model defined by Eqs. 4.152–4.160 can
now be modified. Equation 4.160 is eliminated
and the cost minimization objective Eq. 4.152 is
replaced with:
Maximize

Benefits  ðCostA þ CostBÞ
ð4:174Þ

where
Benefits ¼ 10 þ ½ð45  25Þ=ð21  9ÞQ
linear approximation of BðQÞ
ð4:175Þ
The solution of this model, Eqs. 4.153–4.159,
4.174, and 4.175 (plus the condition that the
fixed benefit of 10 only applies if Q > 0, added
because it is clear the benefits would be 0 with a
Q of 0) indicates that only Wellfield B needs to
be developed, and at a capacity of 10. This would
suggest that Wellfield A can be omitted in any
more detailed modeling exercise. To see if this
assumption, in this example, is valid, consider

the more detailed model that incorporates the red,
solid-line linear approximations of the cost and
benefit functions shown in Figs. 4.25 and 4.28.
Note that the approximation of the generally
concave benefit function in Fig. 4.29 will result
in negative values of the benefits for small values
of Q. For example, when the flow Q, is 0 the
approximated benefits are −10. Yet the actual
benefits are 0 as shown in the left part of
Fig. 4.28. Modeling these initial fixed benefits
the same way as the fixed costs have been
modeled, using another 0, 1 integer variable,
would allow a more accurate representation of
the actual benefits for small values of Q.
Alternatively, to save having to add another
integer variable and constraint to the model, one
can allow the benefits to be negative. If the
model solution shows negative benefits for some
small value of Q, then obviously the more preferred value of Q, and benefits, would be 0. This
more approximate trial-and-error approach is
often preferred in practice, especially when a
model contains a large number of variables and
constraints. This is the approach taken here.

4.5.3.4 Piecewise Linear Model
There are a number of ways of modeling the
piecewise linear concave benefit function shown
on the right side of Fig. 4.28. Several are defined
in the next several sets of equations. Each
method will result in the same model solution.

4.5 Linear Programming

157

One approach to modeling the concave benefit
function is to define a new unrestricted (possibly
negative valued) variable. Let this variable be
Benefits. When being maximized this variable
cannot exceed any of the linear functions that
bound the concave benefit function:
Benefits   10 þ ½ð25  ð10ÞÞ=9Q ð4:176Þ

The linearized benefit function can now be
written as the sum over all three segments of
each segment slope times the variable qi:
Benefits ¼ 10 þ ½ð25 þ 10Þ=9q1
þ ½ð45  25Þ=ð21  9Þq2
þ ½ð50  45Þ=ð30  21Þq3
ð4:183Þ

Benefits  10 þ ½ð45  25Þ=ð21  9ÞQ
ð4:177Þ
Benefits  33 þ ½ð50  45Þ=ð30  21ÞQ
ð4:178Þ
Since most linear programming algorithms
assume the unknown variables are nonnegative
(unless otherwise specified), unrestricted variables, such as Benefits, can be replaced by the
difference between two nonnegative variables,
such as Pben − Nben. Pben will equal Benefits if
its value is greater than 0. Otherwise −Nben will
equal Benefits. Thus in place of Benefits in
Eqs. 4.176–4.178, and those below, one can
substitute Pben − Nben.
Another modeling approach is to divide the
variable Q into parts, qi, one for each segment
i of the function. These parts sum to Q. Each qi,
ranges from 0 to the width of the user-defined
segment i. Thus for the piecewise linear benefit
function shown on the right of Fig. 4.28:
q1  9

ð4:179Þ

q2  21  9

ð4:180Þ

q3  30  21

ð4:181Þ

Q ¼ q1 þ q2 þ q3

ð4:182Þ

and

Since the function being maximized is concave (decreasing slopes as Q increases), we are
assured that each qi + 1 will be greater than 0
only if qi is at its upper limit, as defined by
constraint Eqs. 4.179–4.181.
A third method is to define unknown weights
wi associated with the breakpoints of the linearized function. The value of Q can be expressed as the sum of a weighted combination of
segment endpoint values. Similarly, the benefits
associated with Q can be expressed as a weighted
combination of the benefits evaluated at the
segment endpoint values. The unknown weights
must also sum to 1. Hence, for this example:
Benefits ¼ ð10Þw1 þ 25w2 þ 45w3 þ 50w4
ð4:184Þ
Q ¼ 0w1 þ 9w2 þ 21w3 þ 30w4

ð4:185Þ

1 ¼ w1 þ w2 þ w3 þ w4

ð4:186Þ

For this method to provide the closest
approximation of the original nonlinear function,
the solution must include no more than two
nonzero weights and those nonzero weights must
be adjacent to each other. For concave functions
that are to be maximized, this condition will be
met, since any other situation would yield less
benefits.
The solution to the more detailed model
defined by Eqs. 4.174, 4.162–4.172, and either
4.176–4.178, 4.179–4.183, or 4.184–4.186,

158

4

indicates a value of 10 for Q will result in the
maximum net benefits. This flow is to come from
Wellfield B. This more precise solution is identical to the solution of the simpler model. Clearly
the simpler model could have successfully served
to eliminate Wellfield A from further
consideration.

4.5.4

A Review of Linearization
Methods

This section reviews the piecewise linearization
methods just described and some other approaches for incorporating nonlinear conditions into
linear programming models. All of these methods maintain linearity.
If-then-else conditions
There exist a number of ways “if-then-else” and
“and” and “or” conditions (that is, decision trees)
can be included in linear programming models.
To illustrate some of them, assume X is an
unknown decision variable in a model whose
value may depend on the value of another
unknown decision variable Y. Assume the maximum value of Y would not exceed Ymax and the
maximum value of X would not exceed Xmax.
These upper bounds and all the linear constraints
representing “if-then-else” conditions must not
restrict the values of the original decision variable
Y. Four “if-then-else” (with “and/or”) conditions
are presented below using additional integer 0.1
variables, denoted by Z. All the X, Y, and Z variables in the constraints below are assumed to be
unknown. These constraints would be included in
the any linear programming model where the
particular “if-then-else” conditions apply.
These illustrations are not unique. At the
boundaries of the “if” constraints in the examples
below, either of the “then” or “else” conditions
can apply. All variables (X, Y) are assumed
nonnegative. All variables Z are assumed to be a

An Introduction to Optimization Models and Methods

binary (0, 1) variables. Note the constraints are
all linear.
(a) If Y  50 then X  10; else X  15:
Define constraints:
Y  50 Z þ Ymax ð1  Z Þ
Y  50ð1  Z Þ
X  10Z þ Xmax ð1  Z Þ
X  15ð1  Z Þ
(b) If Y  50 then X  Y; else X  Y:
Define constraints:
Y  50Z
Y  50ð1  Z Þ þ Ymax Z
X  Y þ Xmax Z
X  Y  Ymax ð1  Z Þ
(c) If Y  20 or Y  80 then X ¼ 5; else X  10:
Define constraints:
Y  20Z1 þ 80Z2 þ Ymax ð1  Z1  Z2 Þ
Y  20Z2 þ 80ð1  Z1  Z2 Þ
Z1 þ Z2  1
X  5ðZ1 þ ð1  Z1  Z2 ÞÞ þ Xmax Z2
X  5ðZ1 þ ð1  Z1  Z2 ÞÞ
X  10Z2
(d) If 20 ≤ Y ≤ 50 or 60 ≤ Y ≤ 80, then X ≤ 5,
else X ≥ 10.
Define constraints:
Y  20Z1 þ 50Z2 þ 60Z3 þ 80Z4
þ Ymax ð1  Z1  Z2  Z3  Z4 Þ:
Y  20Z2 þ 50Z3 þ 60Z4
þ 80ð1  Z1  Z2  Z3  Z4 Þ
Z1 þ Z2 þ Z3 þ Z4  1
X  5ðZ2 þ Z4 Þ þ Xmax ð1  Z2  Z4 Þ
X  10ððZ1 þ Z3 Þ þ ð1  Z1  Z2  Z3  Z4 ÞÞ

4.5 Linear Programming

Minimizing the absolute value of the difference between two unknown nonnegative
variables:
Minimize |X − Y| is equivalent to
Minimize D

159

Linearization of convex functions for
maximization or concave function for minimization involves 0, 1 binary variables.
Fixed costs in cost functions

subject to
X  Y  D;
Y  X  D;
X; Y; D  0:
or
Minimize ðPD þ NDÞ
subject to
X  Y ¼ PD  ND;
PD; ND; X; Y  0:
Minimizing the maximum or maximizing
the minimum
Let the set of variables be {X1, X2, X3, …, Xn},
Minimizing the maximum of {X1, X2, X3, …, Xn}
is equivalent to
Minimize U
subject to
U  Xj ;

Cost ¼ C0 þ CX
¼0

Maximize L
subject to
j ¼ 1; 2; 3; . . .; n:

if X [ 0;
otherwise:

To include these fixed costs in a LP model,
define

j ¼ 1; 2; 3; . . .; n:

Maximizing the minimum of {X1, X2, X3, …,
Xn} is equivalent to

L  Xj ;

Consider functions that have fixed components if the argument of the function is greater
than 0.

Cost ¼ C0 I þ CX
Subject to
X  MI
where M is the maximum value of X, and I is an
unknown 0, 1 binary variable.

160

4

An Introduction to Optimization Models and Methods

Minimizing convex functions or maximizing concave functions.

Maximize Gð X Þ ¼ Maximize B
Subject to
I 1 þ S1 X  B
I 2 þ S2 X  B
I 3 þ S3 X  B

Minimize F ð X Þ ¼ S1 x1 þ S2 x2 þ S3 x3
Maximize Gð X Þ ¼ S1 x1 þ S2 x2 þ S3 x3

Subject to
X ¼ x1 þ x2 þ x3
x1  a
x2  b  a

4.5 Linear Programming

161

Minimize Gð X Þ ffi 5x1 þ ð20z2 þ 3x2 Þ þ ð44z3 þ 2x3 Þ
Minimize F ðX Þ ¼ Fð0Þw1 þ F ðaÞw2 þ F ðbÞw3 þ F ðcÞw4
Maximize GðX Þ ¼ Gð0Þw1 þ GðaÞw2 þ GðbÞw3 þ GðcÞw4

Subject to
X ¼ 0w1 þ aw2 þ b w3 þ c w4
w1 þ w2 þ w3 þ w4 ¼ 1
Minimizing concave functions or maximizing convex functions

Subject to
x1 þ ð4z2 þ x2 Þ þ ð12z3 þ x3 Þ ¼ X
zs ¼ 0 or 1 for all segments s
x1  4z1 ;
x2  8z2 ;
x3  99z3 ;
z1 þ z2 þ z3 ¼ 1:

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4

An Introduction to Optimization Models and Methods

Minimizing or maximizing combined concave–convex functions

Maximize
C ð X Þ ¼ ð5z1 þ 6x1 þ 3x2 Þ þ ð17z3 þ 5x3 Þ
Subject to
Maximize
C ð X Þ ¼ ð5z1 þ 6x1 þ 3x2 Þ þ ð53z3 þ 5x3 Þ

ðx1 þ x2 Þ þ x3 ¼ X
z1 ; z3 ¼ 0; 1
x1  4z1
x2  8z1

Subject to
ðx1 þ x2 Þ þ ð12z3 þ x3 Þ ¼ X
x1  4z1

x3  99z3
z1 þ z3 ¼ 1;

x2  8z1
x3  99z3
z1 þ z3 ¼ 1
z1 ; z3 ¼ 0; 1

Minimize
C ð X Þ ¼ ð5z1 þ 6x1 Þ þ ð17z2 þ 3x2 þ 5x3 Þ
Subject to

Minimize
CðXÞ ¼ ð5z1 þ 6x1 Þ þ ð29z2 þ 3x2 þ 5x3 Þ
Subject to
z1 ; z2 ¼ 0; 1:
x1 þ ð4z2 þ x2 þ x3 Þ ¼ X
x1  4z1
x2  8z2
x3  99z2
z1 þ z2  1

x1 þ ð4z2 þ x2 þ x3 Þ ¼ X
z1 ; z2 ¼ 0; 1:
x1  4z1
x2  12z2
x3  99z2
z1 þ z2  1

4.5 Linear Programming

163

counterparts, there are multiple feasible solutions
to each of these problems. Hence, the task is to
F ð X Þ ¼ ð5z1 þ 6x1 Þ þ ð35z2 þ 3x2 Þ þ ð32z3 2x3 Þ þ 22z4
find the best solution, or a number of near-best
solutions. Each solution must satisfy all the
constraints.
Subject to
Constraints can reflect physical conditions,
x1 þ ð4z2 þ x2 Þ þ ð12z3 þ x3 Þ þ ð17z4 þ x4 Þ ¼ X environmental regulations and/or social or economic targets. Especially with respect to envix1  4z1
ronmental or social conditions and goals, it is
x2  8z2
often a matter of judgment to decide what is
x3  5z3
:
considered an objective that is to be minimized
x4  99z4
or maximized and what is considered a constraint
z1 þ z2 þ z3 þ z4 ¼ 1;
that has to be met. For example, do we minimize the costs of meeting specified maximum
zs ¼ 0; 1 for all segments s
levels of pollutant concentrations or minimize
pollutant concentrations without exceeding
specified costs?
4.6 A Brief Review
Except for relatively simple problems, the use
of these optimization models and methods is
Before proceeding to other optimization and primarily for reducing the number of alternatives
simulation methods in the following chapters, it that need to be further analyzed and evaluated
may be useful to review the topics covered so far. using simulation methods. Optimization is genThe focus has been on model development as erally used for preliminary screening—eliminatwell as model solution. Several types of water ing inferior alternatives before more detailed
resources planning and management problems analyses are carried out. Presented were some
have been used to illustrate model development approaches to preliminary screening involving
and solution processes. Like their real-world hill-climbing, calculus-based Lagrange multicounterparts, the example problems all had plier, numerical nonlinear programming, discrete
multiple unknown decision variables and multi- dynamic programming, and linear programming
ple constraints. Also like their real-world methods. Each method has its strengths and
Maximize or Minimize F(X)

164

limitations. Both linear and nonlinear programming models are typically solved using software
packages. Many of these software programs are
free and readily available. But before any model
can be solved, it has to be built. Building models
is an art and that is what this chapter has
attempted to introduce.
The example problems used to illustrate these
modeling and model solution methods have been
relatively simple. However, simple applications
such as these can form the foundation of models
of more complex problems, as will be shown in
following chapters.

Reference
Bellman, R. E. (1957). Dynamic programming. Princeton,
NJ: Princeton University Press.

Additional References
(Further Reading)
Bassson, M. S., Allen, R. B., Pegram, G. G. S., & Van
Rooyen, J. A. (1994). Probabilistic management of
water resource and hydropower systems. Highlands
Ranch, CO: Water Resources Publications.
Beroggi, G. E. G. (1999). Decision modeling in policy
management: An introduction to analytic concepts.
Boston, MA: Kluwer Academic.
Blanchard, B. S., & Fabrycky, W. J. (1998). Systems
engineering and analysis (3rd ed.). Prentice Hall:
Upper Saddle River, NJ.
Buras, N. (1966). Dynamic programming and water
resources development, advances in hydroscience
(Vol. 3). New York: Academic Press.
Dorfman, R., Jacoby, H. D., & Thomas, H. A., Jr. (1972).
Models for managing regional water quality. Cambridge, MA: Harvard University Press.
Esogbue, A. O. (Ed.). (1989). Dynamic programming for
optimal water resources systems analysis. Prentice
Hall: Englewood Cliffs, NJ.
Hall, W. A., & Dracup, J. A. (1970). Water resources
systems engineering. New York: McGraw-Hill.
Hillier, F. S., & Lieberman, G. J. (1990a). Introduction to
operations research (5th ed.). New York:
McGraw-Hill.

4

An Introduction to Optimization Models and Methods
Hillier, F. S., & Lieberman, G. J. (1990b). Introduction to
stochastic models in operations research. New York:
McGraw-Hill.
Hufschmidt, M. M., & Fiering, M. B. (1966). Simulation
techniques for design of water-resource systems.
Cambridge, MA: Harvard University Press.
Kapamoyz, M., Szidarovszky, F., & Zahpaie, B. (2003).
Water resources systems analysis. Boca Raton, FL:
Lewis.
Loucks, D. P., & da Costa, J. R. (Eds.). (1990). Decision
support systems: water resources planning. Berlin:
Springer.
Loucks, D. P., Stedinger, J. S., & Haith, D. A. (1981).
Water resource systems planning and analysis. Prentice Hall: Englewood Cliffs, NJ.
Maass, A., Hufschmidt, M. M., Dorfman, R., Thomas, H.
A., Jr., Marglin, S. A., & Fair, G. M. (1962). Design
of water-resource systems. Cambridge, MA: Harvard
University Press.
Major, D. C., & Lenton, R. L. (1979). Applied water
resources systems planning. Prentice Hall: Englewood
Cliffs, NJ.
Mays, L. W., & Tung, Y.-K. (1992). Hydrosystems
engineering and management. New York:
McGraw-Hill.
Newnan, D. G., Lavelle, J. P., & Eschenbach, T. G.
(2002). Essentials of engineering economic analysis
(2nd ed.). New York: Oxford University Press.
Rardin, R. L. (1998). Optimization in operations
research. Prentice Hall: Upper Saddle River, NJ.
Revelle, C. (1999). Optimizing reservoir resources. New
York: Wiley.
Somlyody, L. (1997). Use of optimization models in river
basin water quality planning: WATERMATEX’97.
In M. B. Beck & P. Lessard (Eds.), Systems analysis
and computing in water quality management:
Towards a new agenda. London: IWA.
Taylor, B. W., III. (1999). Introduction to management
science (6th ed.). Prentice Hall: Upper Saddle River,
NJ.
Willis, R., & Yeh, W. W.-G. (1987). Groundwater
systems planning and management. Prentice Hall:
Englewood Cliffs, NJ.
Winston, W. L. (1994). Operations research: Applications
and algorithms (3rd edn.). Boston, MA: Duxbury.
Winston, W. L., & Albright, S. C. (2001). Practical
management science (2nd edn.). Pacific Grove, CA:
Duxbury.
Wright, J. R., & Houck, M. H. (1995). Water resources
planning and managementm, Chap. 37. In W. F. Chen
(Ed.), Civil engineering handbook. Boca Raton: CRC.
Wurbs, R. A. (1996). Modeling and analysis of reservoir
system operations. Prentice Hall: Upper Saddle River,
NJ.

Exercises

165

Exercises
Engineering economics:
4:1 Consider two alternative water resource
projects, A and B. Project A will cost
$2,533,000 and will return $1,000,000 at
the end of 5 years and $4,000,000 at the
end of 10 years. Project B will cost
$4,000,000 and will return $2,000,000 at
the end of 5 and 15 years, and another
$3,000,000 at the end of 10 years. Project A has a life of 10 years, and B has a
life of 15 years. Assuming an interest rate
of 0.1 (10%) per year:
(a) What is the present value of each
project?
(b) What is each project’s annual net
benefit?
(c) Would the preferred project differ if
the interest rates were 0.05?
(d) Assuming that each of these projects
would be replaced with a similar
project having the same time stream of
costs and returns, show that by
extending each series of projects to a
common
terminal
year
(e.g.,
30 years), the annual net benefits of
each series of projects will be same as
found in part (b).
4:2 Show that A

PT

t¼1

T

ð1 þ rÞt ¼ ð1rð1þ þrÞ rÞ1
T A,

the present value of a series of equal payments, A, at the end of each year for
T years. What is the impact of an increasing
interest rate over time on the present value?
4:3 (a) Show that if compounding occurs at
the end of m equal length periods
within a year in which the nominal
interest rate is r, then the effective
annual interest rate, r′, is equal to

r m
r0 ¼ 1 þ
1
m

(b) Show that when compounding is continuous (i.e., when the number of
periods m → ∞), the compound
interest factor required to convert a
present value to a future value in year
T is erT. [Hint: Use the fact that
lim ð1 þ 1=kÞk ¼ e, the base of natural
k!1

logarithms.]
4:4 The term “capitalized cost” refers to the
present value PV of an infinite series of
end-of-year equal payments, A. Assuming
an interest rate of r, show that as the terminal period T → ∞, PV = A/r.
4:5 The internal rate of return of any project or
plan is the interest rate that equals the present value of all receipts or income with the
present value of all costs. Show that the
internal rate of return of projects A and B in
Exercise 4.1 are approximately 8 and 6%,
respectively. These are the interest rates r,
for each project, that essentially satisfy the
equation
T
X

ðRt  Ct Þð1 þ rÞt ¼ 0

t¼0

4:6 In Exercise 4.1, the maximum annual
benefits were used as an economic criterion for plan selection. The maximum
benefit–cost ratio, or annual benefits divided by annual costs, is another criterion.
Benefit–cost ratios should be no less than
one if the annual benefits are to exceed the
annual costs. Consider two projects, I
and II:
Project
I

II

Annual benefits

20

2

Annual costs

18

1.5

Annual net benefits

2

0.5

Benefit–cost ratio

1.11

1.3

166

What additional information is needed
before one can determine which project is
the most economical project?
4:7 Bonds are often sold to raise money for
water resources project investments. Each
bond is a promise to pay a specified
amount of interest, usually semiannually,
and to pay the face value of the bond at
some specified future date. The selling
price of a bond may differ from its face
value. Since the interest payments are
specified in advance, the current market
interest rates dictate the purchase price of
the bond.
Consider a bond having a face value of
$10,000, paying $500 annually for
10 years. The bond or “coupon” interest
rate based on its face value is 500/10,000,
or 5%. If the bond is purchased for
$10,000, the actual interest rate paid to the
owner will equal the bond or “coupon”
rate. But suppose that one can invest
money in similar quality (equal risk)
bonds or notes and receive 10% interest.
As long as this is possible, the $10,000,
5% bond will not sell in a competitive
market. In order to sell it, its purchase
price has to be such that the actual interest
rate paid to the owner will be 10%. In this
case, show that the purchase price will be
$6927.
The interest paid by the some bonds,
especially municipal bonds, may be
exempt from state and federal income
taxes. If an investor is in the 30% income
tax bracket, for example, a 5% municipal
tax-exempt bond is equivalent to about a
7% taxable bond. This tax exemption
helps reduce local taxes needed to pay the
interest on municipal bonds, as well as
providing attractive investment opportunities to individuals in high tax brackets.
Lagrange Multipliers
4:8 What is the meaning of the Lagrange
multiplier associated with the following
model?

4

An Introduction to Optimization Models and Methods

Maximize Benefitð X Þ  Costð X Þ
Subject to:
X  23
4:9 Assume water can be allocated to three
users. The allocation, xj, to each use j provides
the
following
returns:
R
(x1) = (12x1 − x21), R(x2) = (8x2 − x22) and
R(x3) = (18x3 − 3x23). Assume that the
objective is to maximize the total return, F
(X), from all three allocations and that the
sum of all allocations cannot exceed 10.
(a) How much would each use like to have?
(b) Show that at the maximum total return
solution the marginal values, ∂(R(xj))/∂xj,
are each equal to the shadow price or
Lagrange multiplier (dual variable) λ associated with the constraint on the amount of
water available. (c) Finally, without
resolving a Lagrange multiplier problem,
what would the solution be if 15 units of
water were available to allocate to the three
users and what would be the value of the
Lagrange multiplier?
4:10 In Exercise 4.9, how would the Lagrange
multiplier procedure differ if the objective
function, F(X), were to be minimized?
4:11 Assume that the objective was to minimize
the sum of squared deviations of the actual
allocations xj from some desired or known
target allocations Tj. Given a supply of
water Q less than the sum of all target allocations Tj, structure a planning model
and its corresponding Lagrangian. Will a
global minimum be obtained from solving
the partial differential equations derived
from the Lagrangian? Why?
4:12 Using Lagrange multipliers, prove that the
least-cost design of a cylindrical storage
tank of any volume V > 0 has one-third of
its cost in its base and top and two-thirds
of its cost in its side, regardless of the cost
per unit area of its base or side. (It is these
types of rules that end up in handbooks in
engineering design.)
4:13 An industrial firm makes two products,
A and B. These products require water and

Exercises

167

other resources. Water is the scarce
resource—they have plenty of other needed resources. The products they make are
unique, and hence they can set the unit
price of each product at any value they
want to. However experience tells them
that the higher the unit price for a product,
the less amount of that product they will
sell. The relationship between unit price
and quantity that can be sold is given by
the following two demand functions.
Assume for simplicity that the unit price
for product A is (8 − A) and for product B
is (6 − 1.5B).

Po

(d) What is the value of the dual variable,
or shadow price, associated with the
10 units of available water?
Dynamic programming
4:14 Solve for the optimal integer allocations x1,
x2, and x3 for the problem defined by
Exercise 4.9 assuming the total available
water is 3 and 4. Also solve for the optimal
allocation policy if the total water available
is 7 and each xj must not exceed 4.
4:15 Consider a three-season reservoir operation problem. The inflows are 10, 50 and
20 in seasons 1, 2, and 3, respectively.

Po
8 – A

Unit
price
Quantity of product A

(a) What are the amounts of A and B, and
their unit prices, that maximize the
total revenue obtained?
(b) Suppose the total amount of A and
B could not exceed some amount
Tmax. What are the amounts of A and
B, and their unit prices, that maximize
total revenue, if
(i) Tmax = 10
(ii) Tmax = 5
Water is needed to make each unit of
A and B. The production functions
relating the amount of water XA needed to make A, and the amount of
water XB needed to make B, are
A = 0.5XA,
and
B = 0.25XB,
respectively.
(c) Find the amounts of A and B and their
unit prices that maximize total revenue
assuming the total amount of water
available is 10 units.

Unit
Price

6–1.5B

Quantity of product B

Find the operating policy that minimizes
the sum of total squared deviations from a
constant storage target of 20 and a constant release target of 25 in each of the
three seasons. Develop a discrete dynamic
programming model that considers only 4
discrete storage values: 0, 10, 20 and 30.
Assume the releases cannot be less than 10
or greater than 40. Show how the model’s
recursive equations change depending on
whether the decisions are the releases or
the final storage volumes. Verify the
optimal operating policy is the same
regardless of whether the decision variables are the releases or the final storage
volumes in each period. Which model do
you think is easier to solve? How would
each model change if more importance
were given to the desired releases than to
the desired storage volumes?
4:16 Show that the constraint limiting a reservoir release, rt, to be no greater than the

168

4

initial storage volume, st, plus inflow, it, is
redundant to the continuity equation
s t + i t − r t = s t + 1.
4:17 Develop a general recursive equation for a
forward-moving dynamic programming
solution procedure for a single
reservoir-operating problem. Define all
variables and functions used. Why is this
not a very useful approach to finding a
reservoir-operating policy?
4:18 The following table provides estimates for
the recent values of the costs of additional
wastewater treatment plant capacity needed at the end of each 5-year period for the
next 20 years. Find the capacity expansion
schedule that minimizes the present values
of the total future costs. If there is more
than one least-cost solution, indicate
which one you think is better, and why?
Discounted cost of
additional capacity

An Introduction to Optimization Models and Methods

4:19 Consider a wastewater treatment plant in
which it is possible to include five different treatment processes in series. These
treatment processes must together remove
at least 90% of the 100 units of influent
waste. Assuming the Ri is the amount of
waste removed by process i, the following
conditions must hold:
20  R1  30
0  R2  30
0  R3  10
0  R4  20
0  R5  30
(a) Write the constrained optimizationplanning model for finding the
least-cost combination of the removals
Ri that together will remove 90% of
the influent waste. The cost of the
various discrete sizes of each unit
process i depend upon the waste
entering the process i as well as the
amount of waste removed, as indicated in the table below.

Total required
capacity at end of
period

Units of additional
capacity
Period
years

2

4

6

8

10

1

1–5

12

15

18

23

26

2

6–
10

8

11

13

15

3

11–
15

6

8

4

16–
20

4

2
6

Process i
Influent, Ii

Removal, Ri

Annual cost = Ci(Ii, Ri)

8

100

20

5

100

30

10

80

10

3

80

20

9

80

30

13

70

10

4

70

20

10

70

30

15

60

10

60

20

60

30

50

10

50

20

50

30

10

The cost in each period t must be paid at
the beginning of the period. What was the
discount factor used to convert the costs at
the beginning of each period t to present
value costs shown above? In other words
how would a cost at the beginning of
period t be discounted to the beginning of
period 1, given an annual interest rate of r?
(Only the algebraic expression of the discount factor is asked, not the numerical
value of r.)

1

2

3

3

4

5

1
2

5

2
3

6

2
4

3
6
9

7

3

4

5

8

10
(continued)

Exercises

169

Process i

1

2

3

4

5

Influent, Ii

Removal, Ri

40

10

40

20

40

30

30

10

8

8

30

20

10

12

20

10

Annual cost = Ci(Ii, Ri)
8

5
7

5
12
18

8

(b) Draw the dynamic programming network and solve this problem by
dynamic programming. Indicate on
the network the calculations required
to find the least-cost path from state
100 at stage 1 to state 10 at stage 6
using both forward- and backwardmoving dynamic programming solution procedures.
(c) Could the following conditions be
included in the original dynamic programming model and still be solved
without requiring R4 to be 0 in the first
case and R3 to be 0 in the second case?
(i) R4 = 0 if R3 = 0, or
(ii) R3 = 0 if R2 ≤ 20.
4:20 The city of Eutro Falls is under a court order
to reduce the amount of phosphorus that
which it discharges in its sewage to Lake
Algae. The city presently has three
wastewater treatment plants. Each plant
i currently discharges Pi kg/day of phosphorus into the lake. Some or all plants must
reduce their discharges so that the total for
the three plants does not exceed P kg/day.
Let Xi be the fraction or percent of the
phosphorus removed by additional treatment at plant i, and the Ci(Xi) the cost of
such treatment ($/year) at each plant i.
(a) Structure a planning model to determine the least-cost (i.e., a cost effective) treatment plant for the city.
(b) Restructure the model for the solution
by dynamic programming. Define the

stages, states, decision variables, and
the recursive equation for each stage.
(c) Now assume P1 = 20; P2 = 15;
P3 = 25; and P = 20. Make up some
cost data and check the model if it
works.
4:21 Find (draw) a rule curve for operating a
single reservoir that maximizes the sum of
the benefits for flood control, recreation,
water supply and hydropower. Assume
the average inflows in four seasons of a
year are 40, 80 60, 20, and the active
reservoir capacity is 100. For an average
storage S and for a release of R in a season, the hydropower benefits are 2 times
the square root of the product of S and R,
2(SR)0.5, and the water supply benefits are
3R0.7 in each season. The recreation benefits are 40 − (70 − S)2 in the third season. The flood control benefits are
20 − (40 − S)2 in the second season.
Specify the dynamic programming recursion equations you are using to solve the
problem.
4:22 How would the model defined in Exercise
4.21 change if there were a water user
upstream of this reservoir and you were to
find the best water-allocation policy for
that user, assuming known benefits associated with these allocations that are to be
included in the overall maximum benefits
objective function?
4:23 Suppose there are four water users along a
river who benefit from receiving water
from the river. Each has a water target,
i.e., each expects and plans for a specified
amount. These known water targets are W
(1), W(2), W(3), and W(4) for the four
users, respectively. Show how dynamic
programming can be used to find two
allocation policies. One is to be based on
minimizing the maximum deficit deviation
from any target allocation. The other is to
be based on minimizing the maximum
percentage
deficit
from
any
target allocation.

170

4

An Introduction to Optimization Models and Methods

Gradient “Hill-climbing” methods
4:24 Solve Exercise 4.13(b) using hill-climbing
techniques and assuming discrete integer
values and Tmax = 5. For example, which
product would you produce if you could
make only 1 unit of either A or B? If you
could make another unit of A or B, which
would you make? Continue this process
up to 5 units of products A and/or B.
4:25 Under what conditions will hill-climbing
methods for maximization or minimization not work?
Linear and nonlinear programming
4:26 Consider the industrial firm that makes
two products A and B as described in
Exercise 4.13(b). Using Lingo (or any
other program you wish):
(a) Find the amounts of A and B and their
unit prices that maximize total revenue
assuming the total amount of water
available is 10 units.
(b) What is the value of the dual variable,
or shadow price, associated with the
10 units of available water?
(c) Suppose the demand functions are not
really certain. How sensitive are the
allocations of water to changes in the
parameter values of those functions?
How sensitive are the allocations to
the parameter values in the production
functions?

into a river. Denote the discharge sites by
the subscript i and let Wi be the kg of
waste discharged into the river each day at
those sites i. To improve the river water
quality downstream, wastewater treatment
plants may be required at each site i. Let xi
be the fraction of waste removed by
treatment at each site i. Develop a model
for estimating how much waste removal is
required at each site to maintain acceptable water quality in the river at a minimum total cost. Use the following
additional notation:
aij
qj

Qj
Ci

decrease in quality at site j per unit of waste
discharged at site i
quality at site j that would result if all
controlled upstream discharges were
eliminated (i.e., W1 = W2 = 0)
minimum acceptable quality at site j
cost per unit (fraction) of waste removed at
site i.

4:28 Assume that there are two sites along a
stream, i = 1, 2, at which waste (BOD) is
discharged. Currently, without any
wastewater treatment, the quality (DO), q2
and q3, at each of sites 2 and 3 is less than
the minimum desired, Q2 and Q3, respectively. For each unit of waste removed at
site i upstream of site j, the quality
improves by Aij. How much treatment is
required at sites 1 and 2 that meets the
standards at a minimum total cost?

4:27 Assume that there are m industries or
municipalities that discharge their wastes

W1
Site 3
Site 2

Stream

Site 1
Park

W2

Exercises

171

Following are the necessary data:
Ci

Ri

cost per unit fraction of waste treatment at
site i (both C1 and C2 are unknown but for
the same amount of treatment, whatever that
amount, C1 > C2)
decision variables, unknown waste removal
fractions at sites i = 1, 2
A12 ¼ 1=20
A13 ¼ 1=40
A23 ¼ 1=30

W 1 ¼ 100
W 2 ¼ 75
q2 ¼ 3

(b)

(c)

Q2 ¼ 6
Q3 ¼ 4
q3 ¼ 1

(d)

4:29 Define a linear programming model for
finding the tradeoff between active storage
capacity and the maximum percentage
deviation from a known target storage
volume and a known target release in each
period. How could the solution of the
model be used to define a reservoir policy?
4:30 Consider the possibility of building a
reservoir upstream of three demand sites
along a river.

(e)

guaranteed in each season) and the
reservoir capacity.
Find the tradeoff between the yield
and the maximum total net benefits
that can be obtained from allocating
that yield among the three users.
Find the tradeoff between the reservoir
capacity and the total net benefits one
can obtain from allocating the total
releases, not just the reliable yield, to
the downstream users.
Assuming a reservoir capacity of 7,
and dividing the release into integer
increments of 2 (i.e., 2, 4, 6 and 8),
using linear programming, find the
optimal operating policy. Assume the
maximum release cannot exceed 8,
and the minimum release cannot be
less than 2. How does this solution
differ from that obtained using
dynamic programming?
If you were maximizing the total net
benefit obtained from the three users
and if the water available to allocate to
3

1

2

The net benefits derived from each use
depend on the reliable amounts of water
allocated to each use. Letting xit be the
allocation to use i in period t, the net
benefits for each period t equal
1. 6x1t − x21t
2. 7x2t − 1.5x22t
3. 8x3t − 0.5x23t
Assume the average inflows to the
reservoir in each of four seasons of the
year equal 10, 2, 8, 12.
(a) Find the tradeoff between the yield
(the expected release that can be

the three users were 15 in a particular
time period, what would be the value
of the Lagrange multiplier or dual
variable associated with the constraint
that you cannot allocate more than 15
to the three uses?
(f) There is the possibility of obtaining
recreational benefits in seasons 2 and
3 from reservoir storage. No recreational benefits can occur in seasons 1
and 4. To obtain these benefits facilities must be built, and the question is
at what elevation (storage volume)
should they be built. This is called the
recreational storage volume target.

172

4

Recreational benefits in each recreation season equal 8 per unit of storage target if the actual storage equals
the storage target. If the actual storage
is less than the target the losses are 12
per unit deficit—the difference
between the target and actual storage
volumes. If the actual storage volume
is greater than the target volume the
losses are 4 per unit excess. What is
the reservoir capacity and recreation
storage target that maximizes the
annual total net benefits obtained from
downstream allocations and recreation
in the reservoir less the annual cost of
the reservoir, 3K1.5, where K is the
reservoir capacity?
(g) In (f) above, suppose the allocation
benefits and net recreation benefits
were given weights indicating their
relative importance. What happens to
the relationship between capacity
K and recreation target as the total
allocation benefits are given a greater
weight in comparison to recreation net
benefits?
4:31 Using the network representation of the
wastewater treatment plant design problem
defined in Exercise 4.19, write a linear
programming model for defining the
least-cost sequence of unit treatment process (i.e., the least-cost path through the
network). [Hint: Let each decision variable
xij indicate whether or not the link between
nodes (or states) i and j connecting two
successive stages is on the least-cost or
optimal path. The constraints for each
node must ensure that what enters the
node must also leave the node.]
4:32 Two types of crops can be grown in particular irrigation area each year. Each unit
quantity of crop A can be sold for a price
PA and requires WA units of water, LA units
of land, FA units of fertilizer, and HA units
of labor. Similarly, crop B can be sold at a
unit price of PB and requires WB, LB, FB
and HB units of water, land, fertilizer, and

An Introduction to Optimization Models and Methods

labor,
respectively,
per
unit
of
crop. Assume that the available quantities
of water, land, fertilizer, and labor are
known, and equal W, L, F, and H,
respectively.
(a) Structure a linear programming model
for estimating the quantities of each of
the two crops that should be produced
in order to maximize total income.
(b) Solve the problem graphically, using
the following data:
Requirements
per unit of
Resource

Crop
A

Crop
B

Maximum available
resources

Water

2

3

60

Land

5

2

80

Fertilizer

3

2

60

Labor

1

2

40

Unit
price

30

25

(c) Define the meaning of the dual variables, and their values, associated with
each constraint.
(d) Write the dual model of this problem
and interpret its objective and
constraints.
(e) Solve the primal and dual models using
an existing computer program, and
indicate the meaning of all output data.
(f) Assume that one could purchase
additional water, land, fertilizer, and
labor with capital that could be borrowed from a bank at an annual
interest rate r. How would this
opportunity alter the linear programming model? The objective continues
to be a maximization of net income.
Assume there is a maximum limit on
the amount of money that can be
borrowed from the bank.
(g) Assume that the unit price Pj of crop
j is a decreasing linear function
(Poj bj xj ) of the quantity, xj, produced. How could the linear model be

Exercises

173

restructured also as to identify not
only how much of each crop to produce, but also the unit price at which
each crop should be sold in order to
maximize total income?
4:33 Using linear programming model, derive
an annual storage-yield function for a
reservoir at a site having the following
record of annual flows:
Year y

Flow Qy

Year y

Flow Qy

1

5

9

3

2

7

10

6

3

8

11

8

4

4

12

9

5

3

13

3

6

3

14

4

7

2

15

9

8

1

Task B:

Task C:

Task
D:
Task E:
Task F:

Task
G:
Task
H:

Determine structural and nonstructural
alternatives that will influence
objectives. Duration: 1 month.
Develop an optimization model for
preliminary screening of alternatives
and for estimating tradeoffs among
objectives. Duration: 1 month.
Identify data requirements and collect
data. Duration: 2 months.
Develop a data management system
for the project. Duration: 3 months.
Develop an interactive shared vision
simulation
model
with
the
stakeholdes. Duration: 2 Months.
Work with stakeholders in an effort to
come to a consensus (a shared vision)
of the best plan. Duration: 4 months.
Prepare, present and submit a report.
Duration: 2 months.

B
A

(a) Find the values of the storage capacity
required for yields of 2, 3, 3.5, 4, 4.5,
and 5.
(b) Develop a flow chart defining a procedure for finding the yields for various increasing values of K.
4:34 Water resources planning usually involves
a set of separate tasks. Let the index i denote each task, and Hi the set of tasks that
immediately precede task i. The duration
of each task i is estimated to be di.
(a) Develop a linear programming model
to identify the starting times of tasks
that minimizes the time, T, required to
complete the total planning project.
(b) Apply the general model to the following planning project:
Task A:

Determine planning objectives and
stakeholder
interests.
Duration:
4 months.

C
E

G

H

F
D

4:35 In Exercise 4.34 suppose the project is
penalized if its completion time exceeds a
target T. The difference between 14 months
and T months is Δ, and the penalty is P(Δ).
You could reduce the time it takes to
complete task E by one month at a cost of
$200, and by two months at a cost of $500.
Similarly, suppose the cost of task A could
be reduced by a month at a cost of $600 and
two months at a cost of $1400. Construct a
model to find the most economical project
completion time. Next modify the linear
programming model to find the minimum
total added cost if the total project time is to
be reduced by 1 or 2 months. What is that
added cost and for which tasks?

174

4:36 Solve the reservoir operation problem
described in Exercise 4.15 using linear
programming. If the reservoir capacity is
unknown, show how a cost function (that
includes fixed costs and economies of
scale) for the reservoir capacity could be
included in the linear programming model.
4:37 An upstream reservoir could be built to
serve two downstream users. Each user
has a constant water demand target. The
first user’s target is 30; the second user’s
target is 50. These targets apply to each of
6 within-year seasons. Find the tradeoff
between the required reservoir capacity
and maximum deficit to any user at any
time, for an average year. The average
flows into the reservoir in each of the six
successive seasons are: 40, 80, 100, 130,
70, 50.
4:38 Two groundwater well fields can be used
to meet the water demands of a single
user. The maximum capacity of the A well
field is 15 units of water per period, and
the maximum capacity of the B well field
is 10 units of water per period. The annual
cost of building and operating each well
field, each period, is a function of the
amount of water pumped and transported
from that well field. Three sets of cost
functions are shown below: Construct a
LP model and use it to define and then plot
the total least-cost function and the associated individual well field capacities
required to meet demands from 0 to 25,
assuming cost functions 1 and 2 apply to
well fields A and B, respectively. Next
define another least-cost function and
associated capacities assuming cost functions 3 and 4 apply to A and B, respectively. Finally define a least-cost function
and associated capacities assuming well
field cost functions 5 and 6 apply. You can
check your model results just using common sense—the least-cost functions
should be obvious, even without using
optimization.

4

An Introduction to Optimization Models and Methods

2

1

5

10

3

4
15

2

8

5

5
5
14

6

12

3

4

20

5
7

4:39 Referring to Exercise 4.38 above, assume
cost functions 5 and 6 represent the cost of
adding additional capacity to well fields
A and B, respectively, in any of the next five
5-year construction periods, i.e., in the next
25 years. Identify and plot the least-cost
capacity expansion schedule (one that
minimizes the total present value of current
and future expansions), assuming demands
of 5, 10, 15, 20 and 25 are to be met at the
end of years 5, 10, 15, 20 and 25, respectively. Costs, including fixed costs, of capacity expansion in each construction
period have to be paid at the beginning of
the construction period. Determine the
sensitivity of your solution to the interest
rate used to compute present value.
4:40 Consider a crop production problem
involving three types of crops. How many
hectares of each crop should be planted to
maximize total income?
Resources

Max limits

Resource requirements
Crops:

Corn

Wheat

Oats

Water

1000/week

3.0

1.0

1.5

units/week/ha

Labor

300/week

0.8

0.2

0.3

person

Land

625 ha

h/week/ha

Yield $/ha

400

200

250

Exercises

Show a graph that identifies the tradeoffs
among crops that can be made without reducing
the total income.
4:41 Releases from a reservoir are used for
water supply or for hydropower. The
benefit per unit of water allocated to
hydropower is BH and the benefit per unit
of water allocated to water supply is BW.
For any given release the difference
between the allocations to the two uses
cannot exceed 50% of the total amount of
water available. Show graphically how to
determine the most profitable allocation of
the water for some assumed values of BH
and BW. From the graph identify which
constraints are binding and what their
“dual prices” mean (in words).
4:42 Suppose there are four water users along a
river who benefit from receiving water.
Each has a known water target, i.e., each
expects and plans for a specified amount.
These known water targets are W1, W2,
W3, and W4 for the four users, respectively. Find two allocation policies. One is
to be based on minimizing the maximum
deficit deviation from any target allocation. The other is to be based on minimizing the maximum percentage deficit
from any target allocation.
Deficit allocations are allocations that are
less than the target allocation. For example
if a target allocation is 30 and the actual
allocation is 20, the deficit is 10. Water in
excess of the targets can remain in the
river. The policies are to indicate what the
allocations should be for any particular
river flow Q. The policies can be expressed on a graph showing the amount of
Q on the horizontal axis, and each user’s
allocation on the vertical axis.
Create the two optimization models that
can be used to find the two policies and
indicate how they would be used to define

175

the policies. What are the unknown variables and what are the known variables?
Specify the model in words as well as
mathematically.
4:43 In Indonesia there exists a wet season
followed by a dry season each year. In one
area of Indonesia all farmers within an
irrigation district plant and grow rice during the wet season. This crop brings the
farmer the largest income per hectare; thus
they would all prefer to continue growing
rice during the dry season. However, there
is insufficient water during the dry season
to irrigate all 5000 ha of available irrigable land for rice production. Assume an
available irrigation water supply of
32 × 106 m3 at the beginning of each dry
season, and a minimum requirement of
7000 m3/ha for rice and 1800 m3/ha for
the second crop.
(a) What proportion of the 5000 ha
should the irrigation district manager
allocate for rice during the dry season
each year, provided that all available
hectares must be given sufficient water
for rice or the second crop?
(b) Suppose that crop production functions are available for the two crops,
indicating the increase in yield per
hectare per m3 of additional water, up
to 10, 000 m3/ha for the second
crop. Develop a model in which the
water allocation per hectare, as well as
the hectares allocated to each crop, is
to be determined, assuming a specified
price or return per unit of yield of each
crop. Under what conditions would
the solution of this model be the same
as in part (a)?
4:44 Along the Nile River in Egypt, irrigation
farming is practiced for the production of
cotton, maize, rice, sorghum, full and short

176

4

berseem for animal production, wheat,
barley, horsebeans, and winter and summer
tomatoes. Cattle and buffalo are also produced, and together with the crops that
require labor, water. Fertilizer, and land
area (feddans). Farm types or management
practices are fairly uniform, and hence in
any analysis of irrigation policies in this
region this distinction need not be made.
Given the accompanying data develop a
model for determining the tons of crops and
numbers of animals to be grown that will
maximize (a) net economic benefits based
on Egyptian prices, and (b) net economic
benefits based on international prices.
Identify all variables used in the model.
Known parameters:
Ci
PEi
PIi
v

g
fP
fN
Yi
a
b
rw
swh
sba
rs
lNi
lPi
lim
wim
him

miscellaneous cost of land preparation per
feddan
Egyptian price per 1000 tons of crop i
international price per 1000 tons of crop i
value of meat and dairy production per
animal
annual labor cost per worker
cost of P fertilizer per ton
cost of N fertilizer per ton
yield of crop i, tons/feddan
feddans serviced per animal
tons straw equivalent per ton of berseem
carryover from winter to summer
berseem requirements per animal in winter
straw yield from wheat, tons per feddan
straw yield from barley, tons per feddan
straw requirements per animal in summer
N fertilizer required per feddan of crop i
P fertilizer required per feddan of crop i
labor requirements per feddan in month m,
man-days
water requirements per feddan in month
m, 1000 m3
land requirements per month, fraction
(1 = full month)
Required Constraints (assume known
resource limitations for labor, water, and
land):

An Introduction to Optimization Models and Methods

(a) Summer and winter fodder (berseem)
requirements for the animals.
(b) Monthly labor limitations.
(c) Monthly water limitations.
(d) Land availability each month.
(e) Minimum number of animals required
for cultivation.
(f) Upper bounds on summer and winter
tomatoes (assume these are known).
(g) Lower bounds on cotton areas (assume this is known).
Other possible constraints:
(a)
(b)
(c)
(d)

Crop balances.
Fertilizer balances.
Labor balance.
Land balance.

4:45 In Algeria there are two distinct cropping
intensities, depending upon the availability
of water. Consider a single crop that can be
grown under intensive rotation or extensive rotation on a total of A hectares.
Assume that the annual water requirements
for the intensive rotation policy are
16,000 m3 per ha, and for the extensive
rotation policy they 4000 m3 per ha. The
annual net production returns are 4000 and
2000 dinars, respectively. If the total water
available is 320,000 m3, show that as the
available land area A increases, the rotation
policy that maximizes total net income
changes from one that is totally intensive to
one that is increasingly extensive.
Would the same conclusion hold if instead
of fixed net incomes of 4000 and
2000 dinars per hectares of intensive and
extensive rotation, the net income depended on the quantity of crop produced?
Assuming that intensive rotation produces
twice as much produced by extensive
rotation, and that the net income per unit
of crop Y is defined by the simple linear
function 5 − 0.05 Y, develop and solve a
linear programming model to determine
the optimal rotation policies if A equals
20, 50, and 80. Need this net income or

Exercises

177

price function be linear to be included in a
linear programming model?
4:46 Current stream quality is below desired
minimum levels throughout the stream in
spite of treatment at each of the treatment
plant and discharge sites shown below.
Currently effluent standards are not being

the amount of wastewater to treat at each
site and then release to the river. Any
wastewater at any site that is not undergoing additional treatment can be piped to
other sites. Identify other issues that could
affect the eventual decision.

met, and minimum desired streamflow
concentrations can be met by meeting
effluent standards. All current wastewater
discharges must undergo additional treatment. The issue is where additional treatment is to occur and how much.
Develop a model to identify cost-effective
options for meeting effluent standards
where ever wastewater is discharged into
the stream. The decisions variables include

Assume known current wastewater flows at site
i = qi .
Additional treatment to meet effluent standards
cost = ai þ bi ðDi Þci where Di is the total
wastewater flow undergoing additional treatment
at site i and ci < 1.
Pipeline and pumping for each pipeline segment
costs approximately αij + β(qij)γ.
where qij is pipeline flow between adjacent sites
i and j and γ < 1.

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reproduce the material.

5

Data-Fitting, Evolutionary,
and Qualitative Modeling

Clearly, all model outputs depend on model
inputs. The optimization and simulation models
discussed in the previous chapters are no
exception. This chapter introduces some alternative modeling approaches that depend on
observed data. These approaches include artificial neural networks and various evolutionary
models. The chapter ends with some qualitative
modeling. These data-driven models can serve as
substitutes for more process-based models in
applications where computational speed is critical or where the underlying relationships are
poorly understood or too complex to be easily
incorporated into calculus-based, linear, nonlinear, or dynamic programming models. Evolutionary algorithms involve random searches
based on evolutionary or biological processes for
finding the values of parameters and decision
variables that best satisfy system performance
criteria. Evolutionary algorithms are popular
methods for analyzing systems that require
complex simulation models to determine values
of performance measures. Qualitative modeling
approaches are useful when performance measures are expressed qualitatively, such as “I want

a reliable supply of clean water at a reasonable
cost,” where there can be disagreements among
different stakeholders and decision makers with
respect to specifying just how reliable, how
clean, and how affordable.

5.1

Introduction

Most models used for water resources planning
and management describe, in mathematical
terms, the interactions and processes that take
place among the various components of the
system. These mechanistically or process-based
models usually contain parameters whose values
are determined from observed data during model
calibration. These types of models are contrasted
to what are typically called “black-box” models,
or statistical models. Such models do not
describe physical processes. They attempt to
convert observed inputs (e.g., rainfall and runoff,
inflows to a reservoir, pollutants entering a
wastewater treatment plant or effluent concentrations discharged to a river) to observed outputs
(e.g., runoff, reservoir releases, pollutant

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_5

179

180

5

concentrations) using any set of mathematical
equations or expressions that does the job. One
type of such models is regression.
Regression equations, such as of the forms
Output variable value ¼ a þ bðinput variable valueÞ

ð5:1Þ
Output variable value ¼ a þ bðinput variable valueÞC

ð5:2Þ
Output variable value
¼ a þ b1 ðinput variable1 valueÞC1

ð5:3Þ

C2

þ b2 ðinput variable2 valueÞ

are examples of such data-fitting or statistical
models.
They depend on observed inputs and observed
outputs for the estimation of the values of their
parameters (a, b, c, etc.) and for further refinement of their structure. They lack an explicit,
well-defined representation of the processes
involved in the transformation of inputs to

Fig. 5.1 Data-fitting models are able to estimate relatively accurately within their calibrated ranges, but not
outside those ranges. The bottom curve represents the

Data-Fitting, Evolutionary, and Qualitative Modeling

outputs. While these statistical models are better
at interpolating within the range of data used to
calibrate them, rather than extrapolating outside
that range (as illustrated in Fig. 5.1), many have
proven quite successful in representing complex
physical systems.
Other examples of data-driven models are
based on biological principles and concepts.
These are a class of probabilistic search procedures known as evolutionary algorithms (EAs).
Such algorithms include genetic algorithms
(GAs), genetic or evolutionary programming (GP
or EP), and evolutionary strategy (ES). Each of
these methods has many varieties but all use
computational methods based on natural evolutionary processes and learning. Perhaps the most
robust and hence the most common of these
methods are genetic algorithms and their varieties used to find the values of parameters and
variables that best satisfy some objective. Alternatively, an extension of regression is artificial
neural networks (ANN). The development and
application of black-box models like GA, GP,

relative density of data used in model calibration. The
arrows point to where the model does not predict well

5.1 Introduction

and ANNs emulate larger, deterministic, processoriented models. Once calibrated, their use may
be advantageous if and when it is quicker to use
them to obtain the information needed rather than
using process-oriented models that typically take
longer to solve. Process-oriented models are
sometimes used to calibrate artificial neural networks, which are then used to more quickly
explore and evaluate the range of solution outputs associated with varying inputs.
Examples of such situations where multiple
solutions of a model must be obtained include
sensitivity or uncertainty analysis, scenario
evaluations, risk assessment, optimization,
inverse modeling to obtain parameter values
given the values of the decision variables, and/or
when model runs must be extremely fast, as for
rapid assessment and decision support systems,
real-time predictions/management/control, and
so on. Examples of the use of data-fitting models
for model emulation are given in the next several
sections.
Genetic algorithms and genetic programming
are automated, domain-independent methods for
evolving solutions to existing models or for
producing new models that emulate actual systems, such as rainfall–runoff relationships in a
watershed, wastewater removal processes in a
treatment plant, or discharges of water from a
system of natural lakes, each subject to random
inputs. Search methods such as genetic algorithms and genetic programming are inspired by
our understanding of biology and natural evolution. They start initially with a number of sets of
randomly created values of the unknown variables or a number of black-box models, respectively. The variable values or structure of each of
these models are progressively improved over a
series of generations. The evolutionary search
uses the Darwinian principal of “survival of the
fittest” and is patterned after biological operations including crossover (sexual recombination),
mutation, gene duplication, and gene deletion.
Artificial neural networks are distributed,
adaptive, generally nonlinear networks built from

181

many different processing elements (PEs) (Principe et al. 2000). Each processing element
receives inputs from other processing elements
and/or from itself. The inputs are scaled by
adjustable parameters called weights. The processing elements sum all of these weighted
inputs to produce an output that is a nonlinear
(static) function of the sum. Learning (calibration) is accomplished by adjusting the weights.
The weights are adjusted directly from the
training data (data used for calibration) without
any assumptions about the data’s statistical distribution or other characteristics (Hagan et al.
1996; Hertz et al. 1991).
The following sections are intended to provide some background helpful to those who may
be selecting one among all the available computer codes for implementing a genetic algorithm, genetic program, or artificial neural
network.

5.2

Artificial Neural Networks

5.2.1 The Approach
Before the development of digital computers, any
information processing necessary for thinking
and reasoning was carried out in our brains.
Much of it still is. Brain-based information processing continues today (e.g., see Fig. 2.1) and
will continue in the future even given our continually improving electronic digital processing
capabilities. While recent developments in
information technology (IT) have mastered and
outperformed much of the information processing one can do just using brain power, IT has not
mastered the reasoning power of our brains.
Perhaps because of this, some computer scientists have been working on creating information
processing devices that mimic the human brain.
This has been termed neurocomputing. It uses
ANNs representing simplified models of the
brain. In reality, it is just a more complex type of
regression or statistical (black-box) model.

182

5

Data-Fitting, Evolutionary, and Qualitative Modeling

Fig. 5.2 A typical multilayer artificial neural network showing the input layer for ten different inputs, the hidden layer
(s), and the output layer having three outputs

An example of the basic structure of an ANN
is shown in Fig. 5.2. There are a number of input
layer nodes on the left side of the figure and a
number of output layer nodes on the right. The
middle column(s) of nodes between these input
and output nodes are called hidden layers. The
number of hidden layers and the number of
nodes in each layer are two of the design
parameters of any ANN. Most applications
require networks that contain at least these three
types of layers:
• The input layer consists of nodes that receive
an input from the external environment.
These nodes do not perform any transformations upon the inputs but just send their
weighted values to the nodes in the immediately adjacent, usually “hidden,” layer.
• The hidden layer(s) consist(s) of nodes that
typically receive the transferred weighted
inputs from the input layer or previous hidden
layer, perform their transformations on it, and

pass the output to the next adjacent layer,
which can be another hidden layer or the
output layer.
• The output layer consists of nodes that
receive the hidden layer output and send it to
the user.
The ANN shown in Fig. 5.2 has links only
between nodes in immediately adjacent layers or
columns and is often referred to as a multilayer
perceptron (MLP) network, or a feedforward
(FF) network. Other architectures of ANNs,
which include recurrent neural networks (RNN),
self-organizing feature maps (SOFMs), Hopfield
networks, radial basis function (RBF) networks,
support vector machines (SVMs), and the like,
are described in more detail in other publications
(for example, Haykin 1999; Hertz et al. 1991).
Essentially, the strength (or weight) of the
connection between adjacent nodes is a design
parameter of the ANN. The output values Oj that
leave a node j on each of its outgoing links are

5.2 Artificial Neural Networks

183

multiplied by a weight, wj. The input Ik to each
node k in each middle and output layer is the sum
of each of its weighted inputs, wjOj, from all
nodes j providing inputs (linked) to node k.
Input value to node k:
Ik ¼

X

wj Oj

ð5:4Þ

Again, the sum in Eq. 5.4 is over all nodes
j providing inputs to node k.
At each node k of hidden and output layers,
the input Ik is an argument to a linear or nonlinear
function fk(Ik + θk), which converts the input Ik to
output Ok. The variable θk represents a bias or
threshold term that influences the horizontal
offset of the function. This transformation can
take on a variety of forms. A commonly used
transformation is a sigmoid or logistic function as
defined in Eq. 5.5 and graphed in Fig. 5.3.
Ok ¼ 1=½1 þ expfðIk þ hk Þg

ð5:5Þ

The process of converting inputs to outputs at
each hidden layer node is illustrated in Fig. 5.4.
The same process also happens at each output
layer node.
The design issues in artificial neural networks
are complex and are major concerns of ANN
developers. The number of nodes in the input as
well as in the output layer is usually predetermined from the problem to be solved. The
number of nodes in each hidden layer and the
number of hidden layers are calibration parameters that can be varied in experiments focused on

Fig. 5.3 The sigmoid or logistic threshold function with
threshold θk

Fig. 5.4 A middle-layer node k converting input values
to an output value using a nonlinear function f (such as
defined by Eq. 5.5) in a multilayer ANN

getting the best fit of observed and predicted
output data based on the same input data. These
design decisions, and most importantly the
determination of the values of the weights and
thresholds of each connection, are “learned”
during the “training” of the ANN using predefined (or measured) sets of input and output data.
Some of the present-day ANN packages provide options for building networks. Most provide
fixed network layers and nodes. The design of an
ANN can have a significant impact on its
data-processing capability.
There are two major connection topologies
that define how data flows between the input,
hidden, and output nodes. These main categories
are:
• Feedforward networks in which the data
flow through the network in one direction
from the input layer to the output layer
through the hidden layer(s). Each output
value is based solely on the current set of
inputs. In most networks, the nodes of one
layer are fully connected to the nodes in the
next layer (as shown in Fig. 5.2); however,
this is not a requirement of feedforward
networks.
• Recurrent or feedback networks in which, as
their name suggests, the data flow not only in
one direction but in the opposite direction as

184

5

well for either a limited or a complete part of
the network. In recurrent networks, information about past inputs is fed back into and
mixed with inputs through recurrent (feedback) connections. The recurrent types of
artificial neural networks are used when the
answer is based on current data as well as on
prior inputs.
Determining the best values of all the weights
is called training the ANN. In a so-called
supervised learning mode, the actual output of
a neural network is compared to the desired
output. Weights, which are usually randomly set
to begin with, are then adjusted so that the next
iteration will produce a closer match between the
desired and the actual output. Various learning
methods for weight adjustments try to minimize
the differences or errors between observed and
computed output data. Training consists of presenting input and output data to the network.
These data are often referred to as training data.
For each input provided to the network, the
corresponding desired output set is provided as
well.
The training phase can consume a lot of time.
It is considered complete when the artificial
neural network reaches a user-defined performance level. At this level the network has
achieved the desired statistical accuracy as it
produces the required outputs for a given
sequence of inputs. When no further learning is
judged necessary, the resulting weights are typically fixed for the application.
Once a supervised network performs well on
the training data, it is important to see what it
can do with data it has not seen before. If a
system does not give a reasonable output for
this test set, this means that the training period
should continue. Indeed, this testing is critical
to ensure that the network has learned the
general patterns involved within an application
and has not simply memorized a given set of
data.
Smith (1993) suggests the following procedure for preparing and training an ANN:

Data-Fitting, Evolutionary, and Qualitative Modeling

1. Design a network.
2. Divide the data set into training, validation,
and testing subsets.
3. Train the network on the training data set.
4. Periodically stop the training and measure the
error on the validation data set.
5. Save the weights of the network.
6. Repeat Steps 2, 3, and 4 until the error on the
validation data set starts increasing. This is
the moment where the overfitting has started.
7. Go back to the weights that produced the
lowest error on the validation data set, and
use these weights for the trained ANN.
8. Test the trained ANN using the testing data
set. If it shows good performance, use it. If
not, redesign the network and repeat entire
procedure from Step 3.
There is a wide selection of available neural
network models. The most popular is probably
the multilayer feedforward network, which is
typically trained with static back propagation.
They are easy to use, but they train slowly, and
require considerable training data. In fact, the
best generalization performance is produced if
there are at least 30 times more training samples
than network weights (Haykin 1999). Adding
local recurrent connections can reduce the
required network size, making it less sensitive to
noise, but it may get stuck on a solution that is
inferior to what can be achieved.

5.2.2 An Example
To illustrate how an ANN might be developed,
consider the simple problem of predicting a
downstream pollutant concentration based on an
upstream concentration and the streamflow.
Twelve measurements of the streamflow quantity, velocity, and pollutant concentrations at two
sites (an upstream and a downstream site) are
available. The travel times between the two
measurement sites have been computed and
these, plus the pollutant concentrations, are
shown in Table 5.1.

5.2 Artificial Neural Networks

185

Table 5.1 Streamflow travel times and pollutant concentrations

Assume at first that the ANN structure consists of two input nodes, a hidden node, and a
single output node. One of the input nodes is for
the upstream concentration and the other input
node is for the travel time. The single output
node represents the downstream concentration
expressed as a fraction of the upstream concentration. This is shown in Fig. 5.5.
The model output is the fraction of the
upstream concentration that reaches the downstream site. That fraction can be any value from 0
to 1. Hence the sigmoid function (Eq. 5.5) is
applied at the middle node and at the output
node. Using two or more data sets to train or
calibrate this ANN (Fig. 5.5) results in a poor fit
as measured by the minimum sum of absolute
deviations between calculated and measured
concentration data. The more data samples used,

the worse the fit. This structure is simply too
simple. Hence, another node was added to the
middle layer. This ANN is shown in Fig. 5.6.
Using only half the data (six data sets) for
training or calibration, the weights obtained
provided a near perfect fit. The weights obtained
are shown in Table 5.2.
Next the remaining six data sets were applied
to the network with weights set to those values
shown in Table 5.2. Again the sum of absolute
deviations was essentially 0. Similar results were
obtained with increasing numbers of data sets.
The values of the weights in Table 5.2 indicate something water quality modelers typically
assume, and that is that the fraction of the
upstream pollutant concentration that reaches a
downstream site is independent of the actual
upstream concentration (see Chap. 4). This ANN

186

5

Data-Fitting, Evolutionary, and Qualitative Modeling

Fig. 5.5 Initial ANN for
example problem

Fig. 5.6 Modified ANN
for example problem

Table 5.2 Weights for each link of the ANN shown in Fig. 5.6 based on six data sets from Table 5.1. All bias
variables (θk in Eq. 5.5) were 0

could have had only one input node, namely that
for travel time. This conforms to the typical
first-order decay function:

where the parameter k is the decay rate constant
having units of 1/travel time (travel time units−1).

Fraction of pollutant concentration downstream
per unitconcentration upstream

5.3

¼ expfkðtravel timeÞg;
ð5:6Þ

Evolutionary Algorithms

Evolutionary algorithms (EA) represent a broad
spectrum of heuristic approaches for simulating
biological evolution in the search for improved

5.3 Evolutionary Algorithms

“fitness,” i.e., the best values of decision variables and parameters based on an objective or
fitness function. Evolutionary algorithms are
broadly based on the repeated mutation and
recombination and selection: in each generation
(iteration) to define new individuals (candidate
solutions). These are generated by variation,
usually in a stochastic way, and then some
individuals are selected for the next generation
based on their relative fitness or objection function value. Over the generation sequence, individuals with increasingly better fitness values are
generated (Simon 2013).
Primary examples include genetic algorithms
(Holland
1975),
evolutionary
strategies
(Rechenberg 1973; Schwefel 1981), evolutionary
programming (Fogel et al. 1966), and genetic
programming (Koza 1992). These methods are
comprised of algorithms that operate using a
population of alternative solutions or designs,
each represented by a potential decision vector.
They rely on randomized operators that simulate
mutation and recombination to create new individuals, i.e., solutions, who then compete to
survive via the selection process, which operates
according to a problem-specific fitness or objective function. In some cases this function can be
a complex simulation model dependent on the
values of its parameters and decision variables
derived from the EA. EA popularity is, at least in
part, due to their potential to solve nonlinear,
nonconvex, multimodal, and discrete problems
for which deterministic gradient-based search
techniques incur difficulty or fail completely. The
growing complexity and scope of environmental
and water resources applications has served to
expand EAs’ capabilities.
Currently, the field of biologically inspired
search algorithms mostly include variations of
evolutionary algorithms and swarm intelligence
algorithms, e.g., ant colony optimization (ACO),
particle swarm optimization (PSO), bees algorithm, bacterial foraging optimization (BFO), and
so on, many of which have been used to analyze
water resources planning and management
problems. This is especially true for application
of genetic algorithms, arguably among the most
popular of the several types of EAs. EAs are

187

flexible tools that can be applied to the solution
of a wide variety of complex water resources
problems. Nicklow et al. (2010) provides a
comprehensive review of state-of-the-art methods and their applications in the field of water
resources planning and management. EAs have
been successfully applied to the study of water
distribution systems, urban drainage and sewer
systems, water supply and wastewater treatment,
hydrologic and fluvial modeling, groundwater
systems, and parameter identification, to name a
few. Nicklow et al. also identify major challenges
and opportunities for the future, including a call
to address larger scale problems that involve
uncertainty and an expanded need for collaboration among multiple stakeholders and disciplines. Evolutionary computation methods will
surely continue to evolve in the future as analysts
encounter increased problem complexities and
uncertainty and as the societal pressure for more
innovative and efficient solutions rises.

5.3.1 Genetic Algorithms
Genetic algorithms are randomized generalpurpose search techniques used for finding the
best values of the parameters or decision variables of existing models. It is not a modelbuilding tool like genetic programming. Genetic
algorithms and their variations are based on the
mechanisms of natural selection (Goldberg
1989). Unlike conventional optimization search
approaches based on gradients, genetic algorithms work on populations of possible solutions,
attempting to find a solution set that either
maximizes or minimizes the value of a function
of those parameters and decision variables. This
function is called an objective function. Some
populations of solutions may improve the value
of the objective function, others may not. The
ones that improve its value play a greater role in
the generation of new populations of solutions
than those that do not. This process continues
until no significant improvement in model output
is apparent. Just how good or “fit” a particular
population of parameter and decision variable
values is must be evaluated using a model of the

188

5

Data-Fitting, Evolutionary, and Qualitative Modeling

Fig. 5.7 Water allocation to three users from a stream having a flow of Q

system that contains these parameters and decision variables. This system model is separated
from the GA model. This model separation
makes GA applicable for estimating the best
parameter and decision variable values of a wide
variety of simulation models used for planning,
design, operation, and management.
Each individual solution set of a GA model
contains the values of all the parameters or
variables whose best values are being sought.
These solutions are expressed as strings of values. For example, if the values of three variables
x, y, and z are to be obtained, these variables are
arranged into a string, xyz. Assuming each variable is expressed using three digits, then the
string 056004876 would represent x = 56, y = 4,
and z = 876. These strings are called chromosomes. A chromosome is an array of numbers.
The numbers of the chromosome are called
genes. Pairs of chromosomes from two parents
join together and produce offspring, who in turn
inherit some of the genes of the parents. Altered
genes may result in improved values of the
objective function. These genes will tend to
survive from generation to generation, while
those that are inferior will tend to die and not
reappear in future population sets.
Chromosomes are usually represented by strings
of binary numbers. While much of the literature on

genetic algorithms focuses on the use of binary
numbers, numbers of any base may be used.
To illustrate the main features of genetic
algorithms, consider the problem of finding the
best allocations of water to the three waterconsuming firms shown in Fig. 5.7. Assume only
integer solutions are to be considered. The
maximum allocation, xi, to any single user i cannot exceed 5, and the sum of all allocations
cannot exceed the value of Q, say 6.
0  xi  5 for

i ¼ 1; 2; and 3:

x1 þ x2 þ x3  6

ð5:7Þ
ð5:8Þ

The objective is to find the values of each
allocation that maximizes the total benefits, B(X),
while satisfying (5.7) and (5.8).
Maximize BðXÞ ¼ ð6x1  x21 Þ þ ð7x2  1:5x22 Þ
þ ð8x3  0:5x23 Þ
ð5:9Þ
A population of possible feasible solutions is generated randomly. The best size of the sample solution population—the number of solutions being
considered—is usually determined by trial and
error.

5.3 Evolutionary Algorithms

Using numbers to the base 10, a sample
individual solution (chromosome) could be 312,
representing the allocations x1 = 3, x2 = 1, and
x3 = 2. Another individual solution, picked at
random, might be 101. These two individuals or
chromosomes, each containing three genes, can
pair up and have two children.
The genes of the children are determined by
crossover and mutation operations. These pairing, crossover and mutation operations are random. Suppose a crossover is to be performed on
the pair of strings, 312 and 101. Crossover
involves splitting the two solution strings into
two parts, each string at the same place. Assume
the location of the split was randomly determined
to be after the first digit,
3j12
1j01
Crossover usually involves switching one part
of one string with the corresponding part of the
other string. After a crossover, the two new
individuals are 301 and 112.
Another crossover approach is to determine
for each corresponding pair of genes whether or
not they will be exchanged. This would be based
on some preset probability. For example, suppose the probability of a crossover was set at
0.30. Thus, an exchange of each corresponding
pair of genes in a string or chromosome has a
30% chance of being exchanged. Assume as the
result of this “uniform” crossover procedure,
only the middle gene in the pair of strings 312
and 101 is exchanged. This would result in 302
and 111. The literature on genetic algorithms
describes many crossover methods for both binary as well as base 10 numbers. The interesting
aspect of GA approaches is that they can be, and
are, modified in many ways to suit the analyst in
the search for the best solution set.
Next consider mutation. Random mutation
operations can apply to each gene in each string.
Mutation involves changing the value of the gene
being mutated. If these strings contain binary
numbers, a 1 would be changed to 0, and a 0
would be changed to 1. If numbers to the base 10

189

are used as they are here, mutation processes
have to be defined. Any reasonable mutation
scheme can be defined. For example, suppose the
mutation of a base 10 number reduces it by 1,
unless the resulting number is infeasible. Hence
in this example, a mutation could be defined such
that if the current value of the gene being
mutated (reduced) is 0, then the new number is 5.
Suppose the middle digit 1 of the second new
individual, 112, is randomly selected for mutation. Thus, its value changes from 1 to 0. The
new string is 102. Mutation could just as well
increase any number by 1 or by any other integer
value. The probability of a mutation is usually
much smaller than that of a crossover.
Suppose these paring, crossover, and mutation
operations have been carried out on numerous
parent strings representing possible feasible
solutions. The result is a new population of
individuals (children). Each child’s fitness, or
objective value, can be determined. Assuming
the objective function (or fitness function) is to
be maximized, the higher the value the better.
Adding up all the objective values associated
with each child in the population, and then
dividing each child’s objective value by this total
sum yields a fraction for each child. That fraction
is the probability of that child being selected for
the new population of possible solutions. The
higher the objective value of a child, the higher
the probability of its being selected to be a parent
in a new population.
In this example, the objective is to maximize
the total benefit derived from the allocation of
water, Eq. 5.9. Referring to Eq. 5.9, the string
301 has a total benefit of 16.5. The string 102 has
a total benefit of 19.0. Considering just these two
children, the sum of these two individual benefits
is 35.5. Thus the child (string) 301 has a probability of 16.5/35.5 = 0.47 of being selected for
the new population, and the other child (string
102) has a probability of 19/35.5 = 0.53 of being
selected. Drawing from a uniform distribution of
numbers ranging from 0 to 1, if a random number
is in the range 0–0.47, then the string 301 would
be selected. If the random number exceeds 0.47,
then the string 102 would be selected. Clearly in

190

a more realistic example the new population size
should be much greater than two, and indeed it
typically involves hundreds of strings.
This selection or reproduction mechanism
tends to transfer to the next generation the better
(more fit) individuals of the current generation.
The higher the “fitness” (i.e., the objective value)
of an individual—in other words, the larger the
relative contribution to the sum of objective
function values of the entire population of individual solutions—the greater will be the chances
of that individual string of solution values being
selected for the next generation.
Genetic algorithms involve numerous iterations of the operations just described. Each iteration (or generation) produces populations that
tend to contain better solutions. The best solution
of all populations of solutions should be saved.
The genetic algorithm process can end when
there is no significant change in the values of the
best solution that has been found. In this search
process, there is no guarantee this best solution
will be the best that could be found, that is, a
global optimum.
This general genetic algorithm process just
described is illustrated in the flow chart in
Fig. 5.8.

5.3.2 Example Iterations
A few iterations with a small population of ten
individual solutions for this example water allocation problem can illustrate the basic processes
of genetic algorithms. In practice, the population
typically includes hundreds of individuals and
the process involves hundreds of iterations. It
would also likely include some procedures the
modeler/programmer may think would help
identify the best solution. Here we will keep the
process relatively simple.
The genetic algorithm process begins with the
random generation of an initial population of
feasible solutions, proceeds with the paring of
these solution strings, performs random crossover and mutation operations, computes the
probability that each resulting child will be
selected for the next population, and then

5

Data-Fitting, Evolutionary, and Qualitative Modeling

randomly generates the new population. This
process repeats itself with the new population
and continues until there is no significant
improvement in the best solution found from all
past iterations.
For this example, we will
1. Randomly generate an initial population of
strings of allocation variable values, ensuring
that each allocation value (gene) is no less
than 0 and no greater than 5. In addition, any
set of allocations A1, A2, and A3 that sum to
more than 6 will be considered infeasible and
discarded.
2. Pair individuals and determine if a crossover
is to be performed on each pair, assuming the
probability of a crossover is 50%. If a crossover is to occur, we will determine where in
the string of numbers it will take place,
assuming an equal probability of a crossover
between any two numbers.
3. Determine if any number in the resulting
individual strings is to be mutated, assuming
the probability of mutation of any particular
number (gene) in any string (chromosome) of
numbers as 0.10. For this example, a mutation
reduces the value of the number by 1, or if the
original number is 0, mutation changes it to 5.
After mutation, all strings of allocation values
(the genes in the chromosome) that sum to
more than 6 are discarded.
4. Using Eq. 5.9, evaluate the “fitness” (total
benefits) associated with the allocations represented by each individual string in the
population. Record the best individual string
of allocation values from this and previous
populations.
5. Return to Step 1 above if the change in the
best solution and its objective function value
is significant; Otherwise terminate the
process.
These steps are performed in Table 5.3 for
three iterations using a population of 10.
The best solution found so far is 222: that is,
x1 = 2, x2 = 2, x3 = 2. This process can and
should continue. Once the process has converged
on the best solution it can find, it may be prudent

5.3 Evolutionary Algorithms

191

Fig. 5.8 Flow chart of genetic algorithm procedure

to repeat the process, but this time, change the
probabilities of crossover or mutation or let
mutation be an increase in the value of a number
rather than a decrease. It is easy to modify the
procedures used by genetic algorithms in an
attempt to derive the best solution in an efficient
manner.
Note that the above description of how genetic algorithms work permits the use of any
“fitness function” for comparing alternative

solutions, and for selecting preferred ones. The
search procedure is independent of the particular
characteristics of the water resource system being
analyzed. This fitness “function” can be a complex groundwater quality model, for example, the
parameter values of which are being suggested
by the outcome of the GA procedure. Thus in
such an application, both simulation and optimization procedures are combined and there are
no restrictions on the features of either. As might

192

5

Data-Fitting, Evolutionary, and Qualitative Modeling

Table 5.3 Several iterations for solving the allocation problem using genetic algorithms

230

23|0

220

210

13.5

0.09

0.09

230

220

22|0

230

230

15.5

0.10

0.19

201

021

021

021

021

15.5

0.10

0.29

211

201

201

201

201

15.5

0.10

0.39

132

301

3|01

321

321

221

2|21

201

101

12.5

0.08

(became infeasible)
0.47

301
132

301

30|1

301

301

16.5

0.11

0.58

301

211

21|1

211

211

21.0

0.14

0.72

021

132

132

132

132*

26.5

0.19

0.91

230

310

310

310

210

13.5

0.09

1.00

132

150.0
230

230

230

220

16.0

0.08

0.08

221

201

201

201

201

15.5

0.08

0.16

132

211

21|1

212

212*

27.5

0.14

0.30

230

0.40

212

132

13|2

131

121

20.5

0.10

301

301

301

301

16.5

0.08

0.48

201

132

132

132

132

26.5

0.14

0.62

132

301

3|01

001

001

7.5

0.04

0.66

301

021

0|21

321

221

23.5

0.12

0.78

221

230

230

230

230

15.5

0.08

0.86

212

132

132

132

132

26.5

0.14

1.00

001

E020820e

195.5

221

22|1

222

222*

30.0

0.15

0.15

221

132

13|2

131

121

20.5

0.10

0.25

222

230

230

230

230

15.5

0.07

0.32

230

212

212

212

112

24.5

0.12

0.44

202

201

20|1

202

202

22.0

0.09

0.53

131

132

13|2

131

131

20.0

0.11

0.64

222

301

301

301

201

15.5

0.08

0.72

012

221

221

221

221

23.5

0.11

0.83

121

0.91

202

1.00

121

212

2|12

201

201

15.5

0.08

001

0|01

012

012

19.5

0.09

206.5

5.3 Evolutionary Algorithms

be expected, this has opened up a wide variety of
planning and management problems that now
can be analyzed in the search for effective
solutions.

5.3.3 Differential Evolution
Differential evolution (DE) operates through
similar computational steps as employed by a
standard evolutionary algorithm (EA) such as
genetic algorithms. DE is an optimization technique that iteratively modifies a population of
candidate solutions to make it converge to an
optimum value. However, unlike traditional EAs,
DE programs create new-generation population
members by adding a weighted difference
between two population vectors to a third vector.
To illustrate, after initializing multiple candidate
solutions with random values, begin an iterative
process where for each candidate solution x you
produce a trial vector v = a + (b − c)/2, where a,
b, c are three distinct candidate solutions picked
randomly among the population of possible
solutions. Next, you randomly swap vector
components between x and v to produce v′. At
least one component from v must be swapped.
Finally, you replace x in your population with v′
only if v′ is a better candidate (i.e., it improves
the value your objective or fitness function). This
process is repeated until no better solution can be
found. No separate probability distribution need
be used for generating the offspring.
Since its inception in 1995, many variants of
the basic algorithm have been developed with
improved performance. Books and web pages are
available that present detailed reviews of the
basic concepts of DE and of its major variants, as
well as its application to multiobjective, constrained, large-scale, and uncertain optimization
problems. Numerous computer software packages are also available for solving problems
using DE. For example, see Das and Suganthan
(2011), Storn and Price (1997), Price et al.
(2006), and Schwefel (1995) to mention a few.

193

5.3.4 Covariance Matrix Adaptation
Evolution Strategy
Covariance Matrix Adaptation Evolution Strategy
(CMA-ES) is
another stochastic,
derivative-free method for numerical solution of
nonlinear or nonconvex continuous optimization
problems. They belong to the class of evolutionary algorithms. Pairwise dependencies
between the variables are represented by a
covariance matrix. The covariance matrix adaptation (CMA) method updates the covariance
matrix in a way that improves the value of the
fitness function. Adaptation of the covariance
matrix is similar to the approximation of the
inverse Hessian matrix in calculus-based optimization. In contrast to most classical methods,
fewer assumptions on the nature of the underlying objective function are made. Only the ranking between candidate solutions is exploited for
learning the sample distribution and neither
derivatives nor even the function values themselves are required by the method (Hansen 2006;
Igel et al. 2007).
Some software programs for DE are at (http://
www1.icsi.berkeley.edu/*storn/code.html), for
CMA-ES
at
(https://www.lri.fr/*hansen/
cmaesintro.html) and for multiobjective EAs at
(http://moeaframework.org/).

5.4

Genetic Programming

One of the challenges in computer science is to
program computers to perform tasks without
telling them how. In other words, how to enable
computers to learn to program themselves for
solving particular problems? Since the 1950s,
computer scientists have tried, with varying
degrees of success, to give computers the ability
to learn. The name for this field of study is
“machine learning” (ML), a phrase used in 1959
by the first person to make a computer perform a
serious learning task, Arthur Samuel. Originally,
“machine learning” meant the ability of

194

computers to program themselves. That goal has,
for many years, proven very difficult. As a consequence, computer scientists have pursued more
modest goals. A good present-day definition of
machine learning is given by Mitchell (1997),
who identifies machine learning as the study of
computer algorithms that improve automatically
through experience.
Genetic programming (GP) aspires to do just
that: to induce a population of computer programs or models (objects that turn inputs to
outputs) that improve automatically as they
experience the data on which they are trained
(Banzhaf et al. 1998). Genetic programming is
one of the many machine-learning methods.
Within the machine-learning community, it is
common to use “genetic programming” as
shorthand for any machine-learning system that
evolves tree structures (Koza 1992).
While there is no GP today that will automatically generate a model to solve any problem,
there are some examples where GP has evolved
programs that are better than the best programs
written by people to solve a number of difficult
engineering problems. Some examples of these
human-competitive GP achievements can be seen
in Koza et al. (1999), as well as in a longer list on
the Internet (www.genetic-programming.com/
humancompetitive.html). Since Babovic (1996)
introduced the GP paradigm in the field of water
engineering, a number of researchers have used
the technique to analyze a variety of water
management problems.
The main distinctive feature of GP is that it
conducts its search for a solution to a given
problem by changing model structure rather than
by finding better values of model parameters or
variables. There is no guarantee, however, that
the resulting structure (which could be as simple
as regression Eqs. 5.1, 5.2, or 5.3) will give us
any insight into the actual workings of the
system.
The task of genetic programming is to find at
the same time both a suitable functional form of a
model and the numerical values of its parameters.
To implement GP, the user must define the basic
building blocks (mathematical operations and
variables) that may be used; the algorithm then

5

Data-Fitting, Evolutionary, and Qualitative Modeling

tries to build the model using sequences of the
specified building blocks.
One of the successful applications of GP in
automatic model building is that of symbolic
regression. Here GP searches for a mathematical
regression expression in symbolic form that best
produces the observed output given the associated input. To perform this task GP uses a
physical symbol system divided into two sets.
The first set contains the symbols for independent variables as well as parameter constants as
appropriate. The content of this set is based on
the nature of the problem to be solved. The
second set contains the basic operators used to
form a function. For example, the second set can
contain the arithmetic operators (+, −, *, /) and
perhaps others such as log, square root, sine, and
cosine as well, again based on the perceived
degree of complexity of the regression.
To produce new expressions (individuals)
GP requires that two “parent” expressions from
the previous generation be divided and recombined into two offspring expressions. An example of this is the parse tree for the expression
a + (b/c) illustrated in Fig. 5.9. The crossover
operation simply exchanges a branch of one
parent with a branch of the other.
Software programs have been written to
implement GP. For example, GPKernel developed by Babovic and Keijzer (2000) at the
Danish Hydraulic Institute (DHI) has been used
in applications such as: rainfall–runoff modeling
(Babovic and Abbott 1997; Drecourt 1999;
Liong et al. 2000), sediment transport modeling,
salt intrusion in estuaries, and roughness estimation for a flow over a vegetation bed (Babovic
and Abbott 1997). More details about GPKernel
can be seen in Aguilera (2000).
The challenge in applying genetic programming for model development is not only getting
a close fit between observed and predicted
outputs, given a set of input data, but also of
interpreting the model that is generated to
obtain additional understanding of the actual
processes taking place. There are also potential
problems in creating a dimensionally correct
model if the input data are not dimensionless.
As a consequence, many applications using

5.4 Genetic Programming

195

Fig. 5.9 An illustration of
a crossover operation and
mutation operation for
genetic programming

GP seem to require some guidance based on a
mix of both physically based and data-driven
approaches.

5.5

Qualitative Functions
and Modeling

So far the discussion in this chapter has been
focused on quantitative data that have numerical
values. The precise quantification of many system performance criteria and parameter and
decision variables is not always possible, nor is it
always necessary. When the values of variables
cannot be precisely specified, they are said to be
uncertain or fuzzy. If the values are uncertain,
probability distributions may be used to quantify
them. (The next chapter describes this approach
in some detail.) Alternatively, if they are best
described by qualitative adjectives, such as dry or
wet, hot or cold, expensive or cheap, clean or
dirty, and high or low, membership functions
indicating the fraction of stakeholders who
believe particular quantitative descriptions of

parameter or decision variable values are indeed
hot, or cold, or clean or dirty, etc., can be used to
quantify these qualitative descriptions. Both
probability distributions and membership functions of these uncertain or qualitative variables
can be included in quantitative models. This
section introduces how qualitative variables can
be included within models used for the preliminary screening of alternative water resources
plans and management policies.

5.5.1 Linguistic Functions
Large, small, pure, polluted, satisfactory, unsatisfactory, sufficient, insufficient, excellent, good,
fair, poor, and so on are words often used to
describe various attributes or performance measures of water resources systems. These
descriptors do not have “crisp,” well-defined
boundaries that separate them from their opposites. A particular mix of economic and environmental impacts may be more acceptable to
some and less acceptable to others. Plan A is

196

better than Plan B. The quality and temperature
of water is good for swimming. These qualitative, or so-called “fuzzy,” statements convey
information despite the imprecision of the italicized adjectives. The next section illustrates how
these linguistic qualitative descriptors can be
incorporated into optimization models using
membership functions.

5.5.2 Membership Functions
Assume a set A of real or integer numbers
ranging from 18 to 25. Thus A = [18, 25]. Any
number x is either in or not in the set A. The
statement “x belongs to A” is either true or false
depending on the value of x. The set A is called a
crisp set. If one is not able to say for certain
whether or not any number x is in the set, then
the set A could be referred to as fuzzy. The degree
of truth attached to that statement is defined by a
membership function. Membership functions
range from 0 (completely false) to 1 (completely
true).
Consider the statement, “The water temperature should be suitable for swimming.” Just what
temperatures are suitable will depend on the
persons asked. It would be difficult for anyone to
define precisely those temperatures that are
suitable if it is understood that temperatures
outside that range are absolutely not suitable.
A function defining the interval or range of
water temperatures suitable for swimming is
shown in Fig. 5.10. Such functions may be
defined on the basis of the responses of many
potential swimmers. There is a zone of imprecision or disagreement at both ends of the range.

Fig. 5.10 A membership
function for suitability of
water temperature for
swimming

5

Data-Fitting, Evolutionary, and Qualitative Modeling

The form or shape of a function depends on the
individual subjective feelings of the “members”
or individuals who are asked their opinions. To
define this particular function, each individual
i could be asked to define his or her comfortable
water temperature interval (T1i, T2i). The value
associated with any temperature value T equals
the number of individuals who place that
T within their range (T1i, T2i), divided by the total
number of individual opinions obtained. It is the
fraction of the total number of individuals that
consider the water temperature T suitable for
swimming. For this reason such functions are
often called membership functions (Figs. 5.10,
5.11 and 5.12).
The assignment of membership values is
based on subjective judgments, but such judgments seem to be sufficient for much of human
communication.
Now suppose the water temperature applied to
a swimming pool where the temperature could be
regulated. The hotter the temperature the more it
will cost. If we could quantify the economic
benefits associated with various temperatures we
could perform a benefit–cost analysis by maximizing the net benefits. Alternatively, we could
maximize the fraction of people who consider the
temperature good for swimming subject to a cost
constraint using a membership function such as
in Fig. 5.10 in place of an economic benefit
function (Chap. 4 discusses ways of doing this.).
Continuing with this example, assume you are
asked to provide the desired temperature at a
reasonable cost. Just what is reasonable can also
be defined by another membership function, but
this time the function applies to cost, not temperature. Both the objective and constraint of this

5.5 Qualitative Functions and Modeling

197

Fig. 5.11 Two membership functions relating to swimming water temperature. Set A is the set defining the
fraction of all individuals who think the water temperature

is too cold, and Set B defines the fraction of all individuals
who think the water temperature is too hot

Fig. 5.12 Membership
function for water
temperatures that are
considered too cold
or too hot

problem are described qualitatively. In this case
one could consider there are in fact two objectives, suitable temperature and acceptable cost.
A model that maximizes the minimum value of
both membership functions is one approach for
finding an acceptable policy for controlling the
water temperature at this swimming pool.

These allocations cannot exceed the amount
of water available, Q, less any that must remain
in the river, R. Assuming the available flow for
allocations, Q − R, as 6, the crisp optimization
problem is to maximize Eq. (5.10) subject to the
resource constraint:
x1 þ x2 þ x3  6

5.5.3 Illustrations of Qualitative
Modeling
5.5.3.1 Water Allocation
Consider the application of qualitative modeling
to the water allocation problem illustrated in
Fig. 5.7. Assume, as in the previous uses of this
example, the problem is to find the allocations of
water to each firm that maximize the total benefits TB(X):

 

Maximize TBðXÞ ¼ 6x1 x21 þ 7x2  1:5x22


þ 8x3  0:5x23
ð5:10Þ

ð5:11Þ

The optimal solution is x1 = 1, x2 = 1, and
x3 = 4 as previously obtained in Chap. 4 using
any of several different optimization methods.
The maximum total benefit, TB(X), from
Eq. (5.10), equals 34.5.
To create a qualitative equivalent of this crisp
model, the objective can be expressed as a
membership function of the set of all possible
objective values. The higher the objective value
the greater the membership function value. Since
membership functions range from 0 to 1, the
objective needs to be scaled so that it also ranges
from 0 to 1.

198

5

Data-Fitting, Evolutionary, and Qualitative Modeling

Fig. 5.13 Membership
function for “about 6 units
more or less”

The highest value of the objective occurs
when there is sufficient water to maximize each
firm’s benefits. This unconstrained solution
would result in a total benefit of 49.17 and this
happens when x1 = 3, x2 = 2.33, and x3 = 8.
Thus, the objective membership function can
be expressed by
mðXÞ ¼



 

6x1 x21 þ 7x2  1:5x22


þ 8x3  0:5x23 =49:17

ð5:12Þ

It is obvious that the two functions (Eqs. 5.10
and 5.12) are equivalent. However, the goal of
maximizing objective function 5.10 is changed to
that of maximizing the degree of reaching the
objective target. The optimization problem
becomes
maximize mðXÞ ¼


 

6x1 x21 þ 7x2  1:5x22


þ 8x3  0:5x23 =49:17
ð5:13Þ

subject to
x1 þ x2 þ x3  6

ð5:14Þ

The optimal solution of (5.13) and (5.14)
results in the same values of each allocation as do
Eqs. (5.10) and (5.11). The optimal degree of
satisfaction is m(X) = 0.70.
Next, assume the total amount of resources
available to be allocated is limited to “about 6
units more or less,” which is a qualitative

constraint. Assume the membership function
describing this constraint is defined by Eq. (5.14)
and is shown in Fig. 5.13.
mC ðXÞ ¼ 1

if

x 1 þ x2 þ x3  5

mC ðXÞ ¼ ½7  ðx1 þ x2 þ x3 Þ=2 if
mC ðXÞ ¼ 0 if x1 þ x2 þ x3  7

5  x1 þ x2 þ x3  7

ð5:15Þ
Let the membership function of (5.12) be
called mG(X). The qualitative optimization
problem becomes one of maximizing the minimum value of the two membership functions
(mG(X), mC(X)) subject to their definitions in
Eqs. (5.12) and (5.15).
This yields x1 = 0.91, x2 = 0.94, x3 = 3.81,
mG(X) = mC(X) = 0.67, and the total net benefit,
Eq. (5.10), is TB(X) = 33.1. Compare this with
the crisp solution of x1 = 1, x2 = 1, x3 = 4, and
the total net benefit of 34.5.

5.5.3.2 Qualitative Reservoir Storage
and Release Targets
Consider the problem of trying to identify a
reservoir storage volume target, TS, for recreation
facilities given a known minimum release target,
TR, and reservoir capacity K. Assume, in this
simple example, these known release and
unknown storage targets must apply in each of
the three seasons in a year. The objective will be
to find the highest value of the storage target, TS,
that minimizes the sum of squared deviations
from actual storage volumes and releases that are
less than the minimum release target.

5.5 Qualitative Functions and Modeling

199

Table 5.4 The solution to the reservoir optimization problem

Given a sequence of inflows, Qt, the optimization model is
Minimize D ¼

i
Xh
ðT s  St Þ2 þ DR2  0:001T S
t

ð5:16Þ
subject to
St þ Qt  Rt ¼ St þ 1 t ¼ 1; 2; 3;
if t ¼ 3; t þ 1 ¼ 1
St  K
Rt  T R  DRt

t ¼ 1; 2; 3
t ¼ 1; 2; 3

ð5:17Þ

ð5:18Þ
ð5:19Þ

Assume K = 20, TR = 25 and the inflows Qt are
5, 50, and 20 for periods t = 1, 2, and 3. The
optimal solution, yielding an objective value of
184.4, is listed in Table 5.4.
Now consider changing the objective function
into maximizing the weighted degrees of

“satisfying” the reservoir storage volume and
release targets.
Maximize

X

ðwS mSt þ wR mRt Þ

ð5:20Þ

t

where wS and wR are weights indicating the relative importance of storage volume targets and
release targets, respectively. The variables mSt
are the degrees of satisfying the storage volume
target in the three periods t, expressed by
Eq. (5.21). The variables mRt are the degrees of
satisfying the release target in periods t, expressed by Eq. (5.22).
mSt ¼ St =TS for St  TS and
ðK  St Þ=ðK  TSÞ for TS  St
mRt ¼ Rt =TR for Rt  TR
1 for Rt [ TR

and

ð5:21Þ

ð5:22Þ

Equations (5.21) and (5.22) are shown in
Figs. 5.14 and 5.15, respectively.

200

5

Data-Fitting, Evolutionary, and Qualitative Modeling

Fig. 5.14 Membership
function for storage
volumes

Fig. 5.15 Membership
function for releases

Box 5.1. Reservoir model written for solution using LINGO.

5.5 Qualitative Functions and Modeling

201

Table 5.5 Solution of qualitative model for reservoir storage volumes and releases based on objective (5.20)

This optimization model written for solution
using LINGO® is as shown in Box 5.1.
Given weights ws = 0.4 and wR = 0.6, the
optimal solution obtained from solving the model
shown in Box 5.1 using LINGO® is listed in
Table 5.5.
If the objective Eq. 5.20 is changed to one of
maximizing the minimum membership function
value, the objective becomes

Maximize
mmin ¼ maximize minimumðmSt ; mRt Þ
ð5:23Þ
To include the objective Eq. 5.23 in the optimization model a common lower bound, mmin, is
set on each membership function, mSt and mRt, and
this variable is maximized. The optimal solution
changes somewhat and is as shown in Table 5.6.

202

5

Data-Fitting, Evolutionary, and Qualitative Modeling

Table 5.6 Optimal solution of reservoir operation model based on objective (5.23)

This solution differs from that shown in
Table 5.5 primarily in the values of the membership functions. The target storage volume
operating variable value, Ts, stays the same value
in this example.

5.5.3.3 Qualitative Water Quality
Management Objectives
and Constraints
Consider the stream pollution problem illustrated
in Fig. 5.12. The stream receives waste, Wi from
sources located at sites i = 1 and 2. Without
some waste treatment at these sites, the pollutant
concentrations at sites 2 and 3 will exceed the
maximum desired concentration. The problem is
to find the fraction of wastewater removal, xi, at
sites i = 1 and 2 required to meet the quality

standards at sites 2 and 3 at a minimum total cost.
The data used for the problem shown in Fig. 5.16
are defined and listed in Table 5.7.
Using the notation defined in Table 5.7, the
crisp model for this problem, as discussed in the
previous chapter, is
Minimize

C1 ðX1 Þ þ C2 ðX2 Þ

ð5:24Þ

subject to
Water quality constraint at site 2:
½P1 Q1 þ W1 ð1  X1 Þa12 =Q2  Pmax
2
½ð32Þð10Þ þ 250;000ð1  X1 Þ=86:4 0:25=12
 20 which; when simplified; is : X1  0:78
ð5:25Þ

5.5 Qualitative Functions and Modeling

203

Table 5.7 Parameter values selected for the water quality management problem illustrated in Fig. 5.12

Water quality constraint at site 3:
f½P1 Q1 þ W1 ð1  X1 Þa13
þ ½W2 ð1  X2 Þa23 Þ=Q3  Pmax
3
f½ð32Þð10Þ þ 250;000ð1  X1 Þ=86:4 0:15
þ ½80;000ð1  X2 Þ=86:40:60g=13  20
ð5:26Þ
which, when simplified, is: X1 + 1.28X2 ≥ 1.79
Restrictions on fractions of waste removal:
0  Xi  1:0

for sites i ¼ 1 and 2

ð5:27Þ

For a wide range of reasonable costs, the
optimal solution found using linear programming
was 0.78 and 0.79, or essentially 80% removal
efficiencies at sites 1 and 2. Compare this
solution with that of the following qualitative
model.

To consider a more qualitative version of this
problem, suppose the maximum allowable pollutant concentrations in the stream at sites 2 and
3 were expressed as “about 20 mg/l more or
less.” Obtaining opinions of individuals of what
they consider to be “20 mg/l more or less,” a
membership function can be defined. Assume it
is as shown in Fig. 5.17.
Next, assume that the government environmental agency expects each polluter to install
best available technology (BAT) or to carry out
best management practices (BMP) regardless of
whether or not this is required to meet stream
quality standards. Asking experts just what BAT
or BMP means with respect to treatment efficiencies could result in a variety of answers.
These responses can be used to define membership functions for each of the two firms in this
example. Assume these membership functions
for both firms are as shown in Fig. 5.18.

204

5

Data-Fitting, Evolutionary, and Qualitative Modeling

Fig. 5.16 A stream pollution problem of finding the waste removal efficiencies (x1, x2) that meet the stream quality
standards at least cost

Fig. 5.17 Membership
function for “about 20 mg/l
more or less”

Finally, assume there is a third concern that
has to do with equity. It is expected that no
polluter should be required to treat at a much
higher efficiency than any other polluter.
A membership function defining just what differences are acceptable or equitable could quantify this concern. Assume such a membership
function is as shown in Fig. 5.19.
Considering each of these membership functions as objectives, a number of fuzzy

optimization models can be defined. One is to find
the treatment efficiencies that maximize the
minimum value of each of these membership
functions.
Maximize

m ¼ max minfmP ; mT ; mE g
ð5:28Þ

If we assume that the pollutant concentrations
at sites j = 2 and 3 will not exceed 23 mg/l, the

5.5 Qualitative Functions and Modeling

205

Fig. 5.18 Membership
function for best available
treatment technology

Fig. 5.19 Equity
membership function in
terms of the absolute
difference between the two
treatment efficiencies
expressed as a percent

pollutant concentration membership functions
mPj are
mPj ¼ 1  P2j =5

ð5:29Þ

The pollutant concentration at each site j is the
sum of two components:
Pj ¼ P1j þ P2j

and 2, the treatment technology membership
functions mTi are
mTi ¼ ðx2i =0:05Þ  ðx4i =0:10Þ

and the treatment efficiencies, expressed as
fractions, are

ð5:30Þ

where

ð5:33Þ

Xi ¼ 0:70 þ x2i þ x3i þ x4i

ð5:34Þ

where
P1j  18

ð5:31Þ

x2i  0:05

ð5:35Þ

P2j  ð23  18Þ

ð5:32Þ

x3i  0:05

ð5:36Þ

If we assume the treatment plant efficiencies
will be between 70 and 90% at both sites i = 1

x4i  0:10

ð5:37Þ

206

5

Finally, assuming the difference between
treatment efficiencies will be no greater than 14,
the equity membership function, mE, is
mE ¼ Z  ð0:5=0:05ÞD1 þ 0:5ð1  Z Þ
 ð0:5=ð0:14  0:05ÞÞD2
ð5:38Þ
where
D1  0:05Z

ð5:39Þ

Data-Fitting, Evolutionary, and Qualitative Modeling

½P1 Q1 þ W1 ð1  X1 Þa12 =Q2 ¼ P2
½ð32Þð10Þ þ 250;000ð1  X1 Þ=86:40:25=12 ¼ P2

ð5:44Þ
Water quality constraint at site 3:
f½P1 Q1 þ W1 ð1  X1 Þa13 þ ½W2 ð1  X2 Þ a23 g=Q3 ¼ P3
f½ð32Þð10Þ þ 250;000ð1  X1 Þ=86:4 0:15
þ ½80;000ð1  X2 Þ=86:40:60g=13 ¼ P3

ð5:45Þ
D2  ð0:14  0:05Þð1  Z Þ

ð5:40Þ

X1  X2 ¼ DP  DM

ð5:41Þ

DP þ DM ¼ D1 þ 0:05ð1  Z Þ þ D2

ð5:42Þ

Z is a binary 0; 1 variable:

ð5:43Þ

The remainder of the water quality model
remains the same: Water quality constraint at site 2:

Restrictions on fractions of waste removal:
0  Xi  1:0

for sites i ¼ 1 and 2: ð5:46Þ

Solving this model using LINGO® yields the
results shown in Table 5.8.
This solution confirms the assumptions made
when constructing the representations of the
membership functions in the model. It is also
very similar to the least-cost solution found
from solving the crisp linear programming (LP)
model containing no membership functions.

Table 5.8 Solution to water quality management model Eqs. 5.28 to 5.46

5.6 Conclusions

5.6

Conclusions

Most computer-based models used for water
resources planning and management are physical, mechanistic, or process-based quantitative
models. Builders of such models attempt to
incorporate the important physical, biological,
chemical, geomorphological, hydrological, and
other types of interactions among all system
components, as appropriate for the problem
being addressed and the system being modeled.
This is done in order to be able to predict possible economic, ecologic, environmental, or
social impacts that might result from the implementation of some plan or policy. These types of
models almost always contain parameters. These
need values, and the values of the parameters
affect the accuracy of the impact predictions.
This chapter has outlined some data-fitting
methods of modeling that do not attempt to
model natural, economic, or social processes.
These have included ANN and two evolutionary
search approaches: genetic algorithms (GA) for
estimating the parameter and decision variable
values, and genetic programming for finding
models that replicate the real system. In some
situations, these biologically motivated search
methods, which are independent of the particular
system model, provide the most practical way to
calibrate model parameters.
Fortunately for potential users of GA, GP, and
ANN methods, software programs implementing
many of these methods are available on the
Internet. Applications of such methods to
groundwater modeling, sedimentation processes
along coasts and in harbors, rainfall runoff prediction, reservoir operation, data classification,
and predicting surge water levels for navigation
represent only a small sample of what can be
found in the current literature.
Not all data are quantitative. In many cases
objectives and constraints are expressed as qualitative expressions. Optimization models incorporating such expressions or functions are
sometimes appropriate when only qualitative
statements apply to a particular water management
problem or issue. This chapter concludes by

207

showing how this can be done using some simple
example problems associated with water allocations, reservoir operation, and pollution control.
The information presented in this chapter
serves only as an introduction. Those interested
in more detailed and complete explanations and
applications may refer to any of the additional
references listed in the next section.

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Exercises
5:1 An upstream reservoir serves as a recreation
site for swimmers, wind surfers, and boaters.
It also serves as a flood storage reservoir in the
second of four seasons or time periods in a
year. The reservoir’s releases can be diverted
to an irrigation area. A wetland area further
downstream receives the unallocated portion
of the reservoir release plus the return flow
from the irrigation area. The irrigation return
flow contains a salinity concentration that can
damage the ecosystem.
(a) Assume there exist recreation lake level
targets, irrigation allocation targets, and
wetland flow and salinity targets. The
challenge is to determine the reservoir
releases and irrigation allocations so as
to “best” meet these targets. This is the
crisp’ problem.

210

5

Data:
Reservoir storage capacity: 30 mcm;
During period 2 the flood storage capacity is 5 mcm;
Irrigation return flow fraction: 0.3 (i.e.,
30% of that diverted for irrigation);
Salinity concentration of reservoir
water: 1 ppt;
Salinity concentration of irrigation
return flow: 20 ppt;
Reservoir average inflows for four seasons, respectively: 5, 50, 20, 10 mcm;
Targets for part (a):
Target maximum salinity concentration
in wetland: 3 ppt;
Target storage target for all four seasons: 20 mcm;
Minimum flow target in wetland in each
season, respectively: 10, 20, 15,
15 mcm;
Maximum flow target in wetland in
each season, respectively: 20, 30, 25,
25 mcm;
Target irrigation allocations: 0, 20, 15,
5 mcm;
(b) Next create fuzzy membership functions
to replace the targets and solve the problem. Assume that the targets used in
(a) above are expressed in qualitative
terms as membership functions. The
membership functions indicate the degree
of satisfaction for these targets. Solve for
the “best” reservoir release and allocation
policy that maximizes the minimum
membership function value. Each membership function defines the relative level
of satisfaction, where a value of 1 indicates complete stakeholder satisfaction.
This is the qualitative problem.
5:2 Develop a flow chart showing how you
would apply genetic algorithms to finding the
parameters, aij, of a water quality prediction
model, such as the one we have used to find
the concentration downstream of an upstream

Data-Fitting, Evolutionary, and Qualitative Modeling

discharge site i. This will be based on
observed values of mass inputs, Wi, and
concentrations, Cj, and flows, Qj, at a
downstream site j.
Cj ¼

X

Wi aij =Qj

i

The objective to be used for fitness is to
minimize the sum of the differences
between the observed Cj and the computed
Cj. To convert this to a maximization
objective you could use something like the
following:
Max1=ð1 þ DÞ
where


D  Cj obsCj calculated


D  Cj calculatedCj obs:
5:3 Use a genetic algorithm program to predict
the parameter values asked for in problem
5.2, and then an artificial neural network
ANN to obtain a predictor of downstream
water quality based on the values of these
parameters. You may use the model and
data presented in Sect. 5.2 of Chap. 4 if you
wish.
5:4. Using a genetic algorithm program to
findthe allocations Xi that maximize the
total benefits to the three water users i along
a stream, whose individual benefits are
Use 1: 6X1  X12
Use 2: 7X2  X22
Use 3: 8X3  X32
Assume the available stream flow is some
known value (ranging from 0 to 20).
Determine the effect of different genetic
algorithm parameter values on the ability to
find the best solution.
5:5 Consider the wastewater treatment problem
illustrated in the drawing below.

Exercises

The initial stream concentration just
upstream of site 1 is 32. The maximum
concentration of the pollutant just upstream
of site 2 is 20 mg/l (g/m3), and at site 3 it is
25 mg/l. Assume the marginal cost per
fraction (or percentage) of the waste load
removed at site 1 is no less than that cost at
site 2, regardless of the amount removed.
Using a suitable genetic algorithm program,
solve for the least-cost wastewater treatment
at sites 1 and 2 that will satisfy the quality
constraints at sites 2 and 3, respectively.
Discuss the sensitivity of the GA parameter
values in finding the best solution. You can
get the exact solution using LINGO as
discussed in Sect. 4.5.3.
5:6 Develop an artificial neural network for flow
routing given the following two sets of
upstream and downstream flows. Use one set
of 5-periods for training (finding the unknown
weights and other variables) and the other set
for validation of the calculated parameter
values (weights and bias constants).
Develop the simplest artificial neural network you can that does an adequate job of
prediction.

Open Access This chapter is distributed under the terms
of the Creative Commons Attribution-NonCommercial
4.0 International License (http://creativecommons.org/
licenses/by-nc/4.0/), which permits any noncommercial
use, duplication, adaptation, distribution and reproduction
in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a
link to the Creative Commons license and indicate if
changes were made.

211

Time period

Upstream flow

Downstream flow

1

450

366

2

685

593

3

830

755

4

580

636

5

200

325

1

550

439

2

255

304

3

830

678

4

680

679

5

470

534

[These outflows come from the following
model, assuming an initial storage in period
1 of 50, the detention storage that will
remain in the reach even if the inflows go to
0. For each period t:
OutflowðtÞ ¼ 1:5ð50 þ initial storageðtÞ þ inflowðtÞÞ0:9 ;

where the outflow is the downstream flow
and inflow is the upstream flow.]

The images or other third party material in this
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license, unless indicated otherwise in the credit line; if
such material is not included in the work’s Creative
Commons license and the respective action is not permitted by statutory regulation, users will need to obtain
permission from the license holder to duplicate, adapt or
reproduce the material.

6

An Introduction to Probability,
Statistics, and Uncertainty

Processes that are not fully understood, and
whose outcomes cannot be precisely predicted,
are often called uncertain. Most of the inputs to,
and processes that occur in, and outputs resulting
from water resource systems are not known with
certainty. Hence so too are the future impacts of
such systems, and even people’s reactions and
responses to these impacts. Ignoring this uncertainty when performing analyses in support of
decisions involving the development and management of water resource systems could lead to
incorrect conclusions, or at least more surprises,
than will a more thorough analysis taking into
account these uncertainties. This chapter introduces some commonly used approaches for
dealing with model input and output uncertainty.
Further chapters incorporate these tools in more
detailed optimization, simulation, and statistical
models designed to identify and evaluate alternative plans and policies for water resource
system development and operation.

6.1

Introduction

Uncertainty is always present when planning and
operating water resource systems. It arises
because many factors that affect the performance
of water resource systems are not and cannot be
known with certainty when a system is planned,
designed, built, and operated. The success and
performance of each component of a system often
depends on future meteorological, demographic,

social, technical, and political conditions, all of
which may influence future benefits, costs, environmental impacts, and social acceptability.
Uncertainty also arises due to the stochastic
(random over time) nature of meteorological and
hydrological processes such as rainfall and
evaporation. Similarly, future populations of
towns and cities, per capita water usage rates,
irrigation patterns, and priorities for water uses,
all of which impact water demand, are not known
with certainty. This chapter introduces methods
for describing and dealing with uncertainty, and
provides some simple examples of their use in
water resources planning. These methods are
extended in the following two chapters.
There are many ways to deal with uncertainty.
The simplest approach is to replace each uncertain quantity either by its expected or average
value or by some critical (e.g., “worst-case”)
value and then proceed with a deterministic
approach. Use of expected values or alternatively
median values of uncertain quantities can be
adequate if the uncertainty or variation in a
quantity is reasonably small and does not critically affect the performance of the system. If
expected values of uncertain parameters or variables are used in a deterministic model, the
planner can then assess the importance of uncertainty with sensitivity and uncertainty analyses,
discussed later in this and subsequent chapters.
Replacement of uncertain quantities by either
expected or worst-case values can adversely
affect the evaluation of project performance

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_6

213

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6 An Introduction to Probability, Statistics, and Uncertainty

Table 6.1 Data for determining reservoir recreation potential

when important parameters are highly variable.
To illustrate these issues, consider the evaluation
of the recreation potential of a reservoir.
Table 6.1 shows that the elevation of the water
surface varies from year to year depending on the
inflow and demand for water. The table indicates
the pool levels and their associated probabilities
as well as the expected use of the recreation
facility with different pool levels.
The average pool level L is simply the sum of
each possible pool level times its probability, or
L ¼ 10ð0:10Þ þ 20ð0:25Þ þ 30ð0:30Þ
þ 40ð0:25Þ þ 50ð0:10Þ ¼ 30
This pool level
visitor-days per day

corresponds

VDðLÞ ¼ 100 visitor-days per day

ð6:1Þ
to

100
ð6:2Þ

A worst-case analysis might select a pool
level of 10 as a critical value, yielding an estimate of system performance equal to 25 visitordays per day
VDðLlow Þ ¼ VDð10Þ ¼ 25 visitor-days per day ð6:3Þ

Neither of these values is a good approximation of the average visitation rate, which is

VD ¼ 0:10VDð10Þ þ 0:25VDð20Þ þ 0:30VDð30Þ
þ 0:25VDð40Þ þ 0:10VDð50Þ
¼ 0:10ð25Þ þ 0:25ð75Þ þ 0:30ð100Þ
þ 0:25ð80Þ þ 0:10ð70Þ
¼ 78:25 visitor-days per day

ð6:4Þ
Clearly, the average visitation rate,
V D ¼ 78:25, the visitation rate corresponding to
the average pool level VD(L) = 100, and the
worst-case assessment VD(Llow) = 25, are very
different.
The median and the most likely are other
measures that characterize a data set. They have
the advantage that they are less influenced by
extreme outliers. For the symmetric data set
shown in Table 6.1, the median, most likely, and
the mean are the same, namely 30. But if instead
the probabilities of the respective pool levels were
0.30, 0.25, 0.20, 0.15, and 0.10, (instead of 0.10,
0.25, 0.30, 0.25, 0.10) the expected value or mean
is 25, the value having the highest probability of
occurring (the most likely) is 10, and the median
or value that is greater or equal to half of the other
values and less than or equal to the other half of
the values in the data set is 20.

6.1 Introduction

Thus using only average values in a complex
model can produce a poor representation of both
the average performance and the possible performance range. When important quantities are
uncertain, one should evaluate both the expected
performance of a project and the risk and possible magnitude of project failures and their
consequences.
This chapter reviews many of the methods of
probability and statistics that are useful in
water resources planning and management.
Section 6.2 reviews the important concepts and
methods of probability and statistics that are
needed in this and other chapters of this book.
Section 6.3 introduces several probability distributions that are often used to model or
describe uncertain quantities. The section also
discusses methods for fitting these distributions
using historical information, and methods of
assessing whether the distributions are adequate
representations of the data. Sections 6.4, 6.5,
and 6.6 expand upon the use of these distributions, and discuss alternative parameter estimation methods.
Section 6.7 presents the basic ideas and concepts of stochastic processes or time series.
These are used to model streamflows, rainfall,
temperature, or other phenomena whose values
change with time. The section contains a
description of Markov chains, a special type of
stochastic process used in many stochastic optimization and simulation models. Section 6.8
illustrates how synthetic flows and other time
series inputs can be generated for stochastic
simulations. The latter is introduced with an
example in Sect. 6.9.
Many topics receive only brief treatment here.
Readers desiring additional information should
consult applied statistical texts such as Benjamin
and Cornell (1970), Haan (1977), Kite (1988),
Stedinger et al. (1993), Kottegoda and Rosso
(1997), Ayyub and McCuen (2002), and
Pishro-Nik (2014).

215

6.2

Probability Concepts
and Methods

This section introduces basic concepts of probability and statistics. These are used throughout
this chapter and in later chapters in the book.

6.2.1 Random Variables
and Distributions
A basic concept in probability theory is that of
the random variable. A random variable is a
function whose value cannot be predicted with
certainty. Examples of random variables are
(1) the number of years until the flood stage of a
river washes away a small bridge, (2) the number
of times during a reservoir’s life that the level of
the pool will drop below a specified level, (3) the
rainfall depth next month, and (4) next year’s
maximum flow at a gage site on an unregulated
stream. The values of all of these quantities
depend on events that are not knowable before
the event has occurred. Probability can be used to
describe the likelihood these random variables
will equal specific values or be within a given
range of specific values.
The first two examples illustrate discrete random variables, random variables that take on
values in a discrete set (such as the positive
integers). The second two examples illustrate
continuous random variables. Continuous random variables take on values in a continuous set.
A property of all continuous random variables is
that the probability that they equal any specific
number is zero. For example, the probability that
the total rainfall depth in a month will be exactly
5.0 cm is zero, while the probability that the total
rainfall will lie between 4 and 6 cm can be nonzero. Some random variables are combinations of
continuous and discrete random variables.
Let X denote a random variable and x a possible value of that random variable X. Random

216

6 An Introduction to Probability, Statistics, and Uncertainty

variables are generally denoted by capital letters
and particular values they take on by lowercase
letters. For any real-valued random variable X, its
cumulative distribution function FX(x), often
denoted as just the cdf, equals probability that the
value of X is less than or equal to a specific value
or threshold x

the probability mass function of a discrete random variable. The probability density function,
often called the pdf, is the derivative of the
cumulative distribution function so that

FX ðxÞ ¼ Pr½X  x

The area under a probability density function
always equals 1.

ð6:5Þ

fX ðxÞ ¼

This cumulative distribution function FX(x) is
a non-decreasing function of x because
Pr½X  x  Pr½X  x þ d for d [ 0

ð6:6Þ

In addition,
lim FX ðxÞ ¼ 1

ð6:7Þ

lim FX ðxÞ ¼ 0

ð6:8Þ

x! þ 1

and
x!1

The first limit equals 1 because the probability
that X takes on some value less than infinity must
be unity; the second limit is zero because the
probability that X takes on no value must be zero.
If X is a real-valued discrete random variable that takes on specific values x1, x2, …,
the probability mass function pX(xi) is the probability X takes on the value xi. Thus one would
write
pX ðxi Þ ¼ Pr½X ¼ xi 

ð6:9Þ

The value of the cumulative distribution
function FX(x) for a discrete random variable is
the sum of the probabilities of all xi that are less
than or equal to x.
FX ðxÞ ¼

X

pX ðxi Þ

ð6:10Þ

xi  x

Figure 6.1 illustrates the probability mass
function pX(xi) and the cumulative distribution
function of a discrete random variable.
The probability density function fX(x) for a
continuous random variable X is the analogue of

dFX ðxÞ
0
dx

ð6:11Þ

Zþ 1
fX ðxÞ ¼ 1

ð6:12Þ

1

If a and b are any two constants, the cumulative distribution function or the density function may be used to determine the probability
that X is greater than a and less than or equal to
b where
Zb
Pr½a\X  b ¼ FX ðbÞ  FX ðaÞ ¼

fX ðxÞdx
a

ð6:13Þ
The probability density function specifies the
relative frequency with which the value of a
continuous random variable falls in different
intervals.
Life is seldom so simple that only a single
quantity is uncertain. Thus, the joint probability
distribution of two or more random variables
can also be defined. If X and Y are two continuous real-valued random variables, their joint
cumulative distribution function is
FXY ðx; yÞ ¼ Pr½X  x and Y  y
Zx Zy
¼
fXY ðu; vÞdudv

ð6:14Þ

1 1

If two random variables are discrete, then
FXY ðx; yÞ ¼

XX

pXY ðxi ; yi Þ

ð6:15Þ

x i  x yi  y

where the joint probability mass function is

6.2 Probability Concepts and Methods

pXY ðxi ; yi Þ ¼ Pr½X ¼ xi ; and Y ¼ yi 

217

ð6:16Þ

If X and Y are two random variables, and the
distribution of X is not influenced by the value
taken by Y, and vice versa, the two random variables are said to be independent. Independence is
an important and useful idea when attempting to
develop a model of two or more random variables.
For independent random variables
Pr½a  X  b and c  Y  d
¼ Pr½a  X  b Pr½c  Y  d

ð6:17Þ

for any a, b, c, and d. As a result,
FXY ðx; yÞ ¼ FX ðxÞFY ðyÞ

(a)

ð6:18Þ

which implies for continuous random variables
that
ð6:19Þ
fXY ðx; yÞ ¼ fX ðxÞfY ðyÞ
and for discrete random variables that
pXY ðx; yÞ ¼ pX ðxÞpY ðyÞ

ð6:20Þ

Other useful concepts are those of the marginal and conditional distributions. If X and Y are
two random variables whose joint cumulative
distribution function FXY(x, y) has been specified,
then FX(x), the marginal cumulative distribution
of X, is just the cumulative distribution of X ignoring Y. The marginal cumulative distribution
function of X equals

(b)

Fig. 6.1 Cumulative distribution and probability density or mass functions of random variables: a continuous
distributions; b discrete distributions

218

6 An Introduction to Probability, Statistics, and Uncertainty

FX ðxÞ ¼ Pr½X  x ¼ lim FXY ðx; yÞ
y!1

ð6:21Þ

where the limit is equivalent to letting Y take on
any value. If X and Y are continuous random
variables, the marginal density of X can be
computed from
Zþ 1
fX ðxÞ ¼

fXY ðx; yÞdy

mass function of a discrete random variable,
allows one to calculate the expected value of any
function of the random variable. Such an expectation may represent the average rainfall depth,
average temperature, average demand shortfall, or
expected economic benefits from system operation. If g is a real-valued function of a continuous
random variable X, the expected value of g(X) is

ð6:22Þ

Zþ 1
E½gðXÞ ¼

1

The conditional cumulative distribution function is the cumulative distribution function for
X given that Y has taken a particular value
y. Thus the value of Y may have been observed
and one is interested in the resulting conditional
distribution, for the so far unobserved value of
X. The conditional cumulative distribution function for continuous random variables is given by
Rx
FXjY ðxjyÞ ¼ Pr½X  xjY ¼ y ¼

1 fxy ðs; yÞds

fy ðyÞ

gðxÞfX ðxÞdx

ð6:26Þ

1

whereas for a discrete random variable
E½gðXÞ ¼

X
xi

gðxi ÞpX ðxi Þ

ð6:27Þ

E[ ] is called the expectation operator. It has
several important properties. In particular, the
expectation of a linear function of X is a linear
function of the expectation of X. Thus if a and
b are two nonrandom constants,

ð6:23Þ
E½a þ bX ¼ a þ bE½X

ð6:28Þ

It follows that the conditional density function
is
fXjY ðxjyÞ ¼

fXY ðx; yÞ
fY ðyÞ

The expectation of a function of two random
variables is given by
ð6:24Þ

Zþ 1 Zþ 1
E½gðX; YÞ ¼

For discrete random variables, the probability
of observing X = x, given that Y = y equals
pxy ðx; yÞ
pXjY ðxjyÞ ¼
py ðyÞ

gðx; yÞfXY ðx; yÞdx dy
1

1

or
ð6:25Þ

E½gðX; YÞ ¼

XX
i

These results can be extended to more than
two random variables. See Kottegoda and Rosso
(1997) for a more advanced discussion.

6.2.2 Expected Values
Knowledge of the probability density function of a
continuous random variable, or of the probability

gðxi ; yi ÞpXY ðxi ; yi Þ ð6:29Þ

j

If X and Y are independent, the expectation of
the product of a function h() of X and a function
g() of Y is the product of the expectations
E½gðXÞhðYÞ ¼ E½gðXÞE½hðYÞ

ð6:30Þ

This follows from substitution of Eqs. 6.19
and 6.20 into Eq. 6.29.

6.2 Probability Concepts and Methods

219

6.2.3 Quantiles, Moments, and Their
Estimators
While the cumulative distribution function provides a complete specification of the properties of
a random variable, it is useful to use simpler and
more easily understood measures of the central
tendency and range of values that a random variable may assume. Perhaps the simplest approach
to describing the distribution of a random variable
is to report the value of several quantiles. The pth
quantile of a random variable X is the smallest
value xp such that X has a probability p of
assuming a value equal to or less than xp
Pr½X\xp   p  Pr½X  xp 

ð6:31Þ

Equation 6.31 is written to insist if at some
point xp, the cumulative probability function
jumps from less than p to more than p, then that
value xp will be defined as the pth quantile even
though FX(xp) ≠ p. If X is a continuous random
variable, then in the region where fX(x) > 0, the
quantiles are uniquely defined and are obtained
by solution of
 
FX xp ¼ p

ð6:32Þ

Frequently reported quantiles are the median
x0.50 and the lower and upper quartiles x0.25 and
x0.75. The median describes the location or central tendency of the distribution of X because the
random variable is, in the continuous case,
equally likely to be above as below that value.
The interquartile range [x0.25, x0.75] provides an
easily understood description of the range of
values that the random variable might assume.
The pth quantile is also the 100 p percentile.
In a given application, particularly when
safety is of concern, it may be appropriate to use
other quantiles. In floodplain management and
the design of flood control structures, the
100-year flood x0.99 is often the selected design
value. In water quality management, a river’s
minimum seven-day-average low flow expected
once in 10 years is often used as the critical
planning value: Here the one-in-ten year value is

the 10 percentile of the distribution of the annual
minima of the seven-day average flows.
The natural sample estimate of the median
x0.50 is the median of the sample. In a sample of
size n where x(1) ≤ x(2) ≤  ≤ x(n) are the
observed observations ordered by magnitude,
and for a nonnegative integer k such that n = 2k
(even) or n = 2k + 1 (odd), the sample estimate
of the median is

^x0:50 ¼

xðk
 þ 1Þ

1
2 xðkÞ þ xðk þ 1Þ

for n ¼ 2k þ 1
for n ¼ 2k
ð6:33Þ

Sample estimates of other quantiles may be
obtained using x(i) as an estimate of xq for q =
i/(n + 1) and then interpolating between observations to obtain ^xp for the desired p. This only
works for 1/(n + 1) ≤ p ≤ n/(n + 1) and can
yield rather poor estimates of xp when (n + 1)p is
near either 1 or n. An alternative approach is to
fit a reasonable distribution function to the
observations, as discussed in Sects. 6.3.1 and
6.3.2, and then estimate xp using Eq. 6.32, where
FX(x) is the fitted distribution.
Another simple and common approach to
describing a distribution’s center, spread, and
shape is by reporting the moments of a distribution. The first moment about the origin μX is
the mean of X and is given by
Zþ 1
lX ¼ E½X ¼

xfX ðxÞdx

ð6:34Þ

1

Moments other than the first are normally
measured about the mean. The second moment
measured about the mean is the variance, denoted Var(X) or r2X , where
r2X ¼ VarðXÞ ¼ E½ðX  lX Þ2 

ð6:35Þ

The standard deviation σX is the square root
of the variance. While the mean μX is a measure
of the central value of X, the standard deviation
σX is a measure of the spread of the distribution
of X about its mean μX.

220

6 An Introduction to Probability, Statistics, and Uncertainty

Another measure of the variability in X is the
coefficient of variation,
rX
CVX ¼
lX

ð6:36Þ

The coefficient of variation expresses the
standard deviation as a proportion of the mean. It
is useful for comparing the relative variability of
the flow in rivers of different sizes, or of rainfall
variability in different regions, which are both
strictly positive values.
The third moment about the mean denoted λX,
measures the asymmetry or skewness of the
distribution
kX ¼ E½ðX  lX Þ3 

ð6:37Þ

Typically, the dimensionless coefficient of
skewness γX is reported rather than the third
moment λX. The coefficient of skewness is the
third moment rescaled by the cube of the standard deviation so as to be dimensionless and
hence unaffected by the scale of the random
variable
cX ¼

kX
r3X

ð6:38Þ

Streamflows and other natural phenomena that
are necessarily nonnegative often have distributions with positive skew coefficients, reflecting
the asymmetric shape of their distributions.
When the distribution of a random variable is
not known, but a set of observations {x1, …, xn}
is available, the moments of the unknown distribution of X can be estimated based on the
sample values using the following equations.
The sample estimate of the mean
X¼

n
X

Xi =n

ð6:39aÞ

i¼1

The sample estimate of the variance
^2X ¼ S2X ¼
r

n
1 X
ðXi  XÞ2
ðn  1Þ i¼1

ð6:39bÞ

The sample estimate of skewness
^kX ¼

n
X
n
ðXi  XÞ3
ðn  1Þðn  2Þ i¼1

ð6:39cÞ

The sample estimate of the coefficient of
variation
d
CV X ¼ SX =X

ð6:39dÞ

The sample estimate of the coefficient of
skewness
^cX ¼ ^kX =S3X

ð6:39eÞ

The sample estimate of the mean and variance
are often denoted x and s2X . All of these sample
estimators only provide estimates. Unless the
sample size n is very large, the difference between
the estimators from the true values of lX ; r2X ; kX ;
CVX ; and cX may be large. In many ways, the field
of statistics is about the precision of estimators of
different quantities. One wants to know how well
the mean of 20 annual rainfall depths describes the
true expected annual rainfall depth, or how large
the difference between the estimated 100-year
flood and the true 100-year flood is likely to be.
As an example of the calculation of moments,
consider the flood data in Table 6.2. These data
have the following sample moments:
x ¼ 1549:2
sX ¼ 813:5
d
CV X ¼ 0:525
^cX ¼ 0:712
As one can see, the data are positively skewed
and have a relatively large coefficient of variance.
When discussing the accuracy of sample
estimates, two quantities are often considered,
bias and variance. An estimator ^h of a known or
unknown quantity h is a function of the values of
the random variable X1, …, Xn that will be
available to estimate the value of θ; ^h may be
written ^h[X1, X2, …, Xn] to emphasize that ^h

6.2 Probability Concepts and Methods

221

E021101b

Table 6.2 Annual Maximum Discharges on Magra River, Italy, at Calamazza, 1930–1970

date

discharge
cu ft/s

date

discharge
cu ft/s

1930
1931
1932
1933
1934
1935

410
1150
899
420
3100
2530

1951
1952
1953
1954
1955
1956

3070
2360
1050
1900
1130
674

1936
1937
1938

758
1220
1330

1957
1958
1959

683
1500
2600

1939
1940

1410
3100

1960
1961

3480
1430

1941
1942
1943

2470
929
586

1962
1963
1964

809
1010
1510

1944
1946
1947
1948
1949
1950

450
1040
1470
1070
2050
1430

1965
1966
1967
1968
1969
1970

1650
1880
1470
1920
2530
1490

The value for 1945 ismissing

itself is a random variable because its value
depends on the sample values of the random
variable that will be observed. An estimator ^h of
a quantity h is biased if E½^h 6¼ h and unbiased if
E½^
h ¼ h. The quantity fE½^h  hg is generally
called the bias of the estimator.
An unbiased estimator has the property that its
expected value equals the value of the quantity to
be estimated. The sample mean is an unbiased
estimate of the population mean μX because
"
#
n
n
 
1X
1X
Xi ¼
E½Xi  ¼ lX
E X ¼E
n i¼1
n i¼1
ð6:40Þ

The estimator S2X of the variance of X is an
unbiased estimator of the true variance r2X for
independent observations (Benjamin and Cornell
1970):
 
E S2X ¼ r2X

ð6:41Þ

However, the corresponding estimator of the
standard deviation, SX, is in general a biased
estimator of σx because
E½SX  6¼ rX

ð6:42Þ

The second important statistic often used to
assess the accuracy of an estimator ^h is the

222

6 An Introduction to Probability, Statistics, and Uncertainty

variance of the estimator Varð^hÞ, which equals
Efð^
h E½^
hÞ2 g. For the mean of a set of independent observations, the variance of the sample
mean is
  r2
Var X ¼ X
ð6:43Þ
n
pffiffiffi
It is common to call rx = n the standard error
of X rather than its standard deviation. The
standard error of an average is the most commonly reported measures of the precision.
The bias measures the difference between the
average value of an estimator and the quantity to
be estimated. The variance measures the spread
or width of the estimator’s distribution. Both
contribute to the amount by which an estimator
deviates from the quantity to be estimated. These
two errors are often combined into the mean
square error. Understanding that θ is fixed, and
the estimator ^
h is a random variable, the mean
squared error is the expected value of the squared
distance (error) between the two



2 
^
^
MSE h ¼ E h  h

n h i
o2
h i 2 
^
^
¼ E h  h þ E h  E ^h

¼ ½Bias2 þ Var ^h
ð6:44Þ
where [Bias] is Eð^hÞ  h. Equation 6.44 shows
that the MSE, equal to the expected average
squared deviation of the estimator ^h from the true
value of the parameter θ, can be computed as the
bias squared plus the variance of the estimator.
MSE is a convenient measure of how closely ^h
approximates θ because it combines both bias
and variance in a logical way.
Estimation of the coefficient of skewness γx
provides a good example of the use of the MSE
for evaluating the total deviation of an estimate
from the true population value. The sample
estimate ^cX of cX is often biased, has a large
variance, and was shown by Kirby (1974) to be
bounded so that

pffiffiffi
j^cX j  n

ð6:45Þ

where n is the sample size. The bounds do not
depend on the true skew γX. However, the bias
and variance of ^cX do depend on the sample size
and the actual distribution of X. Table 6.3 contains the expected value and standard deviation
of the estimated coefficient of skewness ^cX when
X has either a normal distribution, for which
γX = 0, or a gamma distribution with γX = 0.25,
0.50, 1.00, 2.00, or 3.00. These values are
adapted from Wallis et al. (1974a, b) who
employed moment estimators slightly different
than those in Eq. 6.39a.
For the normal distribution, E½^cX  ¼ 0 and
Var½^cX  ≅ 5/n. In this case, the skewness estimator is unbiased but highly variable. In all the
other cases in Table 6.3 it is also biased.
To illustrate the magnitude of these errors,
consider the mean square error of the skew
estimator ^cX calculated from a sample of size 50
when X has a gamma distribution with γX = 0.50,
a reasonable value for annual streamflows. The
expected value of ^cX is 0.45; its variance equals
(0.37)2, its standard deviation squared. Using
Eq. 6.44, the mean square error of ^cX is
MSEð^cX Þ ¼ ð0:45  0:50Þ2 þ ð0:37Þ2
¼ 0:0025 þ 0:1369 ¼ 0:139 ffi 0:14
ð6:46Þ
An unbiased estimate of γX is simply
(0.50/0.45)^cX . Here the estimator provided by
Eq. 6.39a has been scaled to eliminate bias. This
unbiased estimator has mean squared error

MSE

0:50^cX
0:45



¼ ð0:50  0:50Þ2 þ





2
0:50
ð0:37Þ
0:45

¼ 0:169 ffi 0:17

ð6:47Þ
The mean square error of the unbiased estimator of ^cX is larger than the mean square error
of the biased estimate. Unbiasing ^cX results in a
larger mean square error for all the cases listed in
Table 6.3 except for the normal distribution for

6.2 Probability Concepts and Methods

223

Table 6.3 Sampling properties of coefficient of skewness estimator

Source Wallis et al. (1974b) who divided just by n in the estimators of the moments, whereas in Eqs. 6.39b and 6.39c
we use the generally adopted coefficients of 1/(n − 1) and n/(n − 1)(n − 2) for the variance and skew

which γX = 0, and the gamma distribution with
γX = 3.00.
As shown here for the skew coefficient, biased
estimators often have smaller mean square errors
than unbiased estimators. Because the mean
square error measures the total average deviation

of an estimator from the quantity being estimated, this result demonstrates that the strict or
unquestioning use of unbiased estimators is not
advisable. Additional information on the sampling distribution of quantiles and moments is
contained in Stedinger et al. (1993).

224

6 An Introduction to Probability, Statistics, and Uncertainty

6.2.4 L-Moments and Their
Estimators

br ¼ EfX ½FðXÞr g

L-moments are another way to summarize the
statistical properties of hydrologic data based on
linear combinations of the original sample
(Hosking 1990). Recently, hydrologists have
found that regionalization methods (to be
discussed in Sect. 6.5) using L-moments are
superior to methods using traditional moments
(Hosking and Wallis 1997; Stedinger and Lu
1995). L-moments have also proved useful for
construction of goodness-of-fit tests (Hosking
et al. 1985; Chowdhury et al. 1991; Fill and
Stedinger 1995), measures of regional homogeneity and distribution selection methods (Vogel
and Fennessey 1993; Hosking and Wallis 1997).
The first L-moment designated as λ1 is simply
the arithmetic mean
k1 ¼ E½X

ð6:48Þ

Now let X(i|n) be the ith largest observation in
a sample of size n (i = n corresponds to the largest). Then, for any distribution, the second
L-moment, λ2, is a description of scale based
upon the expected difference between two randomly selected observations.


k2 ¼ ð1=2ÞE Xð2j1Þ  Xð1j2Þ

ð6:49Þ

Similarly, L-moment measures of skewness
and kurtosis use three and four randomly selected
observations, respectively.


k3 ¼ ð1=3ÞE Xð3j3Þ  2Xð2j3Þ þ Xð1j3Þ

ð6:50Þ



k4 ¼ ð1=4ÞE Xð4j4Þ  3Xð3j4Þ þ 3Xð2j4Þ  Xð1j4Þ
ð6:51Þ
Sample estimates are often computed using
intermediate statistics called probability weighted
moments (PWMs). The rth probability weighted
moment is defined as

ð6:52Þ

where F(X) is the cumulative distribution function of X. Recommended (Landwehr et al. 1979;
Hosking and Wallis 1995) unbiased PWM estimators, br, of βr are computed as
b0 ¼ X
b1 ¼

n
X
1
ðj  1ÞXðjÞ
nðn  1Þ j¼2

b2 ¼

n
X
1
ðj  1Þðj  2ÞXðjÞ
nðn  1Þðn  2Þ j¼3

ð6:53Þ
These are examples of the general formula for
computing estimators br of βr.


n

1X
i
n1
XðiÞ =
r
n i¼r r


n

X
1
i
n
¼
XðiÞ =
rþ1
r þ 1 i¼r r

br ¼

ð6:54Þ

for r = 1, …, n − 1.
L-moments are easily calculated in terms of
probability weighted moments (PWMs) using
k 1 ¼ b0
k2 ¼ 2b1  b0
k3 ¼ 6b2  6b1 þ b0

ð6:55Þ

k4 ¼ 20b3  30b2 þ 12b1  b0
Formulas for directly calculating L-moment
estimators, b, of β, are provided by Wang (1997).
Measures of the coefficient of variation, skewness, and kurtosis of a distribution can be computed with L-moments, as they can with
traditional product moments. Whereas skew primarily measures the asymmetry of a distribution,
the kurtosis is an additional measure of the
thickness of the extreme tails. Kurtosis is particularly useful for comparing symmetric distributions that have a skewness coefficient of

6.2 Probability Concepts and Methods

225

Table 6.4 Definitions of dimensionless product-moment and L-moment ratios

zero. Table 6.4 provides definitions of the traditional coefficient of variation, coefficient of
skewness, and coefficient of kurtosis, as well as
the L-moment, L-coefficient of variation,
L-coefficient of skewness, and L-coefficient of
kurtosis.
The flood data in Table 6.2 can be used to
provide an example of L-moments. Equation 6.53 yields estimates of the first three
Probability Weighted Moments

^k1 ¼ b0 ¼ 1549
^k2 ¼ 2b1  b0 ¼ 458
^k3 ¼ 6b2  6b1 þ b0 ¼ 80

Thus the sample estimates of the L-Coefficient
of Variation, t2, and L-Coefficient of Skewness,
t3, are
t2 ¼ 0:295
t3 ¼ 0:174

b0 ¼ 1549:20
b1 ¼ 1003:89
b2 ¼ 759:02

ð6:57Þ

ð6:58Þ

ð6:56Þ

Recall b0 is just the sample average x. The
sample L-moments are easily calculated using
the probability weighted moments (PWMs). One
obtains

6.3

Distributions of Random Events

A frequent task in water resources planning is the
development of a model of some probabilistic or
stochastic phenomena such as streamflows, flood

226

6 An Introduction to Probability, Statistics, and Uncertainty

flows, rainfall, temperatures, evaporation, sediment or nutrient loads, nitrate or organic compound concentrations, or water demands. This
often requires that one fit a probability distribution function to a set of observed values of the
random variable. Sometimes, one’s immediate
objective is to estimate a particular quantile of
the distribution, such as the 100-year flood,
50-year 6-h-rainfall depth, or the minimum
seven-day-average expected once-in-10-year
flow. Then the fitted distribution and its statistical parameters can characterize that random
variable. In a stochastic simulation, fitted distributions are used to generate possible values of
the random variable in question.
Rather than fitting a reasonable and smooth
mathematical distribution, one could use the
empirical distribution represented by the data to
describe the possible values that a random variable
may assume in the future and their frequency. In
practice, the true mathematical form for the distribution that describes the events is not known.
Moreover, even if it was, its functional form may
have too many parameters to be of much practical
use. Thus using the empirical distribution represented by the data itself has substantial appeal.
Generally the free parameters of the theoretical
distribution are selected (estimated) so as to make
the fitted distribution consistent with the available
data. The goal is to select a physically reasonable
and simple distribution to describe the frequency
of the events of interest, to estimate that distribution’s parameters, and ultimately to obtain
quantiles, performance indices, and risk estimates
of satisfactory accuracy for the problem at hand.
Use of a theoretical distribution does have several
advantages over use of the empirical distribution
1. It presents a smooth interpretation of the
empirical distribution. As a result quantiles,
performance indices, and other statistics
computed using the fitted distribution should

2.
3.

4.

5.

be more easily estimated compared to those
computed from the empirical distribution.
It provides a compact and easy to use representation of the data.
It is likely to provide a more realistic
description of the range of values that the
random variable may assume and their likelihood; for example, using the empirical distribution one often assumes that no values
larger or smaller than the sample maximum or
minimum can occur. For many situations this
is unreasonable.
Often one needs to estimate the likelihood of
extreme events that lie outside of the range of
the sample (either in terms of x values or in
terms of frequency); such extrapolation makes
little sense with the empirical distribution.
In many cases one is not interested in X, but
instead is interested in derived variables
Y that are functions of X. This could be a
performance function for some system. If Y is
the performance function, interest might be
primarily in its mean value E[Y], or the
probability some standard is exceeded, Pr
{Y > standard}. For some theoretical X-distributions, the resulting Y-distribution may be
available in closed form making the analysis
rather simple. (The normal distribution works
with linear models, the lognormal distribution
with product models, and the gamma distribution with queuing systems.)

This section provides a brief introduction to
some useful techniques for estimating the
parameters of probability distribution functions
and determining if a fitted distribution provides a
reasonable or acceptable model of the data.
Subsections are also included on families of
distributions based on the normal, gamma, and
generalized-extreme-value distributions. These
three families have found frequent use in water
resource planning (Kottegoda and Rosso 1997).

6.3 Distributions of Random Events

227

6.3.1 Parameter Estimation
Given a set of observations to which a distribution is to be fit, one first selects a distribution
function to serve as a model of the distribution of
the data. The choice of a distribution may be
based on experience with data of that type, some
understanding of the mechanisms giving rise to
the data, and/or examination of the observations
themselves. One can then estimate the parameters of the chosen distribution and determine if
the observed data could have been drawn from
the fitted distribution. If not, the fitted distribution is judged to be unacceptable.
In many cases, good estimates of a distribution’s parameters are obtained by the maximum
likelihood-estimation procedure. Given a set of
n independent observations {x1, …, xn} of a
continuous random variable X, the joint probability density function for the observations is
fX1 ;X2 ;X3 ; . . .Xn ðx1 ; . . .; xn jhÞ
¼ fX ðx1 jhÞ  fX ðx2 jhÞ . . . fX ðxn jhÞ

ð6:59Þ

where θ is the vector of the distribution’s
parameters.
The maximum likelihood estimator of θ is that
vector which maximizes Eq. 6.59 and thereby
makes it as likely as possible to have observed
the values {x1, …, xn}.
Considerable work has gone into studying the
properties of maximum likelihood parameter
estimates. Under rather general conditions,
asymptotically the estimated parameters are
normally distributed, unbiased, and have the
smallest possible variance of any asymptotically
unbiased estimator (Bickel and Doksum 1977).
These, of course, are asymptotic properties, valid
for large sample sizes n. Better estimation procedures, perhaps yielding biased parameter estimates, may exist for small sample sizes.
Stedinger (1980) provides such an example. Still,
maximum likelihood procedures are to be highly

recommended with moderate and large samples,
even though the iterative solution of nonlinear
equations is often required.
An example of the maximum likelihood procedure for which closed-form expressions for the
parameter estimates are obtained is provided by
the lognormal distribution. The probability density function of a lognormally distributed random
variable X is


1
1
fX ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp  2 ½lnðxÞ  l2
2r
x 2pr2
ð6:60Þ
Here the parameters μ and σ2 are the mean and
variance of the logarithm of X, and not of X itself.
Maximizing the logarithm of the joint density
for {x1, …, xn} is more convenient than maximizing the joint probability density itself. Hence
the problem can be expressed as the maximization of the log-likelihood function
"
L ¼ ln

n
Y

#
f ðxi jl; rÞ

i¼1

¼

n
X

ln f ðxi jl; rÞ

i¼1

n
 pffiffiffiffiffiffi
X
¼
ln xi 2p

ð6:61Þ

i¼1

 n lnðrÞ 

n
1 X
½lnðxi Þ  l2
2r2 i¼1

The maximum can be obtained by equating to
zero the partial derivatives ∂L/∂μ and ∂L/∂σ
whereby one obtains
0¼

n
@L
1X
¼ 2
½lnðxi Þ  l
@l r i¼1

n
@L
n
1X
¼ þ 3
0¼
½lnðxi Þ  l2
@r
r r i¼1

These equations yield the estimators

ð6:62Þ

228

6 An Introduction to Probability, Statistics, and Uncertainty
n
1X
^¼
lnðxi Þ
l
n i¼1
n
1X
2
^ ¼
^ 2
r
½lnðxi Þ  l
n i¼1



^2 ¼ ln 1 þ s2X =x2
r
ð6:63Þ

The second-order conditions for a maximum
are met and these values do maximize Eq. 6.59. It
is useful to note that if one defines a new random
variable Y = ln(X), then the maximum likelihood
estimates of the parameters μ and σ2, which are the
mean and variance of the Y distribution, are the
sample estimates of the mean and variance of Y
^ ¼ y
l
^2 ¼ ½ðn  1Þ=ns2Y
r

ð6:64Þ

The correction [(n − 1)/n] in this last equation is
often neglected.
The second commonly used parameter estimation procedure is the method of moments. The
method of moments is often a quick and simple
method for obtaining parameter estimates for
many distributions. For a distribution with m = 1,
2, or 3 parameters, the first m moments of postulated distribution in Eqs. 6.34, 6.35, and 6.37 are
equated to the estimates of those moments calculated using Eqs. 6.39a. The resulting nonlinear
equations are solved for the unknown parameters.
For the lognormal distribution, the mean and
variance of X as a function of the parameters μ
and σ are given by


1 2
lX ¼ exp l þ r
2

  

2
rX ¼ exp 2l þ r2 exp r2  1

ð6:65Þ

Substituting x for μX and s2x for r2X and solving
for μ and σ2 one obtains

x
^ ¼ ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l
1 þ s2X =x2

!

1 2
^
¼ ln x  r
2

ð6:66Þ

The data in Table 6.2 provide an illustration
of both fitting methods. One can easily compute
the sample mean and variance of the logarithms
of the flows to obtain
^ ¼ 7:202
l
^2 ¼ 0:3164 ¼ ð0:5625Þ2
r

ð6:67Þ

Alternatively, the sample mean and variance
of the flows themselves are
x ¼ 1549:2
s2X

¼ 661;800 ¼ ð813:5Þ2

ð6:68Þ

Substituting those two values in Eq. 6.66
yields
^ ¼ 7:224
l
^2 ¼ 0:2435 ¼ ð0:4935Þ2
r

ð6:69Þ

Method of moments and maximum likelihood
are just two of many possible estimation methods. Just as method of moments equates sample
estimators of moments to population values and
solves for a distribution’s parameters, one can
simply equate L-moment estimators to population values and solve for the parameters of a
distribution. The resulting method of L-moments
has received considerable attention in the
hydrologic literature (Landwehr et al. 1978;
Hosking et al. 1985; 1987; Hosking 1990; Wang
1997). It has been shown to have significant
advantages when used as a basis for regionalization procedures that will be discussed in
Sect. 6.5 (Lettenmaier et al. 1987; Stedinger and
Lu 1995; Hosking and Wallis 1997).

6.3 Distributions of Random Events

Bayesian procedures provide another approach
that is related to maximum likelihood estimation.
Bayesian inference employs the likelihood function to represent the information in the data. That
information is augmented with a prior distribution
that describes what is known about constraints on
the parameters and their likely values beyond the
information provided by the recorded data available at a site. The likelihood function and the prior
probability density function are combined to
obtain the probability density function that
describes the posterior distribution of the
parameters
fh ðhjx1 ; x2 ; . . .; xn Þ / fX ðx1 ; x2 ; . . .; xn jhÞnðhÞ
ð6:70Þ
The symbol ∝ means “proportional to” and
ξ(θ) is the probability density function for the
prior distribution for θ (Kottegoda and Rosso
1997). Thus, except for a constant of proportionality, the probability density function
describing the posterior distribution of the
parameter vector θ is equal to the product of the
likelihood function fX(x1, x2, …, xn|θ) and the
probability density function for the prior distribution ξ(θ) for θ.
Advantages of the Bayesian approach are that
it allows the explicit modeling of uncertainty in
parameters (Stedinger 1997; Kuczera 1999), and
provides a theoretically consistent framework for
integrating systematic flow records with regional
and other hydrologic information (Vicens et al.
1975; Stedinger 1983; and Kuczera 1983).
Martins and Stedinger (2000) illustrate how a
prior distribution can be used to enforce realistic
constraints upon a parameter as well as providing
a description of its likely values. In their case use
of a prior of the shape parameter κ of a GEV
distribution allowed definition of generalized
maximum likelihood estimators that over the
κ-range of interest performed substantially better
than maximum likelihood, moment, and
L-moment estimators.
While Bayesian methods have been available
for decades, the computational challenge posed
by the solution of Eq. 6.70 has been an obstacle
to their use. Solutions to Eq. 6.70 have been

229

available for special cases such as normal data,
and binomial and Poisson samples (Raiffa and
Schlaifier 1961; Benjamin and Cornell 1970;
Zellner 1971). However, a new and very general
set of Markov Chain Monte Carlo (MCMC)
procedures allow numerical computation of the
posterior distributions of parameters for a very
broad class of models (Gilks et al. 1996). As a
result, Bayesian methods are now becoming
much more popular, and are the standard
approach for many difficult problems that are not
easily addressed by traditional methods (Gelman
et al. 1995; Carlin and Louis 2000). The use of
Monte Carlo Bayesian methods in flood frequency analysis, rainfall-runoff modeling, and
evaluation of environmental pathogen concentrations are illustrated by Wang (2001), Bates
and Campbell (2001) and Crainiceanu et al.
(2002) respectively.
Finally, a simple method of fitting flood frequency curves is to plot the ordered flood values
on special probability paper and then to draw a
line through the data (Gumbel 1958). Even
today, that simple method is still attractive when
some of the smallest values are zero or unusually
small, or have been censored as will be discussed
in Sect. 6.4 (Kroll and Stedinger 1996). Plotting
the ranked annual maximum series against a
probability scale is always an excellent and recommended way to see what the data look like
and for determining whether a fitted curve is or is
not consistent with the data (Stedinger et al.
1993).
Statisticians and hydrologists have investigated which of these methods most accurately
estimates the parameters themselves or the quantiles of the distribution (Stedinger 1997). One also
needs to determine how accuracy should be
measured. Some studies have used average
squared deviations, some have used average
absolute weighted deviations with different
weights on under- and over-estimation, and some
have used the squared deviations of the
log-quantile estimator (Slack et al. 1975; Kroll
and Stedinger 1996). In almost all cases, one is
also interested in the bias of an estimator, which is
the average value of the estimator minus the true
value of the parameter or quantile being estimated.

230

6 An Introduction to Probability, Statistics, and Uncertainty

Special estimators have been developed to compute design events that on average are exceeded
with the specified probability, and have the
anticipated risk of being exceeded (Beard 1960,
1997; Rasmussen and Rosbjerg 1989, 1991a, b;
Stedinger 1997; Rosbjerg and Madsen 1998).

This beta distribution has mean and variance
of
E ½U i  ¼

i
nþ1

ð6:72aÞ

and

6.3.2 Model Adequacy

VarðUi Þ ¼

After estimating the parameters of a distribution,
some check of model adequacy should be made.
Such checks vary from simple comparisons of
the observations with the fitted model using
graphs or tables, to rigorous statistical tests.
Some of the early and simplest methods of
parameter estimation were graphical techniques.
Although quantitative techniques are generally
more accurate and precise for parameter estimation, graphical presentations are invaluable for
comparing the fitted distribution with the observations for the detection of systematic or unexplained deviations between the two. The
observed data will plot as a straight line on
probability graph paper if the postulated distribution is the true distribution of the observation.
If probability graph paper does not exist for the
particular distribution of interest, more general
techniques can be used.
Let x(i) be the ith largest value in a set of
observed values {xi} so that x(1) ≤ x(2) ≤  ≤
x(n). The random variable X(i) provides a reasonable estimate of the pth quantile xp of the true
distribution of X for p = i/(n + 1). In fact, if one
thinks of the cumulative probability Ui associated
with the random variable X(i), Ui = FX(X(i)), then
if the observations X(i) are independent, the Ui
have a beta distribution (Gumbel 1958) with
probability density function
fUi ðuÞ ¼

n!
ui1 ð1  uÞni
ði  1Þ!ðn  1Þ!

0u1
ð6:71Þ

iðn  i þ 1Þ
ðn þ 1Þ2 ðn þ 2Þ

ð6:72bÞ

A good graphical check of the adequacy of a
fitted distribution G(x) is obtained by plotting the
observations x(i) versus G−1[i/(n + 1)] (Wilk and
Gnanadesikan 1968). Even if G(x) exactly
equaled the true X-distribution FX[x], the plotted
points will not fall exactly on a 45-degree line
through the origin of the graph. This would only
occur if FX[x(i)] exactly equaled i/(n + 1) and
therefore each x(i) exactly equaled F−1
X [i/(n + 1)].
An appreciation for how far an individual
observation x(i) can be expected to deviate from
G−1[i/(n + 1)] can be obtained by plotting
G−1[u(0.75)
] and G−1[u(0.25)
], where u(0.75)
and
i
i
i
(0.25)
ui
are the upper and lower quantiles of the
distribution of Ui obtained from integrating the
probability density function in Eq. 6.71. The
required incomplete beta function is also available
in many software packages, including Microsoft
Excel. Stedinger et al. (1993) report that u(1) and
(1 – u(n)) fall between 0.052/n and 3(n + 1) with a
probability of 90%, thus illustrating the great
uncertainty associated with those values.
Figure 6.2a, b illustrate the use of this quantile-quantile plotting technique by displaying the
results of fitting a normal and a lognormal
distribution to the annual maximum flows in
Table 6.2 for the Magra River, Italy, at Calamazza for the years 1930–1970. The observations of X(i), given in Table 6.2, are
plotted on the vertical axis against the quantiles
G−1[i/(n + 1)] on the horizontal axis.
A probability plot is essentially a scatter plot
of the sorted observations X(i) versus some
approximation of their expected or anticipated

6.3 Distributions of Random Events

value, represented by G−1(pi), where, as suggested, pi = i/(n + 1). The pi values are called
plotting positions. A common alternative to i/
(n + 1) is (i − 0.5)/n, which results from a
probabilistic interpretation of the empirical distribution of the data. Many reasonable plotting
position formula have been proposed based upon
the sense in which G−1(pi) should approximate
X(i). The Weibull formula i/(n + 1) and the Hazen
formula (i − 0.5)/n bracket most of the reasonable choices. Popular formulas are summarized
in Stedinger et al. (1993), who also discuss the
generation of probability plots for many distributions commonly employed in hydrology.
Rigorous statistical tests are available for
trying to determine whether or not it is reasonable to assume that a given set of observations
could have been drawn from a particular family
of distributions. Although not the most powerful
of such tests, the Kolmogorov–Smirnov test
provides bounds within which every observation
should lie if the sample is actually drawn from
the assumed distribution. In particular, for
G = FX, the test specifies that

231







i
i1
Pr G1
 Ca  XðiÞ  G1
þ Ca for every i
n
n
¼1a

ð6:73Þ
where Cα is the critical value of the test at significance level α. Formulas for Cα as a function
of n are contained in Table 6.5 for three cases:
(1) when G is completely specified independent
of the sample’s values; (2) when G is the normal
distribution and the mean and variance are estimated from the sample with x and s2x ; and
(3) when G is the exponential distribution and the
scale parameter is estimated as 1=ðxÞ. Chowdhury et al. (1991) provide critical values for the
Gumbel and GEV distribution with known shape
parameter κ. For other distributions, the values
obtained from Table 6.5 may be used to construct approximate simultaneous confidence
intervals for every X(i).
Figures 6.2 contain 90% confidence intervals
for the plotted points constructed in this manner.
For the normal distribution, the critical value of
pffiffiffiffiffi
pffiffiffi
Cα
equals
0:819=ð n  0:01 þ 0:85= nÞ,

Table 6.5 Critical valuesα of Kolmogorov-Smirnov statistic as a function of sample size n

232
Fig. 6.2 Plots of annual
maximum discharges of
Magra River, Italy, versus
quantiles of fitted a normal
and b lognormal
distributions

6 An Introduction to Probability, Statistics, and Uncertainty

(a)

(b)

6.3 Distributions of Random Events

where 0.819 corresponds to α = 0.10. For
n = 40, one computes Cα = 0.127. As can be
seen in Fig. 6.2a, the annual maximum flows are
not consistent with the hypothesis that they were
drawn from a normal distribution; three of the
observations lie outside the simultaneous 90%
confidence intervals for all points. This demonstrates a statistically significant lack of fit. The
fitted normal distribution underestimates the
quantiles corresponding to small and large
probabilities while overestimating the quantiles
in an intermediate range. In Fig. 6.2b, deviations
between the fitted lognormal distribution and the
observations can be attributed to the differences
between FX(x(i)) and i/(n + 1). Generally, the
points are all near the 45-degree line through the
origin, and no major systematic deviations are
apparent.
The Kolmogorov–Smirnov test conveniently
provides bounds within which every observation
on a probability plot should lie if the sample is
actually drawn from the assumed distribution, and
thus is useful for visually evaluating the adequacy
of a fitted distribution. However, it is not the most
powerful test available for evaluating from which
of several families a set of observations is likely
to have been drawn. For that purpose several
other more analytical tests are available (Filliben
1975; Hosking 1990; Chowdhury et al. 1991;
Kottegoda and Rosso 1997).
The Probability Plot Correlation test is a
popular and powerful test of whether a sample
has been drawn from a postulated distribution,
though it is often weaker than alternative tests at
rejecting thin-tailed alternatives (Filliben 1975;
Fill and Stedinger 1995). A test with greater
power has a greater probability of correctly
determining that a sample is not from the postulated distribution. The Probability Plot Correlation Coefficient test employs the correlation
r between the ordered observations x(i) and the
corresponding fitted quantiles wi = G−1(pi),
determined by plotting positions pi for each x(i).
Values of r near 1.0 suggest that the observations
could have been drawn from the fitted distribution: r measures the linearity of the probability

233

plot providing a quantitative assessment of fit. If
x denotes the average value of the observations
 denotes the average value of the fitted
and w
quantiles, then

P
Þ
xðiÞ  x ðwi  w
ð6:74Þ
r ¼ h
2 P
2 i0:5
P
Þ
ðwi  w
xðiÞ  x
Table 6.6 provides critical values for r for the
normal distribution, or the logarithms of lognormal variates, based upon the Blom plotting
position that has pi = (i − 3/8)/(n + 1/4). Values
for the Gumbel distribution are reproduced in
Table 6.7 for use with the Gringorten plotting
position pi = (i − 0.44)/(n + 0.12). The table
also applies to logarithms of Weibull variates
(Stedinger et al. 1993). Other tables are available
for the GEV (Chowdhury et al. 1991), the
Pearson type 3 (Vogel and McMartin 1991), and
exponential and other distributions (D’Agostion
and Stephens 1986).
L-moment ratios appear to provide
goodness-of-fit tests that are superior to both the
Kolmogorov–Smirnov and the Probability Plot
Correlation test (Hosking 1990; Chowdhury
et al. 1991; Fill and Stedinger 1995). For normal
data, the L-skewness estimator ^s3 (or t3) would
have mean zero and Var[^s3 ] = (0.1866 + 0.8/n)/
n, allowing construction of a powerful test of
normality against skewed alternatives using the
normally distributed statistic
Z ¼ t3 =

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð0:1866 þ 0:8=nÞ=n

ð6:75Þ

with a reject region |Z| > zα/2.
Chowdhury et al. (1991) derive the sampling
variance of the L-CV and L-skewness estimators
^s2 and ^s3 as a function of κ for the GEV distribution. These allow construction of a test of
whether a particular data set is consistent with a
GEV distribution with a regionally estimated
value of κ, or a regional κ and CV. Fill and
Stedinger (1995) show that the ^s3 L-skewness
estimator provides a test for the Gumbel versus a
general GEV distribution using the normally
distributed statistic

234

6 An Introduction to Probability, Statistics, and Uncertainty

Table 6.6 Lower critical values of the probability plot correlation test statistic for the normal distribution using
pi = (i − 3/8)/(n + 1/4) (Vogel 1987)

Z ¼ ð^s3  0:17Þ=

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð0:2326 þ 0:70=nÞ=n
ð6:76Þ

with a reject region |Z| > zα/2.
The literature is full of goodness-of-fit tests.
Experience indicates that among the better tests
there is often not a great deal of difference
(D’Agostion and Stephens 1986). Generation of
a probability plot is most often a good idea
because it allows the modeler to see what the
data look like and where problems occur. The
Kolmogorov–Smirnov test helps the eye interpret
a probability plot by adding bounds to a graph
illustrating the magnitude of deviations from a
straight line that are consistent with expected
variability. One can also use quantiles of a beta
distribution to illustrate the possible error in
individual plotting positions, particularly at the
extremes where that uncertainty is largest. The

probability plot correlation test is a popular and
powerful goodness-of-fit statistic. Goodness-offit tests based upon sample estimators of the
L-skewness ^s3 for the normal and Gumbel distribution provide simple and useful tests that are
not based on a probability plot.

6.3.3 Normal and Lognormal
Distributions
The normal distribution and its logarithmic
transformation, the lognormal distribution, are
arguably the most widely used distributions in
science and engineering. The density function of
a normal random variable is



1
1
2
fX ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp  2 ðx  lÞ
2r
2pr2
for  1\x\ þ 1

ð6:77Þ

6.3 Distributions of Random Events

235

Table 6.7 Lower critical values of the probability plot correlation test statistic for the Gumbel distribution using
pi = (i − 0.44)/(n + 0.12) (Vogel 1987)

where μ and σ2 are equivalent to μX and r2X , the
mean and variance of X. Interestingly, the maximum likelihood estimators of μ and σ2 are almost
identical to the moment estimates x and s2X .
The normal distribution is symmetric about its
mean μX and admits values from −∞ to +∞.
Thus it is not always satisfactory for modeling
physical phenomena such as streamflows or
pollutant concentrations, which are necessarily
nonnegative and have skewed distributions.
A frequently used model for skewed distributions
is the lognormal distribution. A random variable
X has a lognormal distribution if the natural
logarithm of X, ln(X), has a normal distribution.
If X is lognormally distributed, then by definition
ln(X) is normally distributed, so that the density
function of X is



1
1
dð‘nxÞ
fX ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp  2 ½‘nðxÞ  l2
2r
dx
2pr2


1
1
¼ pffiffiffiffiffiffiffiffiffiffi exp  2 ½‘nðx=gÞ2
2r
x 2pr2
ð6:78Þ
for x > 0 and μ = ln(η). Here η is the median of the
X-distribution. The coefficient of skewness for the
three-parameter lognormal distribution is given by
c ¼ 3t þ t3

 
where t ¼ ½exp r2  10:5
ð6:79Þ

A lognormal random variable takes on values
in the range [0, +∞]. The parameter μ determines the scale of the X-distribution whereas σ2

236

6 An Introduction to Probability, Statistics, and Uncertainty

Fig. 6.3 Lognormal
probability density
functions with various
standard deviations σ

determines the shape of the distribution. The
mean and variance of the lognormal distribution
are given in Eq. 6.65. Figure 6.3 illustrates
the various shapes the lognormal probability
density function can assume. It is highly skewed
with a thick right-hand tail for σ > 1, and
approaches a symmetric normal distribution as
σ → 0. The density function always has a value
of zero at x = 0. The coefficient of variation
and skew are
CVX ¼ ½expðr2 Þ  11=2
cX ¼ 3CVX þ CV3X

ð6:80Þ

The maximum likelihood estimates of μ and
σ2 are given in Eq. 6.63 and the moment estimates in Eq. 6.66. For reasonable-size samples,
the maximum likelihood estimates are generally
performed as well or better than the moment
estimates (Stedinger 1980).
The data in Table 6.2 were used to calculate
the parameters of the lognormal distribution that
would describe the flood flows and the results are
reported after Eq. 6.66. The two-parameter
maximum likelihood and method of moments
estimators identify parameter estimates for which
the distribution skewness coefficients are 2.06

and 1.72, which is substantially greater than the
sample skew of 0.712.
A useful generalization of the two-parameter
lognormal distribution is the shifted lognormal or
three-parameter lognormal distribution obtained
when ln(X − τ) is described by a normal distribution, where X ≥ τ. Theoretically, τ should be
positive if for physical reasons X must be positive; practically, negative values of τ can be
allowed when the resulting probability of negative values of X is sufficiently small.
Unfortunately, maximum likelihood estimates
of the parameters μ, σ2, and τ are poorly behaved
because of irregularities in the likelihood function (Giesbrecht and Kempthorne 1976). The
method of moments does fairly well when the
skew of the fitted distribution is reasonably
small. A method that does almost as well as the
moment method for low-skew distributions, and
much better for highly skewed distributions,
estimates τ by
^s ¼

xð1Þ xðnÞ  ^x20:50
xð1Þ þ xðnÞ  2^x0:50

ð6:81Þ

provided that x(1) + x(n) − 2^x 0.50 > 0, where x(1)
and x(n) are the smallest and largest observations
and ^x0:50 is the sample median (Stedinger 1980;

6.3 Distributions of Random Events

237

Hoshi et al. 1984). If x(1) + x(n) − 2^x0:50 < 0, the
sample tends to be negatively skewed and a
three-parameter lognormal distribution with a
lower bound cannot be fit with this method.
Good estimates of μ and σ2 to go with ^s in
Eq. 6.81 are (Stedinger 1980)
2

3

x  ^s
6
7
^ ¼ ‘n4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5
l
2
1 þ s2x =ðx  ^sÞ
"
#
s2x
2
^ ¼ ‘n 1 þ
r
ðx  ^sÞ2

ð6:82Þ

For the data in Table 6.2, Eq. 6.82 yields the
hybrid moment-of-moments estimates for the
three-parameter lognormal distribution
^ ¼ 7:606
l
^2 ¼ 0:1339 ¼ ð0:3659Þ2
r
^s ¼ 600:1
This distribution has a coefficient of skewness
of 1.19, which is more consistent with the sample
skewness estimator than was the value obtained
when a two-parameter lognormal distribution
was fit to the data. Alternatively, one can estimate µ and σ2 by the sample mean and variance
of ln(X − ^s) that yields the hybrid maximum
likelihood estimates
^ ¼ 7:605
l
^2 ¼ 0:1407 ¼ ð0:3751Þ2
r
^s ¼ 600:1
The two sets of estimates are surprisingly
close in this instance. In this second case, the
fitted distribution has a coefficient of skewness of
1.22.
Natural logarithms have been used here. One
could have just as well used base 10 common
logarithms to estimate the parameters; however,
in that case the relationships between the log
space parameters and the real-space moments
change slightly (Stedinger et al. 1993,
Eq. 18.2.8).

6.3.4 Gamma Distributions
The gamma distribution has long been used to
model many natural phenomena, including daily,
monthly, and annual streamflows as well as flood
flows (Bobee and Ashkar 1991). For a gamma
random variable X,
j bj
ðbxÞa1 ebx
CðaÞ
a
lX ¼
b
a
r2X ¼ 2
b
2
cX ¼ pffiffiffi ¼ 2CVX
a

fX ðxÞ ¼

bx  0

ð6:83Þ

The gamma function, Γ(α), for integer α is
(α − 1)!. The parameter α > 0 determines the
shape of the distribution; β is the scale parameter.
Figure 6.4 illustrates the different shapes that the
probability density function for a gamma variable
can assume. As α → ∞, the gamma distribution
approaches the symmetric normal distribution,
whereas for 0 < α < 1, the distribution has a
highly asymmetric J-shaped probability density
function whose value goes to infinity as x approaches zero.
The gamma distribution arises naturally in
many problems in statistics and hydrology. It
also has a very reasonable shape for such nonnegative random variables as rainfall and
streamflow. Unfortunately, its cumulative distribution function is not available in closed form,
except for integer α, though it is available in
many software packages including Microsoft
Excel. The gamma family includes a very special
case: the exponential distribution is obtained
when α = 1.
The gamma distribution has several generalizations (Bobee and Ashkar 1991). If a constant τ
is subtracted from X so that (X − τ) has a gamma
distribution, the distribution of X is a
three-parameter gamma distribution. This is also
called a Pearson type 3 distribution, because the
resulting distribution belongs to the third type of
distributions suggested by the statistician

238

6 An Introduction to Probability, Statistics, and Uncertainty

Fig. 6.4 The gamma
distribution functions for
various values of the shape
parameter α

Karl Pearson. Another variation is the
log-Pearson type 3 distribution obtained by fitting the logarithms of X with a Pearson type 3
distribution. The log-Pearson distribution is discussed further in the next section.
The method of moments may be used to
estimate the parameters of the gamma distribution. For the three-parameter gamma distribution

sX
^s ¼ x  2
^cX
4
^
a¼
ð^cX Þ2
^¼ 2
b
s X cX

ð6:84Þ

where x; s2X ; and ^cX are estimates of the mean,
variance, and coefficient of skewness of the distribution of X (Bobee and Robitaille 1977).
For the two-parameter gamma distribution,
ðxÞ2
s2x
^ ¼ x
b
s2x
^a ¼

ð6:85Þ

Again the flood record in Table 6.2 can be
used to illustrate the different estimation

procedures. Using the first three sample
moments, one would obtain for the threeparameter gamma distribution the parameter
estimates
^s ¼ 735:6
a^ ¼ 7:888
^ ¼ 0:003452 ¼ 1=289:7
b
Using only the sample mean and variance
yields the method of moment estimators of the
parameters of the two-parameter gamma distribution (τ = 0)
^a ¼ 3:627
^
b ¼ 0:002341 ¼ 1=427:2
The fitted two-parameter gamma distribution
has a coefficient of skewness γ of 1.05 whereas
the fitted three-parameter gamma reproduces the
sample skew of 0.712. As occurred with the
three-parameter lognormal distribution, the estimated lower bound for the three-parameter
gamma distribution is negative (^s ¼ 735:6)
resulting in a three-parameter model that has a
smaller skew coefficient than was obtained with
the corresponding two-parameter model. The
^ is often reported. While b
^ has
reciprocal of b

6.3 Distributions of Random Events

^ is a natural scale parameter
inverse x-units, 1/b
that has the same units as x and thus can be easier
to interpret.
Studies by Thom (1958) and Matalas and
Wallis (1973) have shown that maximum likelihood parameter estimates are superior to the
moment estimates. For the two-parameter gamma
distribution, Greenwood and Durand (1960) give
approximate formulas for the maximum likelihood estimates (also Haan 1977). However, the
maximum likelihood estimators are often not
used in practice because they are very sensitive
to the smallest observations that sometimes suffer
from measurement error and other distortions.
When plotting the observed and fitted quantiles of a gamma distribution, an approximation
to the inverse of the distribution function is often
useful. For |γ| ≤ 3, the Wilson–Hilferty
transformation
"
3 #
2
cxN c2
2
xG ¼ l þ r
1þ


ð6:86Þ
c
c
6
36
gives the quantiles xG of the gamma distribution
in terms of xN, the quantiles of the standard
normal distribution. Here μ, σ, and γ are the
mean, standard deviation, and coefficient of
skewness of xG. Kirby (1972) and Chowdhury
and Stedinger (1991) discuss this and other more
complicated but more accurate approximations.

Fig. 6.5 Log-Pearson type
3 probability density
functions for different
values of coefficient of
skewness γ

239

Fortunately the availability of excellent approximations of the gamma cumulative distribution
function and its inverse in Microsoft Excel and
other packages has reduced the need for such
simple approximations.

6.3.5 Log-Pearson Type 3
Distribution
The log-Pearson type 3 distribution (LP3)
describes a random variable whose logarithms
have a Pearson type 3 distribution. This distribution has found wide use in modeling flood
frequencies and has been recommended for that
purpose (IACWD 1982). Bobee (1975), Bobee
and Ashkar (1991) and Griffis and Stedinger
(2007a) discuss the unusual shapes that this
hybrid distribution may take allowing negative
values of β. The LP3 distribution has a probability density function given by.
fX ðxÞ ¼ jbjfb½lnðxÞ  nga1
ð6:87Þ
expfb½lnðxÞ  ng=fxCðaÞg
with α > 0, and β either positive or negative. For
β < 0, values are restricted to the range
0 < x < exp(ξ). For β > 0, values have a lower
bound so that exp(ξ) < X. Figure 6.5 illustrates
the probability density function for the LP3

240

6 An Introduction to Probability, Statistics, and Uncertainty

distribution as a function of the skew γ of the P3
distribution describing ln(X), with σlnX = 0.3.
The LP3 density function for |γ| ≤ 2 can assume
a wide range of shapes with both positive and
negative skews. For |γ| = 2, the log-space P3
distribution is equivalent to an exponential distribution function which decays exponentially as
x moves away from the lower bound (β > 0) or
upper bound (β < 0): as a result the LP3 distribution has a similar shape. The space with
−1 < γ may be more realistic for describing
variables whose probability density function
becomes thinner as x takes on large values. For
γ = 0, the 2-parameter lognormal distribution is
obtained as a special case.
The LP3 distribution has mean and variance

a
b
n
lX ¼ e
b1
(
a
2a )
b
b
ð6:88Þ
2
2n
rX ¼ e

b2
b1
for b [ 2; or b\0:
For 0 < β < 2, the variance is infinite.
These expressions are seldom used, but they
do reveal the character of the distribution.
Figures 6.6 and 6.7 provide plots of the
real-space coefficient of skewness and coefficient
Fig. 6.6 Real-space
coefficient of skewness γX
for LP3 distributed X as a
function of log-space
standard deviation σY and
coefficient of skewness γY
where Y = ln(X)

of variation of a log-Pearson type 3 variate X as a
function of the standard deviation σY and coefficient of skew γY of the log-transformation Y = ln
(X). Thus the standard deviation σY and skew γY
of Y are in log space. For γY = 0, the log-Pearson
type 3 distribution reduces to the two-parameter
lognormal distribution discussed above, because
in this case Y has a normal distribution. For the
lognormal distribution, the standard deviation σY
serves as the sole shape parameter, and the
coefficient of variation of X for small σY is just
σY. Figure 6.7 shows that the situation is more
complicated for the LP3 distribution. However,
for small σY, the coefficient of variation of X is
approximately σY.
Again, the flood flow data in Table 6.2 can be
used to illustrate parameter estimation. Using
natural logarithms, one can estimate the
log-space moments with the standard estimators
in Eqs. 6.39a that yield
^ ¼ 7:202
l
^ ¼ 0:5625
r
^c ¼ 0:337
For the LP3 distribution, analysis generally
focuses on the distribution of the logarithms
Y = ln(X) of the flows, which would have a

6.3 Distributions of Random Events

241

Fig. 6.7 Real-space
coefficient of variation CVX
for LP3 distributed X as a
function of log-space
standard deviation σY and
coefficient of skewness γY
where Y = ln(X)

Pearson type 3 distribution with moments µY, σY
and γY (IACWD 1982; Bobée and Ashkar 1991).
As a result, flood quantiles are calculated as
xp ¼ expflY þ rY K p ½cY g

ð6:89Þ

where Kp[γY] is a frequency factor corresponding
to cumulative probability for skewness coefficient γY. (Kp[γY] corresponds to the quantiles of a
three-parameter gamma distribution with zero
mean, unit variance, and skewness coefficient γY.)
Since 1967, the recommended procedure for
flood frequency analysis by federal agencies in
the United States uses this distribution. Current
guidelines in Bulletin 17B (IACWD 1982) suggest that the skew γY be estimated by a weighted
average of the at-site sample skewness coefficient
and a regional estimate of the skewness coefficient. Griffis and Stedinger (2007b) compare a
wide range of methods that have been recommended for fitting the LP3 distribution.

6.3.6 Gumbel and GEV Distributions
The annual maximum flood is the largest flood
flow during a year. One might expect that the
distribution of annual maximum flood flows

would belong to the set of extreme value distributions (Gumbel 1958; Kottegoda and Rosso
1997). These are the distributions obtained in the
limit, as the sample size n becomes large, by
taking the largest of n independent random
variables. The Extreme Value (EV) type I distribution or Gumbel distribution has often been
used to describe flood flows. It has the cumulative distribution function
FX ðxÞ ¼ expf exp½ðx  nÞ=ag

ð6:90Þ

with mean and variance of
lX ¼ n þ 0:5772a
r2X ¼ p2 a2 =6 ffi 1:645a2

ð6:91Þ

Its skewness coefficient has the fixed value
equal to γX = 1.1396.
The generalized extreme value (GEV) distribution is a general mathematical expression that
incorporates the type I, II, and III extreme value
(EV) distributions for maxima (Gumbel 1958;
Hosking et al. 1985). In recent years, it has been
used as a general model of extreme events
including flood flows, particularly in the context
of regionalization procedures (NERC 1975;
Stedinger and Lu 1995; Hosking and Wallis

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6 An Introduction to Probability, Statistics, and Uncertainty

1997). The GEV distribution has cumulative
distribution function
1=j

FX ðxÞ ¼ expf½1  jðx  nÞ=a

g

a ¼ jk2 =½Cð1 þ jÞð1  2j Þ

For κ > 0, floods must be less than the upper
bound for κ < 0, ξ < x < ∞, whereas for κ > 0,
ξ < x < ξ + α/κ (Hosking and Wallis 1987). The
mean, variance, and skewness coefficient are (for
κ > −1/3)
lX ¼ n þ ða=jÞ½1  Cð1 þ jÞ;
r2X ¼ ða=jÞ2 fCð1 þ 2jÞ  ½Cð1 þ jÞ2 g
cX ¼ SignðjÞfCð1 þ 3jÞ þ 3Cð1 þ jÞCð1 þ 2jÞ
 2½Cð1 þ jÞ3 g=fCð1 þ 2jÞ  ½Cð1 þ jÞ2 g3=2

ð6:93Þ
where Γ(1 + κ) is the classical gamma function.
The Gumbel distribution is obtained when κ = 0.
For |κ| < 0.3, the general shape of the GEV distribution is similar to the Gumbel distribution,
though the right-hand tail is thicker for κ < 0, and
thinner for κ > 0, as shown in Figs. 6.8 and 6.9.
The parameters of the GEV distribution are
easily computed using L-moments and the relationships (Hosking et al. (1985)

ð6:94Þ

n ¼ k1 þ ða=jÞ½Cð1 þ jÞ  1

for j 6¼ 0
ð6:92Þ

Fig. 6.8 GEV density
distributions for selected
shape parameter κ values

j ¼ 7:8590c þ 2:9554c2

where
c ¼ 2k2 =ðk3 þ 3k2 Þ  lnð2Þ= lnð3Þ
¼ ½2=ðs3 þ 3Þ  lnð2Þ= lnð3Þ
As one can see, the estimator of the shape
parameter κ will depend only upon the
L-skewness estimator ^s3 . The estimator of the
scale parameter α will then depend on the estimate of κ and of λ2. Finally, one must also use
the sample mean λ1 (Eq. 6.48) to determine the
estimate of the location parameter ξ.
Using the flood data in Table 6.2 and the
sample L-moments computed in Sect. 6.2, one
obtains first
c ¼ 0:000896
that yields
^ ¼ 0:007036
j
^n ¼ 1165:20
^a ¼ 657:29

6.3 Distributions of Random Events

243

Fig. 6.9 Right-hand tails
of GEV distributions
shown in Fig. 6.8

The small value of the fitted κ parameter
means that the fitted distribution is essentially a
Gumbel distribution. Here ξ is a location
parameter, not a lower bound, so its value
resembles a reasonable x value.
Madsen et al. (1997a) show that moment estimators can provide more precise quantile estimators. Yet Martins and Stedinger (2001a, b)
found that with occasional uninformative samples, the MLE estimator of κ could be entirely
unrealistic resulting in absurd quantile estimators.
However the use of a realistic prior distribution on
κ yielded better generalized maximum likelihood
estimators (GLME) than moment and L-moment
estimators over the range of κ of interest.
The GMLE estimators are obtained my
maximizing the log-likelihood function, augmented by a prior density function on κ. A prior
distribution that reflects general worldwide geophysical experience and physical realism is in the
form of a beta distribution
pðjÞ ¼ CðpÞCðqÞð0:5 þ jÞp1
ð0:5  jÞq1 =Cðp þ qÞ

ð6:95Þ

for −0.5 < κ < +0.5 with p = 6 and q = 9.
Moreover, this prior assigns reasonable probabilities to the values of κ within that range. For κ
outside the range −0.4 to +0.2 the resulting GEV
distributions do not have density functions consistent with flood flows and rainfall (Martins and

Stedinger 2000). Other estimators implicitly have
similar constraints. For example, L-moments
restricts κ to the range κ > −1, and the method
of moments estimator employs the sample standard deviation so that κ > −0.5. Use of the sample skew introduces the constraint that κ > −0.3.
Then given a set of independent observations
{x1, …, xn} drawn for a GEV distribution, the
generalized likelihood function is
lnfLðn; a; jjx1 ; . . .; xn Þg


n

X
1
¼ n lnðaÞ þ
 1 lnðyi Þ  ðyi Þ1=j
j
i¼1
þ ln½pðjÞ

with
yi ¼ ½1  ðj=aÞðxi  nÞ

ð6:96Þ

For feasible values of the parameters yi is
greater than 0 (Hosking et al. 1985). Numerical
optimization of the generalized likelihood function is often aided by the additional constraint
that min{y1, …, yn } ≥ ε for some small ε > 0 so
as to prohibit the search generating infeasible
values of the parameters for which the likelihood
function is undefined. The constraint should not
be binding at the final solution.
The data in Table 6.2 again provide a convenient data set for illustrating parameter estimators.
The L-moment estimators were used to generate
an initial solution. Numerical optimization of the

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6 An Introduction to Probability, Statistics, and Uncertainty

likelihood function Eq. 6.96 yielded the maximum likelihood estimators of the GEV
parameters
^ ¼ 0:0359
j
^
n ¼ 1165:4
^
a ¼ 620:2
Similarly, use of the geophysical prior
(Eq. 6.95) yielded the generalized maximum
likelihood estimators
^ ¼ 0:0823
j
^
n ¼ 1150:8
^
a ¼ 611:4
Here the record length of 40 years is too short
to reliably define the shape parameter κ so that
result of using the prior is to increase κ slightly
toward the mean of the prior. The other two
parameters adjust accordingly.

6.3.7 L-Moment Diagrams
This chapter has presented several families of
distributions. The L-moment diagram in
Fig. 6.10 illustrates the relationships between the
L-kurtosis (τ3) and L-skewness (τ2) for a number
of the families of distributions often used in
Fig. 6.10 Relationships
between L-skewness and
L-kurtosis for various
distributions

hydrology. It shows that distributions with the
same coefficient of skewness still differ in the
thickness of their tails, described by their kurtosis. Tail shapes are important if an analysis is
sensitive to the likelihood of extreme events.
The normal and Gumbel distributions have a
fixed shape and thus are presented by single
points that fall on the Pearson type 3 (P3) curve
for γ = 0, and the generalized extreme value
(GEV) curve for κ = 0, respectively. The
L-kurtosis/L-skewness relationships for the
two-parameter and three-parameter gamma or P3
distributions are identical, as they are for the
two-parameter and three-parameter lognormal
distributions. This is because the addition of a
location parameter does not change the range of
fundamental shapes that can be generated.
However, for the same skewness coefficient, the
lognormal distribution has a larger kurtosis than
the gamma or P3 distribution and thus assigns
larger probabilities to the largest events.
As the skewness of the lognormal and gamma
distributions approaches zero, both distributions
become normal and their kurtosis/skewness
relationships merge. For the same L-skewness,
the L-kurtosis of the GEV distribution is generally larger than that of the lognormal distribution. For positive κ yielding almost symmetric or
even negatively skewed GEV distributions, the
GEV has a smaller kurtosis than the threeparameter lognormal distribution. The latter can

6.3 Distributions of Random Events

245

be negatively skewed when τ is used as an upper
bound.
Figurer 6.10 also includes the three-parameter
generalized Pareto distribution, whose cdf is
FX ðxÞ ¼ 1  ½1  jðx  nÞ=a1=j

ð6:97Þ

(Hosking and Wallis 1997). For κ = 0 it corresponds to the exponential distribution (gamma
with α = 1). This point is where the Pareto and
P3 distribution L-kurtosis/L-skewness lines
cross. The Pareto distribution becomes increasing more skewed for κ < 0, which is the range of
interest in hydrology. The generalized Pareto
distribution with κ < 0 is often used to describe
peaks-over-a-threshold and other variables
whose density function has its maximum at their
lower bound. In that range for a given
L-skewness, the Pareto distribution always has a
larger kurtosis than the gamma distribution. In
these cases the α parameter for the gamma distribution would need to be in the range
0 < α < 1, so that both distributions would be
J-shaped.
As shown in Fig. 6.10, the GEV distribution
has a thicker right-hand tail than either the
gamma/Pearson type 3 distribution or the lognormal distribution.

6.4

Analysis of Censored Data

There are many instances in water resources
planning where one encounters censored data.
A data set is censored if the values of observations that are outside a specified range of values
are not specifically reported (David 1981). For
example, in water quality investigations many
constituents have concentrations that are reported
as <T, where T is a reliable detection threshold
(MacBerthouex and Brown 2002). Thus the
concentration of the water quality variable of
interest was too small to be reliably measured.
Likewise, low-flow observations and rainfall
depths can be rounded to or reported as zero.
Several approaches are available for analysis
of censored data sets including probability
plots and probability plot regression, conditional

probability models, and maximum likelihood
estimators (Haas and Scheff 1990; Helsel 1990;
Kroll and Stedinger 1996; MacBerthouex and
Brown 2002).
Historical and physical paleoflood data provide another example of censored data. Before the
beginning of a continuous measurement program
on a stream or river, the stages of unusually large
floods can be estimated based on the memories of
humans who have experienced these events
and/or physical markings in the watershed
(Stedinger and Baker 1987). Before continuous
measurements were taken that provided this
information, the annual maximum floods that
were not unusual were not recorded. These
missing data are censored data. They cover periods between occasionally large floods that have
been recorded or that have left some evidence of
their occurrence (Stedinger and Cohn 1986).
The discussion below addresses probability
plot methods for use with censored data. Probability plot methods have a long history of use
with censored data because they are relatively
simple to use and to understand. Moreover,
recent research has shown that they are relatively
efficient when the majority of values are
observed, and unobserved values are known only
to be below (or above) some detection limit or
perception threshold that serves as a lower (or
upper) bound. In such cases, probability plot
regression estimators of moments and quantiles
are as accurate as maximum likelihood estimators. They are almost as good as estimators
computed with complete samples (Helsel and
Cohn 1988; Kroll and Stedinger 1996).
Perhaps the simplest method for dealing with
censored data is adoption of a conditional probability model. Such models implicitly assume
that the data are drawn from one of two classes of
observations: those below a single threshold, and
those above the threshold. This model is appropriate for simple cases where censoring occurs
because small observations are recorded as
“zero,” as often happens with low-flow, low
pollutant concentration, and some flood records.
The conditional probability model introduces an
extra parameter P0 to describe the probability
that an observation is “zero.” If r of a total of

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6 An Introduction to Probability, Statistics, and Uncertainty

n observations were observed because they
exceeded the threshold, then P0 is estimated as
(n − r)/n. A continuous distribution GX(x) is
derived for the strictly positive “nonzero” values
of X. Then the parameters of the G distribution
can be estimated using any procedure appropriate
for complete uncensored samples. The unconditional
cumulative
distribution
function
(cdf) FX(x) for any value x > 0, is then
FX ðxÞ ¼ P0 þ ð1  P0 ÞGðxÞ

ð6:98Þ

This model completely decouples the value of
P0 from the parameters that describe the
G distribution.
Section 6.3.2 discusses probability plots and
plotting positions useful for graphical displaying
of data to allow a visual examination of the
empirical frequency curve. Suppose that among
n samples a detection limit is exceeded by the
observations r times. The natural estimator of the
exceedance probability P0 of the perception
threshold is again (n − r)/n. If the r values that
exceeded the threshold are indexed by i = 1, …,
r, wherein x(r) is the largest, then reasonable
plotting positions within the interval [P0, 1] arerno
pi ¼ P0 þ ð1  P0 Þ½ði  aÞ=ðr þ 1  2aÞ
ð6:99Þ
where a defines the plotting position that is used.
Helsel and Cohn (1988) show that reasonable
choices for a generally make little difference.
Letting a = 0 is reasonable (Hirsch and Stedinger
1987). Both papers discuss development of
plotting positions when there are different
thresholds, as occurs when the analytical precision of instrumentation changes over time. If
there are many exceedances of the threshold so
that r ≫ (1 − 2a), pi is indistinguishable from
p0i ¼ ½i þ ðn þ rÞ  a=ðn þ 1  2aÞ:

The idea behind the probability plot regression estimators is to use the probability plot for
the observed data to define the parameters of the
whole distribution. And if a sample mean, sample variance, or quantiles are needed, then the
distribution defined by the probability plot is
used to fill in the missing (censored) observations
so that standard estimators of the mean, of the
standard deviation, and of the quantiles can be
employed. Such fill-in procedures are efficient
and relatively robust for fitting a distribution and
estimating various statistics with censored water
quality data when a modest number of the
smallest observations are censored (Helsel 1990;
Kroll and Stedinger 1996).
Unlike the conditional probability approach,
here the below threshold probability P0 is linked
with the selected probability distribution for the
above-threshold observations. The observations
below the threshold are censored but are in all
other respects envisioned as coming from the
same distribution that is used to describe the
observed above-threshold values.
When water quality data are well described by
a lognormal distribution, available values ln
[X(1)] ≤  ≤ ln[X(r)] can be regressed upon
F−1[pi] = µ + σF−1[pi] for i = 1, …, r, where
the r largest observation in a sample of size n are
available. If regression yields constant m and
slope s, corresponding to population moments µ
and σ, a good estimator of the pth quantile is


xp ¼ exp m þ szp

where zp is the pth quantile of the standard normal distribution. To estimate sample means and
other statistics one can fill in the missing observations with
xðjÞ ¼ expfyðjÞg

for j ¼ 1; . . . ; ðn  rÞ
ð6:102Þ

ð6:100Þ

where again, i = 1, …, r. These values correspond to the plotting positions that would be
assigned to the largest r observations in a complete sample of n values.

ð6:101Þ

where
yðjÞ ¼ m þ sF 1 fP0 ½ðj  aÞ=ðn  r þ 1  2aÞg
ð6:103Þ

6.4 Analysis of Censored Data

Once a complete sample is constructed, standard estimators of the sample mean and variance
can be calculated, as can medians and ranges. By
filling in the missing small observations, and
then using complete-sample estimators of statistics of interest, the procedure is relatively
insensitive to the assumption that the observations actually have a lognormal distribution.
Maximum likelihood estimators are quite
flexible, and are more efficient than plotting
position methods when the values of the observations are not recorded because they are below
the perception threshold (Kroll and Stedinger
1996). Maximum likelihood methods allow the
observations to be represented by exact values,
ranges, and various thresholds that either were or
were not exceeded at various times. This can be
particularly important with historical flood data
sets because the magnitudes of many historical
floods are not recorded precisely, and it may be
known that a threshold was never crossed or was
crossed at most once or twice in a long period
(Stedinger and Cohn 1986; Stedinger 2000;
O’Connell et al. 2002). Unfortunately, maximum
likelihood estimators for the LP3 distribution
have proven to be problematic. However,
expected moment estimators seem to do as well
as MLEs with the LP3 distribution (Cohn et al.
1997, 2001).
While often a computational challenge, maximum likelihood estimators for complete samples, and samples with some observations
censored, pose no conceptual challenge. One
need to only write the maximum likelihood
function for the data and proceed to seek the
parameter values that maximizes that function.
Thus if F(x|θ) and f(x|θ) are the cumulative distribution and probability density functions that
should describe the data, and θ are its parameters,
then for the case described above wherein
x1, …, xr are r of n observations that exceeded a
threshold T, the likelihood function would be
(Stedinger and Cohn 1986)

247

Here (n − r) observations were below the
threshold T, and the probability an observation is
below T is F(T|θ) which then appears in
Eq. 6.104 to represent that observation. In addition the specific values of the r observations
x1, …, xr are available. The probability an
observation is in a small interval of width δr
around xi is δr f(xi|θ). Thus strictly speaking the
likelihood function also includes a term δr. Here
what is known of the magnitude of all of the
n observations is included in the likelihood
function in the appropriate way. If all that were
known of some observation was that it exceeded
a threshold M, then that value should be represented by a term [1 − F(M|θ)] in the likelihood
function. Similarly, if all that was known was
that the value was between L and M, then a term
[F(M|θ) − F(L|θ)] should be included in the
likelihood function. Different thresholds can be
used to describe different observations, corresponding to changes in the quality of measurement procedures. Numerical methods can be
used to identify the parameter vector that maximizes the likelihood function for the data
available.

6.5

Regionalization
and Index-Flood Method

Research has demonstrated the potential advantages of “index flood” procedures (Lettenmaier
et al. 1987; Stedinger and Lu 1995; Hosking and
Wallis 1997; Madsen and Rosbjerg 1997a). The
idea behind the index-flood approach is to use
the data from many hydrologically “similar”
basins to estimate a dimensionless flood distribution (Wallis 1980). Thus this method “substitutes space for time” using regional information
to compensate for having relatively short records
at each site. The concept underlying the
index-flood method is that the distributions of
floods at different sites in a “region” are the same
except for a scale or index-flood parameter that
ðnrÞ
Lðhjr; n; x1 ; . . .; xr Þ ¼ FðTjhÞ
f ðx1 jhÞf ðx2 jhÞ . . . f ðxr jhÞ
reflects the size, rainfall, and runoff characterisð6:104Þ tics of each watershed. Research is revealing

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6 An Introduction to Probability, Statistics, and Uncertainty

when this assumption may be reasonable. Often a
more sophisticated multi-scaling model is
appropriate (Gupta and Dawdy 1995a; Robinson
and Sivapalan 1997).
Generally the mean is employed as the index
flood. The problem of estimating the pth quantile
xp is then reduced to estimating the mean for a
site µx, and the ratio xp/µx of the pth quantile to
the mean. The mean can often be estimated
adequately with the record available at a site,
even if that record is short. The indicated ratio is
estimated using regional information. The British
Flood Studies Report (NERC 1975) calls these
normalized flood distributions growth curves.
Key to the success of the index-flood
approach is identification of sets of basins that
have similar coefficients of variation and skew.
Basins can be grouped geographically, as well as
by physiographic characteristics including drainage area and elevation. Regions need not be
geographically contiguous. Each site can potentially be assigned its own unique region consisting of sites with which it is particularly
similar (Zrinji and Burn 1994), or regional
regression equations can be derived to compute
normalized regional quantiles as a function of a
site’s physiographic characteristics and other
statistics (Fill and Stedinger 1998).
Clearly the next step for regionalization procedures, such as the index-flood method, is to
move away from estimates of regional parameters that do not depend upon basin size and other
physiographic parameters. Gupta et al. (1994)
argue that the basic premise of the index-flood
method, that the coefficient of variation of floods
is relatively constant, is inconsistent with the
known relationships between the coefficient of
variation CV and drainage area (see also
Robinson and Sivapalan 1997). Recently, Fill
and Stedinger (1998) built such a relationship
into an index-flood procedure using a regression
model to explain variations in the normalized
quantiles. Tasker and Stedinger (1986) illustrated
how one might relate log-space skew to physiographic basin characteristics (see also Gupta and
Dawdy 1995b). Madsen and Rosbjerg (1997b)
did the same for a regional model of κ for the

GEV distribution. In both studies, only a binary
variable representing “region” was found useful
in explaining variations in these two shape
parameters.
Once a regional model of alternative shape
parameters is derived, there may be some
advantage to combining such regional estimators
with at-site estimators employing an empirical
Bayesian framework or some other weighting
schemes. For example, Bulletin 17B recommends weighting at-site and regional skewness
estimators, but almost certainly places too much
weight on the at-site values (Tasker and
Stedinger 1986). Examples of empirical Bayesian procedures are provided by Kuczera (1982),
Madsen and Rosbjerg (1997b) and Fill and
Stedinger (1998). Madsen and Rosbjerg’s
(1997b) computation of a κ-model with a New
Zealand data set demonstrates how important it
can be to do the regional analysis carefully,
taking into account the cross-correlation among
concurrent flood records.
When one has relatively few data at a site, the
index-flood method is an effective strategy for
deriving flood frequency estimates. However, as
the length of the available record increases it
becomes increasingly advantageous to also use
the at-site data to estimate the coefficient of
variation as well. Stedinger and Lu (1995) found
that the L-moment/GEV index-flood method did
quite well for “humid regions” (CV ≈ 0.5) when
n < 25, and for semiarid regions (CV ≈ 1.0) for
n < 60, if reasonable care is taken in selecting
the stations to be included in a regional analysis.
However, with longer records it became advantageous to use the at-site mean and L-CV with a
regional estimator of the shape parameter for a
GEV distribution. In many cases this would
be roughly equivalent to fitting a Gumbel distribution corresponding to a shape parameter
κ = 0. Gabriele and Arnell (1991) develop the
idea of having regions of different size for different parameters. For realistic hydrologic
regions, these and other studies illustrate the
value of regionalizing estimators of the shape,
and often the coefficient of variation of a
distribution.

6.6 Partial Duration Series

6.6

Partial Duration Series

Two general approaches are available for modeling flood and precipitation series (Langbein
1949). An annual maximum series considers
only the largest event in each year. A partial
duration series (PDS) or peaks-over-threshold
(POT) approach includes all “independent” peaks
above a truncation or threshold level. An objection to using annual maximum series is that it
employs only the largest event in each year,
regardless of whether the second largest event in
a year exceeds the largest events of other years.
Moreover, the largest annual flood flow in a dry
year in some arid or semiarid regions may be
zero, or so small that calling them floods is
misleading. When considering rainfall series or
pollutant discharge events, one may be interested
in modeling all events that occur within a year
that exceed some threshold of interest.
Use of a partial duration series (PDS) framework avoids such problems by considering all
independent peaks that exceed a specified threshold. And, one can estimate annual exceedance
probabilities from the analysis of PDS. Arguments
in favor of PDS are that relatively long and reliable
PDS records are often available, and if the arrival
rate for peaks over the threshold is large enough
(1.65 events/year for the Poisson arrival with
exponential-exceedance model), PDS analyses
should yield more accurate estimates of extreme
quantiles than the corresponding annual maximum
frequency analyses (NERC 1975; Rosbjerg 1985).
However, when fitting a three-parameter distribution, there seems to be little advantage from
using a PDS approach over an annual maximum
approach, even when the partial duration series
includes many more peaks than the maximum
series because both contain the same largest events
(Martins and Stedinger 2001a).
A drawback of PDS analyses is that one must
have criteria to identify only independent peaks
(and not multiple peaks corresponding to the
same event). Thus PDS analysis can be more
complicated than analyses using annual maxima.
Partial duration models, perhaps with parameters
that vary by season, are often used to estimate

249

expected damages from hydrologic events when
more than one damage-causing event can occur
in a season or within a year (North 1980).
A model of a PDS series has at least two
components: first, one must model the arrival rate
of events larger than the threshold level; second,
one must model the magnitudes of those events.
For example, a Poisson distribution has often
been used to model the arrival of events, and an
exponential distribution to describe the magnitudes of peaks that exceed the threshold.
There are several general relationships
between the probability distribution for annual
maximum and the frequency of events in a partial
duration series. For a PDS model, let λ be the
average arrival rate of flood peaks greater than
the threshold x0 and let G(x) be the probability
that flood peaks, when they occur, are less than
x > x0, and thus those peaks fall in the range [x0,
x]. The annual exceedance probability for a
flood, denoted 1/Ta, corresponding to an annual
return period Ta, is related to the corresponding
exceedance probability qe = [1 − G(x)] for level
x in the partial duration series by
1=Ta ¼ 1  expfkqe g ¼ 1  expf1=Tp g
ð6:105Þ
where Tp = 1/(λqe) is the average return period
for level x in the PDS.
Many different choices for G(x) may be reasonable. In particular, the Generalized Pareto
distribution (GPD) is a simple distribution useful
for describing floods that exceed a specified
lower bound. The cumulative distribution function for the generalized three-parameter Pareto
distribution is
FX ðxÞ ¼ 1  ½1  jðx  nÞ=a1=j

ð6:106Þ

with mean and variance
lX ¼ n þ a=ð1 þ jÞj
r2X ¼ a2 =½ð1 þ jÞ2 ð1 þ 2jÞ

ð6:107Þ

where for κ < 0, ξ < x < ∞, whereas for κ > 0,
ξ < x < ξ + α/κ (Hosking and Wallis 1987).

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6 An Introduction to Probability, Statistics, and Uncertainty

A special case of the GPD is the two-parameter
exponential distribution obtain with κ = 0.
Method of moment estimators work relatively
well (Rosbjerg et al. 1992).
Use of a generalized Pareto distribution for G
(x) with a Poisson arrival model yields a GEV
distribution for the annual maximum series
greater than x0 (Smith 1984; Stedinger et al.
1993; Madsen et al. 1997a). The Poisson-Pareto
and Poisson-GPD models are a very reasonable
description of flood risk (Rosbjerg et al. 1992).
They have the advantage that they focus on the
distribution of the larger flood events, and
regional estimates of the GEV distribution’s
shape parameter κ from annual maximum and
PDS analyses can be used interchangeably.
Martins and Stedinger (2001a, b) compare PDS
estimation procedures a well as demonstrating
that use of the three-parameter Poisson-GPD
model instead of a three-parameter GEV distribution generally results in flood quantile estimators with the same precision.
Madsen and Rosbjerg (1997a) use a PoissonGPD model as the basis of a PDS index-flood
procedure. Madsen et al. (1997b) show that the
estimators are fairly efficient. They pooled
information from many sites to estimate the
single shape parameter κ and the arrival rate
where the threshold was a specified percentile of
the daily flow duration curve at each site. Then
at-site information was used to estimate the mean
above-threshold flood. Alternatively one could
use the at-site data to estimate the arrival rate as
well.

6.7

Stochastic Processes and Time
Series

Many important random variables in water
resources are functions whose values change
with time. Historical records of rainfall or
streamflow at a particular site are a sequence of
observations called a time series. In a time series,
the observations are ordered by time, and it is
generally the case that the observed value of the
random variable at one time influences the distribution of the random variable at later times.

This means that the observations are not independent. Time series are conceptualized as being
a single observation of a stochastic process,
which is a generalization of the concept of a
random variable.
This section has three parts. The first presents
the concept of stationarity and the basic statistics
generally used to describe the properties of a
stationary stochastic process. The second presents the definition of a Markov process and the
Markov chain model. Markov chains are a convenient model for describing many phenomena,
and are often used in synthetic flow generation
and optimization models. The third part discusses
the sampling properties of statistics used to
describe the characteristics of many time series.

6.7.1 Describing Stochastic Processes
A random variable whose value changes through
time according to probabilistic laws is called a
stochastic process. An observed time series is
considered to be one realization of a stochastic
process, just as a single observation of a random
variable is one possible value the random variable may assume. In the development here, a
stochastic process is a sequence of random
variables {X(t)} ordered by a discrete time index
t = 1, 2, 3, ….
The properties of a stochastic process must
generally be determined from a single time series
or realization. To do this several assumptions are
usually made. First, one generally assumes that
the process is stationary, at least in the short run.
This means that the probability distribution of the
process is not changing over some specified
interval of time. In addition, if a process is
strictly stationary, the joint distribution of the
random variables X(t1), …, X(tn) is identical to
the joint distribution of X(t1 + t), …, X
(tn + t) for any t; the joint distribution depends
only on the differences ti − tj between the times
of occurrence of the events. In other words, its
shape does not change over time if the distribution is stationary. In the long run, however,
because of climate and land changes, many
hydrologic distributions are not stationary, and

6.7 Stochastic Processes and Time Series

251

just how much they will change in the future is
uncertain.
For a stationary stochastic process, one can
write the mean and variance as
lX ¼ E½XðtÞ

ð6:109Þ

r2X ¼ Var ½XðtÞ

ð6:110Þ

and

Both are independent of time t. The autocorrelations, the correlation of X with itself, are
given by
qX ðkÞ ¼

Cov½XðtÞ; Xðt þ kÞ
r2X

ð6:111Þ

for any positive integer k (the time lag). These
are the statistics most often used to describe
stationary stochastic processes.
When one has available only a single time
series, it is necessary to estimate the values of μX,
r2X , and ρX(k) from values of the random variable
that one has observed. The mean and variance
are generally estimated essentially as they were
in Eq. 6.39a.
^X ¼ X ¼
l

^2X ¼
r

T
1X
Xt
T t¼1

T 
2
1X
Xt  X
T t¼1

ð6:112Þ

ð6:113Þ

while the autocorrelations ρX(k) can be estimated
as (Jenkins and Watts 1968)
^ x ðk Þ ¼ rk ¼
q

PTk

ðxt þ k  xÞðxt  xÞ
ð6:114Þ
PT
xÞ2
t¼1 ðxt  

t¼1

The sampling distribution of these estimators
depends on the correlation structure of the
stochastic process giving rise to the time series.
In particular, when the observations are positively correlated as is usually the case in natural
streamflows or annual benefits in a river basin

simulation, the variances of the estimated
x and r
^2X are larger than would be the case if the
observations were independent. It is sometimes
wise to take this inflation into account.
Section 6.7.3 discusses the sampling distribution
of these statistics.
All of this analysis depends on the assumption
of stationarity for only then do the quantities
defined in Eqs. 6.109–6.111 have the intended
meaning. Stochastic processes are not always
stationary. Urban development, deforestation,
agricultural development, climatic variability,
and changes in regional resource management
can alter the distribution of rainfall, streamflows,
pollutant concentrations, sediment loads, and
groundwater levels over time. If a stochastic
process is not essentially stationary over the time
span in question, then statistical techniques that
rely on the stationary assumption cannot be
employed and the problem generally becomes
much more difficult.

6.7.2 Markov Processes and Markov
Chains
A common assumption in many stochastic water
resources models is that the stochastic process X
(t) is a Markov process. A first-order Markov
process has the property that the dependence of
future values of the process on past values
depends only on the current value. In symbols for
k > 0,
FX ½Xðt þ kÞjXðtÞ; Xðt  1Þ; Xðt  2Þ; . . .
¼ FX ½Xðt þ kÞjXðtÞ
ð6:115Þ
For Markov processes, the current value
summarizes the state of the processes. As a
consequence, the current value of the process is
often referred to as the state. This makes physical
sense as well when one refers to the state or level
of an aquifer or reservoir.
A special kind of Markov process is one
whose state X(t) can take on only discrete values.
Such a processes is called a Markov chain. Often

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6 An Introduction to Probability, Statistics, and Uncertainty

in water resources planning, continuous
stochastic processes are approximated by Markov chains. This is done to facilitate the construction of simpler stochastic models. This
section presents the basic notation and properties
of Markov chains.
Consider a stream whose annual flow is to be
represented by a discrete random variable.
Assume that the distribution of streamflows is
stationary. In the following development, the
continuous random variable representing the
annual streamflows (or some other process) is
approximated by a random variable Qy in year y,
which takes on only n discrete values qi (each
value representing a continuous range or interval
of possible streamflows) with unconditional
probabilities pi where
n
X

pi ¼ 1

ð6:116Þ

i¼1

It is frequently the case that the value of Qy+1
is not independent of Qy. A Markov chain can
model such dependence. This requires specification of the transition probabilities pij,
pij ¼ Pr½Qy þ 1 ¼ qj jQy ¼ qi 

ð6:117Þ

A transition probability is the conditional
probability that the next state is qj, given that the
current state is qi. The transition probabilities
must satisfy
n
X

pij ¼ 1

for all i

ð6:118Þ

j¼1

Figure 6.11a, b show a possible set of transition probabilities in a matrix and as histograms.
Each element pij in the matrix is the probability
of a transition from streamflow qi in one year to
streamflow qj in the next. In this example, a low
flow tends to be followed by a low flow, rather
than a high flow, and vice versa.
Let P be the transition matrix whose elements
are pij. For a Markov chain, the transition matrix
contains all the information necessary to describe
the behavior of the process. Let pyi be the

probability that the process resides in state i in
year y. Then the probability that Qy+1 = qj is the
sum of the probabilities pyi that Qy = qi times the
probability pij that the next state is Qy+1 given that
Qy = qi. In symbols, this relationship is written
pjy þ 1 ¼ py1 p1j þ py2 p2j þ    þ pyn pnj ¼

n
X
i¼1

pyi pij

ð6:119Þ
Letting py be the row vector of state resident


probabilities pyi ; . . .; pyn , this relationship may
be written
pðy þ 1Þ ¼ pðyÞ P

ð6:120Þ

To calculate the probabilities of each streamflow state in year y + 2, one can use p(y+1)
in Eq. 6.120 to obtain p(y+2) = p(y+1)P or
p(y+2) = pyP2
Continuing in this matter, it is possible to
compute the probabilities of each possible
streamflow state for years y + 1, y + 2,
y + 3, …, y + k, … as
pðy þ kÞ ¼ py Pk

ð6:121Þ

Returning to the four-state example in
Fig. 6.11, assume that the flow in year y is in the
interval represented by q2. Hence in year y the
unconditional streamflow probabilities pyi are
(0, 1, 0, 0). Knowing each pyi , the probabilities
pjy þ 1 corresponding to each of the four streamflow states can be determined. From Fig. 6.11,
the probabilities pyj þ 1 are 0.2, 0.4, 0.3, and 0.1
for j = 1, 2, 3, and 4, respectively. The probability vectors for nine future years are listed in
Table 6.8.
As time progresses, the probabilities generally
reach limiting values. These are the unconditional or steady-state probabilities. The quantity
pi has been defined as the unconditional probability of qi. These are the steady-state probabilities which p(y+k) approaches for large k. It is clear
from Table 6.8 that as k becomes larger,
Eq. 6.119 becomes

6.7 Stochastic Processes and Time Series

253

(a)

(b)

Fig. 6.11 a Matrix of streamflow transition probabilities
showing probability of streamflow qj (represented by
index j) in year y + 1 given streamflow qi (represented by
index i) in year y. b Histograms (below) of streamflow

transition probabilities showing probability of streamflow
qj (represented by index j) in year y + 1 given streamflow
qi (represented by index i) in year y

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6 An Introduction to Probability, Statistics, and Uncertainty

Table 6.8 Successive streamflow probabilities based on transition probabilities in Fig. 6.11

pj ¼

n
X

pi pij

ð6:122Þ

i¼1

or in vector notation, Eq. 6.122 becomes
p ¼ pP

ð6:123Þ

where p is the row vector of unconditional
probabilities (p1, …, pn). For the example in
Table 6.8, the probability vector p equals (0.156,
0.309, 0.316, 0.219).
The steady-state probabilities for any Markov
chain can be found by solving simultaneous
Eqs. 6.123 for all but one of the states j together
with the constraint
n
X

pi ¼ 1

ð6:124Þ

i¼1

Annual streamflows are seldom as highly
correlated as the flows in this example. However,
monthly, weekly, and especially daily streamflows generally have high serial correlations.
Assuming that the unconditional steady-state
probability distributions for monthly streamflows are stationary, a Markov chain can be
defined for each month’s steamflow. Since there

are 12 months in a year, there would be 12
transition matrices, the elements of which could
be denoted as ptij . Each defines the probability of

a streamflow ptj þ 1 ðyÞ in month t + 1, given a
streamflow pti ðyÞ in month t. The steady-state
stationary probability vectors for each month can
be found by the procedure outlined above, except
that now all 12 matrices are used to calculate all
12 steady-state probability vectors. However,
once the steady-state vector p is found for one
month, the others are easily computed using
Eq. 6.121 with t replacing y.

6.7.3 Properties of Time Series
Statistics
The statistics most frequently used to describe
the distribution of a continuous-state stationary
stochastic process are the sample mean, variance,
and various autocorrelations. Statistical dependence among the observations, as is frequently
the case in time series, can have a marked effect
on the distribution of these statistics. This part of
Sect. 6.7 reviews the sampling properties of
these statistics when the observations are a realization of a stochastic process.

6.7 Stochastic Processes and Time Series

255

The sample mean
X¼

n
1X
Xi
n i¼1

ð6:125Þ

when viewed as a random variable is an unbiased
estimate of the mean of the process μX, because
E½X ¼

n
1X
E½Xi  ¼ lX
n i¼1

ð6:126Þ

However, correlation among the Xi’s, so that
ρX(k) ≠ 0 for k > 0, affects the variance of the
estimated mean X.
h
 
2 i
Var X ¼ E X  lX
(
)
n X
n
X
1
¼ 2E
ðXt  lX ÞðXs  lX Þ
n
t¼1 s¼1
(
)

n1
X
r2X
k
¼
1þ2
1  qX ðkÞ
n
n
k¼1
ð6:127Þ
The variance of X, equal to r2X =n for independent observations, is inflated by the factor
within the brackets. For ρX(k) ≥ 0, as is often the
case, this factor is a non-decreasing function of n,
so that the variance of X is inflated by a factor
whose importance does not decrease with
increasing sample size. This is an important
observation, because it means the average of a
correlated time series will be less precise than the
average of a sequence of independent random

variables of the same length with the same
variance.
A common model of stochastic series has
qX ðkÞ ¼ ½qX ð1Þk ¼ qk

ð6:128Þ

This correlation structure arises from the
autoregressive Markov model discussed at length
in Sect. 6.8. For this correlation structure
  r2
Var X ¼ X
n

(

2q ½nð1  qÞ  ð1  qn Þ
1þ
n
ð1  qÞ2

)

ð6:129Þ
Substitution of the sample estimates for r2X
and ρX(1) in the equation above often yields a
more realistic estimate of the variance of X than
does the estimate s2X =n if the correlation structure
ρX(k) = ρk is reasonable; otherwise, Eq. 6.127
may be employed. Table 6.9 illustrates the affect
of correlation among the Xt values on the standard error of their mean, equal to the square root
of the variance in Eq. 6.127.
The properties of the estimate of the variance
of X,
^2X ¼ v2X ¼
r

n 
2
1X
Xt  X
n t¼1

ð6:130Þ

are also affected by correlation among the Xt’s.
Here v rather than s is used to denote the variance
estimator because n is employed in the

Table 6.9 Standard error of X when σx = 0.25 and ρX(k) = ρk

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6 An Introduction to Probability, Statistics, and Uncertainty

denominator rather than n − 1. The expected
value of v2x becomes
(
)

n1
X
 2
1
2
k
E vX ¼ r2X 1  
1  qX ðkÞ
n n k¼1
n
ð6:131Þ
The bias in v2X depends on terms involving
ρX(1) through ρX(n − 1). Fortunately, the bias in
v2X decreases with n and is generally unimportant
when compared to its variance.
Correlation among the Xt’s also affects the
variance of v2X . Assuming that X has a normal
distribution (here the variance of v2X depends on
the fourth moment of X), the variance of v2X for
large n is approximately (Kendall and Stuart
1966, Sect. 48.1).
 
r4
Var v2X ffi 2 X
n

(
1þ2

1
X
k¼1

)
q2X ðkÞ

ð6:132Þ

where for ρX(k) = ρk, Eq. 6.132 becomes


 2
r4X 1 þ q2
Var vX ffi 2
n 1  q2

and ρX(k) = ρk. This would be the coefficient of
variation of v2X were it not biased.
A fundamental problem of time series analyses is the estimation or description of the relationship between the random variable at different
times. The statistics used to describe this relationship are the autocorrelations. Several estimates of the autocorrelations have been
suggested; a simple and satisfactory estimate
recommended by Jenkins and Watts (1968) is
^X ðkÞ ¼ rk ¼
q

Pnk

ðxt  xÞðxt þ k  xÞ
Pn
x Þ2
t¼1 ðxt  

t¼1

ð6:134Þ
Here, rk is the ratio of two sums where the
numerator contains n − k terms and the denominator contains n terms. The estimate rk is biased,
but unbiased estimates frequently have larger
mean square errors (Jenkins and Watts 1968).
A comparison of the bias and variance of r1 is
provided by the case when the Xt’s are independent normal variates. Then (Kendall and
Stuart 1966)

ð6:133Þ

Like the variance of X, the variance of v2X is
inflated by a factor whose importance does not
decrease with n. This is illustrated by Table 6.10
that gives the standard deviation of v2X divided by
the true variance r2X as a function of n and ρ
when the observations have a normal distribution

E½r1  ¼ 

1
n

ð6:135aÞ

and
Varðr1 Þ ¼

ð n  2Þ 2
1
ffi
n2 ð n  1Þ n

Table 6.10 Standard deviation of (v2X =r2X Þ when observations have a normal distribution and ρX(k) = ρk

ð6:135bÞ

6.7 Stochastic Processes and Time Series

257

For n = 25, the expected value of r1 is −0.04
rather than the true value of zero; its standard
deviation is 0.19. This results in a mean square
error of (E[r1])2 + Var(r1) = 0.0016 + 0.0353 =
0.0369. Clearly, the variance of r1 is the dominant term.
For Xt values that are not independent, exact
expressions for the variance of rk generally are
not available. However, for normally distributed
Xt and large n (Kendall and Stuart 1966),
Varðrk Þ ffi

þ1 
1 X
q2 ðlÞ þ qx ðl þ kÞqx ðl  kÞ
n l¼1 x

 4qx ðkÞqx ðlÞqx ðk  lÞ þ 2q2x ðkÞq2x ðlÞ
ð6:136Þ
If ρX(k) is essentially zero for k > q, then the
simpler expression (Box et al. 1994)
"

Varðrk Þ ffi

1
1þ2
n

q
X
t¼1

#
q2x ðlÞ

ð6:137Þ

is valid for rk corresponding to k > q; thus for
large n, Var(rk) ≥ l/n and values of rk will frepffiffiffi
quently be outside the range of ±1.65/ n, even
though ρx(k) may be zero.
If ρX(k) = ρk, Eq. 6.137 reduces to





1 ð1 þ q2 Þ 1  q2k
2k

2kq
Varðr k Þ ffi
n
1  q2
ð6:138Þ

In particular for rl, this gives
Varðr1 Þ ffi


1
1  q2
n

ð6:139Þ

Approximate values of the standard deviation
of rl for different values of n and ρ are given in
Table 6.11.
The estimates of rk and rk+j are highly correlated for small j; this causes plots of rk versus k to
exhibit slowly varying cycles when the true
values of ρX(k) may be zero. This increases the
difficulty
of
interpreting
the
sample
autocorrelations.

6.8

Synthetic Streamflow
Generation

6.8.1 Introduction
This section is concerned primarily with ways of
generating sample data such as streamflows,
temperatures, and rainfall that are used in water
resource systems simulation studies (e.g., as
introduced in the next section). The models and
techniques discussed in this section can be used
to generate any number of quantities used as
inputs to simulation studies. For example Wilks
(1998, 2002) discusses the generation of wet and
dry days, rainfall depths on wet days, and associated daily temperatures. The discussion here is
directed toward the generation of streamflows

Table 6.11 Approximate standard deviation of r1 when observations have a normal distribution and ρX(k) = ρk

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6 An Introduction to Probability, Statistics, and Uncertainty

because of the historic development and frequent
use of these models in that context (Matalas and
Wallis 1976). In addition, they are relatively
simple compared to more complete daily weather
generators and many other applications. Generated streamflows have been called synthetic to
distinguish them from historical observations
(Fiering 1967). The field has been called
stochastic hydrologic modeling. More detailed
presentations can be found in Marco et al. (1989)
and Salas (1993).
River basin simulation studies can use many
sets of streamflow, rainfall, evaporation, and/or
temperature sequences to evaluate the statistical
properties of the performance of alternative water
resources systems. For this purpose, synthetic
flows and other generated quantities should
resemble, statistically, those sequences that are
likely to be experienced during the planning
period. Figure 6.12 illustrates how synthetic
streamflow, rainfall, and other stochastic
sequences are used in conjunction with projections of future demands and other economic data
to determine how different system designs and
operating policies might perform.
Use of only the historical flow or rainfall
record in water resource studies does not allow
for the testing of alternative designs and policies
against the range of sequences that are likely to
occur in the future. We can be very confident that
the future historical sequence of flows will not be
the historical one, yet there is important

information in that historical record. That information is not fully used if only the historical
sequence is simulated. By fitting continuous distributions to the set of historical flows and then
using those distributions to generate other
sequences of flows, all of which are statistically
similar and equally likely, gives one a broader
range of inputs to simulation models. Testing
designs and policies against that broader range of
flow sequences that could occur more clearly
identifies the variability and range of possible
future performance indicator values. This in turn
should lead to the selection of more robust system
designs and policies.
The use of synthetic streamflows is particularly useful for water resource systems having
large amounts of over-year storage. Use of only
the historical hydrologic record in system simulation yields only one time history of how the
system would operate from year to year. In water
resource systems having relatively little storage
so that reservoirs and/or groundwater aquifers
refill almost every year, synthetic hydrologic
sequences may not be needed if historical
sequences of a reasonable length are available. In
this second case, a 25-year historic record provides 25 descriptions of the possible within-year
operation of the system. This may be sufficient
for many studies.
Generally, use of stochastic sequences is
thought to improve the precision with which
water resource system performance indices can

Fig. 6.12 Structure of a simulation study, indicating the transformation of a synthetic streamflow sequence, future
demands and a system design and operating policy into system performance statistics

6.8 Synthetic Streamflow Generation

be estimated, and some studies have shown this
to be the case (Vogel and Shallcross 1996; Vogel
and Stedinger 1988). In particular, if the operation of the system and performance indices have
thresholds and shape breaks, then the coarse
description provided by historical series are
likely to provide relative inaccurate estimates of
the expected values of such statistics. For
example, suppose that shortages only invoke a
nonlinear penalty function on average one year in
20. Then in a 60-year simulation there is a 19%
probability that the penalty will be invoked at
most once, and an 18% probability it will be
invoked five or more times. Thus the calculation
of the annual average value of the penalty would
be highly unreliable unless some smoothing of
the input distributions is allowed associated with
a long simulation analysis.
On the other hand, if one is only interested in
the mean flow, or average benefits that are mostly
a linear function of flows, then use of stochastic
sequences will probably add little information to
what is obtained simply by simulating the historical record. After all, the fitted models are ultimately based on the information provided in the
historical record, and their use does not produce
new information about the hydrology of the basin.
If in a general sense one has available N years
of record, the statistics of that record can be used
to build a stochastic model for generating thousands of years of flow. These synthetic data can
now be used to estimate more exactly the system
performance, assuming, of course, that the
flow-generating model accurately represents
nature. But the initial uncertainty in the model
parameters resulting from having only N years of
record would still remain (Schaake and Vicens
1980). An alternative is to run the historical
record (if it is sufficient complete at every site
and contains no gaps of missing data) through
the simulation model to generate N years of
output. That output series can be processed to
produce estimates of system performance. So the
question is: is it better to generate multiple input
series based on uncertain parameter values and
use those to determine average system performance with great precision, or is it sufficient to

259

just model the N-year output series that results
from simulation of the historical series?
The answer seems to depend upon how well
behaved the input and output series are. If the
simulation model is linear, it does not make
much difference. If the simulation model were
highly nonlinear, then modeling the input series
would appear to be advisable. Or if one is
developing reservoir operating policies, there is a
tendency to make a policy sufficiently complex
that it deals very well with the few droughts in
the historical record but at the same time giving a
false sense of security and likely misrepresenting
the probability of system performance failures.
Another
situation
where
stochastic
data-generating models are useful is when one
wants to understand the impact on system performance estimates of the parameter uncertainty
stemming from short historical records. In that
case, parameter uncertainty can be incorporated
into streamflow generating models so that the
generated sequences reflect both the variability
that one would expect in flows over time as well
as the uncertainty of the parameter values of the
models that describe that variability (Valdes et al.
1977; Stedinger and Taylor 1982a, b; Stedinger
Pei and Cohn 1985; Vogel and Stedinger 1988).
If one decides to use a stochastic data generator, the challenge is to use a model that appropriately describes the important relationships, but
does not attempt to reproduce more relationships
than are justified or that can be estimated with
available data sets.
Two basic techniques are used for streamflow
generation. If the streamflow population can be
described by a stationary stochastic process, a
process whose parameters do not change over
time, and if a long historical streamflow record
exists, then a stationary stochastic streamflow
model may be fit to the historical flows. This
statistical model can then generate synthetic
sequences that describe selected characteristics of
the historical flows. Several such models are
discussed below.
The assumption of stationarity is not always
plausible, particularly in river basins that have
experienced marked changes in runoff

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6 An Introduction to Probability, Statistics, and Uncertainty

characteristics due to changes in land cover, land
use, climate, or the use of groundwater during the
period of flow record. Similarly, if the physical
characteristics of a basin will change substantially in the future, the historical streamflow
record may not provide reliable estimates of the
distribution of future unregulated flows. In the
absence of the stationarity of streamflows or a
representative historical record, an alternative
scheme is to assume that precipitation is a stationary stochastic process and to route either
historical or synthetic precipitation sequences
through an appropriate rainfall-runoff model of
the river basin.

6.8.2 Streamflow Generation Models
A statistical streamflow generation model is used
to generate streamflow data that can supplement
or replace historical streamflow data in various
analyses requiring such data. If the past flow
record is considered representative of what the
future one might be, at least for a while, then the
statistical characteristics of the historical flow
record can be used as a basis for generating new
flow data. While this may be a reasonable
assumption in the near future, changing land uses
and climate may lead to entirely different statistical characteristics of future streamflows, if not
now, certainly in the more distant future. By then,
improved global climate models (GCMs) and
downscaling methods together with improved
rainfall-runoff predictions given future land use
scenarios may be a preferred way to generate
future streamflows. This section of the chapter
will focus on the use of historical records.
The first step in the construction of a statistical
streamflow generating model based on historical
flow records is to extract from the historical
streamflow record the fundamental information
about the joint distribution of flows at different sites
and at different times. A streamflow model should
ideally capture what is judged to be the fundamental
characteristics of the joint distribution of the flows.
The specification of what characteristics are fundamental is of primary importance.

One may want to model as closely as possible
the true marginal distribution of seasonal flows
and/or the marginal distribution of annual flows.
These describe both how much water may be
available at different times and also how variable is
that water supply. Also, modeling the joint distribution of flows at a single site in different months,
seasons, and years may be appropriate. The persistence of high flows and of low flows, often
described by their correlation, affects the reliability with which a reservoir of a given size can
provide a given yield (Fiering 1967; Lettenmaier
and Burges 1977a, b; Thyer and Kuczera 2000).
For multicomponent reservoir systems, reproduction of the joint distribution of flows at different
sites and at different times will also be important.
Sometimes, a streamflow model is said to
statistically resemble the historical flows if the
streamflow model produces flows with the same
mean, variance, skew coefficient, autocorrelations, and/or cross-correlations as were observed
in the historic series. This definition of statistical
resemblance is attractive because it is operational
and requires that an analyst need only find a
model that can reproduce the observed statistics.
The drawback of this approach is that it shifts the
modeling emphasis away from trying to find a
good model of marginal distributions of the
observed flows and their joint distribution over
time and over space, given the available data, to
just reproducing arbitrarily selected statistics.
Defining statistical resemblance in terms of
moments may also be faulted for specifying that
the parameters of the fitted model should be
determined using the observed sample moments,
or their unbiased counterparts. Other parameter
estimation techniques, such as maximum likelihood estimators, are often more efficient. Definition of resemblance in terms of moments can also
lead to confusion over whether the population
parameters should equal the sample moments, or
whether the fitted model should generate flow
sequences whose sample moments equal the
historical values—the two concepts are different
because of the biases (as discussed in Sect. 6.7) in
many of the estimators of variances and correlations (Matalas and Wallis 1976; Stedinger 1980,
1981; Stedinger and Taylor 1982a).

6.8 Synthetic Streamflow Generation

For any particular river basin study, one must
determine what streamflow characteristics need
to be modeled. The decision should depend on
what characteristics are important to the operation of the system being studied, the data available, and how much time can be spared to build
and test a stochastic model. If time permits, it is
good practice to see if the simulation results are
in fact sensitive to the generation model and its
parameter values using an alternative model and
set of parameter values. If the model’s results are
sensitive to changes, then, as always, one must
exercise judgment in selecting the appropriate
model and parameter values to use.
This section presents a range of statistical
models for the generation of synthetic data. The
necessary sophistication of a data-generating
model depends on the intended use of the data.
Section 6.8.3 below presents the simple autoregressive Markov model for generating annual
flow sequences. This model alone is too simple
for many practical studies, but is useful for
illustrating the fundamentals of the more complex models that follow. Therefore, considerable
time is spent exploring the properties of this
basic model.
Subsequent sections discuss how flows with
any marginal distribution can be produced and
present models for generating sequences of flows
that can reproduce the persistence of historical
flow sequences. Other parts of this section present models to generate concurrent flows at
several sites and to generate seasonal or monthly
flows while preserving the characteristics of
annual flows. For those wishing to study synthetic streamflow models in greater depth more
advanced material can be found in Marco et al.
(1989) and Salas (1993).

6.8.3 A Simple Autoregressive Model
A simple model of annual streamflows is the
autoregressive Markov model. The historical
annual flows qy are thought of as a particular value

261

of a stationary stochastic process Qy. The generation of annual streamflows and other variables
would be a simple matter if annual flows were
independently distributed. In general, this is not the
case and a generating model for many phenomena
should capture the relationship between values in
different years or in different periods. A common
and reasonable assumption is that annual flows are
the result of a first-order Markov process.
Assume also that annual streamflows are
normally distributed. In some areas, the distribution of annual flows is in fact nearly normal.
Streamflow models that produce nonnormal
streamflows are discussed as an extension of this
simple model.
The joint normal density function of two
streamflows Qy and Qw in years y and w having
mean μ, variance σ2, and year-to-year correlation
ρ between flows is


f qy ; qw ¼

1
2pr2 ð1  q2 Þ0:5
"
#
2


qy  l 2q qy  l ðqw  lÞ þ ðqw  lÞ2
 exp
2r2 ð1  q2 Þ

ð6:140Þ
The joint normal distribution for two random
variables with the same mean and variance
depend only on their common mean μ, variance
σ2, and the correlation ρ between the two (or
equivalently the covariance ρσ2).
The sequential generation of synthetic streamflows requires the conditional distribution of the
flow in one year given the value of the flows in
previous years. However, if the streamflows are a
first-order (lag 1) Markov process, then the
dependence of the distribution of the flow in year
y + 1 on flows in previous years depends entirely
on the value of the flow in year y. In addition, if the
annual streamflows have a multivariate normal
distribution, then the conditional distribution of
Qy+1 is normal with mean and variance
E½Qy þ 1 jQy ¼ qy  ¼ l þ qðqy  lÞ
VarðQy þ 1 jQy ¼ qy Þ ¼ r2 ð1  q2 Þ

ð6:141Þ

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6 An Introduction to Probability, Statistics, and Uncertainty

where qy is the value of Qy in year y. Notice that
the larger the absolute value of the correlation ρ
between the flows, the smaller the conditional
variance of Qy+1, which in this case does not
depend at all on the value qy.
Synthetic normally distributed streamflows
that have mean μ, variance σ2, and year-to-year
correlation ρ, are produced by the model
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Qy þ 1 ¼ l þ qðQy  lÞ þ Vy r 1  q2
ð6:142Þ
where Vy is a standard normal random variable,
meaning that it has zero mean, E[Vy] = 0, and
h i
unit variance, E Vy2 ¼ 1. The random variable
Vy is added here to provide the variability in Qy+1
that remains even after Qy is known. By construction, each Vy is independent of past flows
Qw where w ≤ y, and Vy is independent of Vw for
w ≠ y. These restrictions imply that


E Vw Vy ¼ 0

w 6¼ y

ð6:143Þ

and
E½ðQw  lÞVy  ¼ 0

wy

ð6:144Þ

Clearly, Qy+1 will be normally distributed if
both Qy and Vy are normally distributed because
sums of independent normally distributed random variables are normally distributed.
It is a straightforward procedure to show that
this basic model indeed produces streamflows
with the specified moments, as demonstrated
below.
Using the fact that E[Vy] = 0, the conditional
mean of Qy+1 given that Qy equals qy is
pffiffiffiffiffiffiffiffiffiffiffiffiffi
E½Qy þ 1 jqy  ¼ E½l þ qðqy  lÞ þ Vy r 1  q2 
¼ l þ qðqy  lÞ
ð6:145Þ
Since E{V2y ] = Var[Vy] = 1, the conditional
variance of Qy+1 is

Var½Qy þ 1 jqy  ¼ E½fQy þ 1  E½Qy þ 1 jqy g2 jqy 
pffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ E½fl þ qðqy  lÞ þ Vy r 1  q2
 ½l þ qðqy  lÞg2
pffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ E½Vy r 1  q2 2 ¼ r2 ð1  q2 Þ

ð6:146Þ
Thus this model produces flows with the
correct conditional mean and variance.
To compute the unconditional mean of Qy+1
one first takes the expectation of both sides of
Eq. 6.142 to obtain


 
E Qy þ 1 ¼ l þ qðE Qy  lÞ
  pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ E V y r 1  q2

ð6:147Þ

where E[Vy] = 0. If the distribution of streamflows is independent of time so that for all y,
E[Qy+1] = E[Qy] = E[Q], it is clear that (1 − ρ)
E[Q] = (1 − ρ)μ or
E½Q ¼ l

ð6:148Þ

Alternatively, if Qy for y = 1 has mean μ, then
Eq. 6.147 indicates that Q2 will have mean μ.
Thus repeated application of the Eq. 6.147 would
demonstrate that all Qy for y > 1 have mean μ.
The unconditional variance of the annual
flows can be derived by squaring both sides of
6.142 to obtain
pffiffiffiffiffiffiffiffiffiffiffiffiffi 2
1  q2 g 
pffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
¼ q E½ðQy  lÞ  þ 2qr 1  q2 E½ðQy  lÞVy 
h i
þ r2 ð1  q2 ÞE Vy2

E½ðQy þ 1  lÞ2  ¼ E½fqðQy  lÞ þ Vy r

ð6:149Þ
Because Vy is independent of Qy (Eq. 6.144),
the second term on the right-hand side of
Eq. 6.149 vanishes. Hence the unconditional
variance of Q satisfies
E½ðQy þ 1  lÞ2  ¼ q2 E½ðQy  lÞ2  þ r2 ð1  q2 Þ
ð6:150Þ

6.8 Synthetic Streamflow Generation

263

Assuming that Qy+1 and Qy have the same
variance yields
E½ðQ  lÞ2  ¼ r2

ð6:151Þ

so that the unconditional variance is σ2, as
required.
Again, if one does not want to assume that
Qy+1 and Qy have the same variance, a recursive
argument can be adopted to demonstrate that if
Q1 has variance σ2, then Qy for y ≥ 1 has variance σ2.
The covariance of consecutive flows is
another important issue. After all the whole idea
of building these time series models is to
describe the year-to-year correlation of the flows.
Using Eq. 6.142 one can compute that the
covariance of consecutive flows must be.
E½ðQy þ 1  lÞðQy  lÞ ¼ Ef½qðQy  lÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ Vy r 1  q2 ðQy  lÞg
2

¼ qE½ðQy  lÞ  ¼ qr

2

ð6:152Þ
where E[(Qy − μ)Vy] = 0 because Vy and Qy are
independent (Eq. 6.144).
Over a longer time scale, another property of
this model is that the covariance of flows in year
y and y + k is
E½ðQy þ k  lÞðQy  lÞ ¼ qk r2

ð6:153Þ

This equality can be proven by induction. It
has already been shown for k = 0 and 1. If it is
true for k = j − 1, then
E½ðQy þ j  lÞðQy  lÞ ¼ Ef½qðQy þ j1  lÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ Vy þ j1 r 1  q2 ðQy  lÞg
¼ qE½ðQy  lÞðQy þ j1  lÞ
¼ q½qj1 r2  ¼ q j r2

ð6:154Þ
where E[(Qy − μ)Vy+j−1] = 0 for j ≥ 1. Hence
Eq. 6.153 is true for any value of k.

It is important to note that the results in
Eqs. 6.145 to 6.153 do not depend on the
assumption that the random variables Qy and Vy
are normally distributed. These relationships
apply to all autoregressive Markov processes of
the form in Eq. 6.142 regardless of the distributions of Qy and Vy. However, if the flow Qy in year
y = 1 is normally distributed with mean μ and
variance σ2, and if the Vy are independent normally
distributed random variables with mean zero and
unit variance, then the generated Qy for y ≥ 1 will
also be normally distributed with mean μ and
variance σ2. The next section considers how this
and other models can be used to generate streamflows that have other than a normal distribution.

6.8.4 Reproducing the Marginal
Distribution
Most models for generating stochastic processes
deal directly with normally distributed random
variables. Unfortunately, flows are not always
adequately described by the normal distribution.
In fact, streamflows and many other hydrologic
data cannot really be normally distributed because
of the impossibility of negative values. In general,
distributions of hydrologic data are positively
skewed having a lower bound near zero and, for
practical purposes, an unbounded right-hand tail.
Thus they look like the gamma or lognormal distribution illustrated in Figs. 6.3 and 6.4.
The asymmetry of a distribution is often
measured by its coefficient of skewness. In some
streamflow models, the skew of the random
elements Vy is adjusted so that the models generate flows with the desired mean, variance, and
skew coefficient. For the autoregressive Markov
model for annual flows
h
pffiffiffiffiffiffiffiffiffiffiffiffiffii3
E½ðQy þ 1  lÞ3  ¼ E qðQy  lÞ þ Vy r 1  q2
¼ q3 E½ðQy  lÞ3 

h i
þ r3 ð1  q2 Þ3=2 E Vy3

ð6:155Þ

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6 An Introduction to Probability, Statistics, and Uncertainty

denoted ρX(k; s, u), and their variances r2X ðsÞ and
r2X ðuÞ, where

so that
cQ ¼

E½ðQ  lÞ3  ð1  q2 Þ3=2
¼
r3
1  q3

ð6:156Þ

By appropriate choice of the skew of Vy, the
desired skew coefficient of the annual flows can
be produced. This method has often been used to
generate flows that have approximately a gamma
distribution using Vy’s with a gamma distribution
and the required skew. The resulting approximation is not always adequate (Lettenmaier and
Burges 1977a).
The alternative and generally preferred
method is to generate normal random variables
and then transform these variates to streamflows
with the desired marginal distribution. Common
choices for the distribution of streamflows are the
two-parameter and three-parameter lognormal
distributions or a gamma distribution. If Qy is a
lognormally distributed random variable, then
Qy ¼ s þ expðXy Þ

ð6:157Þ

where Xy is a normal random variable; when the
lower bound τ is zero, Qy has a two-parameter
lognormal distribution. Equation 6.157 transforms the normal variates Xy into lognormally
distributed streamflows. The transformation is
easily inverted to obtain
Xy ¼ lnðQy  sÞ

for Qy  s

ð6:158Þ

where Qy must be greater than its lower bound τ.
The mean, variance, skewness of Xy and Qy
are related by the formulas (Matalas 1967)
1 2
r Þ
2 X
r2Q ¼ expð2lX þ r2X Þ½expðr2X Þ  1

lQ ¼ s þ expðlX þ

cQ ¼

expð3r2X Þ  3 expðr2X Þ þ 2

ð6:159Þ

½expðr2X Þ  13=2

If normal variates Xys and Xyu are used to
generate lognormally distributed streamflows
Qsy and Quy at sites s and u, then the lag-k correlation of the Qy’s, denoted ρQ(k; s, u), is determined by the lag-k correlation of the X variables,

qQ ðk; s; uÞ ¼

exp½qX ðk; s; uÞrX ðsÞrX ðuÞ  1
1=2

fexp½r2X ðsÞ  1g

1=2

fexp½r2X ðuÞ  1g

ð6:160Þ
The correlations of the X sy can be adjusted, at
least in theory, to produce the observed correlations among the Qsy variates. However, more
efficient estimates of the true correlation of the
Qsy values are generally obtained by transforming
the historical flows qsy into their normal equivalent xsy ¼ ‘nðqsy  sÞ and using the historical
correlations of these xsy values as estimators of
ρX(k; s, u) (Stedinger 1981).
Some insight into the effect of this logarithmic
transformation can be gained by considering the
resulting model for annual flows at a single site.
If the normal variates follow the simple autoregressive Markov model
Xy þ 1  l ¼ qX ðXy  lÞ þ Vy rX

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1  q2X
ð6:161Þ

then the corresponding Qy follow the model
(Matalas 1967)
Qy þ 1 ¼ s þ Dy fexp½lx ð1  qX ÞgðQy  sÞqX
ð6:162Þ
where
Dy ¼ exp½ð1  q2X Þ1=2 rX Vy 

ð6:163Þ

The conditional mean and standard deviation
of Qy+1 given that Qy = qy now depend on
ðqy  sÞqX . Because the conditional mean of Qy+1
is no longer a linear function of qy, the streamflows are said to exhibit differential persistence:
low flows are now more likely to follow low
flows than high flows are to follow high flows.
This is a property often attributed to real
streamflow distributions. Models can be constructed to capture the relative persistence of wet
and dry periods (Matalas and Wallis 1976; Salas

6.8 Synthetic Streamflow Generation

265

1993; Thyer and Kuczera 2000). Many weather
generators for precipitation and temperature natural include such differences by employing a
Markov chain description of the occurrence of
wet and dry days (Wilks 1998).

6.8.5 Multivariate Models
If long concurrent streamflow records can be
constructed at the several sites at which synthetic
streamflows are desired, then ideally a general
multisite streamflow model could be employed.
O’Connell (1977), Ledolter (1978), Salas et al.
(1980) and Salas (1993) discuss multivariate
models and parameter estimation. Unfortunately,
model identification (parameter value estimation)
is very difficult for the general multivariate
models.
This section illustrates how the basic univariate annual flow model in Sect. 8.3 can be
generalized to the multivariate case. This exercise reveals how easily multivariate models can
be constructed to reproduce specified variances
and covariances of the flow vectors of interest, or
some transformation of those values. This multisite generalization of the annual AR(1) or
autoregressive Markov model follows the
approach taken by Matalas and Wallis (1976).
This general approach can be further extended to
multisite/multiseason modeling procedures, as is
done in the next section employing what have
been called disaggregation models. However,
while the size of the model matrices and vectors
increases, the models are fundamentally the same
from a mathematical viewpoint. Hence this section starts with the simpler case of an annual flow
model.
For simplicity of presentation and clarity,
vector notation is employed. Let Zy ¼
ðZy1 ;

. . . ; Zyn ÞT

be the column vector of transformed zero-mean annual flows at sites s = 1,
2, …, n, so that
E½Z sy  ¼ 0

ð6:164Þ


T
be a
In addition, let Vy ¼ Vy1 ; . . .; Vyn
column vector of standard normal random variables, where V sy is independent of V rw for (r, w) ≠
(s, y) and independent of past flows Zwr where
y ≥ w. The assumption that the variables have
zero mean implicitly suggests that the mean
value has already been subtracted from all the
variables. This makes the notation simpler and
eliminates the need to include a constant term in
the models. With all the variables having zero
mean, one can focus on reproducing the variances and covariances of the vectors included in
a model.
A sequence of synthetic flows can be generated by the model
Zy þ 1 ¼ AZy þ BVy

ð6:165Þ

where A and B are (n × n) matrices whose elements are chosen to reproduce the lag 0 and lag 1
cross-covariances of the flows at each site. The
lag 0 and lag 1 covariances and cross-covariances
can most economically be manipulated by use of
the two matrices S0 and S1; the lag-zero covariance matrix, denoted S0, is defined as
S0 ¼ E½Zy ZTy 

ð6:166Þ

and has elements
h
i
S0 ði; jÞ ¼ E Z iy Z yj

ð6:167Þ

The lag-one covariance matrix, denoted S1, is
defined as
S1 ¼ E½Zy þ 1 ZTy 

ð6:168Þ

and has elements
h
i
S1 ði; jÞ ¼ E Z iy þ 1 Z yj

ð6:169Þ

The covariances do not depend on y because
the streamflows are assumed to be stationary.
Matrix S1 contains the lag 1 covariances and
lag 1 cross-covariances. S0 is symmetric because

266

6 An Introduction to Probability, Statistics, and Uncertainty

the cross covariance S0(i, j) equals S0(j, i). In
general, S1 is not symmetric.
The variance–covariance equations that define
the values of A and B in terms of S0 and S1 are
obtained by manipulations of Eq. 6.165. Multiplying both sides of that equation by ZTy and
taking expectations yields
h

i

h

E Zy þ 1 ZTy ¼ E AZy ZTy

i

h
i
þ E BVy ZTy

h
i
mean. E Vy VTy equals the identity matrix
because the components of Vy are independently
distributed with unit variance. Thus
S0 ¼ AS0 AT þ BBT

ð6:174Þ

Solving of the B matrix one finds that it
should satisfy

ð6:170Þ

T
BBT ¼ S0  AS0 AT ¼ S0  S1 S1
0 S1 ð6:175Þ

The second term on the right-hand side vanishes because the components of Zy and Vy are
independent. Now the first term in Eq. 6.170,
h
i
E AZy ZTy , is a matrix whose (i, j)th element

The last equation results from substitution of
the relationship for A given in Eq. 6.172 and the
fact that S0 is symmetric; hence S1
is
0
symmetric.
It should not be too surprising that the elements of B are not uniquely determined by
Eq. 6.175. The components of the random vector
Vy may be combined in many ways to produce
the desired covariances as long as B satisfies
Eq. 6.175. A lower triangular matrix that satisfies
Eq. 6.175 can be calculated by Cholesky
decomposition (Young 1968; Press et al. 1986).
Matalas and Wallis (1976) call Eq. 6.165 the
lag-1 model. They did not call the lag-1 model a
Markov model because the streamflows at individual sites do not have the covariances of an
autoregressive Markov process given in
Eq. 6.153. They suggest an alternative model
they call the Markov model. It has the same
structure as the lag-1 model except it does not
preserve the lag-1 cross-covariances. By relaxing
this requirement, they obtain a simpler model
with fewer parameters that generates flows that
have the covariances of an autoregressive Markov process at each site. In their Markov model,
the new A matrix is simply a diagonal matrix
whose diagonal elements are the lag-1 correlations of flows at each site

equals
"
E

n
X
k¼1

#
aik Zyk Zyi

¼

n
X
k¼1

aik E½Zyk Zyi  ð6:171Þ

The matrix with these elements is the same as
h
i
the matrix AE Zy ZTy .
Hence, A—the matrix of constants—can be
pulled through the expectation operator just as is
done in the scalar case where E[aZy + b] = aE
[Zy] + b for fixed constants a and b.
Substituting S0 and S1 for the appropriate
expectations in Eq. 6.170 yields
S1 ¼ AS0

or

A ¼ S1 S1
0

ð6:172Þ

A relationship to determine the matrix B is
obtained by multiplying both sides of Eq. 6.165
by its own transpose (this is equivalent to
squaring both sides of the scalar equation a = b)
and taking expectations to obtain
E½Zy þ 1 ZTy þ 1  ¼ E½AZy ZTy AT  þ E½AZy VTy BT 


þ E BVy Zy AT þ E½BVy VTy BT 
ð6:173Þ
The second and third terms on the right-hand
side of Eq. 6.173 vanish because the components
of Zy and Vy are independent and have zero

A ¼ diag½qð1; i; iÞ

ð6:176Þ

where ρ(1; i, i) is the lag-one correlation of flows
at site i.
The corresponding B matrix depends on the
new A matrix and S0, where as before

6.8 Synthetic Streamflow Generation

BBT ¼ S0  AS0 AT

267

ð6:177Þ

The idea of fitting time series models to each
site separately and then correlating in innovations
in those separate models to reproduce the
cross-correlation between the series is a very
general and powerful modeling idea that has seen
a number of applications with different time
series models (Matalas and Wallis 1976;
Stedinger et al. 1985; Camacho et al. 1985; Salas
1993).

6.8.6 Multiseason, Multisite Models
In most studies of surface water systems it is
necessary to consider the variations of flows
within each year. Streamflows in most areas have
within-year variations, exhibiting wet and dry
periods. Similarly, water demands for irrigation,
municipal, and industrial uses also vary, and the
variations in demand are generally out of phase
with the variation in within-year flows; more
water is usually desired when streamflows are
low and less is desired when flows are high. This
increases the stress on water delivery systems
and makes it all the more important that time
series models of streamflows, precipitation and
other hydrological variables correctly reproduce
the seasonality of hydrological processes.
This section discusses two approaches to
generating within-year flows. The first approach
is based on the disaggregation of annual flows
produced by an annual flow generator to seasonal
flows. Thus the method allows for reproduction
of both the annual and seasonal characteristics of
streamflow series. The second approach generates seasonal flows in a sequential manner, as
was done for the generation of annual flows.
Thus the models are a direct generalization of the
annual flow models already discussed.

6.8.6.1 Disaggregation Model
The disaggregation model proposed by Valencia
and Schaake (1973) and extended by Mejia and
Rousselle (1976) and Tao and Delleur (1976)
allows for the generation of synthetic flows that

reproduce statistics both at the annual level and
at the seasonal level. Subsequent improvements
and variations are described by Stedinger and
Vogel (1984), Maheepala and Perera (1996),
Koutsoyiannis and Manetas (1996) and Tarboton
et al. (1998).
Disaggregation models can be used for either
multiseason single-site or multisite streamflow
generation. They represent a very flexible modeling framework for dealing with different time
or spatial scales. Annual flows for the several
sites in question or the aggregate total annual
flow at several sites can be the input to the model
(Grygier and Stedinger 1988). These must be
generated by another model, such as those discussed in the previous sections. These annual
flows or aggregated annual flows are then disaggregated to seasonal values.

T
Let Zy ¼ Z 1y ; . . .; Z Ny
be the column vector of N transformed normally distributed annual
or aggregate annual flows for N separate sites or

1
1
2
; . . .; XT;y
; X1;y
; . . .;
basins. Next let Xy = X1;y
T
2
n
n
XTy
; . . .; X1y
; . . .; XTy
be the column vector
of nT transformed normally distributed seasonal
flows X sty for season t, year y, and site s.
Assuming that the annual and seasonal series,
Z sy and X sty , have zero mean (after the appropriate transformation), the basic disaggregation
model is
Xy ¼ AZy þ BVy

ð6:178Þ

where Vy is a vector of nT independent standard
normal random variables, and A and B are,
respectively, nT × N and nT × nT matrices. One
selects values of the elements of A and B to
reproduce the observed correlations among the
elements of Xy and between the elements of Xy and
Zy. Alternatively, one could attempt to reproduce
the observed correlations of the untransformed
flows as opposed to the transformed flows,
although this is not always possible (Hoshi et al.
1978) and often produces poorer estimates of the
actual correlations of the flows (Stedinger 1981).
The values of A and B are determined using
the matrices Szz ¼ E½Zy ZTy , Szz ¼ E½Zy ZTy ,

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6 An Introduction to Probability, Statistics, and Uncertainty

Sxx ¼ E½Xy XTy ,

Sxz ¼ E½Xy ZTy ,

and

Szy ¼

E½Zy XTy  where Szz was called S0 earlier. Clearly,
STxz ¼ Szx . If Sxz is to be reproduced, then by
multiplying Eq. 6.178 on the right by ZTy and
taking expectations, one sees that A must satisfy
h
i
h
i
E Xy ZTy ¼ E AZy ZTy

ð6:179Þ

or
Sxz ¼ ASzz

ð6:180Þ

Solving for the coefficient matrix A one
obtains
A ¼ Sxz S1
zz

ð6:181Þ

To obtain an equation that determines the
required value of the matrix B, one can multiply
both sides of Eq. 6.178 by their transpose and
take expectations to obtain
Sxx ¼ ASzz AT þ BBT

ð6:182Þ

Thus to reproduce the covariance matrix Sxx
the B matrix must satisfy
BBT ¼ Sxx  ASzz AT

ð6:183Þ

Equations 6.181 and 6.183 for determining
A and B are completely analogous to Eqs. 6.172
and 6.175 for the A and B matrices of the lag-1
models developed earlier. However, for the disaggregation model as formulated, BBT and hence
the matrix B can actually be singular or nearly so
(Valencia and Schaake 1973). This occurs
because the real seasonal flows sum to the
observed annual flows. Thus given the annual
flow at a site and (T − 1) of the seasonal flows,
the value of the unspecified seasonal flow can be
determined by subtraction.
If the seasonal variables Xtys correspond to
nonlinear transformations of the actual flows Qsty ,
then BBT is generally sufficiently non-singular
that a B matrix can be obtained by Cholesky
decomposition. On the other hand, when the
model is used to generate values of Xtys to be

transformed into synthetic flows Qsty , the constraint that these seasonal flows should sum to
the given value of the annual flow is lost. Thus
the generated annual flows (equal to the sums of
the seasonal flows) will deviate from the values
that were to have been the annual flows. Some
distortion of the specified distribution of the
annual flows results. This small distortion can be
ignored, or each year’s seasonal flows can be
scaled so that their sum equals the specified value
of the annual flow (Grygier and Stedinger 1988).
The latter approach eliminates the distortion in
the distribution of the generated annual flows by
distorting the distribution of the generated seasonal flows. Koutsoyiannis and Manetas (1996)
improve upon the simple scaling algorithm by
including a step that rejects candidate vectors Xy
if the required adjustment is too large and instead
generates another vector Xy. This reduces the
distortion in the monthly flows that results from
the adjustment step.
The disaggregation model has substantial data
requirements. When the dimension of Zy is n and
the dimension of the generated vector Xy is m,
the A matrix has mn elements. The lower diagonal B matrix and the symmetric Sxx matrix,
upon which it depends, each have m(m + 1)/2
nonzero or nonredundant elements. For example,
when disaggregating two aggregate annual flow
series to monthly flows at five sites, n = 2 and
m = 12 × 5 = 60 so that A has 120 elements
while B and Sxx each have 1830 nonzero or
nonredundant parameters. As the number of sites
included in the disaggregation increases, the size
of Sxx and B increases rapidly. Very quickly the
model can become over parameterized, and there
will be insufficient data to estimate all parameters
(Grygier and Stedinger 1988).
In particular, one can think of Eq. 6.178 as a
series of linear models generating each monthly
flow X kty for k = 1, t = 1, …, 12; k = 2,
t = 1, …, 12 up to k = n, t = 1, …, 12 that
reproduces the correlation of each X kty with all
n annual flows, Z ky , and all previously generated
monthly flows. Then when one gets to the last flow
in the last month, the model will be attempting to
reproduce n + (12n − 1) = 13n − 1 annual to

6.8 Synthetic Streamflow Generation

monthly and cross-correlations. Because the
model implicitly includes a constant, this means
one needs k* = 13n years of data to obtain a
unique solution for this critical equation. For
n = 3, k* = 39. One could say that with a record
length of 40 years, there would be only 1 degree of
freedom left in the residual model error variance
described by B. That would be unsatisfactory.
When flows at many sites or in many seasons
are required, the size of the disaggregation model
can be reduced by disaggregation of the flows in
stages and not attempting to explicitly reproduce
every season-to-season correlation by constructing what have been called condensed and contemporaneous models (Lane 1979; Stedinger and
Vogel 1984; Gryier and Stedinger 1988;
Koutsoyiannis and Manetas 1996). Condensed
models do not attempt to reproduce the
cross-correlations among all the flow variates at
the same site within a year (Lane 1979; Stedinger
et al. 1985), whereas contemporaneous models
are like the Markov model developed earlier in
Sect. 8.5 and are essentially models developed
for individual sites whose innovation vectors Vy
have the needed cross-correlations to reproduce
the cross-correlations of the concurrent flows
(Camacho et al. 1985), as was done in Eq. 6.177.
Grygier and Stedinger (1991) describe how this
can be done for a condensed disaggregation
model without generating inconsistencies.

6.8.6.2 Aggregation Models
One can start with annual or seasonal flows, and
break them down into flows in shorter periods
representing months or weeks. Or instead one
can start with a model that describes flows and
the shortest time step included in the model; this
latter approach has been referred to as aggregation model to distinguish it from the disaggregation approach.
One method for generating multiseason flow
sequences is to convert the time series of seasonal flows Qty into a homogeneous sequence of
normally distributed zero-mean unit-variance

269

random variables Zty. These can then be modeled by an extension of the annual flow generators that have already been discussed. This
transformation can be accomplished by fitting a
reasonable marginal distribution to the flows in
each season so as to be able to convert the
observed flows qsty into their transformed counterparts zsty , and vice versa. Particularly, when
shorter streamflow records are available, these
simple approaches may yield a reasonable model
of some streams for some studies. However, it
implicitly assumes that the standardized series is
stationary in the sense that the season-to-season
correlations of the flows do not depend on the
seasons in question. This assumption seems
highly questionable.
This theoretical difficulty with the standardized series can be overcome by introducing a
separate streamflow model for each month. For
example, the classic Thomas-Fiering model
(Thomas and Fiering 1962) of monthly flows
may be written
qffiffiffiffiffiffiffiffiffiffiffiffiffi
Zt þ 1;y ¼ bt Zty þ 1  b2t Vty
ð6:184Þ
where the Zty’s are standard normal random variables corresponding to the streamflow in season
t of year y, βt is the season-to-season correlation of
the standardized flows, and Vty are independent
standard normal random variables. The problem
with this model is that it often fails to reproduce
the correlation among months during a year and
thus misrepresents the risk of multi-month and
multi-year droughts (Hoshi et al. 1978).
For an aggregation approach to be attractive it
is necessary to use a model with greater persistence than the Thomas-Fiering model. Time
series models that allow reproduction of different
correlation structures are the Box-Jenkins
Autoregressive-Moving
average
models
(Box et al. 1994). These models are presented by
the notation ARMA(p, q) for a model which
depends on p previous flows, and q extra innovations Vty. For example, Eq. 6.142 would be

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6 An Introduction to Probability, Statistics, and Uncertainty

called an AR(1) or AR(1, 0) model. A simple
ARMA(1, 1) model is
Zt þ 1 ¼ /1  Zt þ Vt þ 1  h1  Vt

ð6:185Þ

The correlations of this model have the values


q1 ¼ ð1  h1 /1 Þð/1  h1 Þ= 1 þ h21  2/1 h1
ð6:186Þ
for the first lag. For i > 1
qi ¼ /i1 q1

ð6:187Þ

For / values near one and 0 < θ1 < ϕ1 the
autocorrelations ρk can decay much slower than
those of the standard AR(1) model.
The correlation function ρk of general ARMA
(p, q) model
Zt þ 1 ¼

p
X
i¼1

/i  Zt þ 1i þ Vt þ 1 

q
X

better job of reproducing the correlation of seasonal flows beyond lag one (see for example,
Bartolini and Salas 1993).

hj  Vt þ 1j

j¼1

ð6:188Þ
is a complex function that must satisfy a number
of conditions to ensure the resultant model is
stationary and invertible (Box et al. 1994).
ARMA(p, q) models have been extended to
describe seasonal flow series by having their
coefficients depend upon the season—these are
called periodic Autoregressive-Moving average
models, or PARMA. Salas and Obeysekera
(1992), Salas and Fernandez (1993), and Claps
et al. (1993) discuss the conceptual basis of such
stochastic streamflow models. For example, Salas
and Obeysekera (1992) found that low-order
PARMA models, such as a PARMA(2,1), arise
from reasonable conceptual representations of
persistence in rainfall, runoff, and groundwater
recharge and release. Claps et al. (1993, p. 2553)
observe that the PARMA(2, 2) model which may
be needed if one wants to preserve year-to-year
correlation poses a parameter estimation challenge
(see also Rasmussen et al. 1996). The PARMA (1,
1) model is more practical and easy to extend to the
multivariate case (Hirsch 1979; Stedinger et al.
1985; Salas 1993; Rasmussen et al. 1996). Experience has shown that PARMA(1, 1) models do a

6.9

Stochastic Simulation

This section introduces stochastic simulation.
Much more detail on simulation is contained in
later parts of this chapter and in the next chapter.
Simulation is the most flexible and widely used
tool for the analysis of complex water resources
systems. Simulation is trial and error. One must
define the system being simulated, both its
design and operating policy, and then simulate it
to see how it works. If the purpose is to find the
best design and policy, many such alternatives
must be simulated.
As with optimization models, simulation
models may be deterministic or stochastic. One
of the most useful tools in water resource
systems planning is stochastic simulation.
While optimization can be used to help define
reasonable design and operating policy alternatives to be simulated, it takes those simulations to obtain the best insights of just how
each such alternative will perform. Stochastic
simulation of complex systems on digital
computers provides planners with a way to
define the probability distribution of performance indices of complex stochastic water
resources systems.
When simulating any system, the modeler
designs an experiment. Initial flow, storage,
and water quality conditions must be specified
if these are being simulated. For example,
reservoirs can start full, empty, or at random
representative conditions. The modeler also
determines what data are to be collected on
system performance and operation and how
they are to be summarized. The length of time
the simulation is to be run must be specified
and, in the case of stochastic simulations, the
number of runs to be made must also be
determined. These considerations are discussed
in more detail by Fishman (2001) and in other
books on simulation. The use of stochastic

6.9 Stochastic Simulation

simulation and the analysis of the output of
such models are introduced here primarily in
the context of an example to illustrate what
goes into a simulation model and how one can
deal with the information that is generated.

271

Fu ðuÞ ¼ 0 for u  0;
u for 0  u  1 and

1

if u  1
ð6:189Þ

and

6.9.1 Generating Random Variables
Included in any stochastic simulation model is
some provision for the generation of sequences of
random numbers that represent particular values
of events such as rainfall, streamflows, or floods.
To generate a sequence of values for a random
variable, probability distributions for the variables must be specified. Historical data and an
understanding of the physical processes are used
to select appropriate distributions and to estimate
their parameters (as discussed in Sect. 6.3.2).
Most computers have algorithms for generating random numbers uniformly distributed
between zero and one. This uniform distribution
is defined by its cdf and pdf;

Fig. 6.13 The probability
distribution of a random
variable can be inverted to
produce values of the
random variable

fU ðuÞ ¼ 1

if 0  u  1

and 0 otherwise
ð6:190Þ

These uniform random variables can then be
transformed into random variables with any
desired distribution. If FQ(qt) is the cumulative
distribution function of a random variable Qt in
period t, then Qt can be generated using the
inverse function, as
Qt ¼ FQ1 ½Ut 

ð6:191Þ

Here Ut is the uniform random number used
to generate Qt. This is illustrated in Fig. 6.13.
Analytical expressions for the inverse of many
distributions, such as the normal distribution, are
not known, so that special algorithms are

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6 An Introduction to Probability, Statistics, and Uncertainty

employed to efficiently generate deviates with
these distributions (Fishman 2001).

6.9.2 River Basin Simulation
An example will demonstrate the use of
stochastic simulation in the design and analysis
of water resource systems. Assume that farmers
in a particular valley have been plagued by frequent shortages of irrigation water. They currently draw water from an unregulated river to
which they have water rights. A government
agency has proposed construction of a
moderate-size dam on the river upstream from
Table 6.12 Projected water demand for irrigation water

points where the farmers withdraw water. The
dam would be used to increase the quantity and
reliability of irrigation water available to the
farmers during the summer growing season.
After preliminary planning, a reservoir with an
active capacity of 4 × 107 m3 has been proposed
for a natural dam site. It is anticipated that because
of the increased reliability and availability of irrigation water, the quantity of water desired will
grow from an initial level of 3 × 107 m3/yr after
construction of the dam to 4 × 107 m3/yr within
6 years. After that, demand will grow more
slowly, to 4.5 × 107 m3/yr, the estimated maximum reliable yield. The projected demand for
summer irrigation water is shown in Table 6.12.

6.9 Stochastic Simulation

273

Table 6.13 Characteristics of the river flow

A simulation study will evaluate how the
system can be expected to perform over a
20-year planning period. Table 6.13 contains
statistics that describe the hydrology at the dam
site. The estimated moments are computed from
the 45-year historic record. .
Using the techniques discussed in the previous section, a Thomas-Fiering model is used to
generate 25 lognormally distributed synthetic
streamflow sequences. The statistical characteristics of the synthetic flows are those listed in
Table 6.14. Use of only the 45-year historic flow
sequence would not allow examination of the
system’s performance over the large range of
streamflow sequences which could occur during
the 20-year planning period. Jointly, the synthetic sequences should be a description of the
range of inflows that the system might experience. A larger number of sequences could be
generated.

6.9.3 The Simulation Model
The simulation model is composed primarily of
continuity constraints and the proposed operating
policy. The volume of water stored in the reservoir at the beginning of seasons 1 (winter) and 2

(summer) in year y are denoted by S1y and S2y.
The reservoir’s winter operating policy is to store
as much of the winter’s inflow Q1y as possible.
The winter release R1y is determined by the rule
8
< S1y þ Q1y  K
R1y ¼ Rmin
:
S1y þ Q1y

if S1y þ Q1y  Rmin > K
if K  S1y þ Q1y  Rmin  0
otherwise

ð6:192Þ
where K is the reservoir capacity of 4 × 107 m3
and Rmin is 0.50 × 107 m3, the minimum release
to be made if possible. The volume of water in
storage at the beginning of the year’s summer
season is
S2y ¼ S1y þ Q1y  R1y

ð6:193Þ

The summer release policy is to meet each
year’s projected demand or target release Dy, if
possible, so that
R2y ¼ S2y þ Q2y  K
¼ Dy
¼ S2y þ Q2y

if S2y þ Q2y  Dy [ K
if 0  S2y þ Q2y  Dy  K
otherwise
ð6:194Þ

This operating
Fig. 6.14.

policy

is

illustrated

in

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6 An Introduction to Probability, Statistics, and Uncertainty

Fig. 6.14 Summer reservoir operating policy. The shaded area denotes the feasible region of reservoir releases

The volume of water in storage at the beginning of the next winter season is
S1;y þ 1 ¼ S2y þ Q2y  R2y

ð6:195Þ

6.9.4 Simulation of the Basin
The question to be addressed by this simulation
study is how well will the reservoir meet the
farmers’ water requirements. Three steps are
involved in answering this question. First, one
must define the performance criteria or indices to
be used to describe the system’s performance.
The appropriate indices will, of course, depend
on the problem at hand and the specific concerns
of the users and managers of a water resource
system. In our reservoir-irrigation system, several
indices will be used relating to the reliability with
which target releases are met and the severity of
any shortages.

The next step is to simulate the proposed
system to evaluate the specified indices. For our
reservoir-irrigation system, the reservoir’s operation was simulated 25 times using the 25 synthetic streamflow sequences, each 20 years in
length. Each of the 20 simulated years consisted
of first a winter and then a summer season. At the
beginning of the first winter season, the reservoir
was taken to be empty (S1y = 0 for y = 1)
because construction would just have been
completed. The target release or demand for
water in each year is given in Table 6.12.
After simulating the system, one must proceed to interpret the resulting information so as
to gain an understanding of how the system
might perform both with the proposed design
and operating policy and with modifications in
either the system’s design or its operating policy. To see how this may be done, consider the
operation of our example reservoir-irrigation
system.

6.9 Stochastic Simulation

275

Fig. 6.15 Number of
failures in each year of 25
twenty-year simulations

The reliability py of the target release in year
y is the probability that the target release Dy is
met or exceeded in that year:
Py ¼ Pr½R2y  Dy 

ð6:196Þ

The system’s reliability is a function of the
target release Dy, the hydrology of the river, the
size of the reservoir, and the operating policy of
the system. In this example, the reliability also
depends on the year in question. Figure 6.15
shows the total number of failures that occurred in
each year of the 25 simulations. In 3 of the 25
simulations, the reservoir did not contain sufficient
water after the initial winter season to meet the
demand the first summer. After year 1, few failures
occur in years 2 through 9 because of the low
demand. Surprisingly few failures occur in years
10 and 13, when demand has reached its peak; this
results because the reservoir was normally full at
the beginning of this period as a result of lower
demand in the earlier years. Starting in years 14
and after, failures occurred more frequently
because of the high demand placed on the system.
Thus one has a sense for how the reliability of the
target releases changes over time.

6.9.5 Interpreting Simulation Output
Table 6.14 contains several summary statistics of
the 25 simulations. Column 2 of the table contains the average failure frequency in each

simulation, which equals the number of years the
target release was not met divided by 20, the
number of years simulated. At the bottom of
column 2 and the other columns are several
statistics that summarize the 25 values of the
different performance indices. The sample estimates of the mean and variance of each index are
given as one way of summarizing the distribution
of the observations. Another approach is specification of the sample median, the approximate
interquartile range [x(6) − x(20)], and/or the range
[x(1) − x(25)] of the observations, where x(i) is the
ith largest observation. Either set of statistics
could be used to describe the center and spread of
each index’s distribution.
Suppose that one is interested in the distribution of the system’s failure frequency or,
equivalently, the reliability with which the
target can be met. Table 6.14 reports that the
mean failure rate for the 25 simulations is
0.084, implying that the average reliability over
the 20-year period is 1 − 0.084 = 0.916 or
92%. The median failure rate is 0.05, implying
a median reliability of 95%. These are both
reasonable estimates of the center of the distribution of the failure frequency. Note that the
actual failure frequency ranged from 0 (seven
times) to 0.30. Thus the system’s reliability
ranged from 100% to as low as 70, 75, and
80% in runs 17, 8, and 11. Certainly, the
farmers are interested not only in knowing the
mean or median failure frequency but also the

276
Table 6.14 Results of 25 20-year simulations

6 An Introduction to Probability, Statistics, and Uncertainty

6.9 Stochastic Simulation

277

range of failure frequencies they are likely to
experience.
If one knew the form of the distribution
function of the failure frequency, one could use
the mean and standard deviation of the observations to determine a confidence interval within
which the observations would fall with some
prespecified probability. For example, if the
observations are normally distributed, there is a
90% probability that the index falls within the
interval μx ± 1.65σx. Thus, if the simulated
failure rates are normally distributed, there is
about a 90% probability the actual failure rate is
within the interval x ± 1.65sx. In our case this
interval would be [0.084 − 1.65(0.081),
0.084 + 1.65(0.081)] = [−0.050, 0.218].
Clearly, the failure rate cannot be less than
zero, so that this interval makes little sense in our
example.
A more reasonable approach to describing the
distribution of a performance index whose
probability distribution function is not known is
to use the observations themselves. If the
observations are of a continuous random variable, the interval [x(i) − x(n+1−i)] provides a reasonable estimate of an interval within which the
random variable falls with probability
P¼

nþ1  i
i
n þ 1  2i

¼
nþ1
nþ1
nþ1

ð6:197Þ

In our example, the range [x(1) − x(25)] of the
25 observations is an estimate of an interval in
which a continuous random variable falls with
probability (25 + 1 − 2)/(25 + 1) = 92%, while
[x(6) − x(20)]
corresponds
to
probability
(25 + 1 – 2 × 6)/(25 + 1) = 54%.
Table 6.14 reports that for the failure frequency, [x(1) − x(25)] equals [0 − 0.30], while
[x(6) − x(20)] equals [0 − 0.15]. Reflection on
how the failure frequencies are calculated
reminds us that the failure frequency can only
take on the discrete, nonnegative values 0, 1/20,
2/20, …, 20/20. Thus, the random variable
X cannot be less than zero. Hence, if the lower
endpoint of an interval is zero, as is the case here,
then [0 − x(k)] is an estimate of an interval within
which the random variable falls with a

probability of at least k/(n + 1). For k equal to 20
and 25, the corresponding probabilities are 77
and 96%.
Often, the analysis of a simulated system’s
performance centers on the average value of
performance indices, such as the failure rate. It is
important to know the accuracy with which the
mean value of an index approximates the true
mean. This is done by the construction of confidence intervals. A confidence interval is an
interval that will contain the unknown value of a
parameter with a specified probability. Confidence intervals for a mean are constructed using
the t statistic,
t¼

x  lx
pffiffiffi
sx = n

ð6:198Þ

which for large n has approximately a standard
normal distribution. Certainly, n = 25 is not very
large, but the approximation to a normal distribution may be sufficiently good to obtain a rough
estimate of how close the average frequency of
failure x is likely to be to μx. A 100(1 − 2α)%
confidence interval for μx is, approximately,
sx
sx
x  ta pffiffiffi  lx  x þ ta pffiffiffi
n
n
or

0:084  ta




0:081
0:081
pffiffiffiffiffi  lx  0:084 þ ta pffiffiffiffiffi
25
25
ð6:199Þ

If α = 0.05, then tα = 1.65 and Eq. 6.199
becomes 0.057 ≤ μx ≤ 0.11.
Hence, based on the simulation output, one
can be about 90% sure that the true mean
failure frequency lies between 5.7 and 11%.
This corresponds to a reliability between 89
and 94%. By performing additional simulations
to increase the size of n, the width of this
confidence interval can be decreased. However,
this increase in accuracy may be an illusion,
because the uncertainty in the parameters of the
streamflow model has not been incorporated
into the analysis.

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6 An Introduction to Probability, Statistics, and Uncertainty

Failure frequency or system reliability
describes only one dimension of the system’s
performance. Table 6.14 contains additional
information on the system’s performance related
to the severity of shortages. Column 3 lists the
frequencies with which the shortage exceeded
20% of that year’s demand. This occurred in
approximately 2% of the years, or in 24% of the
years in which a failure occurred. Taking another
point of view, failures in excess of 20% of
demand occurred in 9 out of 25, or in 36% of the
simulation runs
Columns 4 and 5 of Table 6.14 contain two
other indices that pertain to the severity of the
failures. The total shortfall in Column 4 is calculated as
TS 

20 
X
þ
D2y  R2y
y1

where
þ

½Q ¼



Q
0

if Q [ 0
otherwise

ð6:200Þ

The total shortfall equals the total amount by
which the target release is not met in years in
which shortages occur.
Related to the total shortfall is the average
deficit. The deficit is defined as the shortfall in
any year divided by the target release in that year.
The average deficit is


20
D2y  R2y
1X
AD ¼
m y¼1
D2y

ð6:201Þ

where m is the number of failures (deficits) or
nonzero terms in the sum.
Both the total shortfall and the average deficit
measure the severity of shortages. The mean total
shortfall TS, equal to 1.00 for the 25 simulation
runs, is a difficult number to interpret. While no
shortage occurred in seven runs, the total shortage was 4.7 in run 8, in which the shortfall in two
different years exceeded 20% of the target. The
median of the total shortage values, equal to
0.76, is an easier number to interpret in that one

knows that half the time the total shortage was
greater and half the time less than this value.
 is 0.106, or
The mean average deficit AD
11%. However, this mean includes an average
deficit of zero in the seven runs in which no
shortages occurred. The average deficit in the
18 years in which shortages occurred is (11%)
(25/18) = 15%. The average deficit in individual
simulations in which shortages occurred ranges
from 4 to 43%, with a median of 11.5%.
After examining the results reported in
Table 6.14, the farmers might determine that the
probability of a shortage exceeding 20% of a
year’s target release is higher than they would
like. They can deal with more frequent minor
shortages, not exceeding 20% of the target, with
little economic hardship, particularly if they are
warned at the beginning of the growing season
that less than the targeted quantity of water will
be delivered. Then they can curtail their planting
or plant crops requiring less water.
In an attempt to find out how better to meet
the farmers’ needs, the simulation program was
rerun with the same streamflow sequences and a
new operating policy in which only 80% of the
growing season’s target release is provided (if
possible) if the reservoir is less than 80% full at
the end of the previous winter season. This gives
the farmers time to adjust their planting schedules and may increase the quantity of water
stored in the reservoir to be used the following
year if the drought persists.
As the simulation results with the new policy
in Table 6.15 demonstrate, this new operating
policy appears to have the expected effect on the
system’s operation. With the new policy, only six
severe shortages in excess of 20% of demand
occur in the 25 twenty-year simulations, as
opposed to 10 such shortages with the original
policy. In addition, these severe shortages are all
less severe than the corresponding shortages that
occur with the same streamflow sequence when
the original policy is followed.
The decrease in the severity of shortages is
obtained at a price. The overall failure frequency
has increased from 8.4 to 14.2%. However, the
latter figure is misleading because in 14 of the 25

6.9 Stochastic Simulation
Table 6.15 Results of 25 20-Year simulations with modified operating policy to avoid severe shortages

279

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6 An Introduction to Probability, Statistics, and Uncertainty

simulations, a failure occurs in the first simulation year with the new policy, whereas only three
failures occur with the original policy. Of course,
these first-year failures occur because the reservoir starts empty at the beginning of the first
winter and often does not fill that season.
Ignoring these first-year failures, the failure
rates with the two policies over the subsequent
19 years are 8.2 and 12.0%. Thus the frequency
of failures in excess of 20% of demand is
decreased from 2.0 to 1.2% by increasing the
frequency of all failures after the first year from
8.2 to 12.0%. Reliability increases while vulnerability decreases. If the farmers are willing to
put up with more frequent minor shortages, it
appears they can reduce their risk of experiencing
shortages of greater severity.
The preceding discussion has ignored the
statistical issue of whether the differences
between the indices obtained in the two simulation experiments are of sufficient statistical reliability to support the analysis. If care is not
taken, observed changes in a performance index
from one simulation experiment to another may
be due to sampling fluctuations rather than to
modifications of the water resource system’s
design or operating policy.
As an example, consider the change that
occurred in the frequency of shortages. Let X1i
and X2i be the simulated failure rates using the ith
streamflow sequence with the original and
modified operating policies. The random variables Yi = X1i − X2i for i equal 1 through 25 are
independent of each other if the streamflow
sequences are generated independently, as they
were.
One would like to confirm that the random
variable Y tends to be negative more often than it
is positive and hence that policy 2 indeed results
in more failures overall. A direct test of this
theory is provided by the sign test. Of the 25
paired simulation runs, yi < 0 in 21 cases and
yi = 0 in four cases. We can ignore the times
when yi = 0. Note that if yi < 0 and yi > 0 were
equally likely, then the probability of observing
yi < 0 in all 21 cases when yi ≠ 0 is 2−21 or
5 × 10−7. This is exceptionally strong proof that

the new policy has increased the failure
frequency.
A similar analysis can be made of the frequency with which the release is less than 80% of
the target. Failure frequencies differ in the two
policies in only four of the 25 simulation runs.
However, in all 4 cases where they differ, the
new policy resulted in fewer severe failures. The
probability of such a lopsided result, were it
equally likely that either policy would result in a
lower frequency of failures in excess of 20% of
the target, is 2−4 = 0.0625. This is fairly strong
evidence that the new policy indeed decreases
the frequency of severe failures.
Another approach to this problem is to ask if
the difference between the average failure rates
x1 and x2 is statistically significant; that is, can
the difference between x1 and x2 be attributed
to the fluctuations that occur in the average of
any finite set of random variables? In this
example, the significance of the difference
between the two means can be tested using the
random variable Yi defined as X1i − X2i for
i equal 1 through 25. The mean of the observed
yi’s is
25
1 X
ðx1i  x2i Þ ¼ x1  x2
25 i1
¼ 0:084  0:142 ¼ 0:058

y ¼

ð6:202Þ

and their variance is
s2y ¼

25
1 X
ðx1i  x2i  yÞ2 ¼ ð0:0400Þ2
25 i¼1

ð6:203Þ
Now if the sample size n, equal to 25 here, is
sufficiently large, then t defined by
t¼

y  lY
pffiffiffi
sY = n

ð6:204Þ

has approximately a standard normal distribution.
The closer the distribution of Y is to that of the
normal distribution, the faster the convergence of
the distribution of t is to the standard normal
distribution with increasing n. If X1i − X2i is

6.9 Stochastic Simulation

281

normally distributed, which is not the case here,
then each Yi has a normal distribution and t has
Student’s t-distribution.
If E[x1i] = E[x2i], then μY equals zero and
upon substituting the observed values of y and s2Y
into Eq. 6.204, one obtains
0:0580
pffiffiffiffiffi ¼ 7:25
t¼
0:0400= 25

ð6:205Þ

The probability of observing a value of t equal
to −7.25 or smaller is less than 0.1% if n is
sufficiently large that t is normally distributed.
Hence it appears very improbable that μY equals
zero.
This example provides an illustration of the
advantage of using the same streamflow
sequences when simulating both policies. Suppose that different streamflow sequences were
used in all the simulations. Then the expected
value of Y would not change, but its variance
would be given by
VarðYÞ ¼ E½X1  X2  ðl1  l2 Þ2
¼ E½ðX1  l1 Þ2  2E½ðX1  l1 ÞðX2  l2 Þ
E½ðX2  l2 Þ2 
¼ r2x1  2 CovðX1 ; X2 Þ þ r2X2

ð6:206Þ
where Cov(X1, X2) = E[(X1 − μ1)(X2 − μ2)] and
is the covariance of the two random variables.
The covariance between X1 and X2 will be zero if
they are independently distributed as they would
be if different randomly generated streamflow
sequences were used in each simulation. Estimating r2x1 and r2x2 by their sample estimates, an
estimate of what the variance of Y would be if
Cov(X1, X2) were zero is
^2Y ¼ s2x1 þ s2x2 ¼ ð0:081Þ2 þ ð0:087Þ2 ¼ ð0:119Þ2
r
ð6:207Þ
The actual sample estimate sY equals 0.040; if
independent streamflow sequences are used in all
simulations, sY will take a value near 0.119 rather

than 0.040 (Eq. 6.203). A standard deviation of
0.119 yields a value of the test statistic
t¼

y  lY 
pffiffiffiffiffi lY¼0 ¼ 2:44
0:119= 25

ð6:208Þ

If t is normally distributed, the probability of
observing a value less than −2.44 is about 0.8%.
This illustrates that use of the same streamflow
sequences in the simulation of both policies
allows one to better distinguish the differences in
the policies’ performance. Using the same
streamflow sequences, or other random inputs,
one can construct a simulation experiment in
which variations in performance caused by different random inputs are confused as little as
possible with the differences in performance
caused by changes in the system’s design or
operating policy.

6.10

Conclusions

This chapter has introduced some approaches
analysts can consider and use when working with
the randomness or uncertainty of their data. Most
of the data water resource systems analysts use is
uncertain. This uncertainty comes from not
understanding as well as we would like how our
water resource systems (including its ecosystems) function as well as not being able to
forecast, perfectly, the future. It is that simple.
We do not know the exact amounts, qualities,
and their distributions over space and time of
both the supplies of water we manage and the
water demands we try to meet. We also do not
know the benefits and costs, however measured,
of any actions we take to manage both water
supply and water demand.
The chapter began with an introduction to
some probability concepts and methods for
describing random variables and parameters of
their distributions. It then reviewed some of the
commonly used probability distributions and
how to determine the distributions of sample
data, how to work with censored and partial

282

6 An Introduction to Probability, Statistics, and Uncertainty

duration series data, methods of regionalization,
stochastic processes and time series analyses.
The chapter concluded with an introduction to
a range of univariate and multivariate stochastic
models that are used to generate stochastic
streamflow, precipitation depths, temperatures,
and evaporation. These methods have been
widely used to generate temporal and spatial
stochastic process that serve as inputs to
stochastic simulation models for system design,
for system operations studies, and for the evaluation of the reliability and precision of different
estimation algorithms. The final section of this
chapter provides an example of stochastic simulation, and the use of statistical methods to
summarize the results of such simulations.
This chapter is merely an introduction to some
of the tools available for use when dealing with
uncertain data. Many of the concepts introduced
in this chapter will be used in the chapters that
follow on constructing and implementing various
types of optimization, simulation, and statistical
models. The references provided in the next
section provide additional and more detailed
information.
Although many of the methods presented in
this and the following two chapters can describe
many of the characteristics and consequences of
uncertainty, it is unclear as to whether or not
society knows exactly what to do with that
information. Nevertheless there seems to be an
increasing demand from stakeholders involved in
planning processes for information related to the
uncertainty associated with the impacts predicted
by models. The challenge is not only to quantify
that uncertainty, but also to communicate it in
effective ways that inform, and not confuse, the
decision-making process.

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(2), 503–510.
Stedinger, J. R. (1997). Expected probability and annual
damage estimators. Journal of Water Resources
Planning and Management, 123(2), 125–35 [With
discussion, Leo R. Beard, Journal of Water Resources
Planning and Management, 124(6), 365–366, 1998].
Stedinger, J. R. (2000). Flood frequency analysis and
statistical estimation of flood risk (Chapter 12). In E.
E. Wohl (Ed.), Inland flood hazards: Human, Riparian and aquatic communities. Stanford, United Kingdom: Cambridge University Press.
Stedinger, J. R., & Baker, V. R. (1987). Surface water
hydrology: Historical and paleoflood information.
Reviews of Geophysics, 25(2), 119–124.
Stedinger, J. R., & Cohn, T. A. (1986). Flood frequency
analysis with historical and paleoflood information.
Water Resources Research, 22(5), 785–793.
Stedinger, J. R., & Lu, L. (1995). Appraisal of regional
and index flood quantile estimators. Stochastic
Hydrology and Hydraulics, 9(1), 49–75.
Stedinger, J. R., Pei, D., & Cohn, T. A. (1985).
A disaggregation model for incorporating parameter
uncertainty into monthly reservoir simulations. Water
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Stedinger, J. R., & Taylor, M. R. (1982a). Synthetic
streamflow generation, part I. Model verification and
validation. Water Resources Research, 18(4), 919–
924.
Stedinger, J. R., & Taylor, M. R. (1982b). Synthetic
streamflow generation, part II. Effect of parameter
uncertainty. Water Resources Research, 18(4), 919–
924.
Stedinger, J. R., & Vogel, R. (1984). Disaggregation
procedures for the generation of serially correlated flow
vectors. Water Resources Research, 20(1), 47–56.
Stedinger, J. R., Vogel, R. M., & Foufoula-Georgiou, E.
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18). In D. Maidment (Ed.), Handbook of hydrology.
New York: McGraw-Hill, Inc.
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disaggregation. Water Resources Research, 12(6),
1303–1311.
Tarboton, D. G., Sharma, A., & Lall, U. (1998).
Disaggregation procedures for stochastic hydrology

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6 An Introduction to Probability, Statistics, and Uncertainty

based on nonparametric density estimation. Water
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Tasker, G. D., & Stedinger, J. R. (1986). Estimating
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Management, 112(2), 225–237.
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M. Hufschmidt, R. Dorfman, H. A. Thomas Jr., S.
A. Marglin, & G. M. Fair (Eds.), Design of water
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Bayesian generation of synthetic streaamflows, 2.
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diagrams should replace product moment diagrams.
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blocks bootstrap versus parametric time series models.
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stochastic streamflow models in over-year reservoir
design applications. Water Resources Research, 24(9),
1483–1490.
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goodness-of-fit and skewness estimation procedures
for the Pearson type 3 distribution. Water Resources
Research, 27(42), 3149–3158. December.
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evaluation of flood events for the design of hydraulic
structures. In E. Guggino, G. Rossi, & E. Todini
(Eds.), Piene e Siccita (pp. 3–36). Catania, Italy:
Fondazione Politecnica del Mediterraneo.
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moment! Water Resources Research, 10(2), 211–219.
Wallis, J. R., Matalas, N. C., & Slack, J. R. (1974b). Just
a moment! Appendix, National Technical Information
Service, PB-231 816.

Wang, Q. (2001). A bayesian joint probability approach
for flood record augmentation. Water Resources
Research, 37(6), 1707–1712.
Wang, Q. J. (1997). LH moments for statistical analysis of
extreme events. Water Resources Research, 33(12),
2841–2848.
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plotting methods for the analysis of data. Biometrika,
55(1), 1–17.
Wilks, D. S. (1998). Multisite generalization of a daily
stochastic precipitation generation model. Journal of
Hydrology, 210, 178–191.
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forecast seasonal climate. Journal of Hydrometeorology, 3, 195–207.
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assessment of synthetic hydrology” by N.C. Matalas,
and Reply. Water Resources Research, 4(3), 681–683.
Zellner, A. (1971). An introduction to bayesian inference
in econometrics. New York: Wiley.
Zrinji, Z., & Burn, D. H. (1994). Flood frequency analysis
for ungauged sites using a region of influence
approach. Journal of Hydrology, 13, 1–21.

Additional References
(Further Reading)
Claps, P. (1993). Conceptual basis for stochastic models of
monthly streamflows. In J. B. Marco, R. Harboe, &
J. D. Salas (Eds.), Stochastic hydrology and its use in
water resources systems simulation and optimization
(pp. 229–235). Dordrecht: Kluwer Academic Publishers.
Kumar, D. N., Lall, U., & Peterson, M. R. (2000).
Multisite disaggregation of monthly to daily streamflow. Water Resources Research, 36(7), 1823–1833.
Pegram, G. G. S. (1989). Why stochastic hydrology?
Developments in Water Engineering (Die Siviele
Ingenieur in Suid-Afrika, October 1989).
Young, G. K., & Pisano, W. C. (1968). Operational
hydrology using residuals. Journal of the Hydraulics
Division, ASCE, 94(HY4), 909–923.

Exercises
6:1 Identify a water resources planning study
with which you have some familiarity.
Make a list of the basic information used
in the study and the methods used transform that information into decisions, recommendations, and conclusions.
(a) Indicate the major sources of uncertainty and possible error in the basic

Exercises

287

information and in the transformation
of that information into decisions,
recommendations, and conclusions.
(b) In systems studies, sources of error
and uncertainty are sometimes
grouped into three categories
1. Uncertainty due to the natural
variability of rainfall, temperature,
and stream flows which affect a
system’s operation.
2. Uncertainty due to errors made in
the estimation of the models’
parameters with a limited amount
of data.
3. Uncertainty or errors introduced
into the analysis because conceptual and/or mathematical models
do not reflect the true nature of the
relationships being described.
Indicate, if applicable, into which category
each of the sources of error or uncertainty
you have identified falls.
6:2 The following matrix displays the joint
probabilities of different weather conditions
and of different recreation benefit levels
obtained from use of a reservoir in a state
park:
Weather

Possible recreation benefits
RB1

RB2

RB3

Wet

0.10

0.20

0.10

Dry

0.10

0.30

0.20

(a) Compute the probabilities of recreation levels RB1, RB2, RB3, and of dry
and wet weather.
(b) Compute the conditional probabilities P
(wet∣RB1), P(RB3∣dry), and P(RB2∣wet).
6:3 In flood protection planning, the 100-year
flood, which is an estimate of the quantile
x0.99, is often used as the design flow.
Assuming that the floods in different years
are independently distributed

(a) Show that the probability of at least
one 100-year flood in a 5-year period
is 0.049.
(b) What is the probability of at least
one 100-year flood in a 100-year
period?
(c) If floods at 1000 different sites occur
independently, what is the probability
of at least one 100-year flood at some
site in any single year?
6:4 The price to be charged for water by an
irrigation district has yet to be determined. Currently it appears as if there is
as 60% probability that the price will be
$10 per unit of water and a 40% probability that the price will be $5 per unit.
The demand for water is uncertain. The
estimated probabilities of different
demands given alternative prices are as
follows:
Price/Quantity

Prob. of quantity demanded given
price
30

55

80

100

120

$5

0.00

0.15

0.30

0.35

0.20

$10

0.20

0.30

0.40

0.10

0.00

(a) What is the most likely value of future
revenue from water sales?
(b) What are the mean and variance of
future water sales?
(c) What is the median value and interquartile range of future water sales?
(d) What price will maximize the revenue
from the sale of water?
6:5 Plot the following data on possible recreation losses and irrigated agricultural
yields. Show that use of the expected
storage level or expected allocation
underestimates the expected value of the
convex function describing reservoir losses while it overestimates the expected
value of the concave function describing

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6 An Introduction to Probability, Statistics, and Uncertainty

crop yield. A concave function f(x) has the
property that f(x) ≤ f(x0) + f′(xo)(x – x0)
for any x0; prove that use of f(E[X]) will
always overestimate the expected value of
a concave function f(X) when X is a random variable.
Irrigation
water
allocation

Crop
yield/Hectare

Probability of
allocation

10

6.5

0.2

20

10

0.3

30

12

0.3

40

11

0.2

Summer
storage level

Decrease in
recreation
benefits

Probability of
storage level

200

5

0.1

250

2

0.2

300

0

0.4

350

1

0.2

400

4

0.1

discount rate is high and the life is relatively long, the uncertainty may not very
important. However, if the life of a reservoir, or of a wastewater treatment facility,
or any other such project, relatively short,
the practice of using the expected life to
calculate present costs or benefits may be
misleading.In this problem, assume that a
project results in $1000 of net benefits at
the end of each year is expected to last
between 10 and 30 years. The probability
of ending at the end of each year within the
range of 11–30 is the same. Given a discount rate of 10%
(a) Compute the present value of net
benefits NB0, assuming a 20-year
project life.
(b) Compare this with the expected present net benefits E[NB0] taking
account of uncertainty in the project
lifetime.
(c) Compute the probability that the
actual present net benefits is at least
$1000 less than NB0, the benefit esti-

12
5
Decrease
in
Benefits

Yield

10

20

30

40

Irrigation water allocation

6:6 Complications can be added to the economic evaluation of a project by uncertainty concerning the usefulness life of the
project. For example, the time at which the
useful life of a reservoir will end due to
silting is never known with certainty when
the reservoir is being planned. If the

200

250

300

350

400

Summer Storage Level

mate based on a 20-year life.
(d) What is the chance of getting $1000
more than the original estimate NB0?
6:7 A continuous random variable that could
describe the proportion of fish or other
animals in different large samples which

Exercises

289

have some distinctive features is the beta
distribution whose density is (a > 0, b > 0)

b1
a1
0x1
fX ðxÞ ¼ cx ð1  xÞ
0
otherwise
(a) Directly calculate the value of c and
the mean and variance of X for
α = β = 2.
(b) In general, c = Γ(α + β)/Γ(α)Γ(β),
where Γ(α) is the gamma function
equal to (α − 1)! for integer α. Using
this information, derive the general
expression for the mean and variance of
X. To obtain a formula which gives the
values of the integrals of interest, note
that the expression for c must be such
that the integral over (0, 1) of the density function is unity for any α and β.
6:8 The joint probability density of rainfall at
two places on rainy days could be described by

2=ðx þ y þ 1Þ3 x; y  0
fX;Y ðx; yÞ
0
otherwise
Calculate and graph
(a) FXY(x, y), the joint distribution function of X and Y.
(b) FY(y), the marginal cumulative distribution function of Y, and fY(y), the
density function of Y.
(c) fY|X(y|x), the conditional density function of Y given that X = x, and FY|X(y|
x), the conditional cumulative distribution function of Y given that
X = x (the cumulative distribution
function is obtained by integrating the
density function).
Show that
FYjX ðyjx ¼ 0Þ [ FY ðyÞ

for y [ 0

Find a value of x0 and y0 for which
FYjX ðy0 jx0 Þ\FY ðy0 Þ
6:9 Let X and Y be two continuous independent random variables. Prove that
E½gðXÞhðYÞ ¼ E½gðXÞE½hðYÞ
for any two real-valued functions g and
h. Then show that Cov(X, Y) = 0 if X and
Y are independent.
6:10 A frequent problem is that observations
(X, Y) are taken on such quantities as
flow and concentration and then a
derived quantity g(X, Y) such as mass
flux is calculated. Given that one has
estimates of the standard deviations of
the observations X and Y and their correlation, an estimate of the standard
deviation of g(X, Y) is needed. Using a
second-order Taylor series expansion for
the mean of g(X, Y) as a function of its
partial derivatives and of the means,
variances, covariance of the X and
Y. Using a first-order approximation of g
(X, Y), obtained an estimates of the
variances of g(X, Y) as a function of its
partial derivatives and the moments of
X and Y. Note, the covariance of X and
Y equals
E½ðX  lX ÞðY  lY Þ ¼ r2XY
6:11 A study of the behavior of water waves
impinging upon and reflecting off a
breakwater located on a sloping beach was
conducted in a small tank. The height
(crest-to-trough) of the waves was measured a short distance from the wave
generator and at several points along the
beach different distances from the breakwater were measured and their mean and
standard error recorded.

290

6 An Introduction to Probability, Statistics, and Uncertainty

Location

Mean wave
height (cm)

Standard error of
mean (cm)

Near wave
generator

3.32

0.06

1.9 cm from
breakwater

4.42

0.09

1.9 cm from
breakwater

2.59

0.09

1.9 cm from
breakwater

3.26

0.06

At which points were the wave heights
significantly different from the height near
wave generator assuming that errors were
independent?
Of interest to the experimenter is the ratio of
the wave heights near the breakwater to the
initial wave heights in the deep water. Using
the results in Exercise 6.10, estimate the
standard error of this ratio at the three points
assuming that errors made in measuring the
height of waves at the three points and near
the wave generator are independent. At
which point does the ratio appear to be
significantly different from 1.00?
Using the results of Exercise 6.10, show
that the ratio of the mean wave heights is
probably a biased estimate of the actual
ratio. Does this bias appear to be
important?
6:12 Derive Kirby’s bound, Eq. 6.45, on the
estimate of the coefficient of skewness by
computing the sample estimates of the
skewness of the most skewed sample it
would be possible to observe. Derive also
the upper bound (n − 1)1/2 for the estimate
of the population coefficient of variation mxx
when all the observations must be
nonnegative.
6:13 The errors in the predictions of water
quality models are sometimes described
by the double exponential distribution
whose density is
a
f ðxÞ ¼ expðajx  bjÞ
2
 1\x\ þ 1

What are the maximum likelihood estimates of α and β? Note that
d
jx  bj ¼
db



1
þ1

x[b
x\b

Is there always a unique solution for β?
6:14 Derive the equations that one would need
to solve to obtain maximum likelihood
estimates of the two parameters α and β of
the gamma distribution. Note an analytical
expression for dΓ(α)/dα is not available so
that a closed-form expression for maximum likelihood estimate of α is not
available. What is the maximum likelihood estimate of β as a function of the
maximum likelihood estimate of α?
6:15 The log-Pearson Type-III distribution is
often used to model flood flows. If X has a
log-Pearson Type-III distribution then
Y ¼ lnðXÞ  m
has a two-parameter gamma distribution
where em is the lower bound of X if β > 0
and em is the upper bound of X if β < 0.
The density function of Y can be written
ðbyÞa1
expðbyÞdðbyÞ
CðaÞ
0\by\ þ 1

fY ðyÞdy ¼

Calculate the mean and variance of X in
terms of α, β and m. Note that
E½X r  ¼ E½ðexpðY þ mÞÞr 
¼ expðrmÞE½expðrYÞ
To evaluate the required integrals
remember that the constant terms in the
definition of fY(y) ensure that the integral
of this density function must be unity for
any values of α and β so long as α > 0 and
βy > 0. For what values of r and β does
the mean of X fail to exist? How do the
values of m, α and β affect the shape and
scale of the distribution of X?

Exercises

291

6:16 When plotting observations to compare
the empirical and fitted distributions of
streamflows, or other variables, it is necessary to assign a cumulative probability
to each observation. These are called
plotting positions. As noted in the text, for
the ith largest observation Xi,

Date

Maximum
discharge
(m3/s)

Date

03/26/33

445

12/11/33

1410

02/27/55

603

11/17/34

475

04/08/56

880

03/13/36

978

01/03/57

485

E½FX ðXi Þ ¼ i=ðn þ 1Þ

12/18/36

461

12/15/58

812

12/15/37

362

12/23/59

1420

Thus the Weibull plotting position
i/(n + 1) is one logical choice. Other
commonly used plotting positions are the
Hazen
plotting
position
(i – 3/8)/
(n + 1/4). The plotting position (i − 3/8)/
(n + 1/4) is a reasonable choice because
its use provides a good approximation to
the expected value of Xi. In particular for
standard normal variables

04/08/39

530

01/16/60

4090

02/04/40

350

01/26/61

376

02/21/41

1100

03/24/62

904

02/25/42

980

01/07/63

4120

12/20/42

575

12/21/63

1740

02/29/44

694

03/02/65

973

12/21/44

612

02/23/66

378

12/24/45

540

10/11/66

827

05/15/47

381

04/01/68

626

05/11/48

334

02/28/69

3170

05/11/49

670

01/13/70

2790

01/01/50

769

04/04/71

1130

12/30/50

1570

01/18/72

437

01/26/52

512

02/16/73

312

01/20/53

613

E½Xi  ffi U1 ½ði  3=8Þ=ðn þ 1=4Þ
where Φ() is the cumulative distribution
function of a standard normal variable.
While much debate centers on the appropriate plotting position to use to estimate
pi = FX(Xi), often people fail to realize
how imprecise all such estimates must be.
Noting that
Varðpi Þ ¼

iðn  i  1Þ
ðn þ 1Þ2 ðn þ 2Þ

;

contrast the difference between the estimates ^
pi of pi provided by these three
plotting positions and the standard deviation of pi. Provide a numerical example.
What do you conclude?
6:17 The following data represent a sequence of
annual flood flows, the maximum flow
rate observed each year, for the Sebou
River at the Azib Soltane gaging station in
Morocco.

03/13/54

Maximum
discharge
(m3/s)
750

(a) Construct a histogram of the Sebou
flood flow data to see what the flow
distribution looks like.
(b) Calculate the mean, variance, and
sample skew. Based on Table 6.3,
does the sample skew appear to be
significantly different from zero?
(c) Fit a normal distribution to the data
and use the Kolmogorov–Smirnov test
to determine if the fit is adequate.
Draw a quantile-quantile plot of the
fitted
quantiles
F−1[(i – 3/8)/
(n + 1/4)] versus the observed quantiles xi and include on the graph the

292

6 An Introduction to Probability, Statistics, and Uncertainty

(d)
(e)

(f)

(g)

Kolmogorov–Smirnov bounds on
each xi, as shown in Figs. 6.2a, b.
Repeat part (c) using a two-parameter
lognormal distribution.
Repeat part (c) using a three-parameter
lognormal distribution. The Kolmogorov–Smirnov test is now
approximate if applied to loge[Xi − τ],
where τ is calculated using Eq. 6.81 or
some other method of your choice.
Repeat part (c) for two- and threeparameter versions of the gamma
distribution. Again, the KolmogorovSmirnov test is approximate.
A powerful test of normality is provided by the correlation test. As
described by Filliben (1975), one
should approximate pi = FX(xi) by
8
< 1  ð0:5Þ1=n
^pi ¼ ði  0:3175Þ=ðn þ 0:365Þ
:
ð0:5Þ1=n

i¼1
i ¼ 2; . . .; n  1
i¼n

Then one obtains a test for normality
by calculation of the correlation
r between the ordered observations Xi
and mi the median value of the ith
largest observation in a sample of
n standard normal random variables
so that
mi ¼ U1 ð^pi Þ
where Φ(x) is the cumulative distribution function of the standard normal
distribution. The value of r is then
Pn
 2
xÞ2 ðmi  mÞ
i¼1 ðxi  
r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pn
P
 2
xÞ2 ni¼1 ðmj  mÞ
i¼1 ðxi  
Some significance levels for the value
of r are (Filliben 1975)

Significance level
n

1%

5%

10%

10

0.876

0.917

0.934

20

0.925

0.950

0.960

30

0.947

0.964

0.970

40

0.958

0.972

0.977

50

0.965

0.977

0.981

60

0.970

0.980

0.983

The probability of observing a value of
r less than the given value, where the
observations actually drawn from a
normal distribution, equals the specified
probability. Use this test to determine
whether a normal or two-parameter
lognormal distribution provides an
adequate model for these flows.
6:18 A small community is considering the
immediate expansion of its wastewater
treatment facilities so that the expanded
facility can meet the current deficit of 0.25
MGD and the anticipated growth in demand
over the next 25 years. Future growth is
expected to result in the need of an additional 0.75 MGD. The expected demand for
capacity as a function of time is


Demand ¼ 0:25MGD þ G 1  e0:23t
where t is the time in years and G = 0.75
MGD. The initial capital costs and maintenance and operating costs related to
capital are $1.2 × 106 C0.70 where C is the
plant capacity (MGD). Calculate the
loss Primary>Loss of economic efficiency
LEE and the misrepresentation of minimal
costs (MMC) that would result if a
designer incorrectly assigned G a value of
0.563 or 0.938 (±25%) when determining
the required capacity of the treatment

Exercises

293

plant. [Note: When evaluating the true
cost of a nonoptimal design which provides insufficient capacity to meet demand
over a 25-year period, include the cost of
building a second treatment plant; use an
interest rate of 7% per year to calculate the
present value of any subsequent expansions.] In this problem, how important is
an error in G compared to an error in the
elasticity of costs equal to 0.70?
One MGD, a million gallons per day, is
equivalent to 0.0438 m3/s.
6:19 A municipal water utility is planning the
expansion of their water acquisition system over the next 50 years. The demand
for water is expected to grow and is given
by
D ¼ 10tð1  0:006tÞ
where t is the time in years. It is expected
that two pipelines will be installed along
an acquired right-of-way to bring water to
the city from a distant reservoir. One pipe
will be installed immediately and then a
second pipe when the demand just equals
the capacity C in year t is
PV ¼ ða þ bCc Þert
where
a ¼ 29:5
b ¼ 5:2
c ¼ 0:5
r ¼ 0:07=year
Using a 50-year planning horizon, what is
the capacity of the first pipe which minimizes the total present value of the
construction of the two pipelines? When
is the second pipe built? If a ± 25% error

is made in estimating γ or r, what are the
losses of economic efficiency (LEE) and
the misrepresentation of minimal costs
(MMC)? When finding the optimal decision with each set of parameters, find the
time of the second expansion to the
nearest year; a computer program that
finds the total present value of costs as a
function of the time of the second
expansion t for t = 1, …, 50 would be
helpful. (A second pipe need not be
built.)
6:20 A national planning agency for a small
country must decide how to develop the
water resources of a region. Three development plans have been proposed, which
are denoted d1, d2, and d3. Their respective
costs are 200f, 100f, and 100f where f is a
million farths, the national currency. The
national benefits which are derived from
the chosen development plan depend, in
part, on the international market for the
goods and agricultural commodities that
would be produced. Consider three possible international market outcomes, m1, m2,
and m3. The national benefits if development plan 1 selected would be, respectively, 400, 290, 250. The national benefits
from selection of plan 2 would be 350,
160, 120, while the benefits from selection
of plan 3 would be 250, 200, 160.
(a) Is any plan inferior or dominated?
(b) If one felt that probabilities could not
be assigned to m1, m2, and m3 but
wished to avoid poor outcomes, what
would be an appropriate decision
criterion, and why? Which decisions
would be selected using this
criterion?
(c) If Pr[m1] = 0.50 and Pr[m2] = Pr
[m3] = 0.25, how would each of the

294

6 An Introduction to Probability, Statistics, and Uncertainty

expected net benefits and expected
regret criteria rank the decisions?
6:21 Show that if one has a choice between two
water management plans yielding benefits
X and Y, where X is stochastically smaller
than Y, then for any reasonable utility
function, plan Y is preferred to X.
6:22 A reservoir system was simulated for
100 years and the average annual benefits
and their variance were found to be
B ¼ 4:93
s2B ¼ 3:23
The correlation of annual benefits was also
calculated and is:
k

rk

0

1.000

1

0.389

2

0.250

3

0.062

4

0.079

5

0.041

(a) Assume that ρ(l) = 0 for l > k, compute (using Eq. 6.137) the standard
error of the calculated average benefits
for k = 0, 1, 2, 3, 4, and 5. Also calculate the standard error of the calculated benefits, assuming that annual
benefits may be thought of as a
stochastic process with a correlation
structure ρB(k) = [ρB(1)]k. What is the
effect of the correlation structure
among the observed benefits on the
standard error of their average?
(b) At the 90 and 95% levels, which of the
rk are significantly different from zero,
assuming that ρB(l) = 0 for l > k?

Replicate

Policy 2

1

6.27

4.20

2

3.95

2.58

3

4.49

3.87

4

5.10

5.70

5

5.31

4.02

6

7.15

6.75

7

6.90

4.21

8

6.03

4.13

9

6.35

3.68

10

6.95

7.45

11

7.96

6.86

Mean, Xi

6.042

1.570

Standard deviation
of values, sxi

1.217

4.859

(a) Construct a 90% confidence limits for
each of the two means X i .
(b) With what confidence interval can you
state that Policy 1 produces higher
benefits than Policy 2 using the sign
test and using the t-test?
(c) If the corresponding replicate with
each policy were independent, estimate with what confidence one could
have concluded that Policy 1 produces
higher benefits with the t-test.
6:24 Assume that annual streamflow at a gaging
site have been grouped into three categories or states. State 1 is 5–15 m3/s, state
2 is 15–25 m3/s, and state 3 is 25–35 m3/s,
and these grouping contain all the flows on
records. The following transition probabilities have been computed from record:
Pij
i

6:23 Replicated reservoir simulations using two
operating policies produced the following
results:

Benefits
Policy 1

j
1

2

3

1

0.5

0.3

0.2

2

0.3

0.3

0.4

3

0.1

0.5

0.4

Exercises

295

(a) If the flow for the current year is
between 15 and 25 m3/s, what is the
probability that the annual flow
2 years from now will be in the range
25–35 m3/s?
(b) What is the probability of a dry, an
average, and a wet year many years
from now?
6:25 A Markov chain model for the streamflows in two different seasons has the
following transition probabilities
Streamflow in Season
1

Streamflow next Season 2
0–3
m3/s

3–6
m3/s

≥6 m3/s

0–10 m3/s

0.25

0.50

0.25

≥10 m3/s

0.05

0.55

0.40

Streamflow in Season
2

6:27 Assume that there exist two possible discrete flows Qit into a small reservoir in
each of two periods t each year having
probabilities Pit. Find the steady-state
operating policy (release as a function of
initial reservoir volumes and current period’s inflow) for the reservoir that minimizes the expected sum of squared
deviations from storage and release targets.
Limit the storage volumes to integer values
that vary from 3 to 5. Assume a storage
volume target of 4 and a release target of 2
in each period t. (Assume only integer
values of all states and decision variables
and that each period’s inflow is known at
the beginning of the period.) Find the
annual expected sum of squared deviations
from the storage and release targets.

Streamflow next Season 1
0–10 m3/s

≥10 m3/s

0–3 m3/s

0.70

0.30

3–6 m3/s

0.50

0.50

Calculate the steady-state probabilities of
the flows in each interval in each season.
6:26 Can you modify the deterministic discrete
DP reservoir operating model to include
the uncertainty, expressed as Ptij, of the
inflows, as in Exercise 6.25?
(Hints: The operating policy would define
the release (or final storage) in each season
as a function of not only the initial storage
but also the inflow. If the inflows change,
so might the release or final storage volume. Hence you need to discretize the
inflows as well as the storage volumes.
Both storage and inflow are state variables.
Assume for this model you can predict
with certainty the inflow in each period at
the beginning of the period. So, each node
of the network represents a known initial
storage and inflow value. You cannot
predict with certainty the following period’s flows, only their probabilities. What
does the network look like now?

Period, t

Flows, Qit

Probabilities,
Pit

i=1

i=2

i=1

i=2

1

1

2

0.17

0.83

2

3

4

0.29

0.71

This is an application of Exercise 6.26
except the flow probabilities are independent of the previous flow.
6:28 Assume that the streamflow Q at a particular site has cumulative distribution
function FQ(q) = q/(1 + q) for q ≥ 0.
Show how to compute the mean streamflow, and the probability that any specified
value of streamflow, q, will be exceeded.
6:29 Assume that a potential water user can
withdraw water from an unregulated
stream, and that the probability distribution function FQ() of the available
streamflow Q is known. Calculate the
value of the withdrawal target T that will
maximize the expected net benefits from
the water’s use given the two short-run
benefit functions specified below.
(a) The benefits from streamflow Q when
the target is T are

296

6 An Introduction to Probability, Statistics, and Uncertainty


BðQjTÞ ¼

B0 þ bT þ cðQ  TÞ Q  T
B0 þ bT þ dðQ  TÞ Q\T

where δ > β > γ. In this case, the
optimal target T* can be expressed as
a function of P* = FQ(T) = Pr{Q ≤
T}, the probability that the random
streamflow Q will be less than or
equal to T. Prove that
P ¼ ðb  cÞ=ðd  cÞ:
(b) The benefits from streamflow Q when
the target is T are
BðQjTÞ ¼ B0 þ bT  dðQ  TÞ2

fact that observations are paired (i.e.,
simulation j for 11 ≤ j ≤ 20 in both tables
were obtained with the same streamflow
sequence) to perform the analysis with the
sign test.
(b) Use the sign test to demonstrate that the
average deficit with Policy 1 (Table 6.14)
is stochastically smaller than with Policy 2
(Table 6.15); use all simulations.
6:32 The accompanying table provides an
example of the use of non-parametric
statistics for examining the adequacy of
synthetic streamflow generators. Here the
maximum yield that can be supplied with
a given size reservoir is considered. The
following table gives the rank of the
maximum yield obtainable with the historic flows among the set consisting of the

Benefits

β
γ
Bo

δ
T

Flow q

fQ(q)
P*

6:30 If a random variable is discrete, what effect
does this have on the specified confidence
of a confidence interval for the median or
any other quantile? Give an example.
6:31 (a) Use Wilcoxon test for unpaired samples to test the hypothesis that the distribution of the total shortage TS in
Table 6.14 is stochastically less than the
total shortage TS reported in Table 6.15.
Use only the data from the second 10
simulations reported in the table. Use the

1-P*

historic yield and the maximum yield
achievable with 1000 synthetic sequences
of 25 different rivers in North America.
(a) Plot the histogram of the ranks for
reservoir sizes S/μQ = 0.85, 1.35,
2.00. (Hint: Use the intervals 0–100,
101–200, 201–300, etc.) Do the ranks
look uniformly distributed?
Rank of the Maximum Historic Yield
among 1000 Synthetic Yields

Exercises
River number

297
Normalized active storage, S/μQ
0.35

0.85

1.35

2.00

1

47

136

128

235

2

296

207

183

156

3

402

146

120

84

4

367

273

141

191

5

453

442

413

502

6

76

92

56

54

7

413

365

273

279

8

274

191

86

51

9

362

121

50

29

10

240

190

188

141

11

266

66

60

118

12

35

433

562

738

13

47

145

647

379

14

570

452

380

359

15

286

392

424

421

16

43

232

112

97

17

22

102

173

266

18

271

172

260

456

19

295

162

272

291

20

307

444

532

410

21

7

624

418

332

22

618

811

801

679

23

1

78

608

778

24

263

902

878

737

25

82

127

758

910

Source A.I. McLeod and K.W. Hipel, Critical Drought
Revisited, Paper presented at the international Symposium
on Risk and Reliability in Water Resources, Waterloo,
Ont., June 26–28, 1978

(b) Do you think this streamflow generation model produces streamflows
which are consistent with the historic
flows when one uses as a criterion the
maximum possible yield? Construct a
statistical test to support your conclusion and show that it does support
your conclusion. (Idea: You might
want to consider if it is equally likely
that the rank of the historical yield is

500 and below 501 and above. You
could then use the binomial distribution to determine the significance of
the results.)
(c) Use the Kolmogrov–Smirnov test to
check if the distribution of the yields
obtainable with storage S/μQ = 1.35 is
significantly different from uniform
FU(u) = u for 0 ≤ u ≤ 1. How
important do you feel this result is?
6:33 Section 7.3 dismisses the bias in ν2x for
correlated X’s as unimportant to its variance.
(a) Calculate the approximate bias in ν2x
for the cases corresponding to
Table 6.10 and determine if this
assertion is justified.
(b) By numerically evaluating the bias
and variance of ν2x , when n = 25,
determine if the same result holds if
ρx(k) = 0.5(0.9)k, which is the autocorrelation function of an ARMA(1,
1) process sometimes used to describe
annual streamflow series.
6:34 Consider the crop irrigation problem in
Exercise 4.31. For the given prices 30 and
25 for crop A and B, the demand for each
crop varies over time. Records of demands
show for crop A the demand ranges from 0
to 10 uniformly. There is an equal probability of that the demand will be any value
between 0 and 10. For crop B the demand
ranges from 5 units to 15 units, and the
most likely demand is 10. At least 5 units
and no more than 15 units of crop B will
be demanded. The demand for crop B can
be defined by a triangular density function,
beginning with 5, having a mean of 10 and
an upper limit of 15. Develop and solve a
model for finding the maximum expected
net revenue from both crops, assuming the
costs of additional resources are 2/unit of
water, 4/unit of land, 8/unit of fertilizer,

298

6 An Introduction to Probability, Statistics, and Uncertainty

and 5/unit of labor. The cost of borrowed
money, i.e., the borrowing interest rate, is
8 percent per growing season. How does
the solution change if the resource costs
are 1/10th of those specified above?
6:35 In Sect. 6.9.2 generated synthetic streamflows sequences were used to simulate a
reservoir’s operation. In the example, a
Thomas-Fiering model was used to generate ln(Q1y) and ln(Q2y,) the logarithms of
the flows in the two seasons of each year
y, so as to preserve the season-to-season
correlation of the untransformed flows.
Noting that the annual flow is the sum of
the untransformed seasonal flows Q1y and
Q2y, calculate the correlation of annual
flows produced by this model. The
required data are given in Table 6.13.
(Hint: You need to first calculate the
covariance of ln(Q1y) and ln(Q1,y+1) and
then of Q1y and Q2,y+1).
6:36 Part of New York City’s municipal water
supply is drawn from three parallel reservoirs in the upper Delaware River basin.
The covariance matrix and lag-1 covariance matrix, as defined in Eqs. 6.166 and
6.168, were estimated based on the
50-year flow record to be (in m3/s):
2

20:002
6
S0 ¼ 4 21:436
6:618
2
6:487
6
S1 ¼ 4 7:500
2:593

21:436
25:141
6:978
6:818
7:625
2:804

3
6:618
7
6:978 5 ¼ ½CovðQiy ; Qyj Þ
2:505
3

1:638

7
1:815 5 ¼ ½CovðQiy þ 1 ; Qyj Þ
0:6753

Other statistics of the annual flow are
Site

Reservoir

Mean
flow

Standard
deviation

r1

1

Pepacton

20.05

4.472

0.3243

2

Cannosville

23.19

5.014

0.3033

3

Neversink

7.12

1.583

0.2696

(a) Using these data, determine the values
of the A and B matrices of the lag 1
model defined by Eq. 6.165. Assume

that the flows are adequately modeled
by a normal distribution. A lower triangular B matrix that satisfies M = BBT
may be found by equating the elements
of BBT to those of M as follows:
pffiffiffiffiffiffiffiffi
M11
M21
M21
¼ b11 b21 ! b21 ¼
¼ pffiffiffiffiffiffiffiffi
b11
M11
M31
M31
¼ b11 b31 ! b31 ¼
¼ pffiffiffiffiffiffiffiffi
b11
M11
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ b221 þ b222 ! b222 ¼ M22  b221
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 =M
¼ M22  M21
11

M11 ¼ b211 ! b11 ¼
M21
M31
M22

and so forth for M23 and M33. Note
that bij = 0 for i < j and M must be
symmetric because BBT is necessarily
symmetric.
(b) Determine A and BBT for the Markov
model which would preserve the
variances and cross-covariances of the
flows at each site, but not necessarily
the lag 1 cross covariances of the
flows. Calculate the lag 1 crosscovariances of flows generated with
your calculated A matrix.
(c) Assume that some model has been
built to generate the total annual flow
into the three reservoirs. Construct and
calculate the parameters of a disaggregation model that, given the total
annual inflow to all three reservoirs,
will generate annual inflows into each
of the reservoirs preserving the variances and cross-covariances of the
flows. [Hint: The necessary statistics
of the total flows can be calculated
from those of the individual flows.]
6:37 Derive the variance of an ARMA(1, 1)
process in terms of ϕ1, θ1, and σv2. [Hint:
Multiply both sides of the equation to
obtain a second. Be careful to remember
which Vt’s are independent of which Zt’s.]
6:38 The accompanying table presents a
60-year flow record for the normalized

Exercises

299

flows of the Gota River near SjotopVannersburg in Sweden.
(a) Fit an autoregressive Markov model to
the annual flow record.
(b) Using your model, generate a 50-year
synthetic flow record. Demonstrate the
mean, variance, and correlation of your
generated flows deviate from the specified values no more than would be
expected as a result of sampling error.
(c) Calculate the autocorrelations and partial autocovariances of the annual flows
for a reasonable number of lags. Calculate the standard errors of the calculated
values. Determine reasonable value of
p and q for an ARMA(p, q) model of the
flows. Determine the parameter values
for the selected model.
Annual
Flows,
Gota
River
Sjotop-Vannersburg, Sweden

near

1898

1.158

1918

0.948

1938

0.892

1899

1.267

1919

0.907

1939

1.020

1900

1.013

1920

0.991

1940

0.869

1901

0.935

1921

0.994

1941

0.772

1902

0.662

1922

0.701

1942

0.606

1903

0.950

1923

0.692

1943

0.739

1904

1.120

1924

1.086

1944

0.813

1905

0.880

1925

1.306

1945

1.173

1906

0.802

1926

0.895

1946

0.916

1907

0.856

1927

1.149

1947

0.880

1908

1.080

1928

1.297

1948

0.601

1909

0.959

1929

1.168

1949

0.720

1910

1.345

1930

1.218

1950

0.955

1911

1.153

1931

1.209

1951

1.186

1912

0.929

1932

0.974

1952

1.140

1913

1.158

1933

0.834

1953

0.992

1914

0.957

1934

0.638

1954

1.048

1915

0.705

1935

0.991

1955

1.123

1916

0.905

1936

1.198

1956

0.774

1917

1.000

1937

1.091

1957

0.769

Source V.M. Yevdjevich, Fluctuations of Wet and Dry Years,
Part I, Hydrology Paper No. 1, Colorado State University,
Fort Collins, Colo., 1963

(d) Using the estimated model in (c),
generate a 50-year synthetic streamflow record and demonstrate that the
mean, variance, and show that first
autocorrelations of the synthetic flows
deviate from the modeled values by no
more than would be expected as a
result of sampling error.
6:39 (a) Assume that one wanted to preserve
the covariance matrices S0 and S1 of
the flows at several site Zy using the
multivariate or vector ARMA(0, 1)
model
Zy þ 1 ¼ AVy  BVy1
where Vy contains n independent standard normal random variables. What is
the relationship between the values of
S0 and S1 and the matrices A and B?
(b) Derive estimates of the matrices A, B,
and C of the multivariate AR(2) model
Zy þ 1 ¼ AZy þ BZy1 þ CVy
using the covariance matrices S0, S1,
and S2.
6:40 Create a model for the generation of
monthly flows. The generated monthly
flows should have the same marginal
distributions as were fitted to the
observed flows of record and should
reproduce (i) the month-to-month correlation of the flows, (ii) the month-toseason correlation between each monthly
flow and the total flow the previous
season, and (iii) the month-to-year correlation between each monthly flow and
the total 12-month flow in the previous
year. Show how to estimate the model’s
parameters. How many parameters
does your model have? How are the
values of the seasonal model? How
do you think this model could be
improved?

300

6 An Introduction to Probability, Statistics, and Uncertainty

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reproduce the material.

7

Modeling Uncertainty

Decision-makers are increasingly willing to
consider the uncertainty associated with model
predictions of the economic, environmental, or
social impacts associated with possible decisions.
Information on uncertainty does not make
decision-making easier, but to ignore it is to
ignore reality. Incorporating what is known about
the uncertainty of input parameters and variables
used in optimization and simulation models can
help in quantifying the uncertainty in the resulting model output. This chapter outlines and
illustrates some approaches for doing this.

7.1

Introduction

Water resource planners and managers work in
an environment of change and uncertainty. Water
supplies are always uncertain, if not in the short
term at least in the long term. Water demands and
the multiple purposes and objectives water serve
tend to change over time, and these changes
cannot always be predicted. Many of the
parameters of models used to predict the multiple
hydrologic, economic, environmental, ecological, and social impacts are also uncertain. Indeed,
models used to predict these impacts are, at least
in part, based on many uncertain assumptions.
This uncertainty associated with planning and
managing cannot be avoided (WWAP 2012).
To the extent that probabilities can be included where appropriate in models and their inputs
at least some of the uncertainty of their outputs

can be identified and quantified. These models
are called probabilistic or stochastic models.
Most probabilistic models provide a range of
possible values for each output variable along
with their probabilities. Stochastic models
attempt to model the random processes that occur
over time, and provide alternative time series of
outputs along with their probabilities. In other
cases sensitivity analyses (solving models under
different assumptions) can be carried out to
estimate the impact of any uncertainty on the
decisions being considered. In some situations
uncertainty may not significantly impact the
decisions that should be made. In other situations
it will. Sensitivity analyses can help guide efforts
needed to reduce that uncertainty. Model sensitivity and uncertainty analysis are discussed in
more detail in the next chapter.
This chapter is divided into two main sections.
The first section introduces a number of
approaches to probabilistic optimization and
simulation modeling. Probabilistic models will
be developed and applied to some of the same
water resources management problems used to
illustrate deterministic modeling in previous
chapters. These modeling methods could be, and
have been, applied to numerous other water
resources planning and management problems as
well. The purpose here, however, is simply to
illustrate some of these commonly used approaches to probabilistic modeling and show how
they can be applied to water resources system
design and operating problems.

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_7

301

302

7.2

7

Generating Values from Known
Probability Distributions

As discussed in the previous chapter, variables
whose values cannot be predicted with certainty
are called random variables. Often inputs to
hydrologic simulation models are observed or
synthetically generated values of rainfall or
streamflow. Other examples of such random
variables could be evaporation losses, point and
nonpoint source wastewater discharges, demands
for water, spot prices for energy that may impact
the amount of hydropower to produce, etc.
Random processes are considered stationary if
the statistical attributes of the process are not
changing. If there is no serial correlation in the
spatial or temporal sequence of observed values,
then such stationary random processes can be
characterized by single probability distributions.
These probability distributions are often based on
past observations of the values of the random
variable. These past observations or measurements are used either to define the probability
distribution itself or to estimate parameter values
of an assumed type of distribution.
Let R be a random variable whose probability
density distribution, fR(r), is as shown in Fig. 7.1.
This distribution indicates the probability or
likelihood of an observed value of the random
variable R being between any two values of r on
the horizontal axis. For example, the probability
of an observed value of R being between 0 and r*
is p*, the shaded area to the left of r* in Fig. 7.1.
The entire area under a probability density distribution, as shown in Fig. 7.1, is 1.

Modeling Uncertainty

Integrating this function over the entire range
of r, converts the density function to a cumulative distribution function, FR(r*), ranging from 0
to 1, as illustrated in Fig. 7.2.
Zr


fR ðrÞdr ¼ Prðr
 RÞ ¼ FR ðr
Þ

ð7:1Þ

0

Given any value of p* from 0 to 1, one can
find its corresponding random variable value r*
from the inverse of the cumulative distribution
function.
FR1 ðp
Þ ¼ r


ð7:2Þ

From the distribution shown in Fig. 7.1 it is
obvious that the likelihood of different values of
the random variable varies; values in the vicinity
of r* are much more likely to occur than are
values at the tails of the distribution. A uniform
distribution is one that looks like a rectangle; any
value of the random variable between its lower
and upper limits is equally likely. Using Eq. 7.2,
together with a series of uniformly distributed
(all equally likely) values of p* over the range
from 0 to 1 (i.e., along the vertical axis of
Fig. 7.2), a corresponding series of random
variable values, r*, associated with any distribution can be generated. These random variable
values will have a cumulative distribution as
shown in Fig. 7.2, and hence a density distribution as shown in Fig. 7.1, regardless of the
types or shapes of those distributions. The mean
and variance of the distributions will be
maintained.
The mean and variance of continuous distributions are
Z
rfR ðrÞdr ¼ E½R
Z

Fig. 7.1 Probability density distribution of a random
variable R. The probability that r is less than or equal r* is p*

ðrE½RÞ2 fR ðrÞdr ¼ Var½R

ð7:3Þ

ð7:4Þ

The mean, variance and serial correlations of
discrete distributions having possible values
denoted by ri with probabilities pi are

7.2 Generating Values from Known Probability Distributions

303

Fig. 7.2 Cumulative
distribution function of a
random variable R showing
the probability of any
observed random value of
R being less than or equal
to a given value r. The
probability of an observed
value of R being less than
or equal to r* is p*

X

ri pi ¼ E½R

ð7:5Þ

ðri E½RÞ2 pi ¼ Var[R

ð7:6Þ

i

X
i

If a time series of T random variable values, rt,
generated from the distribution of the same stationary random variable, R, exist, then the serial
or autocorrelations of rt and rt+k in this time
series for any positive integer k are
qR ðkÞ ¼

X
t¼1;Tk

½ðrt E½RÞðrt þ k E½RÞ=

X

ðrt E½RÞ2

t¼1;T

ð7:7Þ
The probability density and corresponding
cumulative probability distributions can be of any
shape, not just those named distributions commonly found in probability and statistics books.
The process of generating a time sequence
t = 1, 2, … of inputs, rt, from the probability
distribution of a random variable R where the lag
1 serial correlation, ρR(1) = ρ, is to be preserved
is a little more complex. The expected value of
the random variable Rt+1 depends on the
observed value, rt, of the random variable Rt,

together with the mean of the distribution, E[R],
and the correlation coefficient ρ. If there is no
correlation (ρ is 0), the expected value of Rt+1 is
the mean of the population, E[R]. If there is
perfect correlation (ρ is 1), the expected value of
Rt+1 is rt. In general, the expected value of Rt+1
given an observed value rt of Rt is
E½Rt þ 1 jRt ¼ rt  ¼ E½R þ qðrt E½RÞ:

ð7:8Þ

The variance of the random variable Rt+1
depends on the variance of the distribution, Var
[R], and the lag one correlation coefficient, ρ.
Var½Rt þ 1 jRt ¼ rt  ¼ Var½Rð1  q2 Þ:

ð7:9Þ

If there is perfect correlation (ρ = 1) the process is deterministic and there is no variance. The
value for rt+1 is rt. If there is no correlation, i.e.,
serial correlation does not exist (ρ = 0), the
generated value for rt+1 is its mean, E[R], plus
some randomly generated deviation from a normal distribution having a mean of 0 and a standard deviation of 1, denoted as N(0, 1). In this
case the value rt+1 is not dependent on rt.
When the serial correlation is more than 0 but
less than 1, then both the correlation and the
standard deviation (the square root of the

304

7

Modeling Uncertainty

Fig. 7.3 Diagram showing the calculation of a sequence of values of the random variable R from a multivariate normal
distribution in a way that preserves the mean, variance and correlation of the random variable

variance) influence the value of rt+1. A sequence
of random variable values from a multivariate
normal distribution that preserves the mean, E
[R], overall variance, Var[R], and lag one correlation ρ, can be obtained from Eq. 7.10.
rt þ 1 ¼ E½R þ qðrt  E½RÞ þ Nð0; 1Þrð1  q2 Þ1=2 :

ð7:10Þ
The term N(0, 1) in Eq. 7.10 is a random
number generated from a normal distribution
having a mean of 0 and a variance of 1. The
process involves selecting a random number
from a uniform distribution ranging from 0 to 1,
and using it in Eq. 7.2 for a N(0, 1) distribution
to obtain a value of random number for use in
Eq. 7.10. This positive or negative number is
substituted for the term N(0, 1) in Eq. 7.10 to
obtain a value rt+1. This is shown on the graph in
Fig. 7.3.
Simulation models that have random inputs,
such as a series of rt values, will generally produce random outputs. After many simulations,
the probability distributions of each random
output variable value can be defined. These then
can be used to estimate reliabilities and other
statistical characteristics of those output distributions. This process of generating multiple
random inputs for multiple simulations to obtain
multiple random outputs is called Monte Carlo
simulation.

7.3

Monte Carlo Simulation

To illustrate Monte Carlo simulation, consider
the allocation problem involving three firms,
each of which receives a benefit, Bi(xit), from the
amount of water, xit, allocated to it in each period
t. This situation is shown in Fig. 7.4. Monte
Carlo simulation can be used to find the probability distribution of the benefits to each firm
associated with the firm’s allocation policy.
Suppose the policy is to keep the first two
units of flow in the stream, to allocate the next 3
units to Firm 3, and the next 4 units to firms 1
and 2 equally. The remaining flow is to be allocated to each of the three firms equally up to the
limits desired by each firm, namely 3.0, 2.33, and
8.0, respectively. Any excess flow will remain in
the stream. The plots in Fig. 7.5 illustrate this
policy. Each allocation plot reflects the priorities
given to the three firms and the users further
downstream.
A simulation model can now be created. In
each of a series of discrete time periods t, the
flows Qt are drawn from a probability distribution, such as from Fig. 7.2 using Eq. 7.2. Once
this flow is determined, each successive allocation, xit, is computed. Once an allocation is made
it is subtracted from the streamflow and the next
allocation is made based on that reduced
streamflow, in accordance with the allocation

7.3 Monte Carlo Simulation

305

Fig. 7.4 Stream flow allocations in each period t result in benefits, Bi(xit), to each firm i. The flows, Qit, at each
diversion site i are the random flows Qt less the upstream withdrawals, if any

(a)

(b)

(c)

(d)

Fig. 7.5 a Water allocation policy for Firm 1 based on
the flow at its diversion site. This policy applies for each
period t. b Water allocation policy for Firm 2 based on the
flow at its diversion site for that firm. This policy applies
for each period t. c Water allocation policy for Firm 3

based on the flow at its diversion site. This policy applies
for each period t. d Streamflow downstream of site 3
given the streamflow Q3t at site 3 before the diversion.
This applies for each period t

policy defined in Fig. 7.5. After numerous time
steps the probability distributions of the allocations to each of the firms can be defined.
Figure 7.6 shows a flow chart for this simulation model.

Having defined the
the allocations, based
one can now consider
random variables, X1,
and 3, respectively.

probability distribution of
on the allocation policy,
each of the allocations as
X2, and X3 for firms 1, 2

306

7

Modeling Uncertainty

Fig. 7.6 Monte Carlo simulation to determine probability distributions of allocations to each of three water users, as
illustrated in Fig. 7.4. The dash lines represent information (data) flows

7.4

Chance Constrained Models

For models that include random variables it may
be appropriate in some situations to consider
constraints that do not have to be satisfied all the

time. Chance constraints specify the probability
of a constraint being satisfied, or the fraction of
the time a constraint has to apply. Consider, for
example, the allocation problem shown in
Fig. 7.4. For planning purposes, the three firms
may want to set allocation targets, not expecting

7.4 Chance Constrained Models

307

to have those targets met 100% of the time. To
insure, for example, that an allocation target, Ti,
of firm i will be met at least 90% of the time, one
could write the chance constraint
PrfTi  Xi g  0:90

i ¼ 1; 2; and 3

ð7:11Þ

In this constraint, the allocation target Ti is an
unknown decision variable, and Xi is a random
variable whose distribution has just been computed and is known.
To include chance constraints in optimization
models, their deterministic equivalents must be
defined. The deterministic equivalents of these
three chance constraints in Eq. 7.11 are
Ti  x0:10
it

i ¼ 1; 2; and 3

ð7:12Þ

is the particular value of the random
where x0:10
it
variable Xi that is equaled or exceeded 90% of
the time. This value is shown on the probability
distribution for Xi in Fig. 7.7.
To modify the allocation problem somewhat,
assume the benefit obtained by each firm is a
function of its target allocation and that the same
allocation target applies in each time period
t. The equipment and labor used in the firm is
presumably based on the target allocations. Once
the target is set assume there are no benefits
gained by excess allocations of water. If the
benefits obtained are to be based on the target allocations, rather than the actual allocations,
then the optimization problem is one of finding

Fig. 7.7 Probability density distribution of the random
allocation Xi to firm i. The particular allocation value x0:10
it
has a 90% chance of being equaled or exceeded, as
indicated by the shaded region

the values of the three targets that maximize the
total benefits obtained with a reliability of, say, at
least 90%.

 
 

Maximize 6T1 T12 þ 7T2  1:5T22 þ 8T3  0:5T32

ð7:13Þ
Subject to:
PrfT1 þ T2 þ T3  ½Qt  minðQt ; 2Þg  0:90
for all periods t
ð7:14Þ
where Qt is the random streamflow variable
upstream of all diversion sites. If the same
unconditional probability distribution of Qt applies
for each period t then only one Eq. 7.14 is needed.
Assuming the value of the streamflow, q0:10
,
t
that is equaled or exceeded 90% of the time, is
greater than 2 (the amount that must remain in
the stream), the deterministic equivalent of
chance constraint Eq. 7.14 is



T1 þ T2 þ T3  q0:10
 min q0:10
;2
ð7:15Þ
t
t
The value of the flow that is equaled or
exceeded 90% of the time, q0:10
, can be obtained
t
from the cumulative distribution of flows as
illustrated in Fig. 7.8.
Assume this 90% reliable flow is 8. The deterministic equivalent of the chance constraint
Eq. 7.14 for all periods t is simply T1 + T2 +
T3 ≤ 6. The optimal solution of the chance

Fig. 7.8 Example cumulative probability distribution
showing the particular value of the random variable,
q0:10
, that is equaled or exceeded 90% of the time
t

308

7

constrained target allocation model, Eqs. 7.13
and 7.15, is, as seen before, T1 = 1, T2 = 1, and
T3 = 4. The next step would be to simulate this
problem to see what the actual reliabilities might
be for various sequences of flows qt.

7.5

Markov Processes
and Transition Probabilities

Var½Q ¼

31
X

Modeling Uncertainty

ðqt  3:155Þ2 =31 ¼ 1:95 ð7:17Þ

t¼1

Lag-one correlation coefficient ¼ q
"
#
30
31
X
X
¼
ðqt þ 1  3:155Þðqt  3:155Þ =
ðqt  3:155Þ2
t¼1

t¼1

¼ 0:50

ð7:18Þ
Time series correlations can be incorporated into
models using transition probabilities. To illustrate this process, consider the observed flow
sequence shown in Table 7.1.
The estimated mean, variance and correlation
coefficient of the observed flows shown in
Table 7.1 can be calculated using Eqs. 7.16, 7.17
and 7.18.
E½Q ¼

31
X

qt =31 ¼ 3:155

ð7:16Þ

t¼1

Table 7.1 Sequence of flows for 31 time periods t

The probability distribution of the flows in
Table 7.1 can be approximated by a histogram.
Histograms can be created by subdividing the
entire range of random variable values, e.g.,
flows, into discrete intervals. For example, let
each interval be 2 units of flow. Counting the
number of flows in each interval and then
dividing those interval counts by the total number of counts results in the histogram shown in
Fig. 7.9. In this case, just to compare this with
what will be calculated later, the first flow, q1, is
ignored.

7.5 Markov Processes and Transition Probabilities

Fig. 7.9 Histogram showing an equal 1/3 probability
that the values of the random variable Qt will be in any
one of the three 2-flow unit intervals

Figure 7.9 shows a uniform unconditional
probability distribution of the flow being in any
of the possible discrete flow intervals. It does not
show the possible dependency of the probabilities of the random variable value, qt+1, in period
t + 1 on the observed random variable value, qt,
in period t. It is possible that the probability of
being in a flow interval j in period t + 1 depends
on the actual observed flow interval i in period t.
To see if the probability of being in any given
interval of flows is dependent on the past flow
interval one can create a matrix. The rows of the
matrix are the flow intervals i in period t. The
columns are the flow intervals j in the following
period t + 1. Such a matrix is shown in
Table 7.2. The numbers in the matrix are based
on the flows in Table 7.1 and indicate the
Table 7.2 Matrix showing the number of times a flow in
interval i in period t was followed by a flow in interval j in
period t + 1

309

number of times a flow in interval j followed a
flow in interval i.
Given an observed flow in an interval i in
period t, the probabilities of being in one of the
possible intervals j in the next period t + 1 must
sum to 1. Thus each number in each row of the
matrix in Table 7.2 can be divided by the total
number of flow transitions in that row (the sum of
the number of flows in the row) to obtain the
probabilities of being in each interval j in t + 1
given a flow in interval i in period t. In this case
there are 10 flows that followed each flow
interval i, hence by dividing each number in each
row of the matrix by 10 defines the transition
probabilities Pij.
Pij ¼ PrfQt þ 1 in interval jjQt in interval ig
ð7:19Þ
These conditional or transition probabilities,
shown in Table 7.3, correspond to the number of
transitions shown in Table 7.2.
Table 7.3 is a matrix of transition probabilities. The sum of the probabilities in each row
equals 1. Matrices of transition probabilities
whose rows sum to one are also called stochastic
matrices or first-order Markov chains.
If each row’s probabilities were the same, this
would indicate that the probability of observing
any flow interval in the future is independent of
the value previous flows. Each row would have
Table 7.3 Matrix showing the probabilities Pij of having
a flow in interval j in period t + 1 given an observed flow
in interval i in period t

310

the same probabilities as the unconditional distribution shown in Fig. 7.9. In this example the
probabilities in each row differ, showing that low
flows are more likely to follow low flows, and
high flows are more likely to follow high flows.
Thus the flows in Table 7.1 are positively correlated, as indeed has already determined from
Eq. 7.18.
Using the information in Table 7.3, one can
compute the probability of observing a flow in
any interval at any period on into the future given
the present flow interval. This can be done one
period at a time. For example assume the flow in
the current time period t = 1 is in interval i = 3.
The probabilities, PQj,2, of being in any of the
three intervals in the following time period t + 1
are the probabilities shown in the third row of the
matrix in Table 7.3.
The probabilities of being in an interval j in
the following time period t = 3 is the sum over
all intervals i of the joint probabilities of being in

7

Modeling Uncertainty

interval i in period t = 2 and making a transition
to interval j in period t = 3.
PrfQ3 in interval jg ¼ PQj;3
X
¼
PrfQ2 in interval ig PrfQ3 in interval jjQ2 in interval ig
i

ð7:20Þ
The last term in Eq. 7.20 is the transition
probability, from Table 7.3, that in this example
remains the same for all time periods t. These
transition probabilities, Pr{Qt+1 in interval j | Qt
in interval i} can be denoted as Pij.
Referring to Eqs. 7.19, 7.20 can be written in
a general form as
PQj;t þ 1 ¼

X
i

PQit Pij

for all intervals j and periods t

ð7:21Þ

This operation can be continued to any future
time period. Table 7.4 illustrates the results of

Table 7.4 Probabilities of observing a flow in any flow interval i in a future time period t given a current flow in
interval i = 3

These probabilities are derived using the transition probabilities Pij in Table 7.3 in Eq. 7.21 and assuming the flow
interval observed in period 1 is in interval 3

7.5 Markov Processes and Transition Probabilities

such calculations for up to six future periods,
given a present period (t = 1) flow in interval
i = 3.
Note that as the future time period t increases,
the flow interval probabilities are converging to
the unconditional probabilities, in this example
1/3. 1/3, 1/3, as shown in Fig. 7.9. The predicted
probability of observing a future flow in any
particular interval at some time in the future
becomes less and less dependent on the current
flow interval as the number of time periods
increases between the current period and that
future time period.
When these unconditional probabilities are
reached, PQit will equal PQi,t+1 for each flow
interval i. To find these unconditional probabilities directly, Eq. 7.21 can be written as
PQj ¼

X

PQi Pij

for all intervals j less one

i

ð7:22Þ
Equation 7.22 along with Eq. 7.23 can be
used to calculate all the unconditional probabilities PQi directly.
X

PQi ¼ 1

ð7:23Þ

i

Conditional or transition probabilities can be
incorporated into stochastic optimization models
of water resource systems.

7.6

Stochastic Optimization

To illustrate the development and use of
stochastic optimization models consider first the
allocation of water to a single user. Assume the
flow in the stream where the diversion takes
place is not regulated and can be described by a
known probability distribution based on historical records. Clearly the user cannot divert more
water than is available in the stream. A deterministic model would include the constraint that
the diversion x cannot exceed the available water
Q. But Q is a random variable. Some discrete
value, q, of the random variable Q will have to be

311

selected, knowing that there is some probability
that in reality, or in a simulation model, the
actual flow may be less than the selected value
q. Hence if the constraint x ≤ q is binding
(x = q), the actual allocation may be less than the
value of the allocation or diversion variable
x produced by the optimization model.
If the value of x affects one of the system’s
performance indicators, e.g., the net benefits, B
(x), to the user, a more accurate estimate of the
user’s net benefits will be obtained from considering a range of possible allocations x,
depending on the range of possible values of the
random flow Q. One way to do this is to divide
the known probability distribution of flows q into
discrete ranges, i, each range having a known
probability PQi. Designate a discrete flow qi for
each range. Associated with each known flow qi
is an unknown allocation xi. Now the deterministic constraint x ≤ q can be replaced with the set
of constraints xi ≤ qi and the term B(x) in the
original objective function can be replaced by its
P
expected value, i POi Bðxi Þ.
Note, when dividing a continuous known
probability distribution into discrete ranges, the
discrete flows qi selected to represent each range
i having a given probability PQi, should be
selected so as to maintain at least the mean and
variance of that known distribution as defined by
Eqs. 7.5 and 7.6.
To illustrate this consider a slightly more
involved example involving the allocation of
water to consumers upstream and downstream of
a reservoir. Both the policies for allocating water
to each user and the reservoir release policy are
to be determined. This example problem is
shown in Fig. 7.10.
If the allocation of water to each user is to be
based on a common objective, such as the minimization of the total sum, over time, of squared
deviations from prespecified target allocations,
each allocation in each time period will depend
in part on the reservoir storage volume.
Consider first a deterministic model of the
above problem, assuming known flows Qt and
upstream and downstream allocation targets UTt
and DTt in each of T within-year periods t in a

312

7

Modeling Uncertainty

Fig. 7.10 Example water
resource system involving
water diversions from a
river both upstream and
downstream of a reservoir
of known capacity

year. Assume the objective is to minimize the
sum of squared deviations of actual allocations,
ut and dt, from their respective target allocations,
UTt and DTt in each within-year period t.
Minimize

T n
o
X
ðUTt ut Þ2 þ ðDTt dt Þ2
t

ð7:24Þ
The constraints include:
(a) Continuity of storage involving initial storage volumes St, net inflows Qt − ut, and
releases Rt. Assuming no losses

St þ Qt  ut  Rt ¼ St þ 1
for each period t; T þ 1 ¼ 1

ð7:25Þ

(b) Reservoir capacity limitations. Assuming a
known active storage capacity K,

St  K

for each period t

ð7:26Þ

Allocation restrictions for each period t:
ut  Qt

ð7:27Þ

dt  Rt

ð7:28Þ

Equations 7.25 and 7.28 could be combined
to eliminate the release variable Rt since in this
problem knowledge of the total release in each
period t is not required. In this case Eq. 7.25
would become an inequality.
The solution for this model, Eqs. 7.24–7.28,
would depend on the known variables (the targets UTt and DTt, flows Qt and reservoir capacity
K). It would identify the particular upstream and
downstream allocations and reservoir releases in
each period t. It would not provide a policy that
defines what allocations and releases to make for
a range of different inflows and initial storage
volumes in each period t. A backward-moving
dynamic programming model can provide such a
policy. This policy will identify the allocations
and releases to make based on various initial
storage volumes, St, as well flows, Qt, as discussed in Chap. 4.
This deterministic discrete dynamic programming allocation and reservoir operation model

7.6 Stochastic Optimization

313

can be written for different discrete values of
storage volumes St from 0 ≤ St ≤ capacity K as
n
o
Ftn ðSt ; Qt Þ ¼ min ðUTt  ut Þ2 þ ðDTt  dt Þ2 þ Ftn1
þ 1 ðSt þ 1 ; Qt þ 1 Þ
ut ; Rt ; dt
ut  Qt
Rt  St þ Qt  ut
Rt  St þ Qt  ut  K
dt  Rt
St þ 1 ¼ St þ Qt  ut  Rt

ut  Qt
dt  St þ Qt  ut  St þ 1

ð7:29Þ
There are three variables to be determined at
each stage or time period t in the above dynamic
programming model. These three variables are
the allocations ut and dt and the reservoir release
Rt. Each decision involves three discrete decision
variable values. The functions Ftn ðSt ; Qt Þ define
the minimum sum of squared deviations given an
initial storage volume St and streamflow Qt in
time period or season t with n time periods
remaining until the end of reservoir operation.
One can reduce this three-decision variable
model to a single variable model by realizing that
for any fixed discrete initial and final storage
volume states, there can be a direct tradeoff
between the upstream and downstream allocations given the particular streamflow in each
period t. Increasing the upstream allocation will
decrease the resulting reservoir inflow and this in
turn will reduce the release by the same amount.
This reduces the amount of water available to
allocate to the downstream use.
Hence for this example problem involving
these upstream and downstream allocations, a
local optimization can be performed at each time
step t for each combination of storage states St
and St+1. This optimization finds the allocation
decision variables ut and dt that
minimizeðUTt  ut Þ2 þ ðDTt  dt Þ2

where

ð7:30Þ

ð7:31Þ
ð7:32Þ

This local optimization can be solved to
identify the ut and dt allocations for each feasible
combination of St and St+1 in each period t.
Given these optimal allocations, the dynamic
programming model can be simplified to include
only one discrete decision variable, either Rt or
St+1. If the decision variable St+1 is used in each
period t, the releases Rt in those periods t do not
need to be considered. Thus the dynamic programming model expressed by Eq. 7.29 can be
written for all discrete storage volumes St from 0
to K and for all discrete flows Qt as

2
ðUTt ut ðSt ;St þ 1 ÞÞ þ
Ftn ðSt ; Qt Þ ¼ min
2
St þ 1
ðDTt dt ðSt ;St þ 1 ÞÞ
St þ 1  K

ð7:33Þ
where the functions ut(St, St+1) and dt(St, St+1)
have been determined using Eqs. 7.30–7.32.
As the total number of periods remaining, n,
increases, the solution of this dynamic programming model will converge to a steady or
stationary state. The best final storage volume
St+1 given an initial storage volume St will likely
differ for each within-year period or season t, but
for a given season t it will be the same in successive years. In addition, for each storage volume St, streamflow, Qt, and within-year period
t the difference between Ftn þ T ðSt ; Qt Þ and
Ftn ðSt ; Qt Þ will be the same constant regardless of
the storage volume St, flow Qt and period t. This
constant is the optimal, in this case minimum,
annual value of the objective function, Eq. 7.24.
There could be additional limits imposed on
storage variables and release variables, such as for

314

flood control storage or minimum downstream
flows, as might be appropriate in specific situations.
The above deterministic dynamic programming model (Eq. 7.33) can be converted to a
stochastic model. Stochastic models consider
multiple discrete flows as well as multiple discrete storage volumes, and their probabilities, in
each period t. A common way to do this is to
assume that the sequence of flows follow a
first-order Markov process. Such a process
involves the use of transition or conditional
probabilities of flows as defined by Eq. 7.20.
To develop these stochastic optimization
models it is convenient to introduce some additional indices or subscripts. Let the index k denote different initial storage volume intervals.
These discrete intervals divide the continuous

7

Modeling Uncertainty

range of storage volume values from 0 to the
active reservoir capacity K. Each Skt is a discrete
storage volume that represents the range of
storage volumes in interval k at the beginning of
each period t.
Let the index l denote different final storage
volume intervals. Each Sl,t+1 is a discrete volume
that represents the storage volume interval l at the
end of in each period t or equivalently at the
beginning of period t + 1. As previously defined,
let the indices i and j denote the different flow
intervals, and each discrete qit and qj,t+1 represent
the flows in those flow intervals i and j in periods
t and t + 1, respectively.
These subscripts and the volume or flow
intervals they represent are illustrated in
Fig. 7.11.

Fig. 7.11 Discretization of streamflows and reservoir storage volumes. The area within each flow interval i below the
probability density distribution curve is the unconditional probability, PQit, associated with the discrete flow qit

7.6 Stochastic Optimization

315

With this notation it is now possible to develop
a stochastic dynamic programming model that
will identify the allocations and releases that are
to be made given both the initial storage volume,
Skt, and the flow, qit. It follows the same structure
as the deterministic models defined by Eqs. 7.30–
7.32, and 7.33.
To identify the optimal allocations in each
period t for each pair of feasible initial and final
storage volumes Skt and Sl,t+1, and inflows qit, one
can solve Eqs. 7.34–7.36.
minimizeðUTt  ukit Þ2 þ ðDTt  dkilt Þ2 ð7:34Þ
where
ukit  qit

for all k; i; t:

ð7:35Þ

dkilt  Skt þ qit  ukit  Sl;t þ 1
for all feasible k; i; l; t:

ð7:36Þ

The solution to these equations for each feasible combination of intervals k, i, l, and period
t defines the optimal allocations that can be
expressed as ut(k, i) and dt(k, i, l).
The stochastic version of Model 7.33, again
expressed in a form suitable for backward moving discrete dynamic programming, can be
written for different discrete values of Skt from 0
to K and for all qit as
Ftn ðSkt ; qit Þ ¼ minfðUTt  ut ðk; iÞÞ2 þ ðDTt  dt ðk; i; lÞÞ2
X
þ
Pt F n1 ðS
;q
Þg
j ij t þ 1 l;t þ 1 j;t þ 1
Sl;t þ 1
Sl;t þ 1  K

Ftn ðk; iÞ ¼ minfðUTt  ut ðk; iÞÞ2 þ ðDTt  dt ðk; i; lÞÞ2
X
þ
Pt F n1 ðl; jÞg
j ij t þ 1
l such that
Sl;t þ 1  K
Sl;t þ 1  Skt þ qit

ð7:38Þ
The steady-state solution of this dynamic
programming model will identify the preferred
final storage volume Sl,t+1 in period t given the
particular discrete initial storage volume Skt and
flow qit. This optimal policy can be expressed as
a function ‘ that identifies the best interval
l given intervals k, i and period t.
l ¼ ‘ðk; i; tÞ

Sl;t þ 1  Skt þ qit

ð7:37Þ
Each Ptij in the above recursive equation is
the known conditional or transition probability of
a flow qj,t+1 within interval j in period t + 1 given
a flow of qit within interval i in period t.

Ptij ¼ Pr flow qj;t þ 1 within interval j in t þ 1jflow of qit
within interval i in tg

The sum over all flow intervals j of these
conditional
probabilities
times
the


n1
Ft þ 1 Sl;t þ 1 ; qj;t þ 1 values is the expected minimum sum of future squared deviations from
allocation targets with n − 1 periods remaining
given an initial storage volume of Skt and flow of
qit and final storage volume of Sl,t+1. The value
Ftn ðSkt ; qit Þ is the expected minimum sum of
squared deviations from the allocation targets
with n periods remaining given an initial storage
volume interval of Skt and flow interval of qit.
Stochastic models such as these provide expected
values of objective functions.
Another way to write the recursion equations
of this model, Eq. 7.37, is by using just the
indices k and l to denote the discrete storage
volume variables Skt and Sl,t+1 and indices i and
j to denote the discrete flow variables qit and qj,t+1:

ð7:39Þ

All values of l given k, i, and t, defined by
Eq. 7.39, can be expressed in a matrix, one for
each period t.
Knowing the best final storage volume interval
l given an initial storage volume interval k and
flow interval i, the optimal downstream allocation, dt(k, i), can, like the upstream allocation, be
expressed in terms of only k and i in each period t.
Thus knowing the initial storage volume Skt and

316

7

flow qit is sufficient to define the optimal allocations ut(k, i) and dt(k, i), final storage volume
Sl,t+1, and hence the release Rt(k, i).
Skt þ qit  ut ðk; iÞ  Rt ðk; iÞ ¼ Sl;t þ 1
8k; i; t where l ¼ ‘ðk; i; tÞ

7.6.1

ð7:40Þ

Probabilities of Decisions

Knowing the function l = ‘(k, i, t) permits a
calculation of the probabilities of the different
discrete storage volumes, allocations, and flows.
Let

result in a final volume interval l. This relationship is defined by Eq. 7.39 (l = ‘(k, i, t)).
While Eq. 7.41 must apply, it is not sufficient.
The joint probability of a final storage volume in
interval l in period t and an inflow j in period
t + 1 must equal the joint probability of an initial
storage volume in the same interval l and an
inflow in the same interval j in period t + 1.
Multiplying the joint probability Pkit times the
conditional probability Ptij and then summing
over all k and i that results in a final storage
interval l defines the former, and the joint probability Pl,j,t+1 defines the latter.
Pl;j;t þ 1 ¼

PSkt = the unknown probability of an initial
storage volume Skt being within some interval
k in period t.
PQit = the steady-state unconditional probability
of flow qit within interval i in period t.
Pkit = the unknown probability of the upstream
and downstream allocations ut(k, i) and
dt(k, i) and reservoir release Rt(k, i) in period t.

Ptij = the known conditional or transition probability of a flow within interval j in period t + 1
given a flow within interval i in period t.
These transition probabilities Ptij can be displayed in matrices, similar to Table 7.3, but as a
separate matrix (Markov chain) for each period t.
The joint probabilities of an initial storage
interval k, an inflow in the interval i, Pkit in each
period t must satisfy two conditions. Just as the
initial storage volume in period t + 1 is the same
as the final storage volume in period t, the
probabilities of these same respective discrete
storage volumes must also be equal. Thus
X
j

Pl;j;t þ 1 ¼

XX
k

XX
i

k

Pkit Ptij

8l; j; t

l ¼ ‘ðk; i; tÞ
ð7:42Þ

Once again the sums in Eq. 7.42 are over all
combinations of k and i that result in the designated storage volume interval l as defined by the
policy ‘(k, i, t).
Finally, the sum of all joint probabilities Pkit
in each period t must equal 1.
XX

As previously defined

Modeling Uncertainty

k

Pkit ¼ 1

8t

ð7:43Þ

i

Note the similarity of Eqs. 7.42 and 7.43 to
the Markov steady-state flow Eqs. 7.22 and 7.23.
Instead of only one flow interval index considered in Eqs. 7.22 and 7.23, Eqs. 7.42 and 7.43
include two indices, one for storage volume
intervals and the other for flow intervals. In both
cases, one of Eqs. 7.22 and 7.42 can be omitted
in each period t since it is redundant with that
period’s Eqs. 7.23 and 7.43, respectively.
The unconditional probabilities PSkt and PQit
can be derived from the joint probabilities Pkit.
PSkt ¼

X

Pkit

8k; t

ð7:44Þ

Pkit

8i; t

ð7:45Þ

i

Pkit

8l; t

ð7:41Þ

i

where the sums in the right hand side of Eq. 7.41
are over only those combinations of k and i that

PQit ¼

X
k

7.6 Stochastic Optimization

317

Each of these unconditional joint or marginal
probabilities, when summed over all their volume and flow indices, will equal 1. For example
X

PSkt ¼

X
i

k

PQit ¼ 1

ð7:46Þ

Note that these probabilities are determined
based only on the relationships among flow and
storage intervals as defined by Eq. 7.39, l = ‘(k,
i, t) in each period t, and the Markov chains
defining the flow interval transition or conditional probabilities, Ptij . It is not necessary to
know the actual discrete storage values representing those intervals. Thus assuming any relationship among the storage volume and flow
interval indices, l = ‘(k, i, t) and a knowledge of
the flow interval transition probabilities Ptij , one
can determine the joint probabilities Pkit and their
marginal or unconditional probabilities PSkt. One
does not need to know what those storage
intervals are to calculate their probabilities.
(Amazing, isn’t it?)
Given the values of these joint probabilities
Pkit, the deterministic model defined by Eqs. 7.24–
7.28 can be converted to a stochastic model to
identify the best storage and allocation decision
variable values associated with each storage
interval k and flow interval i in each period t.
Minimize

T
XXX
k

i

t

n
o
Pkit ðUTt  ukit Þ2 þ ðDTt  dkit Þ2

ð7:47Þ

The constraints include
(a) Continuity of storage involving initial storage volumes Skt, net inflows qit − ukit, and at
least partial releases dkit. Again assuming no
losses:
Skt þ qit  ukit  dkit  Sl;t þ 1
l ¼ ‘ðk; i; tÞ

8k; i; t

ð7:48Þ

Reservoir capacity limitations.
Skit  K

8k; i; t

ð7:49Þ

8k; i; t

ð7:50Þ

Allocation restrictions.
ukit  qit

7.6.2

A Numerical Example

A simple numerical example may help to illustrate how these stochastic models can be developed without getting buried in detail. Consider,
for simplicity, two within-year periods each year.
The random flows Qt in each period t are divided
into two intervals. These flow intervals are represented by discrete flows of 1 and 3 volume
units per second in the first period and 3 and 6
volume units per second in the second period.
Their transition probabilities are shown in
Table 7.5.
Assuming equal within-year period durations,
these three discrete flow rates are equivalent to

Table 7.5 Transition probabilities for two ranges of flows in two within-year periods

318

7

about 16, 47 and 95 million volume units per
period.
Assume the active storage volume capacity
K in the reservoir equals 50 million volume
units. This capacity can be divided into different
intervals of storage. For this simple example
assume three storage volume intervals represented by 10, 25, and 40 million volume units.
Assume the allocation targets remain the same in
each period at both the upstream and downstream
sites. The upstream allocation target is approximately 2 volume units per second or 30 million
volume units in each period. The downstream
allocation target is approximately 5 volume units
per second or 80 million volume units in each
period.
With these data we can use Eqs. 7.34–7.36 to
determine the allocations that minimize the sum
of squared deviations from targets and what that
sum is, for all feasible combinations of initial and
final storage volumes, and flows. Table 7.6
shows the results of these optimizations. These
results will be used in the dynamic programming
model to determine the best final storage volumes given initial volumes and flows.
With the information in Tables 7.5 and 7.6,
the dynamic programming model, Eq. 7.38 or as
expressed in Eq. 7.51, can be solved to find the
optimal final storage volumes given an initial
storage volume and flow. The iterations of the
recursive equation, sufficient to reach a steady
state, are shown in Table 7.7.
X
Ftn ðk; iÞ ¼minfSDkil þ
Ptij Ftn1
þ 1 ðl; jÞg
j

over all l such that
Sl;t þ 1  K
Sl;t þ 1  Skt þ Qit
ð7:51Þ
This process can continue until a steady-state
policy is defined. Table 7.8 summarizes the next
five iterations. At this stage, the annual differences in the objective values associated with a

Modeling Uncertainty

particular state and season have come close to a
common constant value.
While the differences between corresponding
nþT
and Ftn have not yet reached a common
Ft
constant value to the nearest unit deviation (they
range from, 3475.5 to 3497.1 for an average of
3485.7), the policy has converged to that shown
in Tables 7.8 and 7.9.
Given this policy, the probabilities of being in
any of these volume and flow intervals can be
determined
by
solving
Eqs. 7.42–7.45.
Table 7.10 shows the results of these equations
applied to the data in Tables 7.5 and 7.8. It is
obvious that if the policy from Table 7.9 is followed, the steady-state probabilities of being in
storage interval 1 in period 1 and in interval 3 in
period 2 are 0.
Multiplying these joint probabilities by the
corresponding SDkit values in the last column of
Table 7.6 provides the annual expected squared
deviations, associated with the selected discrete
storage volumes and flows. This is done in
Table 7.11 for those combinations of k, i, and
l that are contained in the optimal solution as
listed in Table 7.9.
The sum of products of the last two columns
in Table 7.11 for each period t equals the
expected squared deviations in the period. For
period t = 1 the expected sum of squared deviations are 1893.3 and for t = 2 they are 1591.0.
The total annual expected squared deviations are
3484.3. This compares with the expected squared
deviations derived from the dynamic programming model, after nine iterations, ranging from
3475.5 to 3497.1 (as calculated from data in
Table 7.8).
The policy for reservoir releases is a function
not only of the initial storage volumes, but
also of the current inflow, i.e., the total water
available in the period. Reservoir release rule
curves now must become two-dimensional.
However, the inflow for each period usually
cannot be predicted at the beginning of each
period. Thus the reservoir release policy has to be

7.6 Stochastic Optimization

319

Table 7.6 Optimal allocations associated with given initial storage, Sk, flow, Qi, and final storage, Sl, volumes

These allocations uki and dkil minimize the sum of squared deviations, DSkil = (30 − uki)2 + (80 − dkil)2, from upstream
and downstream targets, 30 and 80, respectively, subject to uki ≤ flow Qi, and dkil ≤ release (Sk + Qi − uki − Sl)

320

7

Modeling Uncertainty

Table 7.7 First four iterations of dynamic programming model, Eq. 7.51, moving backward in successive periods n,
beginning in season t = 2 with n = 1

(continued)

7.6 Stochastic Optimization

321

Table 7.7 (continued)

(continued)

322

7

Modeling Uncertainty

Table 7.7 (continued)

(continued)

7.6 Stochastic Optimization

323

Table 7.7 (continued)

The iterations stop when the final storage policy given any initial storage volume and flow repeats itself in two
successive years. Initially, with no more periods remaining, F10 ðk; iÞ ¼ 0 for all k and i

324

7

Modeling Uncertainty

Table 7.8 Summary of objective function values Ftn ðk; iÞ and optimal decisions for stages n = 5–9 periods remaining

Table 7.9 Optimal reservoir policy l = ‘(k, i, t) for the example problem

7.6 Stochastic Optimization

325

Table 7.10 Probabilities of flow and storage volume intervals associated with policy as defined in Table 7.9 for the
example problem

expressed in a way that it can be followed
without knowledge of the current inflow. One
way to do this is to compute the expected value
of the release for each discrete storage volume,
and show it in a release rule. This is done in
Fig. 7.12. The probability of each discrete
release associated with each discrete river flow is
the probability of the flow itself. Thus in period 1
when the storage volume is 40, the expected
release is 46(0.41) + 56(0.59) = 52. These

discrete expected releases can be used to define a
continuous range of releases for the continuous
range of storage volumes from 0 to full capacity,
50. Figure 7.12 also shows the hedging that
might take place as the reservoir storage volume
decreases.
These and modifications of these policies can
be simulated to determine improved release
rules. Simulation modeling is the subject of the
following chapter.

326

7

Modeling Uncertainty

k, i, l

10
10

16
47

10
10

1,1,1
1,2,1

25
25
40
40

16
47
16
47

10
10
10
25

2,1,1
2,2,1
3,1,1
3,2,2

ukit

dkit

SDkit

P kit

1
1

0.0
0.0

16.0
47.0

4996.0
1989.0

1
1
1
1

0.0
6.0
0.0
6.0

31.0
56.0
46.0
56.0

3301.0
1152.0
2056.0
1152.0

0.0
0.0
0.2964706
0.1270588
0.1152941
0.4611765

sum = 1.0
10
10
25
25
40
40

47
95
47
95
47
95

25
25
25
40
40
40

1,2,2
1,3,2
2,2,2
2,3,2
3,2,3
3,3,3

2
2
2
2
2
2

0.0
7.5
0.0
15.0
0.0
22.5

32.0
57.5
47.0
65.0
47.0
72.5

3204.0
1012.5
1989.0
450.0
1989.0
112.5

0.2851765
0.2536471
0.1383529
0.3228235
0.0
0.0

sum = 1.0

Fig. 7.12 Reservoir release rule showing an interpolated release, increasing as storage volumes increase

E020829k

joint
probability

interval
indices

S1

sum squared
deviations

final
storage

Qi

optimal
allocation

flow

Sk

time period

initial
storage

Table 7.11 The optimal operating policy and the probability of each state and decision

7.7 Summary

7.7

Summary

This chapter has introduced some approaches for
including risk into optimization and simulation
models. The discussion began with ways to
obtain values of random variables whose probability distributions are known. These values, for
example streamflows or parameter values, can be
inputs to simulation models. Monte Carlo simulation involves the use of multiple simulations
using these random variable values to obtain the
probability distributions of outputs, including
various system performance indicators.
Two methods were reviewed for introducing
random variables along with their probabilities
into optimization models. One involves the use
of chance constraints. These are constraints that
must be met, as all constraints must be, but now
with a certain probability. As in any method there
are limits to the use of chance constraints. These
limitations were not discussed, but in cases
where chance constraints are applicable, and if
their deterministic equivalents can be defined,
they are probably the only method of introducing
risk into otherwise deterministic models that do
not add to the model size.
Alternatively, the range of random variable
values can be divided into discrete ranges. Each
range can be represented by a specific or discrete
value of the random variable. These discrete
values and their probabilities can become part of
an optimization model. This was demonstrated
using transition probabilities incorporated into
both linear and dynamic programming models.
The examples used in this chapter to illustrate
the development and application of stochastic
optimization and simulation models are relatively
simple. These and similar probabilistic and
stochastic models have been applied to numerous
water resources planning and management
problems. They can be a much more effective
screening tool than deterministic models based
on the mean or other selected values of random
variables. But sometimes they are not. Clearly if
the system being analyzed is very complex, or
just very big in terms of the number of variables

327

and constraints, the use of deterministic models
for a preliminary screening of alternatives prior
to a more precise probabilistic screening is often
warranted.

Reference
WWAP (United Nations World Water Assessment Programme) World Water Development Report 4. (2012).
Managing water under uncertainty and risk (867 pp).
Paris: UNESCO.

Additional References
(Further Reading)
Bassson, M. S., Allen, R. B., Pegram, G. G. S., & van
Rooyen, J. A. (1994). Probabilistic management of
water resource and hydropower systems. Highlands
Ranch, CO: Water Resources Publications.
Birge, J. R., & Louveaux, F. (1997). Introduction to
stochastic programming. NY: Springer.
DeGroot, M. H. (1980). Optimal statistical decisions. NY:
McGraw-Hill.
Dempster, M. A. H. (1980). Stochastic programming.
London: Academic Press.
Dorfman, R., Jacoby, H. D., & Thomas, H. A., Jr. (1972).
Models for managing regional water quality. Cambridge, MA: Harvard University Press.
Ermoliev, Y. (1970). Methods of stochastic programming.
Moscow: Nauka. (in Russian).
Ermoliev, Y., & Wets, R. (Eds.). (1988). Numerical
techniques for stochastic optimization. Berlin:
Springer.
Heyman, D. P., & Sobel, M. J. (1984). Stochastic models
in operations research, Volume II. Stochastic optimization. NY: McGraw-Hill.
Hillier, F. S., & Lieberman, G. J. (1990). Introduction to
stochastic models in operations research. NY:
McGraw-Hill.
Howard, R. A. (1960). Dynamic programming and
Markov processes. Cambridge, MA: MIT Press.
Hufschmidt, M. M., & Fiering, M. B. (1966). Simulation
techniques for design of water-resource systems.
Cambridge, MA: Harvard University Press.
Loucks, D. P., Stedinger, J. S., & Haith, D. A. (1981).
Water resource systems planning and analysis. Englewood Cliffs, NJ: Prentice Hall.
Maass, A., Hufschmidt, M. M., Dorfman, R., Thomas, H.
A., Jr., Marglin, S. A., & Fair, G. M. (1962). Design of
water-resource systems. Cambridge, MA: Harvard
University Press.

328
Prékopa, A. (1995). Stochastic programming. Dordrecht,
NL: Kluwer Academic Publishers.
Raiffa, H., & Schlaifer, R. (1961). Applied statistical
decision theory. Cambridge, MA: Harvard University
Press.
ReVelle, C. (1999). Optimizing reservoir resources. NY:
Wiley.
Ross, S. M. (1983). Introduction to stochastic dynamic
programming. NY: Academic Press.
Somlyódy, L., & Wets, R. J- B. Stochastic optimization
models for lake eutrophication management. Operations Research, 36, 660–681.
Wets, R. J- B. (1989). Stochastic programming. In G.
L. Nemhauser, A. H. G. Rinnooy Kan, & M. J. Todd
(Eds.), Handbooks in Operations Research and Management Science (Vol. 1, pp. 573–629). North-Holland,
Amsterdam: Elsevier Science Publishers.
Wurbs, R. A. (1996). Modeling and analysis of reservoir
system operations. Upper Saddle River, NJ: Prentice
Hall.

Exercises

7

Modeling Uncertainty

policy (release as a function of initial
reservoir volumes and current period’s
inflow) for the reservoir that minimizes the
expected sum of squared deviations from
storage and release targets. Limit the storage volumes to integer values that vary
from 3 to 5. Assume a storage volume target
of 4 and a release target of 2 in each period
t. (Assume only integer values of all states
and decision variables and that each period’s inflow is known at the beginning of the
period.) Find the annual expected sum of
squared deviations from the storage and
release targets.
Period, t

Flows, Qit

Probabilities,
Pit

i=1

i=2

i=1

i=2

1

1

2

0.17

0.83

2

3

4

0.29

0.71

This is an application of Exercise 6.27 except
7:1 Can you modify the deterministic discrete
DP reservoir operating model to include the the flow probabilities are independent of the
uncertainty, expressed as Ptij , of the inflows, previous flow.
as in Exercise 6.25?
7:3 Develop a linear model for defining the
(Hints: The operating policy would define
optimal joint probabilities of predefined disthe release (or final storage) in each season
crete initial storage volumes, discrete
as a function of not only the initial storage
inflows, and discrete final storage volumes in
but also the inflow. If the inflow changes, so
a reservoir in each period t. Let values of the
might the release or final storage volume.
index k represent the different discrete initial
Hence you need to discretize the inflows as
storage volumes, Skt. Similarly, let the index
well as the storage volumes. Both storage
i represent the inflows, Qit, and the index
and inflow are state variables. Assume, for
l represent the final storage volumes, Sl,t+1, in
this model, you can predict with certainty
period t. Let the index j represent the discrete
the inflow in each period at the beginning of
inflows, Qj,t+1, and m represent the discrete
the period. So, each node of the network
final storage volumes, Sm,t+2, in period t + 1.
represents a known initial storage and
Let PRkilt be the unknown joint probability of
inflow value. You cannot predict with cera discrete initial storage, Skt, an inflow, Qit,
tainty the following period’s flows, only
and a final storage volume, Sl,t+1, in period
their probabilities. What does the network
t. It is also the probability of a release assolook like now?
ciated with a particular combination of k, i,
7:2 Assume that there exist two possible disand l in period t. The objective is to maxicrete flows Qit into a small reservoir in each
mize the expected net benefits, however,
of two periods t each year having probameasured. The net benefits associated with
bilities Pit. Find the steady-state operating
any combination represented by k, i, and l in

Exercises

329

period t is Bkilt. These net benefits and the
conditional inflow probabilities. Ptij ¼


Pr Qj;t þ 1 jQit ; are known. Show how the
optimal operating policy can be determined
once the values of the joint probabilities,
PRkilt, are known.
The same policy can be found by
DP. Develop a DP model to find the optimal
operating policy.
7:4 Referring to Exercise 7.3, instead of defining
a final volume subscript l and m for computing joint probabilities PRkilt, assume that
subscripts d and e were used to denote different reservoir release volumes. How would
the linear programming model developed be
altered to include d and e in place of l and m?
How would the dynamic programming
recursion equation be altered?
7:5 Given joint probabilities PRkilt found from
Exercise 7.3, how would one derive the
probability distribution of reservoir releases
and storage volumes in each period t?
7:6 Assume that the streamflow Q at a particular
site has cumulative distribution function
FQ(q) = q/(1 + q) for q ≥ 0. The withdrawal
x at that location must satisfy a chance constraint of the form Pr[x ≥ Q] ≤ 1 − α. Write
the deterministic equivalent for each of the
following chance constraints:
Pr½x  Q  0:90

Pr½x  Q  0:80

Pr½x  Q  0:95
Pr½x  Q  0:75

Pr½x  Q  0:10

7:7 Monte Carlo Simulation:
Consider the symmetric triangular probability density function that ranges from 0 to 10
whose mean and most likely value is 5
fX x)

0

5

10

x

(a) Generate values of x that come from this
distribution.
To do this you need to
• Determine the equations of the cumulative
distribution.
• Generate uniformly distributed random values of probabilities p.
• For each p find corresponding value of x. The
inverse of the cumulative probability function
FX(x) denoted as FX1 ð pÞ.
• Using this inverse function, generate a series
of 100 random variable values x that would
have a probability distribution as shown
above.
(b) Calculate the mean, variance, and standard
deviation of this distribution based on the
random values you computed. What is the
effect on these statistics of increasing the
number of sample values of the random
variable, say from 100 to 1000 to 9000?
(c) Calculate and compare with the true mean
and variance.
(d) Next, suppose these random values of X are
flows entering a reservoir having a capacity
of 6. The purpose of the reservoir is to
release a target flow of 5 in each time period.
Simulate the operation of the reservoir
assuming that if there is insufficient water to
meet the target release of 5, release what is
available, leaving an empty reservoir. Find
the mean, variance, and standard deviation of
reservoir storage and release values.
(e) Finally assume the reservoir releases are to
be allocated to three water users whose target
allocations are 3, 2.33, and 8. Actual allocations should not exceed these target allocations. Make the allocations such that in
each time period the maximum percentage
deficit allocation is minimized. Find the
mean, variance and standard deviation of
each user’s percentage deficit allocations.

330

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7

Modeling Uncertainty

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8

System Sensitivity and Uncertainty
Analysis

The usefulness of any model is in part dependent
on the accuracy and reliability of its output data.
Yet, because all models are abstractions of reality, and because precise input data are rarely if
ever available, all output values are subject to
imprecision. The input data and modeling
uncertainties are not independent of each other.
They can interact in various ways. The end result
is imprecision and uncertainty associated with
model output. This chapter focuses on ways of
identifying, quantifying, and communicating the
uncertainties in model outputs.

8.1

Introduction

Models are the primary way we have to estimate
the multiple impacts of alternative water resource
system design and operating policies. Models are
used to estimate the values of various system
performance indicators resulting from specific
design and/or operating policy decisions. Model
outputs are based on model structure, hydrologic
and other time series inputs and a host of
parameters whose values characterize the system
being simulated. Even if these assumptions and
input data reflect, or are at least representative of,
conditions believed to be true, we know the
model outputs or results will be wrong. Our
models are always simplifications of the real
systems we are analyzing. Furthermore, we
simply cannot forecast the future with precision.
So we know the model outputs defining future
conditions are uncertain estimates, at best.

Some input data uncertainties can be reduced
by additional research and further data collection
and analysis. Before spending money and time to
gather and analyze additional data, it is reasonable to ask what improvement in estimates of
system performance or what reduction in the
uncertainty associated with those estimates
would result if all data and model uncertainties
could be reduced if not eliminated. Such information helps determine how much one would be
willing to “pay” to reduce model output uncertainty. If the uncertainty on average is costing a
lot, it may pay to invest in additional data collection, in more studies, or in developing better
models, all aimed at reducing that uncertainty. If
that uncertainty only a very modest, impact on
the likely decision that is to be made, one should
find other issues to worry about.
If it appears that reducing uncertainty is
worthwhile, then the question is how best to do
it. If doing this involves obtaining additional
information, then it is clear that the value of this
additional information, however measured,
should exceed the cost of obtaining it. The value
of such information will be the benefits of more
precise estimates of system performance, or the
reduction of the uncertainty, that one can expect
from obtaining such information. If additional
information is to be obtained, it should be
focused on that which reduces the uncertainties
considered important, not the unimportant ones.
This chapter reviews some methods for identifying and communicating model output uncertainty. The discussion begins with a review of the

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_8

331

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8

causes of risk and uncertainty in model output. It
then examines ways of measuring or quantifying
uncertainty and model output sensitivity to
model input imprecision, concentrating on
methods that seem most relevant or practical for
analyses of large-scale regional systems. It builds
on some of the statistical and stochastic modeling
methods reviewed in the previous two chapters.

8.2

Issues, Concerns,
and Terminology

Outcomes or events that cannot be predicted with
certainty are often called risky or uncertain.
Some individuals draw a special and interesting
distinction between risk and uncertainty. In particular, the term risk is often reserved to describe
situations for which probabilities are available to
describe the likelihood of various possible events
or outcomes. Often risk refers to these probabilities times the magnitude of the consequences of
these events or outcomes. If probabilities of
various events or outcomes cannot be quantified,
or if the events themselves are unpredictable,
some would say the problem is then one of
uncertainty, and not of risk. In this chapter what
is not certain is considered uncertain, and
uncertainty is often estimated or described using
probability distributions. When the ranges of
possible events are known and their probabilities
are measurable, risk is called objective risk. If the
probabilities are based solely on human judgment, the risk is called subjective risk.
Such distinctions between objective and subjective risk, and between risk and uncertainty,
rarely serve any useful purpose to those developing and using models. Likewise the distinctions
are often unimportant to those who should be
aware of the risks or uncertainties associated with
system performance indicator values. If the probabilities associated with possible events or outcomes are unknown, and especially if the events
themselves are unknown, then the approaches for

System Sensitivity and Uncertainty Analysis

performing sensitivity and uncertainty analyses
will differ from those that are based on assumed
known events and their probabilities.
Uncertainty in information is inherent in
future-oriented planning efforts. Uncertainty
stems from inadequate information and incorrect
assumptions, as well as from the variability and
possibly the nonstationarity of natural processes.
Water managers often need to identify both the
uncertainty as well as the sensitivity of system
performance due to any changes in possible input
data. They are often obligated to reduce any
uncertainty to the extent practicable. Finally, they
need to communicate the residual uncertainties
clearly so that decisions can be made with this
knowledge and understanding.
Sensitivity analysis can be distinguished from
uncertainty analysis. Sensitivity analysis procedures explore and quantify the impact of possible
changes (errors) in input data on predicted model
outputs and system performance indices. Simple
sensitivity analysis procedures can be used to
illustrate either graphically or numerically the
consequences of alternative assumptions about
the future. Uncertainty analyses employing
probabilistic descriptions of model inputs can be
used to derive probability distributions of model
outputs and system performance indices. Figure 8.1 illustrates the impact of both input data
sensitivity and input data uncertainty on model
output uncertainty.
It is worthwhile to explore the transformation of
uncertainties in model inputs and parameters into
uncertainty in model outputs when conditions
differ from those reflected by the model inputs.
Historical records of system characteristics are
typically used as a basis for model inputs. Yet
conditions in the future may change. There may be
changes in the frequency and amounts of precipitation, changes in land cover and topography, and
changes in the design and operation of control
structures, all resulting in changes of water stages
and flows, and their qualities, and consequently
changes in the impacted ecosystems.

8.2 Issues, Concerns, and Terminology

333

Fig. 8.1 Schematic diagram showing relationship among model input parameter uncertainty and sensitivity to model
output variable uncertainty (Lal 1995)

If asked how the system would operate with
inputs similar to those observed in the past, the
model should be able to provide a fairly precise
estimate. Still that estimate will not be perfect.
This is because our ability to reproduce current
and recent operations is not perfect, though it
should be fairly good. If asked to predict system
performance for situations very different from
those in the past, or when the historical data are
not considered representative of what might
happen in the future, say due to climate or
technology change, such predictions become
much less precise. There are two reasons why.
First, our description of the characteristics of
those different situations or conditions may be
imprecise. Second, our knowledge base may not
be sufficient for calibrating model parameters in
ways that would enable us to reliably predict how
the system will operate under conditions unlike
those that have been experienced historically.
The more conditions of interest are unlike those
in the past, the less confidence we have that the
model is providing a reliable description of systems operation. Figure 8.2 illustrates this issue.

Clearly a sensitivity analysis needs to consider
how well a model can replicate current operations, and how similar the target conditions or
scenarios are to those that existed in the past. The
greater the required extrapolation from what has
been observed, the greater will be the importance
of parameter and model uncertainties.
The relative and absolute importance of different parameters will depend on the system
performance indicators of interest. Seepage rates
may have a very large local effect, but a small
global effect. Changes in system-wide evapotranspiration rates will likely impact system-wide
flows. The precision of model projections and the
relative importance of errors in different parameters will depend upon the:
(1) precision with which the model can reproduce observed conditions,
(2) difference between the conditions predicted
in the future and the those that occurred in
the past, and the
(3) system performance characteristics of
interest.

334

8

System Sensitivity and Uncertainty Analysis

Fig. 8.2 The precision of
model predictions is
affected by the difference
between the conditions or
scenarios of interest and the
conditions or scenarios for
which the model was
calibrated

Errors and approximations in input data
measurement, parameter values, model structure
and model solution algorithms, are all sources of
uncertainty. While there are reasonable ways of
quantifying and reducing these errors and the
resulting range of uncertainty of various system
performance indicator values they are impossible
to eliminate. Decisions will still have to be made
in the face of a risky and uncertain future. Some
decisions may be able to be modified as new data
and knowledge are obtained in a process of
adaptive management.
There is also uncertainty with respect to
human behavior and reaction related to particular
outcomes and their likelihoods, i.e., to their risks
and uncertainties. As important as risks and
uncertainties associated with human reactions are
to particular outcomes, they are not usually part
of the models themselves. Social uncertainty may
often be the most significant component of the
total uncertainty associated with just how a water
resource system will perform. For this reason, we
should seek designs and operating policies that
are flexible and adaptable.
When uncertainties associated with system
operation under a new operating regime are
large, one should anticipate the need to make
changes and improvements as experience is

gained and new information accumulates. When
predictions are highly unreliable, responsible
managers should favor actions that are robust
(e.g., good under a wide range of situations), gain
information through research and experimentation, monitor results to provide feedback for the
next decision, update assessments and modify
policies in the light of new information, and
avoid irreversible actions and commitments.

8.3

Variability and Uncertainty
in Model Output

Differences between model output and observed
values can result from either natural variability,
say caused by unpredictable rainfall, evapotranspiration, water consumption, and the like, and/or
by both known and unknown errors in the input
data, the model parameters, or the model itself.
The later is sometimes called knowledge uncertainty but it is not always due to a lack of
knowledge. Models are always simplifications of
reality and hence “imprecision” can result.
Sometimes imprecision occurs because of a lack
of knowledge, such as just how much rainfall,
evapotranspiration and consumption will occur,
or just how a particular species will react to

8.3 Variability and Uncertainty in Model Output

various environmental and other habitat conditions. Other times known errors are introduced
simply for practical reasons.
Imperfect representation of processes in a
model constitutes model structural uncertainty.
Imperfect knowledge of the values of parameters
associated with these processes constitutes model
parameter uncertainty. Natural variability
includes both temporal variability and spatial
variability, to which model input values may be
subject.
Figure 8.3 illustrates these different types of
uncertainty. For example, the rainfall measured
at a weather station within a particular model grid
cell may be used as an input value for that cell,
but the rainfall may actually vary at different
points within that cell and its mean value will
vary across the landscape. Knowledge uncertainty can be reduced through further measurement and/or research. Natural variability is a
property of the natural system, and is usually not
reducible. Decision uncertainty is simply an
acknowledgement that we cannot predict ahead
of time just what decisions individuals and
organizations will make, or even just what

Fig. 8.3 One way of classifying types of uncertainty

335

particular set of goals or objectives will be considered in the future and the relative importance
of each of them.
Rather than contrasting “knowledge” uncertainty versus natural variability versus decision
uncertainty, one can classify uncertainty in
another way based on specific sources of uncertainty, such as those listed below, and address
ways of identifying and dealing with each source
of uncertainty.
Informational Uncertainties:
• imprecision in specifying the boundary and
initial conditions that impact the output variable values
• imprecision in measuring observed output
variable values
Model Uncertainties:
• uncertain model structure and parameter
values
• variability of observed input and output values over a region smaller than the spatial
scale of the model

336

• variability of observed model input and output values within a time smaller than the
temporal scale of the model. (e.g., rainfall and
depths and flows within a day)
• errors in linking models of different spatial
and temporal scales
Numerical Errors:
• errors in the model solution algorithm

8.3.1 Natural Variability
The main source of hydrologic model output
value variability is the natural variability in
hydrological and meteorological input series.
Periods of normal precipitation and temperature
can be interrupted by periods of extended
drought and intense meteorological events such
as hurricanes and tornadoes. There is reason to
think such events will continue to occur and
become even more frequent and extreme.
Research has demonstrated that climate has been
variable in the past and concerns about anthropogenic activities that may increase that variability increase each year. Sensitivity analysis
can help assess the effect of errors in predictions
if those predictions are based only on past
records of historical time series data describing
precipitation, temperature, and other exogenous
forces in and on the border of the regions being
studied.
Time series input data are often actual, or at
least based on, historical data. The time series
values typically describe historical conditions
including droughts and wet periods. What is
distinctive about natural uncertainty, as opposed
to errors and uncertainty due to modeling limitations, is that natural variability in meteorological forces cannot be reduced by improving the
model’s structure, increasing the resolution of the
simulation, or by better calibration of model
parameters.

8

System Sensitivity and Uncertainty Analysis

Errors result if meteorological values are not
measured or recorded accurately, or if mistakes
are made when creating computer data files.
Furthermore, there is no assurance the statistical
properties of historical data will accurately represent the statistical properties of future data.
Actual future precipitation and temperature scenarios will be different from those in the past, and
this difference in many cases may have a larger
affect than the uncertainty due to incorrect
parameter values. However, the effects of uncertainties in the parameter values used in stochastic
generation models are often much more significant than the effects of using different stochastic
generation models (Stedinger and Taylor 1982).
While variability of model output is a direct
result of variability of model input (e.g., hydrologic and meteorological data), the extent of the
variability, and the lower and upper limits of that
variability, may also be affected by errors in the
inputs, the values of parameters, initial boundary
conditions, model structure, processes and solution algorithms.
Figure 8.4 illustrates the distinction between
the variability of a system performance indicator
due to input data variability, and the extended
range of variability due to the total uncertainty
associated with any combination of the causes
listed in the previous section. This extended
range is what is of interest to water resource
planners and managers.
In practice a time series of system performance
indicator values can range anywhere within or
even outside the extended range, assuming the
confidence level of that extended range is less
than 100%. The confidence one can have that
some future value of a time series will be within a
given range is dependent on two factors. The first
is the number of measurements used to compute
the confidence limits. The second is on the
assumption that those measurements are representative of—come from the same statistical or
stochastic process yielding—future measurements. Figure 8.5 illustrates this point. Note that
the time series may even contain values outside

8.3 Variability and Uncertainty in Model Output

337

Fig. 8.4 Time series of model output or system performance showing variability over time. Range “a” results
from the natural variability of input data over time. The
extended range “b” results from the variability of natural
input data as well as from imprecision in input data

measurement, parameter value estimation, model structure
and errors in model solution algorithms. The extent of this
range will depend on the confidence level associated with
that range

Fig. 8.5 Typical time series of model output or system
performance indicator values that are the result of input
data variability and possible imprecision in input data

measurement, parameter value estimation, model structure
and errors in model solution algorithms

the range “b” defined in Fig. 8.4 if the confidence
level of that range is less than 100%. Confidence
intervals associated with less than 100% certainty
will not include every possible value that might
occur. Furthermore, it is unlikely one will ever
know the 100% confident interval that includes
all values that could ever occur.

uncertainties. First, we consider parameter value
uncertainty including boundary condition uncertainty, and then model and solution algorithm
uncertainty.

8.3.2 Knowledge Uncertainty
Referring to Fig. 8.3, knowledge uncertainty
includes model structure and parameter value

8.3.2.1 Parameter Value Uncertainty
A possible source of uncertainty in model output
results from uncertain estimates of various model
parameter values. If the model calibration procedure was repeated using different data sets, it
would have been resulted in different parameter
values. Those values would yield different simulated system behavior, and thus different

338

predictions. We can call this parameter uncertainty in the predictions because it is caused by
imprecise parameter values. If such parameter
value imprecision was eliminated, then the prediction would always be the same and so the
parameter value uncertainty in the predictions
would be zero. But this does not mean that predictions would be perfectly accurate.
In addition to parameter value imprecision,
uncertainty in model output can result from
imprecise specification of boundary conditions.
These boundary conditions can be either fixed or
variable. However, because they are not being
computed based on the state of the system, their
values can be uncertain. These uncertainties can
affect the model output, especially in the vicinity
of the boundary, in each time step of the
simulation.

8.3.2.2 Model Structural
and Computational Errors
Uncertainty in model output can also result from
errors in the model structure compared to the real
system, and approximations made by numerical
methods employed in the simulation. No matter
how good our parameter value estimates, our
models are not perfect and there is a residual
model error. Increasing model complexity to
more closely represent the complexity of the real
system may not only add to the cost of data
collection, but also introduce even more parameters, and thus even more potential sources of
error in model output. It is not an easy task to
judge the appropriate level of model complexity,
and to estimate the resulting levels of uncertainty
associated with various assumptions regarding
model structure and solution methods. Kuczera
(1988) provides an example of a conceptual
hydrologic modeling exercise with daily time
steps where model uncertainty dominated
parameter value uncertainty.

8.3.3 Decision Uncertainty
Uncertainty in model predictions can result from
unanticipated changes in what is being modeled.
These can include changes in nature, human

8

System Sensitivity and Uncertainty Analysis

goals, interests, activities, demands, and impacts.
An example of this is the deviation from standard
or published operating policies by operators of
infrastructure such as canal gates, pumps, and
reservoirs in the field, as compared to what is
specified in documents and incorporated into the
water systems models. Comparing field data with
model data for model calibration may yield
incorrect calibrations if operating policies actually implemented in the field differ significantly
from those built into the models. What do operators do in times of stress? And can anyone
identify a place where deviations from published
policies do not occur? Policies implemented in
practice tend to address short-term changes in
policy objectives.
What humans will want to achieve in the
future may not be the same as what they want
today. Predictions of what people will want in
the future are clearly sources of uncertainty.
A perfect example of this is in the very flat
Greater Everglades region of south Florida in the
US. Sixty years ago, folks wanted the swampy
region protected from floods and drained for
agricultural and urban development. Today,
many want just the opposite at least where there
are no human settlements. They want a return to
a more natural hydrologic system with more
wetlands and unobstructed flows, but now for
ecological restoration objectives that were not a
major concern or much appreciated half a century
ago. Once the mosquitoes return and if the sea
level continues to rise, future populations who
live there may want more flood control and
drainage again. Who knows? Complex changing
social and economic processes influence human
activities and their demands for water resources
and environmental amenities over time.
Sensitivity scenarios that include human
activities can help define the effects of those
human activities within an area. It is important
that these alternative scenarios realistically capture the forces or stresses that the system may
face. The history of systems studies is full of
examples where the issues studied were rapidly
overwhelmed by much larger social forces
resulting from, for example, the relocation of
major economic activities, an oil embargo,

8.3 Variability and Uncertainty in Model Output

changes in national demand for natural resources,
economic recession, sea-level rise, an act of terrorism, or even war. One thing is sure: the future
will be different than the past, and no one can be
certain just how.

8.3.3.1 Surprises
Water resource managers may also want to
consider how vulnerable a system is to undesirable environmental surprises. What havoc might
an introduced species like the zebra mussel
invading the Great Lakes of North America have
in a particular watershed? Might some introduced
disease suddenly threaten key plant or animal
species? Might management plans have to be
restructured to address the survival of some
species such as salmon in the Rhine River in
Europe or in the Columbia River in North
America? Such uncertainties are hard to anticipate when by their nature they are truly surprises.
But surprises should be expected. Hence system
flexibility and adaptability should be sought to
deal with changing management demands,
objectives, and constraints.

8.4

Sensitivity and Uncertainty
Analyses

An uncertainty analysis is not the same as a
sensitivity analysis. An uncertainty analysis
attempts to describe the entire set of possible
outcomes, together with their associated probabilities of occurrence. A sensitivity analysis
attempts to determine the relative change in
model output values given modest changes in
model input values. A sensitivity analysis thus
measures the change in the model output in a
localized region of the space of inputs. However,
one can often use the same set of model runs for
both uncertainty analyses and sensitivity analyses. It is possible to carry out a sensitivity analysis
of the model around a current solution and then
use it as part of a first-order uncertainty analysis.
This discussion begins by focusing on some
methods of uncertainty analysis. Then various
ways of performing and displaying sensitivity
analyses are reviewed.

339

8.4.1 Uncertainty Analyses
Recall that uncertainty involves the notion of
randomness. If a value of a performance indicator or performance measure, or in fact any variable (like the phosphorus concentration or the
depth of water at a particular location) varies and
this variation over space and time cannot be
predicted with certainty, it is called a random
variable. One cannot say with certainty what the
value of a random variable will be but only the
likelihood or probability that it will be within
some specified range of values. The probabilities
of observing particular ranges of values of a
random variable are described or defined by a
probability distribution. Here we are assuming
we know, or can compute, or can estimate, this
distribution.
Suppose the random variable is X. If the
observed values of this random variable can be
only discrete values, the probability distribution
of X can be expressed as a histogram, as shown
in Fig. 8.6a. The sum of the probabilities for all
possible outcomes must equal 1. If the random
variable is a continuous variable that can assume
any real value over a range of values, the probability distribution of X can be expressed as a
continuous distribution as shown in Fig. 8.6b.
The shaded area under the density function for
the continuous distribution is 1. The area
between two values of the continuous random
variable, such as between u and v in Fig. 8.6c,
represents the probability that the observed value
x of the random variable value X will be within
that range of values.
The probability distribution, PX(x) shown in
Fig. 8.6a is called a probability mass function.
The probability distributions shown in Fig. 8.6b,
c are called probability density functions
(pdf) and are denoted by fX(x). The subscript X of
PX and fX represents the random variable, and the
variable x (on the horizontal axes in Fig. 8.6) is
some value of that random variable X.
Uncertainty analyses involve identifying
characteristics of various probability distributions
of model input and output variables, and subsequently functions of those random output
variables that are performance indicators or

340

8

(a)

(b)

System Sensitivity and Uncertainty Analysis

(c)

Fig. 8.6 Probability distributions for a discrete or continuous random variable X. The area under the distributions (shaded areas in a and b) is 1, and the shaded area in

c is the probability that the observed value x of the
random variable X will be between u and v

measures. Often targets associated with
these indicators or measures are themselves
uncertain.
A complete uncertainty analysis would
involve a comprehensive identification of all
sources of uncertainty that contribute to the joint
probability distributions of each input or output
variable. Assume such analyses were performed
for two alternative project plans, A and B, and
that the resulting probability density distributions
for a specified performance measure were as
shown in Fig. 8.7. Figure 8.7 also identifies the
costs of these two projects. The introduction of
two performance criteria, cost and probability of

exceeding a performance measure target (e.g., a
pollutant concentration standard) introduces a
conflict where a tradeoff must be made.

Fig. 8.7 Tradeoffs
involving cost and the
probability that a maximum
desired target value will be
exceeded. In this
illustration, we want the
lowest cost (B is best) and
the lowest probability of
exceedance (A is best)

8.4.1.1 Model and Model Parameter
Uncertainties
Consider a situation as shown in Fig. 8.8, in
which for a specific set of model inputs, the
model outputs differ from the observed values,
and for those model inputs, the observed values
are always the same. Here nothing randomly
occurs. The model parameter values or model
structure needs to be changed. This is typically
done in a model calibration process.

8.4 Sensitivity and Uncertainty Analyses

341

Fig. 8.8 A deterministic system and a simulation model of that system needing calibration or modification in its
structure. There is no randomness, only parameter value or model structure errors to be identified and if found, corrected

Given specific inputs, the outputs of deterministic models are always going to be the same
each time those inputs are simulated. If for
specified inputs to any simulation model the
predicted output does not agree with the
observed value, as shown in Fig. 8.8, this could
result from imprecision in the measurement of
observed data. It could also result from imprecision in the model parameter values, the model
structure, or the algorithm used to solve the
model.
Next consider the same deterministic simulation model but now assume at least some of the
inputs are random, i.e., not predictable, as may
be case when random outputs of one model are
used as inputs into another model. Random
inputs will yield random outputs. The model
input and output values can be described by
probability distributions. If the uncertainty in the
output is due only to the uncertainty in the input,
the situation is similar to that shown in Fig. 8.8.
If the distribution of performance measure output
values does not fit or is not identical to the distribution of observed performance measure values, then calibration of model parameter values
or modification of model structure may be
needed.

If a model calibration or “identification” exercise finds the “best” values of the parameters to be
outside reasonable ranges of values based on scientific knowledge, then the model structure or
algorithm might be in error. Assuming the algorithms used to solve the models are correct and
observed measurements of system performance
vary for the same model inputs, as shown in
Fig. 8.9, it can be assumed that the model structure
does not capture all the processes that are taking
place and that impact the value of the performance
measures. This is often the case when relatively
simple and low-resolution models are used to
estimate the hydrological and ecological impacts
of water and land management policies. However,
even large and complex models can fail to include
or adequately describe important phenomena.
In the presence of informational uncertainties,
there may be considerable uncertainty about the
values of the “best” parameters during calibration. This problem becomes even more pronounced with increases in model complexity.
An example: Consider the prediction of a pollutant
concentration at some site downstream of a pollutant discharge site. Given a streamflow Q (in
units of 1000 m3/day), the distance between the
discharge site and the monitoring site, X (m), the

342

8

System Sensitivity and Uncertainty Analysis

Fig. 8.9 A deterministic simulation model of a “random
or stochastic” system. To produce the variability in the
model output that is observed in the real system, even
given the same input values, the model’s parameter values

may need to vary over distributions of values and/or the
model structure may need modification along with
additional model inputs

pollutant decay rate constant k (day−1), and the
pollutant discharge W (kg/day), we can use the
following simplified model to predict the concentration of the pollutant C (g/m3 = mg/l) at the
downstream monitoring site:

the assumed parameter distribution of k was simply
the result of the distribution of streamflow temperatures on the term kθT−20.

C ¼ ðW=QÞexpfkðX=U Þg
In the above equation assume the velocity
U (m/day) is a known function of the streamflow Q.
In this case the observed value of the pollutant
concentration C may differ from the computed
value of C even for the same inputs of W, Q, k, X,
and U. Furthermore, this difference varies in different time periods. This apparent variability, as
illustrated in Fig. 8.9, can be simulated using the
same model but by assuming a distribution of
values for the decay rate constant k. Alternatively
the model structure can be modified to include the
impact of streamflow temperature T on the prediction of C.
C ¼ ðW=QÞexpfkhT20 ðX=U Þg
Now there are two model parameters, the decay
rate constant k and the dimensionless temperature
correction factor θ, and an additional model input,
the streamflow temperature, T. It could be that the
variation in streamflow temperature was the sole
cause of the first equation’s “uncertainty” and that

If the output were still random given constant
values of all the inputs, then another source of
uncertainty exists. This uncertainty might be due to
additional random loadings of the pollutant, possibly from nonpoint sources. Once again the model
could be modified to include these additional
loadings if they are knowable. Assuming these
additional loadings are not known, a new random
parameter could be added to the input variable
W or to the right hand side of the equations above
that would attempt to capture the impact on C of
these additional loadings. A potential problem,
however, might be the likely correlation between
those additional loadings and the streamflow Q.

While adding model detail removed some
“uncertainty” in the above example, increasing
model complexity will not always eliminate
or reduce uncertainty in model output. Adding
complexity is generally not a good idea when
the increased complexity is based on processes whose parameters are difficult to measure, the right equations are not known at the
scale of application, or the amount of data for
calibration is small compared to the number of
parameters.

8.4 Sensitivity and Uncertainty Analyses

Even if more detailed models requiring more
input data and more parameter values were to be
developed, the likelihood of capturing all the
processes occurring in a complex system is small.
Hence those involved will have to make decisions
taking this uncertainty into account. Imprecision
will always exist due to less than a complete
understanding of the system and the hydrologic
processes being modeled. A number of studies
have addressed model simplification, but only in
some simple cases have statisticians been able to
identify just how one might minimize model
output uncertainty due to model structure.
The problem of determining the “optimal”
level of modeling detail is particularly important
when simulating the hydrologic events at many
sites over large areas. Perhaps the best approach
for these simulations is to establish confidence
levels for alternative sets of models and then
statistically compare simulation results. But even
this is not a trivial or costless task. Increases in
the temporal or spatial resolution typically
require considerable data collection and/or processing, model recalibrations, and possibly the
solution of stability problems resulting from the
numerical methods used in the models. Obtaining and implementing alternative hydrologic
simulation models will typically involve

343

considerable investments of money and time for
data preparation and model calibration.
What is needed is a way to predict the variability evident in the system shown in Fig. 8.9.
Instead of a fixed output vector for each fixed
input vector, a distribution of outputs is needed
for each performance measure based on fixed
inputs (Fig. 8.9) or a distribution of inputs
(Fig. 8.10). Furthermore, the model output distribution for each performance measure should
“match” as well as possible the observed distribution of that performance measure.

8.4.1.2 What Uncertainty Analysis Can
Provide
An uncertainty analysis takes a set of randomly
chosen input values (that can include parameter
values), passes them through a model (or transfer
function) to obtain the distributions (or statistical
measures of the distributions) of the resulting
outputs. As illustrated in Fig. 8.11, the output
distributions can be used to
• Describe the range of potential outputs of the
system at some probability level.
• Estimate the probability that the output will
exceed a specific threshold or performance
measure target value.

Fig. 8.10 Simulating variable inputs to obtain probability distributions of predicted performance indices that match the
probability distributions of observed performance values

344

8

System Sensitivity and Uncertainty Analysis

Fig. 8.11 The distribution of performance measures
defines range of potential values and the likelihood that
a specified target value will be exceeded. The shaded area
under the density function on the left represents the

probability that the target value will be exceeded. This
probability is shown in the probability of exceedance plot
on the right

Common uses for uncertainty analyses are to
make general inferences, such as the following:

when used to assess system performance for
alternative scenarios, as well as detailed information addressing the relative significance of
errors in various parameters. As a result, sensitivity results should be of interest to the general
public, federal and state management agencies,
local watershed planners and managers, model
users, and model developers.
Clearly, upper level management and the
public may be interested in more general statements of model precision, and should be provided such information along with model
predictions. On the other hand, detailed studies
addressing the significance and interactions
among individual parameters would likely be
meaningful to model developers and some model
users. They can use such data to interpret model
results and to identify where efforts to improve
models and their input values should be directed.
Initial sensitivity analysis studies could focus
on two products:

• Estimating the mean and standard deviation
of the outputs.
• Estimating the probability the performance
measure will exceed a specific threshold.
• Putting a reliability level on a function of the
outputs, e.g., the range of function values that
is likely to occur with some probability.
• Describing the likelihood of different potential outputs of the system.
Implicit in any uncertainty analysis are the
assumptions that statistical distributions for the
input values are correct and that the model is a
sufficiently realistic description of the processes
taking place in the system. Neither of these
assumptions is likely to be entirely correct.

8.4.2 Sensitivity Analyses
“Sensitivity analysis” is aimed at describing how
much model output values are affected by changes in model input values. It is the investigation
of the importance of imprecision or uncertainty
in model inputs in a decision-making or modeling process. The exact character of sensitivity
analysis depends upon the particular context and
the questions of concern. Sensitivity studies can
provide a general assessment of model precision

(1) detailed results to guide research and assist
model development efforts, and
(2) calculation of general descriptions of uncertainty associated with model predictions so
that policy decisions can reflect both the
predicted system performance and the precision of such predictions.
In the first case, knowing the relative uncertainty in model projections due to possible errors

8.4 Sensitivity and Uncertainty Analyses

in different sets of parameters and input data
should assist in efforts to improve the precision
of model projections. This knowledge should
also contribute to a better understanding of the
relationships between model assumptions,
parameters, data and model predictions.
For the second case, knowing the relative
precision associated with model predictions
should have a significant effect on policy development. For example, the analysis may show
that, given data inadequacies, there are very large
error bands associated with some model variable
values. When such large uncertainties exist,
predictions should be used with appropriate
skepticism. Incremental strategies should be
explored along with monitoring so that greater
experience can accumulate to resolve some of
those uncertainties.
Sensitivity analysis features are available in
many linear and nonlinear programming (optimization) packages. They identify the changes in
the values of the objective function and unknown
decision variables given a change in the model
input values, and a change in levels set for various constraints (Chap. 4). Thus sensitivity
analysis addresses the change in “optimal” system performance associated with changes in
various parameter values, and also how “optimal” decisions would change with changes in
resource constraint levels, or target output
requirements. This kind of sensitivity analysis
provides estimates of how much another unit of
resource would be worth, or what “cost” a proposed change in a constraint places on the optimal solution. This information should be of value
to those making investment decisions.
Various techniques have been developed to
determine how sensitive model outputs are to
changes in model inputs. Most approaches
examine the effects of changes in a single
parameter value or input variable assuming no
changes in all the other inputs. Sensitivity analyses can be extended to examine the combined
effects of multiple sources of error as well.
Changes in particular model input values can
affect model output values in different ways. It is
generally true that only a relatively few input
variables dominate or substantially influence the

345

values of a particular output variable or performance indicator at a particular location and time.
If the range of uncertainty of only some of the
output data is of interest, then undoubtedly only
those input data that significantly impact the
values of those output data need be included in
the sensitivity analysis.
If input data estimates are based on repeated
measurements, a frequency distribution can be
estimated that characterizes input data variability.
The shorter the record of measurements, the
greater will be the uncertainty regarding the
long-term statistical characteristics of that variability. If obtaining a sufficient number of replicate measurements is not possible, subjective
estimates of input data ranges and probability
distributions are often made. Using a mixture of
subjective estimates and actual measurements
does not affect the application of various sensitivity analysis methods that can use these sets or
distributions of input values, but it may affect the
conclusions that can be drawn from the results of
these analyses.
It would be nice to have available accurate
and easy-to-use analytical methods for relating
errors in input data to errors in model outputs,
and to errors in system performance indicator
values that are derived from model outputs. Such
analytical methods do not exist for complex
simulation models. However, methods based on
simplifying assumptions and approximations can
be used to yield useful sensitivity information.
Some of these are reviewed in the remainder of
this chapter.

8.4.2.1 Sensitivity Coefficients
One measure of sensitivity is the sensitivity
coefficient. This is the derivative of a model
output variable with respect to an input variable
or parameter. A number of sensitivity analysis
methods use these coefficients. First-order and
approximate first-order sensitivity analyses are
two such methods that will be discussed later.
The difficulty of
1. obtaining the derivatives for many models,
2. needing to assume mathematical (usually
linear) relationships when obtaining estimates

346

8

of derivatives by making small changes of
input data values near their nominal or most
likely values, and
3. having large variances associated with most
hydrologic process models have motivated
the replacement of analytical methods by
numerical and statistical approaches to sensitivity analysis.
By varying the input probability distributions,
one can determine the sensitivity of these distributions on the output distributions. If the output
distributions vary significantly, then the output is
sensitive to the specification of the input distributions and hence they should be defined with
care. A relatively simple deterministic sensitivity
analysis can be of value here (Benaman 2002).
A sensitivity coefficient can be used to measure
the magnitude of change in an output variable
Q per unit change in the magnitude of an input
parameter value P from its base value Po. Let
SIPQ be the sensitivity index for an output variable Q with respect to a change ΔP in the value
of the input variable P from its base value Po.
Noting that the value of the output Q(P) is a
function of P, a sensitivity index could be
defined as
SIPQ ¼ ½QðPo þ DPÞ  QðPo  DPÞ=2DP
ð8:1Þ
Other sensitivity indices could be defined
(McCuen 1973). Let the index i represent a
decrease and j represent an increase in the
parameter value from its base value Po, the sensitivity index SIPQ for parameter P and output
variable Q could be defined as
SIPQ ¼ fjðQo  Qi Þ=ðPo



Pi j þ j Qo  Qj =ðPo  Pj Þjg=2 ð8:2Þ
or
SIPQ ¼ maxfjðQo  Qi Þ=ðPo





Pi Þj;  Qo  Qj = Po  Pj g

ð8:3Þ

System Sensitivity and Uncertainty Analysis

A dimensionless expression of sensitivity is
the elasticity index, EIPQ that measures the relative change in output Q for a relative change in
input P.
EIPQ ¼ ½Po =QðPo ÞSIPQ

ð8:4Þ

8.4.2.2 A Simple Deterministic
Sensitivity Analysis
Procedure
This deterministic sensitivity analysis approach
is very similar to those most often employed in
the engineering economics literature. It is based
on the idea of varying one uncertain parameter
value, or set of parameter values, at a time, and
observing the results.
The output variable of interest can be any
performance measure or indicator. Thus one does
not know if more or less of a given variable is
better or worse. Perhaps too much and/or too
little is undesirable. The key idea is that, whether
employing physical measures or economic metrics of performance, various parameters (or sets
of associated parameters) are assigned high and
low values. Such ranges may reflect either the
differences between the minimum and maximum
values for each parameter, the 5 and 95 percentiles of a parameter’s distribution, or points
corresponding to some other criteria. The system
model is then run with the various alternatives,
one at a time, to evaluate the impact of those
errors in various sets of parameter values on the
output variable.
Table 8.1 illustrates the character of the
results that one would obtain. Here Y0 is the
nominal value of the model output when all
parameters assume the estimated best values, and
Yi,L and Yi,H are the values obtained by increasing
or decreasing the values of the ith set of
parameters.
A simple water quality example is employed
to illustrate this deterministic approach to sensitivity analysis. The analysis techniques illustrated
here are just as applicable to complex models.
The primary difference is that more work would
be required to evaluate the various alternatives

8.4 Sensitivity and Uncertainty Analyses

347

Table 8.1 Sensitivity of model output Y to possible
errors in four parameter sets containing a single parameter
or a group of parameters that vary together

with a more complex model, and the model responses might be more complicated.
The simple water quality model is provided
by Vollenweider’s empirical relationship for the
average phosphorus concentration in lakes
(Vollenweider 1976). He found that the phosphorus concentration, P (mg/m3), is a function of
the annual phosphorus loading rate, L (milligrams per square meter per year, mg/m2 a), the
annual hydraulic loading, q (m/a or more exactly
m3/m2 a), and the mean water depth, z (m).
h
i
P ¼ ðL=qÞ= 1 þ ðz=qÞ0:5

ð8:5Þ

L/q and P have the same units; the denominator
is an empirical factor that compensates for
nutrient recycling and elimination within the
aquatic lake environment.

Data for Lake Ontario in North America
would suggest that reasonable values of the
parameters are L = 680 mg/m2a; q = 10.6 m/a;
and z = 84 m, yielding P = 16.8 mg/m3. Values
of phosphorus concentrations less than 10 mg/m3
are considered oligotrophic, whereas values
greater than 20 mg/m3 generally correspond to
eutrophic conditions. Reasonable ranges reflecting possible errors in the three parameters yield
the values in Table 8.2.
One may want to display these results so they
can be readily visualized and understood. A tornado diagram (Eschenback 1992) would show
the lower and upper values of P obtained from
variation of each parameter, with the parameter
with the widest limits displayed on top, and the
parameter having smallest limits on the bottom.
Tornado diagrams (Fig. 8.12) are easy to construct and can include a large number of
parameters without becoming crowded.
These error bars shown in Fig. 8.12 indicate
there is substantial uncertainty associated with
the phosphorus concentration P, primarily due to
uncertainty in the loading rate L.
An alternative to tornado diagrams is a Pareto
chart showing the width of the uncertainty range
associated with each variable, ordered from largest to smallest. A Pareto chart is illustrated in
Fig. 8.13.
Another visual presentation is a spider plot
showing the impact of uncertainty in each
parameter on the variable in question, all on the
same graph (Eschenback 1992; DeGarmo 1993,
p. 401). A spider plot, Fig. 8.14, shows the

Table 8.2 Sensitivity of estimates of phosphorus concentration (mg/m3) to model parameter values

The two right most values in each row correspond to the low and high values of the parameter, respectively

348

Fig. 8.12 A Tornado diagram showing the range of the
output variable representing phosphorus concentrations
for high and low values of each of the parameter sets.

8

System Sensitivity and Uncertainty Analysis

Parameters are sorted so that the largest range is on top,
and the smallest on the bottom

Fig. 8.13 A Pareto Chart showing the range of the output variable representing phosphorus concentrations resulting
from high and low values of each parameter set considered

Fig. 8.14 Spider Plot
illustrates the relationships
between phosphorus
concentrations and
variations in each of the
parameter sets, expressed
as a percentage deviation
from their nominal values

8.4 Sensitivity and Uncertainty Analyses

particular functional response of the output to
each parameter on a common scale, so one needs
a common metric to represent changes in all of
the parameters. Here we use percentage change
from the nominal or best values.
Spider plots are a little harder to construct
than tornado diagrams, and can generally include
only 4–5 variables without becoming crowded.
However, they provide a more complete view of
the relationships between each parameter and the
performance measure. In particular, a spider plot
reveals nonlinear relationships and the relative
sensitivity of the performance measure to (percentage) changes in each variable.
In the spider plot, the linear relationship
between P and L and the gentle nonlinear relationship between P and q is illustrated. The range
for z has been kept small given the limited
uncertainty associated with that parameter.

8.4.2.3 Multiple Errors and Interactions
An important issue that should not be ignored is
the impact of simultaneous errors in more than
one parameter. Probabilistic methods directly
address the occurrence of simultaneous errors,
but the correct joint distribution needs to be
employed. With simple sensitivity analysis procedures, errors in parameters are generally
investigated one at a time, or in groups. The idea
of considering pairs or sets of parameters is
discussed here.
Groups of factors. It is often the case that
reasonable error scenarios would have several
parameters changing together. For example,
possible errors in water depth would be accompanied with corresponding variations in aquatic
vegetation and chemical parameters. Likewise,
alternatives related to changes in model structure
might be accompanied with variations in several
parameters. In other cases, there may be no
causal relationship among possible errors (such
as model structure versus inflows at the boundary
of the modeled region), but they might still
interact to affect the precision of model
predictions.
Combinations. If one or more non-grouped
parameters interact in significant ways, then
combinations of one or more errors should be

349

investigated. However, one immediately runs
into a combinatorial problem. If each of m parameters can have three values (high, nominal,
and low) there are 3m combinations, as opposed
to 2m + 1 if each parameter is varied separately.
[For m = 5, the differences are 35 = 243 versus 2
(5) + 1 = 11.] These numbers can be reduced by
considering instead only combinations of
extremes so that only 2m + 1 cases need be
considered [25 + 1 = 33], which is a more
manageable number. However, all of the
parameters would be at one extreme or the other,
and such situations would be unlikely.
Two factors at a time. A compromise is to
consider all pairs of two parameters at a time.
There are m(m − 1)/2 possible pairs of m
parameters. Each parameter has a high and low
value. Since there are four combinations of high
and low values for each pair, there are a total of
2m(m − 1) combinations. [For m = 5 there are
40 combinations of two parameters each having
two values.]
The presentation of these results could be
simplified by displaying for each case only the
maximum error, which would result in m
(m − 1)/2 cases that might be displayed in a
Pareto diagram. This would allow identification
of those combinations of two parameters that
might yield the largest errors and thus are of most
concern.
For the water quality example, if one plots the
absolute value of the error for all four combinations of high (+) and low (−) values for each pair
of parameters, they obtain Fig. 8.15.
Considering only the worst error for each pair
of variables yields Fig. 8.16.
Here we see, as is no surprise, the worst error
results from the most unfavorable combination of
L and q values. If both parameters have their
most unfavorable values, the predicted phosphorus concentration would be 27 mg/m3.
Looking for nonlinearities. One might also
display in a Pareto diagram the maximum error
for each pair as a percentage of the sum of the
absolute values of the maximum error from each
parameter separately. The ratio of the joint error
to the individual errors would illustrate potentially important nonlinear interactions. If the

350

Fig. 8.15 Pareto diagram showing errors in phosphorus
concentrations for all combinations of pairs of input
parameters errors. A “+” indicates a high value, and a “−”

8

System Sensitivity and Uncertainty Analysis

indicates a low value for indicated parameter. L is the
phosphorus loading rate, q is the hydraulic loading, and
z is the mean lake depth

Fig. 8.16 Pareto diagram
showing worst error
combinations for each pair
of input parameters. A “+”
indicates a high value, and
a “−” indicates a low value
for indicated parameter

model of the system and the physical measure or
economic metric were strictly linear, then the
individual ratios should add to one.

8.4.2.4 First-Order Sensitivity Analysis
The above deterministic analysis has trouble
representing reasonable combinations of errors in
several parameter sets. If the errors are independent, it is highly unlikely that any two sets
would actually be at their extreme ranges at the
same time. By defining probability distributions
of the values of the various parameter sets, and
specifying their joint distributions, a probabilistic
error analysis can be conducted. In particular, for

a given performance indicator, one can use
multivariate linear analyses to evaluate the
approximate impact on the performance indices
of uncertainty in various parameters. As shown
below, the impact depends upon the square of the
sensitivity coefficients (partial derivatives) and
the variances and covariances of the parameter
sets.
For a performance indicator I = F(Y), which
is a function F(•) of model outputs Y, that are in
turn a function g(P) of input parameters P, one
can use a multivariate Taylor series approximation of F to obtain the expected value and variance of the indicator:

8.4 Sensitivity and Uncertainty Analyses

351

E½I  ¼ F ðbased on mean values of input parametersÞ
XX
þ ð1=2Þ
½@F2 =@Pi @Pj  Cov ½Pi ; Pj g
i

j

ð8:6Þ
and
Var½I  ¼

XX
i






ð@F=@Pi Þ @F=@Pj Cov Pi ; Pj

j

ð8:7Þ
where (∂F/∂Pi) are the partial derivative of the
function F with respect to Pi evaluated at the
mean value of the input parameters Pi, and ∂F2/
∂Pi∂Pj are the second partial derivatives. The
covariance of two random input parameters Pi
and Pj is the expected value of the product of
differences between the values and their means.


Cov Pi ; Pj ¼ E½ðPi  E½Pi ÞðPj  E½Pj Þ
ð8:8Þ
If all the parameters are independent of each
other, and the second-order terms in the expression for the mean E[I] are neglected, one obtains
E½I  ¼ F ðbased on mean values of input parametersÞ

ð8:9Þ
and
Var ½I  ¼

X



½@F=@Pi 2 Var Pi 

ð8:10Þ

i

Table 8.3 Approximate parameter sensitivity coefficients

Benjamin and Cornell (1970). Equation 8.6
for E[I] shows that in the presence of substantial
uncertainty, the mean of the output from nonlinear systems is not simply the system output
corresponding to the mean of the parameters
(Gaven and Burges 1981, p. 1523). This is true
for any nonlinear function.
Of interest in the analysis of uncertainty is the
approximation for the variance Var[I] of indicator I. In Eq. 8.10 the contribution of Pi to the
variance of I equals Var[Pi] times [∂F/∂Pi]2,
which are the squares of the sensitivity coefficients for indicator I with respect to each input
parameter value Pi.
An Example of First-Order Sensitivity Analysis
It may appear that first-order analysis is difficult
because the partial derivatives of the performance indicator I are needed with respect to the
various parameters. However, reasonable
approximations of these sensitivity coefficients
can be obtained from the simple sensitivity
analysis described in Table 8.3. In that table,
three different parameter sets, Pi, are defined in
which one parameter of the set is at its high
value, PiH, and one is at its low value, PiL, to
produce corresponding values (called high, IiH,
and low, IiL) of a system performance indicator I.
It is then necessary to estimate some representation of the variances of the various parameters with some consistent procedure. For a
normal distribution, the distance between the 5
and 95 percentiles is 1.645 standard deviations

352

8

on each side of the mean, or 2(1.645) = 3.3
standard deviations. Thus, if the high/low range
is thought of as approximately a 5–95 percentile
range for a normally distributed variate, a reasonable approximation of the variance might be
Var½Pi  ¼ f½PiH  PiL =3:3g2 :

ð8:11Þ

This is all that is needed. Use of these average
sensitivity coefficients is very reasonable for
modeling the behavior of the system performance indicator I over the indicated ranges.
As an illustration of the method of first-order
uncertainty analysis, consider the lake quality
problem described above. The “system performance indicator” in this case is the model output,
the phosphorus concentration P, and the input
parameters, now denoted as X = L, q, and z. The
standard deviation of each parameter is assumed
to be the specified range divided by 3.3. Average
sensitivity coefficients ∂P/∂X were calculated.
The results are reported in Table 8.4.
Assuming the parameter errors are
independent:
Var½P ¼ 9:18 þ 2:92 þ 0:02 ¼ 12:12 ð8:12Þ
The square root of 12.12 is the standard
deviation and equals 3.48. This agrees well with
a Monte Carlo analysis reported below.
Note that 100 * (9.18/12.12), or about 76% of
the total parameter error variance in the

System Sensitivity and Uncertainty Analysis

phosphorus concentration P is associated in the
phosphorus loading rate L and the remaining
24% is associated with the hydrologic loading
q. Eliminating the uncertainty in z would have a
negligible impact on the overall model error.
Likewise, reducing the error in q would at best
have a modest impact on the total error.
Due to these uncertainties, the estimated
phosphorus concentration has a standard deviation of 3.48. Assuming the errors are normally
distributed, and recalling that ±1.645 standard
deviations around the mean define a 5–95 percentile interval, the 5–95 percentile interval
would be about
16:8  1:645ð3:48Þmg=m3 ¼ 16:8  5:7 mg=m3
¼ 11:1 to 22:5 mg=m3 :
ð8:13Þ
The upper bound of 22.5 mg/m3 is considerably less than the 27 mg/m3 that would be
obtained if both L and q had their most unfavorable
values. In a probabilistic analysis with independent errors, such a combination is highly unlikely.
Warning on Accuracy
First-order uncertainty analysis is indeed an
approximate method based upon a linearization
of the response function represented by the full
simulation model. It may provide inaccurate
estimates of the variance of the response variable
for nonlinear systems with large uncertainty in

Table 8.4 Calculation of approximate parameter sensitivity coefficients

8.4 Sensitivity and Uncertainty Analyses

the parameters. In such cases, Monte Carlo
simulation (discussed below and in the previous
chapter) or the use of higher order approximation
may be required. Beck (1987, p. 1426) cites
studies that found that Monte Carlo and
first-order variances were not appreciably different, and a few studies that found specific differences. Differences are likely to arise when the
distributions used for the parameters are bimodal
(or otherwise unusual), or some rejection algorithm is used in the Monte Carlo analysis to
exclude some parameter combinations. Such
errors can result in a distortion in the ranking of
predominant sources of uncertainty. However, in
most cases very similar results were obtained.

8.4.2.5 Fractional Factorial Design
Method
An extension of first-order sensitivity analysis
would be a more complete exploration of the
response surface using a careful statistical design.
First consider a complete factorial design. Input
data are divided into discrete “levels.” The simplest case is two levels. These two levels can be
defined as a nominal value, and a high (low) value.
Simulation runs are made for all combinations of
parameter levels. For n different inputs, this would
require 2n simulation runs. Hence for a three-input
variable or parameter problem, 8 runs would be
required. If four discrete levels of each input

Table 8.5 A three-input factorial design

353

variable or parameter were allowed to provide a
more reasonable description of a continuous
variable, the three-input data problem would
require 43 or 64 simulation runs. Clearly, this is not
a useful tool for large regional water resources
simulation models.
A fractional factorial design involves simulating only a fraction of what is required from a
full factorial design method. The loss of information prevents a complete analysis of the
impacts of each input variable or parameter on
the output.
To illustrate the fractional factorial design
method, consider the two-level with three-input
variable or parameter problem. Table 8.5 shows
the 8 simulations required for a full factorial
design method. The “+” and the “−” show the
upper and lower levels of each input variable or
parameter Pi where i = 1, 2, 3. If all eight simulations were performed, seven possible effects
could be estimated. These are the individual
effects of the three inputs P1, P2, and P3, the
three two-input variable or parameter interactions, (P1)(P2), (P1)(P3), and (P2)(P3), and the
one three-input variable or parameter interaction
(P1)(P2)(P3).
Consider an output variable Y, where Yj is the
value of Y in the jth simulation run. Then an
estimate of the effect, denoted δ(Y|Pi) that input
variable or parameter Pi has on the output

354

8

variable Y, is the average of the four separate
effects of varying Pi:
For i = 1:
dðYjP1 Þ ¼ 0:25½ðY2  Y1 Þ þ ðY4  Y3 Þ
þ ðY6  Y5 Þ þ ðY8  Y7 Þ

ð8:14Þ

Each difference in parentheses is the difference
between a run in which P1 is at its upper level and
a run in which P1 is at its lower level, but the other
two parameter values, P2 and P3, are unchanged.
If the effect is equal to 0, then, in this case, P1 has
no impact on the output variable Y.
Similarly the effects of P2 and P3, on variable
Y can be estimated as:
d ðYjP2 Þ ¼ 0:25fðY3  Y1 Þ þ ðY4  Y2 Þ
þ ðY7  Y5 Þ þ ðY8  Y6 Þg

ð8:15Þ

and
dðYjP3 Þ ¼ 0:25fðY5  Y1 Þ þ ðY6  Y2 Þ
þ ðY7  Y3 Þ þ ðY8  Y4 Þg

ð8:16Þ

Consider next the interaction effects between
P1 and P2. This is estimated as the average of the
difference between the average P1 effect at the
upper level of P2, and the average P1 effect at the
lower level of P2. This is the same as the difference between the average P2 effect at the
upper level of P1 and the average P2 effect at the
lower level of P1:
dðYjP1 ; P2 Þ ¼ ð1=2Þf½ðY8  Y7 Þ þ ðY4  Y3 Þ=2
½ðY2  Y1 Þ þ ðY6  Y5 Þ=2g ¼ ð1=4Þf½ðY8  Y6 Þ
þ ðY4  Y2 Þ  ½ðY3  Y1 Þ þ ðY7  Y5 Þg

ð8:17Þ
Similar equations can be derived for looking
at the interaction effects between P1 and P3, and
between P2 and P3 and the interaction effects
among all three inputs P1, P2, and P3.
Now assume only half of the simulation runs
were performed, perhaps runs 2, 3, 5, and 8 in
this example. If only outputs Y2, Y3, Y5, and Y8
are available, for our example:

System Sensitivity and Uncertainty Analysis

dðYjP3 Þ ¼ dðYjP1 ; P2 Þ
¼ 0:5fðY8  Y3 Þ  ðY2  Y5 Þg ð8:18Þ
The separate effects of P3 and of P1P2 are not
available from the output. This is the loss in
information resulting from fractional instead of
complete factorial design.

8.4.2.6 Monte Carlo Sampling Methods
The Monte Carlo method of performing sensitivity analyses, illustrated in Fig. 8.17, first
selects a random set of input data values drawn
from their individual probability distributions.
These values are then used in the simulation
model to obtain some model output variable
values. This process is repeated many times, each
time making sure the model calibration is valid
for the input data values chosen. The end result is
a probability distribution of model output variables and system performance indices that results
from variations and possible errors in all of the
input values.
Using a simple Monte Carlo analysis, values
of all of the parameter sets are selected randomly
from distributions describing the individual and
joint uncertainty in each, and then the modeled
system is simulated to obtain estimates of the
selected performance indices. This must be done
many times (often well over 100) to obtain a
statistically significant description of system
performance variability. The number of replications needed is generally not dependent on the
number of parameters whose errors are to be
analyzed. One can include in the simulation the
uncertainty in parameters as well as natural
variability. This method can evaluate the impact
of single or multiple uncertain parameters.
A significant problem that arises in such
simulations is that some combinations of
parameter values may result in unreasonable
models. For example, model output based on
calibrated data sets might be inconsistent with
available data sets. The calibration process places
interesting constraints on different sets of
parameter values. Thus, such Monte Carlo
experiments often contain checks that exclude
combinations of parameter values that are

8.4 Sensitivity and Uncertainty Analyses

Fig. 8.17 Monte Carlo sampling and simulation procedure for finding distributions of output variable values
based on distributions, for specified reliability levels, of
input data values. This technique can be applied to one or

355

more uncertain input variables at a time. The output
distributions will reflect the combined effects of this input
uncertainty over the specified ranges

356

8

System Sensitivity and Uncertainty Analysis

Table 8.6 Monte Carlo analysis of lake phosphorus levels

unreasonable. In these cases the generated results
are conditioned on this validity check.
Whenever sampling methods are used, one
must consider possible correlations among input
data values. Sampling methods can handle spatial
and temporal correlations that may exist among
input data values, but the existence of correlation
requires defining appropriate conditional
distributions.
One major limitation of applying Monte Carlo
methods to estimate ranges of risk and uncertainty for model output variable values, and
system performance indicator values based on
these output variable values, is the computing
time required. To reduce the computing times
needed to perform sensitivity analyses using
sampling methods, some tricks and as well as
stratified sampling methods are available. The
discussion below illustrates the idea of a simple
modification (or trick) using a “standardized”
Monte Carlo analysis. The more general Latin
Hypercube Sampling procedure is also discussed.

were generated independently from normal distributions with the means and variances as shown
in Table 8.6.
The table contains the specified means and
variances for the generated values of L, q, and z,
and also the actual values of the means and
variances of the 200 generated values of L, q,
z and also of the 200 corresponding generated
output phosphorus concentrations, P. Figure 8.18 displays the distribution of the generated values of P.
One can see that given the estimated levels of
uncertainty, phosphorus levels could reasonably
range from below 10 to above 25. The probability of generating a value greater than
20 mg/m3 was 12.5%. The 5% to 95 percentile
range was 11.1–23.4 mg/m3. In the figure, the
cumulative probability curve is rough because
only 200 values of the phosphorus concentration
were generated, but these are clearly enough to
give a good impression of the overall impact of
the errors.

Simple Monte Carlo Sampling
To illustrate the use of Monte Carlo sampling
methods consider again Vollenweider’s empirical relationship, Eq. 8.5, for the average phosphorus concentration in lakes (Vollenweider
1976). Two hundred values of each parameter

Sampling Uncertainty
In this example, the mean of the 200 generated
values of the phosphorus concentration, P, was
17.07. However, a different set of random values
would have generated a different set of P values
as well. Thus it is appropriate to estimate the

8.4 Sensitivity and Uncertainty Analyses

357

Fig. 8.18 Distribution of
lake phosphorus
concentrations from Monte
Carlo analysis

standard error, SE, of this average. The standard
error equals the standard deviation σ of the
P values divided by the square root of the sample
size n:
SE ¼ r=ðnÞ0:5 ¼ 3:61=ð200Þ0:5 ¼ 0:25: ð8:19Þ
From the central limit theorem of mathematical
statistics, the average of a large number of independent values should have very nearly a normal
distribution. Thus, 95% of the time, the true mean
of P should be in the interval 17.1 ± 1.96 (0.25),
or 16.6–17.6 mg/m3. This level of uncertainty
reflects the observed variability of P and the fact
that only 200 values were generated.

Table 8.7 Correlation analysis of Monte Carlo results

Making Sense of the Results
A significant challenge with complex models is
to determine from the Monte Carlo simulation
which parameter errors are important. Calculating the correlation between each generated input
parameter value and the output variable value is
one way of doing this. As Table 8.7 shows,
based upon the magnitudes of the correlation
coefficients, errors in L were most important, and
those in q second in importance.
One can also use regression to develop a
linear model defining variations in the output
based on errors in the various parameters. The
results are shown in Table 8.8. The fit is very
good, and R2 = 98%. If the model for P had been

358

8

System Sensitivity and Uncertainty Analysis

Table 8.8 Results of regression analysis on Monte Carlo Results

linear, a R2 value of 100% should have resulted.
All of the coefficients are significantly different
from zero.
Note that the correlation between P and z was
positive in Table 8.7, but the regression coefficient for z is negative. This occurred because
there is a modest negative correlation between
the generated z and q values. Use of partial
correlation coefficients can also correct for such
spurious correlations among input parameters.
Finally we display a plot, Fig. 8.19, based on
this regression model illustrating the reduction in
the variance of P that is due to dropping each
variable individually. Clearly L has the biggest
impact on the uncertainty in P, and z the least.

Standardized Monte Carlo Analysis
Using a “standardized” Monte Carlo analysis,
one could adjust the generated values of L, q, and
z above so that the generated samples actually
have the desired mean and variance. While
making that correction, one can also shuffle their
values so that the correlations among the generated values for the different parameters are near
zero, as is desired. This was done for the 200
generated values to obtain the statistics shown in
Table 8.9.
Repeating the correlation analysis from before
(shown in Table 8.10) now yields much clearer
results that are in agreement with the regression
analysis. The correlation between P and both

Fig. 8.19 Reduction in the variance of P that is due to dropping from the regression model each variable individually.
Clearly L has the biggest impact on the uncertainty in P, and z the least

8.4 Sensitivity and Uncertainty Analyses

359

Table 8.9 Standardized Monte Carlo analysis of lake phosphorus levels

Table 8.10 Correlation analysis of standardized Monte Carlo results

q and z are now negative as they should be.
Because the generated values of the three
parameters have been adjusted to be uncorrelated, the signal from one is not confused with
the signal from another.
The mean phosphorus concentration changed
very little. It is now 17.0 instead of 17.1 mg/m3.
Using control variates with a linear predictive
model in conjunction with the standardized
Monte Carlo variates, the standard deviation of
the errors associated with the 200 observations is
only 0.45. Thus the standard error for this estimate of the mean of P is 0.45/(200)0.5 or just
0.03. Thus this is a highly accurate result. The

regressions were also repeated and yielded very
similar results. The only real difference was that
the parameter estimates had small standard errors
and were more significant because of the elimination of correlation between the generated
parameters.
Generalized Likelihood Estimation
Beven (1993) and Binley and Beven (1991) suggest a Generalized Likelihood Uncertainty Estimation (GLUE) technique for assessment of
parameter error uncertainty using Monte Carlo
simulation. It is described as a “formal methodology for some of the subjective elements of

360

8

model calibration” (Beven 1989, p. 47). The basic
idea is to begin by assigning reasonable ranges for
the various parameters and then to draw parameter
sets from those ranges using a uniform or some
similar (and flat) distribution. These generated
parameter sets are then used on a calibration data
set so that unreasonable combinations can be
rejected, while reasonable values are assigned a
posterior probability based upon a likelihood
measure which may reflect several dimensions
and characteristics of model performance.
Let L(Pi) > 0 be the value of the likelihood
measure assigned to the ith parameter set’s calibration sequence. Then the model predictions
generated with parameter set/combination Pi are
assigned posterior probability, p(Pi).
pðPi Þ ¼ LðPi Þ=

X

L Pj

ð8:20Þ

j

These probabilities reflect the form of Bayes
theorem, which is well supported by probability
theory (Devore 1991). This procedure should
capture reasonably well the dependence or correlation among parameters, because reasonable
sequences will all be assigned larger probabilities, whereas sequences that are unable to
reproduce the system response over the calibration period will be rejected or assigned small
probabilities.
However, in a rigorous probabilistic framework, the L would be the likelihood function for the
calibration series for particular error distributions.
(This could be checked with available goodnessof-fit procedures; for example, Kuczera 1988.)
When relatively ad hoc measures are adopted for
the likelihood measure with little statistical validity, the p(Pi) probabilities are best described as
pseudo-probabilities or “likelihood” weights.
Another concern with this method is the
potential efficiency. If the parameter ranges are
too wide, a large number of unreasonable or very
unlikely parameter combinations will be generated. These will either be rejected or else will
have small probabilities and thus little effect on
the analysis. In this case the associated processing would be a waste of effort. A compromise is

System Sensitivity and Uncertainty Analysis

to use some data to calibrate the model and to
generate a prior or initial distribution for the
parameters that is at least centered in the best
range (Beven 1993, p. 48). Then use of a different calibration period to generate the p(Pi)
allows an updating of those initial probabilities to
reflect the information provided by the additional
calibration period with the adopted likelihood
measures.
After the accepted sequences are used to generate sets of predictions, the likelihood weights
would be used in the calculation of means, variances and quantiles, rather than the customary
procedure of giving all the generated realizations
equal weight. The resulting conditional distribution of system output reflects the initial probability
distributions assigned to parameters, the rejection
criteria, and the likelihood measure adopted to
assign “likelihood” weights.

8.4.2.7 Latin Hypercube Sampling
For the simple Monte Carlo simulations described above, with independent errors, a probability
distribution is assumed for each input parameter
or variable. In each simulation run, values of all
input data are obtained from sampling those
individual and independent distributions. The
value generated for an input parameter or variable is usually independent of what that value
was in any previous run, or what other input
parameter or variable values are in the same run.
This simple sampling approach can result in a
clustering of parameter values and hence a
redundancy of information from repeated sampling in the same regions of a distribution and a
lack of information from no sampling in other
regions of the distributions.
A stratified sampling approach ensures more
even coverage of the range of input parameter or
variable values with the same number of simulation runs. This can be accomplished by dividing the input parameter or variable space into
sections and sampling from each section with the
appropriate probability.
One such approach, Latin hypercube sampling
(LHS), divides each input distribution into sections of equal probability for the specified

8.4 Sensitivity and Uncertainty Analyses

Fig. 8.20 Schematic representation of a Latin hypercube sampling procedure for six simulation runs

361

362

8

System Sensitivity and Uncertainty Analysis

probability distribution, and draws one observation randomly from each range. Hence the ranges
of input values within each section actually occur
with equal frequency in the experiment. These
values from each interval for each distribution
are randomly assigned to those from other
intervals to construct sets of input values for the
simulation analysis. Figure 8.20 shows the steps
in constructing a LHS for six simulations
involving three inputs Pj (P1, P2, and P3) and six
intervals of their respective normal, uniform and
triangular probability distributions.

8.5

Performance Indicator
Uncertainties

Fig. 8.21 Combining the probability distribution of
performance measure values with the probability distribution of performance measure target values to estimate

the confidence one has in the probability of exceeding a
maximum desired target value

8.5.1 Performance Measure Target
Uncertainty
Another possible source of uncertainty is the
selection of performance measure target values.
For example, consider a target value for a pollutant concentration based on the effect of
exceeding it in an ecosystem. Which target value
is best or correct? When this is not clear, there

8.5 Performance Indicator Uncertainties

are various ways of expressing the uncertainty
associated with any target value. One such
method is the use of qualitative approaches
involving membership functions (Chap. 5). Use
of “grey” numbers or intervals instead of “white”
or fixed target values is another. When some
uncertainty or disagreement exists over the
selection of the best target value for a particular
performance measure, it seems to us the most
direct and transparent way to do this is to subjectively assume a distribution over a range of
possible target values. Then this subjective
probability distribution can be factored into the
tradeoff analysis, as outlined in Fig. 8.21.
One of the challenges associated with defining
and including in an analysis the uncertainty
associated with a target or threshold value for a
performance measure is that of communicating
just what the result of such an analysis means.
Referring to Fig. 8.20, suppose the target value
represents some maximum limit of a pollutant,
say phosphorus, concentration in the flow during
a given period of time at a given site or region,
and it is not certain just what that maximum limit
should be. Subjectively defining the distribution
of that maximum limit, and considering that
uncertainty along with the uncertainty (probability of exceedance function) of pollutant concentrations—the performance measure—one can
attach a confidence to any probability of
exceeding the maximum desired concentration
value.
The 95% probability of exceedance shown on
Fig. 8.20, say P0.95, should be interpreted as “we
can be 95% confident that the probability of the
maximum desired pollutant concentration being
exceeded will be no greater than P0.95.” We can
be only 5% confident that the probability of
exceeding the desired maximum concentration
will be no greater than the lower P0.05 value.
Depending on whether the middle line through
the subjective distribution of target values in

363

Fig. 8.20 represents the most likely or median
target value, the associated probability of
exceedance is either the most likely, as indicated
in Fig. 8.20, or that for which we are only 50%
confident.
Figure 8.21 attempts to show how to interpret
the reliabilities when the uncertain performance
targets are
• minimum acceptable levels that are to be
maximized,
• maximum acceptable levels that are to be
minimized or
• optimum levels.
An example of a minimum acceptable target
level might be the population of wading birds in
an area. An example of a maximum acceptable
target level might be, again, the phosphorus
concentration of the flow in a specific wetland or
lake. An example of an optimum target level
might be the depth of water most suitable for
selected species of aquatic vegetation during a
particular period of the year.
For performance measure targets that are not
expressed as minimum or maximum limits but
that are the “best” values, referring to Fig. 8.22,
one can state that one is 90% confident that the
probability of achieving the desired target is no
more than B. The 90% confidence level probability of not achieving the desired target is at
least A + C. The probability of the performance
measure being too low is at least A and the
probability of the performance measure being too
high is at least C, again at the 90% confidence
levels. As the confidence level decreases the
bandwidth decreases, and the probability of not
meeting the target increases.
Now, clearly there is uncertainty associated
with each of these uncertainty estimations, and
this raises the question of how valuable is the
quantification of the uncertainty of each

364

8

System Sensitivity and Uncertainty Analysis

Fig. 8.22 Interpreting the results of combining performance measure probabilities with performance measure
target probabilities depends on the type of performance

measure. The letters A, B, and C represent proportions of
the probability density function of performance measure
values. (Hence probabilities A + B + C = 1)

additional component of the plan in an evaluation
process. Will plan evaluators and decision makers benefit from this additional information, and
just how much additional uncertainty information
is useful?
Now consider again the tradeoffs that need to
be made as illustrated in Fig. 8.7. Instead of
considering a single target value as shown on
Fig. 8.7, assume there is a 90% confidence range
associated with that single performance measure
target value. Also assume that the target is a

maximum desired upper limit (e.g., of some
pollutant concentration).
In the case shown in Fig. 8.23, the tradeoff is
clearly between cost and reliability. In this
example, no matter what confidence one chooses,
Plan A is preferred to Plan B with respect to
reliability, but Plan B is preferred to Plan A with
respect to cost. The tradeoff is only between
these two performance indicators or measures.
Consider however a third plan, as shown in
Fig. 8.24. This situation adds to the complexity

8.5 Performance Indicator Uncertainties

365

Fig. 8.23 Two plans showing ranges of probabilities,
depending on one’s confidence, that an uncertain desired
maximum (upper limit) performance target value will be
exceeded. The 95% confidence levels are associated with
the higher probabilities of exceeding the desired

maximum target. The 5% confident levels are associated
with the more desirable lower probabilities of exceeding
the desired maximum target. Plan A with reduced
probabilities of exceeding the upper limit costs more than
Plan B

Fig. 8.24 Tradeoffs among cost, reliabilities, and the
confidence level of those reliabilities. The relative ranking
of plans with respect to the probability of exceeding the

desired (maximum limit) target may depend on the
confidence given to that probability

of making appropriate tradeoffs. Now there are
three criteria: cost, probability of exceedance
(reliability) and the confidence in those reliabilities or probabilities. Add to this the fact that

there will be multiple performance measure targets, each expressed in terms of their maximum
probabilities of exceedance and the confidence in
those probabilities.

366

In Fig. 8.23, in terms of cost the plans are
ranked, from best to worst, B, C, and A. In terms
of reliability at the 95% confidence level, they
are ranked A, B, and C but at the 5% confidence
level the ranking is A, C, and B.
If the plan evaluation process has difficulty
handling all this it may indicate the need to focus
the uncertainty analysis effort on just what is
deemed important, achievable, and beneficial.
Then when the number of alternatives has been
narrowed down to only a few that appear to be the
better ones, a more complete uncertainty analysis
can be performed. There is no need nor benefit in
performing sensitivity and uncertainty analyses
on all possible management alternatives. Rather
one can focus on those alternatives that look the
most promising, and then carry out additional
uncertainty and sensitivity analyses only when
important uncertain performance indicator values
demand more scrutiny. Otherwise the work is not
likely to affect the decision anyway.

8.5.2 Distinguishing Differences
Between Performance
Indicator Distributions
Simulations of alternative water management
infrastructure designs and operating policies
require a comparison of the simulation outputs—
the performance measures or indicators—associated with each alternative. Now the question is
whether or not the observed differences are statistically significant. Can one really tell if one
alternative is better than another or are the
observed differences explainable by random
variations attributable to variations in the inputs
and how the system responds?
This is a common statistical issue that is
addressed by standard hypothesis tests (Devore
1991; Benjamin and Cornell 1970). Selection of
an appropriate test requires that one first resolve
what type of change one expects in the variables.
To illustrate, consider the comparison of two
different operating policies. Let Y1 denote the set
of output performance variable values with the
first policy, and Y2 the set of output performance
variable values of the second policy. In many

8

System Sensitivity and Uncertainty Analysis

cases, one would expect one policy to be better
than the other. One measure might be the difference in the mean of the variables. For example, is E[Y1] < E[Y2]? Alternatively one could
check the difference in the median (50 percentile)
of the two distributions.
In addition, one could look for a change in the
variability or variance, or a shift in both the mean
and the variance. Changes described by a difference in the mean or median often make the
most sense and many statistical tests are available
that are sensitive to such changes. For such
investigations parametric and nonparametric tests
for paired and unpaired data can be employed.
Consider the differences between “paired” and
“unpaired” data. Suppose that the meteorological
data for 1941–1990 is used to drive a simulation
model generating data as described in
Table 8.11.
Here there is one sample, Y1(1) through Y1(50),
for policy 1, and another sample, Y2(1) through
Y2(50), for policy 2. However, the two sets of
observations are not independent. For example, if
1943 was a very dry year, then we would expect
both Y1(3) for policy 1 in that year and Y2(3) for
policy 2 to be unusually small. With such paired
data, one can use a paired hypothesis test to check
for differences. Paired tests are usually easier than
the corresponding unpaired tests that are appropriate in other cases. (For example, if one were
checking for a difference in average rainfall depth
between 1941–1970, and 1971–2000, they would
have two sets of independent measurements for the
Table 8.11 Possible flow data from a 50-year simulation

8.5 Performance Indicator Uncertainties

two periods. With such data, one should use a
two-sample unpaired test.)
Paired tests are generally based on the differences between the two sets of output,
Y1(i) − Y2(i). These are viewed as a single
independent sample. The question is then: are
the differences positive (say Y1 tends to be larger
then Y2), or negative (Y1 tends to be smaller), or
are positive and negative differences are equally
likely (there is no difference between Y1 and Y2).
Both parametric and nonparametric families
of statistical tests are available for paired data.
The common parametric test for paired data (a
one-sample T test) assumes that the mean of the
differences
X ðiÞ ¼ Y1 ðiÞ  Y2 ðiÞ
ð8:21Þ
is normally distributed. Then the hypothesis of
no difference is rejected if the T statistic is sufficiently large, given the sample size n.
Alternatively, one can employ a nonparametric test and avoid the assumption that the
differences X(i) are normally distributed. In such
a case, one can use the Wilcoxon Signed Rank
test. This nonparametric test ranks the absolute
values |X(i)| of the differences. If the sum S of the
ranks of the positive differences deviates sufficiently from its expected value, n(n + 1)/4 (were
there no difference between the two distributions), one can conclude that there is a statistically significant difference between the Y1(i) and
Y2(i) series. Standard statistical texts have tables
of the distribution of the sum S as a function of
the sample size n, and provide a good analytical
approximation for n > 20 (for example, Devore
1991). Both the parametric t test and the nonparametric Wilcoxon Signed Rank test require
that the differences between the simulated values
for each year be computed.

8.6

Communicating Model Output
Uncertainty

Spending money on reducing uncertainty would
seem preferable to spending it on ways of calculating and describing it better. Yet attention to
uncertainty
communication
is
critically

367

important if uncertainty analyses and characterizations are to be of value in a decision-making
process. In spite considerable efforts by those
involved in risk assessment and management, we
know very little about how to ensure effective
risk communication to gain the confidence of
stakeholders, incorporate their views and
knowledge, and influence favorably the acceptability of risk assessments and risk management
decisions.
The best way to communicate concepts of
uncertainty may well depend on what the audiences already know about risk and the various
types of probability distributions (e.g., density,
cumulative, exceedance) based on objective and
subjective data, and the distinction between
mean or average values and the most likely
values. Undoubtedly graphical representations of
these ways of describing uncertainty considerably facilitate communication.
The National Research Council (NRC 1994)
addressed the extensive uncertainty and variability associated with estimating risk and concluded that risk characterizations should not be
reduced to a single number or even to a range of
numbers intended to portray uncertainty. Instead,
the report recommended managers and the
interested public should be given risk characterizations that are both qualitative and quantitative and both verbal and mathematical.
In some cases, communicating qualitative
information about uncertainty to stakeholders
and the public in general may be more effective
than quantitative information. There are, of
course, situations in which quantitative uncertainty analyses are likely to provide information
that is useful in a decision-making process. How
else can tradeoffs such as illustrated in Figs. 8.10
and 8.27 be identified? Quantitative uncertainty
analysis often can be used as the basis of qualitative information about uncertainty, even if the
quantitative information is not what is communicated to the public.
One should acknowledge to the public the
widespread confusion regarding the differences
between variability and uncertainty. Variability
does not change through further measurement or
study, although better sampling can improve our

368

knowledge about variability. Uncertainty reflects
gaps in information about scientifically observable phenomena.
While it is important to communicate uncertainties and confidence in predictions, it is equally
important to clarify who or what is at risk, possible consequences, and the severity and irreversibility of an adverse effect should a target
value, for example, not be met. This qualitative
information is often critical to informed
decision-making. Risk and uncertainty communication is always complicated by the reliability
and amounts of available relevant information as
well as how that information is presented. Effective communication between people receiving
information about who or what is at risk, or what
might happen and just how severe and irreversible an adverse effect might be should a target
value not be met, is just as important as the level
of uncertainty and the confidence associated with
such predictions. A two-way dialog between
those receiving such information and those giving
it can help identify just what seems best for a
particular audience.
Risk and uncertainty communication is a
two-way street. It involves learning and teaching.
Communicators dealing with uncertainty should
learn about the concerns and values of their
audience, their relevant knowledge, and their
experience with uncertainty issues. Stakeholders’
knowledge of the sources and reasons for
uncertainty needs to be incorporated into
assessment and management and communication
decisions. By listening, communicators can craft
risk messages that better reflect the perspectives,
technical knowledge, and concerns of the
audience.
Effective communication should begin before
important decisions have been made. It can be
facilitated in communities by citizen advisory
panels. Citizen advisory panels can give planners
and decision-makers a better understanding of
the questions and concerns of the community and
an opportunity to test its effectiveness in communicating concepts and specific issues regarding uncertainty.

8

System Sensitivity and Uncertainty Analysis

Fig. 8.25 Reducing risk by orders of magnitude is not
equivalent to linear reductions

One approach to make uncertainty more
meaningful is to make risk comparisons. For
example, a ten-parts-per-billion target for a particular pollutant concentration is equivalent to
10 s in over 31 years. If this is an average daily
concentration target that is to be satisfied “99%,”
of the time, this is equivalent to an expected
violation of less than one day every three
months.
Many perceive the reduction of risk by an
order of magnitude as though it were a linear
reduction. An alternative way to illustrate orders
of magnitude of risk reduction is shown in
Fig. 8.25, in which a bar graph depicts better
than words that a reduction in risk from one in a
1000 (10−3) to one in 10,000 (10−4) is a reduction of 90% and that a further reduction to one in
100,000 (10−5) is a reduction 10-fold less than
the first reduction of 90%. The percent of the risk
that is reduced by whatever measures is an easier
concept to communicate than reductions expressed in terms of estimated absolute risk levels,
such as 10−5.
Risk comparisons can be helpful, but they
should be used cautiously and tested if possible.
There are dangers in comparing risks of diverse
character, especially when the intent of the
comparison is seen as minimizing a risk (NRC

8.6 Communicating Model Output Uncertainty

369

Fig. 8.26 Different types
of displays used to show
model output Y or system
performance indicator
values F(Y)

1989). One difficulty in using risk comparisons is
that it is not always easy to find risks that are
sufficiently similar to make a comparison meaningful. How is someone able to compare two
alternatives having two different costs and two
different risk levels, for example, as is shown in
Fig. 8.7? One way is to perform an indifference
analysis (as discussed in the next chapter), but
that can lead to different results depending who
Fig. 8.27 Plots of ranges
of possible model output
Y or system indicator
values F(Y) for different
types of displays

performs it. Another way is to develop utility
functions using weights, where, for example
reduced phosphorus load by half is equivalent to
a 25% shorter hydroperiod in that area, but again
each person’s utility or preferred tradeoff may
differ.
At a minimum, graphical displays of uncertainty can be helpful. Consider the common
system performance indicators that include:

370

8

• Time series plots for continuous timedependent indicators (Fig. 8.26 upper left)
• Probability exceedance distributions for continuous indicators (Fig. 8.26 upper right),
• Histograms for discrete event indicators
(Fig. 8.26 lower left), and
• Overlays on maps for space-dependent discrete events (Fig. 8.26 lower right).
The first three graphs in Fig. 8.26 could show,
in addition to the single curve or bar that represents the most likely output, a range of outcomes
associated with a given confidence interval. For
overlays of information on maps, different colors
could represent the spatial extents of events
associated with different ranges of risk or
uncertainty. Figure 8.27, corresponding to
Fig. 8.26, illustrates these approaches for displaying these ranges.

8.7

Conclusions

This chapter provides an overview of uncertainty
and sensitivity analyses in the context of hydrologic or water resources systems simulation
modeling. A broad range of tools are available to
explore, display, and quantify the sensitivity and
uncertainty in predictions of key output variables
and system performance indices with respect to
imprecise and random model inputs and to
assumptions concerning model structure. They
range from relatively simple deterministic sensitivity analysis methods to more involved
first-order analyses and Monte Carlo sampling
methods.
Because of the complexity of many watersheds or river basins, Monte Carlo methods for
uncertainty analyses may be a very major and
unattractive undertaking. Therefore it is often
prudent to begin with the relatively simple
deterministic procedures. This coupled with a
probabilistically based first-order uncertainty
analysis method can help quantify the uncertainty in key output variables and system performance indices, and the relative contributions

System Sensitivity and Uncertainty Analysis

of uncertainty in different input variables to the
uncertainty in different output variables and
system performance indices. These relative contributions may differ depending upon which
output variables and indices are of interest.
A sensitivity analysis can provide a systematic
assessment of the impact of parameter value
imprecision on output variable values and performance indices, and of the relative contribution
of errors in different parameter values to that
output uncertainty. Once the key variables are
identified, it should be possible to determine the
extent to which parameter value uncertainty can
be reduced through field investigations, development of better models, and other efforts.
Model calibration procedures can be applied to
individual catchments and subsystems, as well as to
composite systems. Automated calibration procedures have several advantages including the explicit
use of an appropriate statistical objective function,
identification of those parameters that best reproduce the calibration data set with the given objective
function, and the estimations of the statistical precision of the estimated parameters.
All of these tasks together can represent a
formidable effort. However, knowledge of the
uncertainty associated with model predictions
can be as important to management decision and
policy formulation as are the predictions
themselves.
No matter how much attention is given to
quantifying and reducing uncertainties in model
outputs, uncertainties will remain. Professionals
who analyze risk, managers and decision-makers
who must manage risk, and the public who must
live with risk and uncertainty, have different
information needs and attitudes regarding risk
and uncertainty. It is clear that information needs
differ among those who model or use models,
those who make substantial investment or social
decisions, and those who are likely to be impacted
by those decisions. Meeting those needs should
result in more informed decision-making. But it
comes at a cost that should be considered along
with the benefits of having this sensitivity and
uncertainty information.

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Exercises
8:1 Distinguish between sensitivity analysis and
uncertainty analysis.
8:2 Consider the allocation model used in previous chapters involving three water consumers i. Allocations xi of water can be made
from a given total amount Q to the three
consumers. The respective benefits are
(6x1 − x21), (7x2 − 1.5x22) and (8x3 − 0.5x23).
Discuss possible sources of uncertainty in
model structure and model output, and
identify and display parameter sensitivity.
8:3 Discuss how model output uncertainty is
impacted by both model input uncertainty as
well as parameter sensitivity.
8:4 In many water resources studies considerable
attention is given to the uncertainty of water
supplies (precipitation, streamflows, evaporation, infiltration, etc.) and much less
attention is given to the uncertainty of the
management objectives, the costs and benefits of infrastructure, the political support
associated with alternative possible decisions, and the like. Develop a simple water
resources planning model involving the
management of water quantity and quality
and show how these management objective
uncertainties may actually dominate the
hydrologic uncertainties.

374

8:5 Perform a deterministic sensitivity analysis
for the consumer 1 in Exercise 8.2. Consider
the three parameters, Q, 6 and 1; the latter
two numbers are the parameters of the benefit function. Low values of these three
parameters are 3, 3, and 0.5, respectively.
Most likely values are 6, 6, and 1. High
values are 12, 9, and 1.5. Display the results
using a Pareto chart, a tornado diagram, and
a spider plot.
8:6 Referring to water allocation problem defined
in Exercise 9.2, assume the available amount
of water Q is uncertain. Its cumulative probability distribution is defined by q/(6 + q) for
values q ≥ 0 of the random variable Q. The
expected value of Q does not exist. Perform
an uncertainty analysis showing how to
define, at least approximately:

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8

System Sensitivity and Uncertainty Analysis

• Estimating the mean and standard deviation of the outputs.
• Estimating the probability the performance
measure will exceed a specific threshold.
• Assigning a reliability level on a function
of the outputs, e.g., the range of function
values that is likely to occur with some
probability.
• Describing the likelihood of different
potential outputs of the system.
Show the application of Monte Carlo sampling and analysis, Latin hypercube sampling,
generalized likelihood uncertainty estimation and
factorial design methods.

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reproduce the material.

9

Performance Criteria

Water resource systems typically provide a
variety of economic, environmental, and ecological services. They also serve a variety of
purposes (e.g., water supply, flood protection,
hydropower production, navigation, recreation,
and waste assimilation and transport). Performance criteria provide measures of just how well
a plan or management policy performs. There are
a variety of criteria one can use to judge and
compare alternative system performances. Some
of these performance criteria may be conflicting.
In these cases tradeoffs exist among conflicting
criteria and these tradeoffs should be considered
when searching for the best compromise. This
chapter presents ways of identifying and working
with these performance criteria in the political
process of selecting the best decision.

9.1

Introduction

Decision-makers and those who influence them
are people, and people’s opinions and experiences and goals may differ. These differences
force one to think in terms of tradeoffs. Decisions
in water resources management inevitably
involve making tradeoffs among competing
opportunities, goals or objectives. One of the
tasks of water resource system planners or
managers involved in evaluating alternative designs and management plans or policies is to
identify the tradeoffs, if any, among competing
opportunities, goals or objectives. It is then up to
the largely political process involving all

interested stakeholders to find the best compromise decision.
If every system performance measure or
objective could be expressed in the same units,
and if there was only one decision-maker and one
objective or goal, then decision-making would be
relatively straight forward. Such is not the case
when dealing with the public’s water resources.
A cost-benefit framework, used for many
decades in water resources planning and management, converted the different types of impacts
into a single monetary metric. Once that was
done, the task was to find the plan or policy that
maximized the net benefits, i.e., the benefits less
the costs. If the maximum difference between
benefits and costs was positive, that was the best
plan or policy. But not all system performance
criteria we consider today can be easily expressed in monetary units. Even if monetary units
could be used for each objective, that in itself
does not address the distributional issues
involving who benefits and who pays and by
how much. While all stakeholders may agree that
maximizing total net benefits is a desirable
objective, not everyone, if indeed anyone, will
likely agree on to how best to allocate or share
those net benefits among them.
Clearly, water resource planning and management takes place in a multiple criteria environment. A key element of many problems
facing designers and managers is the need to deal
explicitly with multiple ecological, economical,
and social impacts, expressed in multiple metrics
that may result from the design, management,

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_9

375

376

9

and operation of water resource systems.
Approaches that fail to recognize and explicitly
include ways of handling conflict among multiple system performance measures and objectives
and among multiple stakeholders are not likely to
be very useful (Hipel et al. 2015).
Successful decision-making involves creating
a consensus among multiple participants in the
planning and management process. These
include stakeholders—individuals or interest
groups who have an interest in the outcome of
any management plan or policy. The relatively
recent acknowledgement that stakeholders need
to be fully included in the decision-making processes only complicates the life of professional
planners and managers. Increasingly important
sources of information come from discussion
groups, public hearings, negotiations, and
dispute-resolution processes. Using the types of
modeling methods discussed in this chapter can
potentially inform the debates that occur in these
meetings. In addition, the application of game
theory may be helpful in reaching a consensus
among stakeholders having different objectives
and desires (Madani 2010; Madani and Hipel
2011).
Eventually someone or some organization
must make a decision. Even if water resource
analysts view their job as one of providing
options or tradeoffs for someone else to consider,
even they are making decisions that define or
limit the range of those options or tradeoffs. The
importance of making informed effective decisions applies to everyone.

9.2

Informed Decision-Making

Informed decision-making involves both, qualitative thinking and analysis as well as quantitative modeling (Hammond et al. 1999).
Qualitative thinking and analysis typically follows quantitative analyses. Qualitative analyses
are useful for identifying
• the real objectives of concern to all stakeholders (which may be other than those
expressed by them),

Performance Criteria

• the likelihoods of events for which decisions
are needed,
• the socially acceptable alternatives that
address and meet each objective, and
• the key tradeoffs among all interests and
objectives involved.
When the objectives and general alternative
ways of meeting these objectives are not well
defined, no amount of quantitative modeling and
analysis will make up for this weakness. The
identification of objectives and general alternatives are the foundation upon which water
resource managers can develop appropriate
quantitative models. These models will provide
additional insight and definition of alternatives
and their expected impacts. Models cannot
identify new ideas or so called ‘out-of-the-box’
alternatives that no one has thought of before.
Neither can they identify the best criteria that
should be considered in specific cases. Only our
minds can do this, individually, and then
collectively.
For example, periodic municipal water supply
shortages might be reduced by increasing surface
water reservoir storage capacity or by increasing
groundwater pumping. Assuming the objective
in this case is to increase the reliability of some
specified level of water supply, models can be
developed to identify the tradeoffs between the
cost of increased reservoir storage and pumping
capacities and the increased reliability of meeting
the supply target. However these models will not
identify and evaluate completely different alternatives, such as importing water by trucks or the
use of canals or pipelines from other river basins,
implementing water use restrictions, or water
reuse, unless of course those options are included
in those models. Someone has to think of these
general alternatives before models can help
identify just how many trucks or the capacities of
canals or pipes or how to best implement water
reuse and for what uses.
Time needs to be taken to identify the relevant
objectives and general alternatives that then can
be included in quantitative analyses. Clearly
some preliminary screening of general alternatives is and should be carried out at the

9.2 Informed Decision-Making

qualitative level, but the alternatives that remain
(assuming the best is not obvious) should then be
further analyzed using quantitative modeling
methods. This includes the methods of quantifying qualitative objectives and constraints discussed in Chap. 5.
Being creative in the identification of possible
objectives and general alternatives is helped by
addressing the following questions: What is an
ideal decision? What does each stakeholder think
other stakeholders’ ideal decisions would be?
What is to be avoided? What makes a great
alternative, even an infeasible one, so great?
What makes a terrible alternative so terrible?
How would each individual’s best alternative be
justified to someone else? When each manager
and stakeholder has gone through such thinking,
the combined set of responses may be more
comprehensive and less limited by what others
say or believe. They can become a basis for
group discussions and consensus building.
General statements defining objectives can be
converted to ones that are short and include the
words maximize or minimize. For example,
minimize cost, maximize net benefits, maximize
reliability, maximize water quality, maximize
ecosystem biodiversity, minimize construction
time, or minimize the maximum deviations from
some target storage volume or a target water
allocation. Economic, environmental, ecological,
and purely physical objectives such as these are
able to become the objective functions that drive
the solutions of optimization models, as illustrated in many of the previous chapters. Social
objectives should also be considered. Examples
might include maximize employment, maximize
interagency coordination, maximize stakeholder
participation, and minimize legal liability and the
potential of future legal action and costs.
Quantitative modeling is employed to identify
more precisely the design and operation of
structural and nonstructural alternatives that best
satisfy system performance criteria, and the
impacts and tradeoffs among these various performance criteria. Once such analyses have been
performed, it is always wise to question whether
or not the results are reasonable. Are they as
expected? If not, why not? If the results are

377

surprising, are the analyses providing new
knowledge and understanding or have errors
been made? How sensitive are the results to
various assumptions with respect to the input
data and models themselves?
Thus the modeling process ends with some
more qualitative study. Models do not replace
human judgment. Humans, not quantitative
models, are responsible for water resources
planning and management decisions as well as
the decision-making process itself that identifies
performance criteria and general alternatives.

9.3

Performance Criteria
and General Alternatives

There is a way to identify performance criteria
that matter most to stakeholders (Gregory and
Keeney 1994). One can begin with very broad
fundamental goals, such as public health,
national as well as individual security, economic
development, happiness, and general wellbeing.
Almost anyone would include these as worthy
objectives. Just how these goals are to be met can
be expressed by a host of other more specific
objectives or criteria.
By asking “how” any specified broad fundamental criterion or objective can be achieved
usually leads, eventually, to more specific system
performance criteria or objectives and to the
means of improving these criteria, i.e., to the
general system design and operating policy
alternatives themselves. As one gets further from
the fundamental objective that most will agree to,
there is a greater chance of stakeholder disagreement. For example we might answer the
question “How can we maximize public health?”
by suggesting the maximization of surface and
groundwater water quality. How? By minimizing
wastewater discharges into surface water bodies
and groundwater aquifers. How? By minimizing
wastewater production, or by maximizing
removal rates at wastewater treatment facilities,
or by minimizing the concentrations of pollutants
in runoff or by increasing downstream flow
augmentation from upstream reservoirs or by a
combination of flow augmentation and

378

wastewater treatment. How? By increasing
reservoir storage capacity and subsequent releases upstream and by upgrading wastewater
treatment to a tertiary level. And so on.
Each ‘how’ question can have multiple
answers. This can lead to a tree of branches, each
branch representing a different and more specific
performance criterion or an alternative way to
accomplish a higher level objective. In the
example just illustrated, the answer to how to
improve water quality might include a combination of water and wastewater treatment, flow
augmentation, improved wastewater collection
systems and reduced applications of fertilizers
and pesticides on land to reduce nonpoint source
pollutant discharges. There are others. If the
lower level objective of minimizing point source
discharges had been the first objective considered, many of those other alternatives may not
have been identified. The more fundamental the
objective, the greater will be the range of alternatives that might be considered.
If any of the alternatives identified for meeting
some objective are not considered desirable, it is
a good sign that there are other objectives that
should be considered. For example, if flow augmentation is not desired, it could mean that in
addition to water quality considerations, the
regime of water flows, or the existing uses of the
water are also being considered and flow augmentation may detract from those other objectives. If a stakeholder has trouble explaining why
some alternative will not work, it is a possible
sign there are other objectives and alternatives
waiting to be identified and evaluated. What are
they? Get them identified. Consider them along
with the others that have been already defined.
More fundamental objectives can be identified
by asking the question “why?” Answers to the
question “why?” will lead one back to increasingly more fundamental objectives. A fundamental objective is reached when the only
answer to why is “it is what everyone really
wants” or something similar. Thinking hard
about “why” will help clarify what is considered
most important.

9

Performance Criteria

9.3.1 Constraints on Decisions
Constraints limit alternatives. There are some
laws of physics that obviously we cannot change.
Water will naturally flow downhill unless of
course we pump it uphill. Society can limit what
can be done to satisfy any performance criterion
as well, but it is not the time to worry about them
during the process of objective and general
alternative identification. Be creative, and don
not get stuck in the status quo, i.e., carrying on as
usual or depending on a default (and often
politically risk-free) alternative. It may turn out
these default or risk-free alternatives are the best
alternatives, but that can be determined later.
When performing qualitative exercises to identify objectives and general alternatives enlarge
the number of options and think creatively. If
something seems really worth further consideration, then assume it can happen. If it is really
worth it, engineers can do it. Lawyers can change
laws. Society can and will want to adapt—again
if it is really important.
Consider for example the water that is
pumped uphill at all pumped storage hydroelectric facilities, or the changing objectives over
time related to managing water and ecosystems
in the Mekong, Rhine and Senegal Rivers, or in
the Great Lakes, the Sacramento-San Joaquin
Delta, and Florida Everglades regions in the US.
There are other traps to avoid in the planning
and management process. Do not become
anchored to any initial feasible or best-case or
worst-case scenario. Be creative when identifying scenarios. Consider the whole spectrum of
possible events that your decision responds to.
Do not focus solely on extreme events to the
exclusion of the more likely events. While toxic
spills and floods bring headlines, and suffering,
they are not the usual more routine events that
one must also plan for.
It is tempting to consider past or sunk costs
when determining where additional investments
should be made. If past investments were a
mistake, it is only our egos that motivate us to
justify those past investments by spending more

9.3 Performance Criteria and General Alternatives

money on them instead of taking more effective
actions.
Finally, target values of objectives and goals
should not be set too low (easy to meet). The
chances of finding good unconventional alternatives are increased if targets are set high, even
beyond reach. High aspirations often force individuals to think in entirely new ways. Politically
it may end up that only marginal changes to the
status quo will be acceptable, but it is not the
time to worry about that when identifying
objectives and general alternatives.

9.3.2 Tradeoffs Among Performance
Criteria
Tradeoffs are inevitable when there are conflicting multiple objectives, and multiple objectives
are inevitable when there are many stakeholders
or participants in the planning and management
process. We all want clean water in our aquifers,
lakes, rivers and streams, but if it costs money to
achieve that desired quality there are other
activities or projects involving education, health
care, security, or even other environmental
restoration or pollution prevention and reduction
efforts, that compete for those same, and often
limited, amounts of money.
Tradeoffs arise because we all want more of
good things, and many of these good things are
conflicting. We cannot have cleaner water in our
homes or in our environment without spending
money, and minimizing costs is always worthy
of consideration. Identifying these tradeoffs is
one of the tasks of qualitative as well as quantitative analyses. Qualitatively we can identify
what the tradeoffs will involve, e.g., cleaner
water in our streams and rivers require increased
costs. Just how much money it will cost to
increase the minimum dissolved oxygen concentration in a specific lake by 3, or 4, or 5 mg/l
is the task of quantitative analyses.
Finding a balance among all conflicting performance criteria characterizes water management
decision-making. Understanding the technical
information that identifies the efficient or nondominated tradeoffs is obviously helpful. Yet if the

379

technical information fails to address the real
objectives or system performance measures of
interest to stakeholders, significant time and
resources can be wasted in the discussions that
take place in stakeholder meetings, as well as in the
analyses that are performed in technical studies.
Finding the best compromise among competing decisions is a political or social process.
The process is helped by having available the
tradeoffs among competing objectives. Models
that can identify these tradeoffs among quantitative objectives cannot go the next step, i.e.,
identify the best compromise decision. While
models can help identify and evaluate alternatives, they cannot take the place of human
judgments that are needed to make the final
decision—the selection of the best tradeoffs.
Models can help identify nondominated
tradeoffs among competing objectives or system
performance indicators. These are sometimes
called efficient tradeoffs. Efficient decisions are
those that cannot be altered so as to gain more of
one objective without worsening one or more
other objective values. If one of multiple
conflicting objective values is to be improved,
some worsening of one or more other objective
values will likely be necessary. Dominated or so
called inferior decisions are those in which it is
possible to improve at least one objective value
without worsening any of the others.
Many will argue that these dominated decisions can be eliminated from further considerations, since why would any rational individual
prefer such decisions when there are better ones
available with respect to all the objectives being
considered. However, some may consider a
quantitatively nondominated or efficient solution
inferior and dominated with respect to one or
more other objectives that were omitted from the
analysis. Eliminating inferior or dominated
alternatives from the political decision-making
process may not always be very helpful. One of
these inferior alternatives could be viewed as the
best by some who are considering other criteria
either not included or, in their opinions, not
properly quantified in such analyses.
Tradeoffs exist not only among conflicting
outcome objectives, system performance

380

9

indicators, or impacts. Tradeoffs can also exist
among alternative processes of decision-making.
Some of the most important objectives—and
toughest tradeoffs—involve process decisions
that establish how a decision is going to be made.
What is, for example, the best use of time and
financial resources in performing a quantitative
modeling study, who should be involved in such
a study, who should be advising such a study,
and how should stakeholders be involved? Such
questions often lie at the heart of water and other
resource use disputes and can significantly
influence the trust and cooperation among all
who participate in the process. They can also
influence the willingness of stakeholders to
support any final decision or selected management policy or plan.

9.4

Quantifying Performance
Criteria

So far this chapter has focused on the critical
qualitative aspects of identifying objectives and
general alternatives. The remainder of this
chapter will focus on quantifying various criteria
and how to use them to compare various alternatives and to identify the tradeoffs among them.
What is important here, however, is to realize
that considerable effort is worth spending on
getting the general objectives and alternatives
right before spending any time on their quantification. There are no quantitative aids for this,
just hard thinking, perhaps keeping in mine some
of the advice just presented.
Quantification of an objective is the adoption of
some quantitative (numerical) scale that provides
an indicator for how well the objective would be
achieved. For example, one of the objectives of a
watershed conservation program might be protection or preservation of wildlife. In order to rank
how various plans meet this objective, a numerical
criterion is needed, such as acres of preserved
habitat or populations of key wildlife species.
Quantification does not require that all
objectives be described in comparable units. The
same watershed conservation program could
have a flood control objective quantified as the

Performance Criteria

height of the protected flood stage. It could have
a regional development objective quantified as
increased income. Quantification does not
require that monetary costs and benefits be
assigned to all objectives.
The following subsections review various
economic, environmental, ecological, and social
criteria.

9.4.1 Economic Criteria
Water resource system development and management is often motivated by economic criteria.
It goes without saying that money is important;
it’s completing uses often makes it a limiting
resource Most reservoirs, canals, hydropower
facilities, groundwater-pumping systems, locks,
and flood control structures have been built and
are operated for economic reasons. The benefits
and costs of such infrastructure can be expressed
in monetary units. Typical objectives have been
either to maximize the present value of the net
benefits (total benefits less the costs) or minimize
the costs of providing some purpose or service.
To achieve the former involves benefit-cost
analyses and to achieve the latter involves
cost-effectiveness analyses.
Applied to water resources, maximization of
net benefits requires the efficient and reliable
allocation, over both time and space, of water (in
its two dimensions: quantity and quality) to its
many uses, including hydropower, recreation,
water supply, flood control, navigation, irrigation, cooling, waste disposal and assimilation and
habitat enhancement. The following example
illustrates the maximization of net benefits from a
multiple-purpose reservoir.
Consider a reservoir that can be used for
irrigation and recreation. Irrigation and recreation
are not very compatible. Recreation benefits are
greater when reservoir elevations remain high
throughout the summer recreation season, just
when satisfying an irrigation demand that
exceeds the inflow would normally cause a drop
in the reservoir storage level. Thus the project
has two conflicting purposes: provision of irrigation water and of recreation opportunities.

9.4 Quantifying Performance Criteria

Fig. 9.1 A two-purpose planning problem showing the
feasible and efficient combinations of irrigation water
X and recreation visitor days Y

Let X be the quantity of irrigation water to be
delivered to farmers each year and Y the number
of visitor days of recreation use on the reservoir.
Possible levels of irrigation and recreation are
shown in Fig. 9.1. The solid line in Fig. 9.1,
termed the production-possibility frontier or efficiency frontier, is the boundary of the feasible
combinations of X and Y.
Any combination of X and Y within the shaded blue area can be obtained by operation of the
reservoir (i.e., by regulating the amount of water
released for irrigation and other uses). Obviously
the more of both X and Y the better. Thus
attention
generally
focuses
on
the
production-possibility frontier which comprises
those combinations of X and Y that are on the
frontier. They are said to be technologically
efficient in the sense that more of either X or
Y cannot be obtained without a decrease in the
other. The shape and location of the
production-possibility frontier is determined by
the quantity of available resources (water, reservoir storage capacity, recreation facilities, etc.)
and their ability to satisfy various demands for
both X and Y.
Assume that a private entrepreneur owns this
two-purpose reservoir in a competitive environment (i.e., there are a number of competing irrigation water suppliers and reservoir recreation
sites). Let the unit market prices for irrigation

381

Fig. 9.2 Two-purpose project involving tradeoffs
between irrigation and recreation income and various
constant total income levels, I1 < I2 < I3

water and recreation opportunities be pX and pY,
respectively. Also assume that the entrepreneur’s
costs are fixed and independent of X and Y. In
this case the total income is
I ¼ pxX þ pyY

ð9:1Þ

Values of X and Y that result in fixed income
levels I1 < I2 < I3 are plotted in Fig. 9.2. The
value of X and Y that maximizes the entrepreneur’s income is indicated by the point on the
production-possibility frontier yielding an
income of I3. Incomes greater than I3 are not
possible.
Now assume the reservoir is owned and
operated by a public agency and that competitive
conditions prevail. The prices pX and pY reflect
the value of the irrigation water and recreation
opportunities to the users. The aggregated value
of the project is indicated by the user’s willingness to pay for the irrigation and recreation outputs. In this case, this willingness to pay is pX
X + pY Y, which is equivalent to the entrepreneur’s income. Private operation of the reservoir
to maximize income or government operation to
maximize user benefits both should, under competitive conditions, produce the same result.
When applied to water resources planning,
benefit-cost analysis presumes a similarity
between decision-making in the private and
public sectors. It also assumes that the income

382

resulting from the project is a reasonable surrogate for the project’s social value.

9.4.1.1 Benefit and Cost Estimation
In a benefit-cost analysis, one may need to estimate the monetary value of irrigation water,
shoreline property, land inundated by a lake, lake
recreation, fishing opportunities, scenic vistas,
hydropower production, navigation, or the loss of
a wild river. The situations in which benefits and
costs may need to be estimated are sometimes
grouped into four categories, reflecting the way
prices can be determined. These situations are
1. market prices exist and are an accurate
reflection of marginal social values (i.e.,
marginal willingness to pay for all individuals). This situation often occurs in the presence of competitive market conditions. An
example would be agricultural commodities
that are not subsidized, i.e., do not have
supported prices (some do exist!).
2. market prices exist but for various reasons do
not reflect marginal social values. Examples
include price-supported agricultural crops,
labor that would otherwise be unemployed, or
inputs whose production generates pollution,
the economic and social cost of which is not
included in its price.
3. market prices are essentially nonexistent, but
for which it is possible to infer or determine
what users or consumers would pay if a
market existed. An example is outdoor
recreation.
4. no real or simulated market-like process is
easily conceived. This category may be relatively rare. Although scenic amenities and
historic sites are often considered appropriate
examples, both are sometimes privately
owned and managed to generate income.
For the first three categories, benefits and
costs can be measured as the aggregate net
willingness to pay of those affected by the project. Assume, for example, that alternative water
resources projects X1, X2, …, are being considered. Let B(Xj) equal the amount beneficiaries of

9

Performance Criteria

the plan Xj are willing to pay rather than forego
the project. This represents the aggregate value
of the project to the beneficiaries. Let D(Xj) equal
the amount the non-beneficiaries of plan Xj are
willing to pay to prevent it from being implemented. This includes the social value of the
resources that will be unavailable to society if
project or plan Xj is implemented. The aggregate
net willingness to pay W(Xj) for plan Xj is equal
to the difference between B(Xj) and D(Xj),





W Xj ¼ B Xj D Xj

ð9:2Þ

Plans Xj can be ranked according to the aggregate
net willingness to pay, W(Xj). If, for example, W
(Xj) > W(Xk), it is inferred that plan Xj is
preferable or superior to plan Xk.
One rationale for the willingness to pay criterion is that if B(Xj) > D(Xj), the beneficiaries
could compensate the losers and everyone would
benefit from the project. However, this compensation rarely happens. There is usually no
mechanism established for this compensation to
be paid. Those who lose favorite scenic sites or
the opportunity to use a wild river or who must
hear the noise or breathe dirtier air or who suffer
a loss in their property value because of the
project are seldom compensated.
This compensation criterion also ignores the
resultant income redistribution, which should be
considered during the plan selection process. The
compensation criterion implies that the marginal
social value of income to all affected parties is
the same. If a project’s benefits accrue primarily
to affluent individuals and the costs are borne by
lower-income groups, B(X) may be larger than D
(X) simply because the beneficiaries can pay
more than the non-beneficiaries. It matters who
benefits and who pays, i.e., who gets to eat the
pie and how much of it as well as how big the pie
is. Traditional benefit-cost analyses typically
ignore these distributional issues.
In addition to these and other conceptual difficulties related to the willingness-to-pay criterion, practical measurement problems also exist.
Many of the products of water resources plans
are public or collective goods. This means that

9.4 Quantifying Performance Criteria

they are essentially indivisible, and once provided to any individual it is very difficult not to
provide them to others. Collective goods often
also have the property that their nonconsumptive use by one person does not prohibit or infringe upon their use by others.
Community flood protection is an example of
a public good. Once protection is provided for
one individual, it is often simultaneously provided for many others. As a result, it is not in an
individual’s self-interest to volunteer to help pay
for the project by contributing an amount equal
to his or her actual benefits if others are willing to
pay for the project. However, if others are going
to pay for the project (such as the taxpayers),
individuals may exaggerate their own benefits to
ensure that the project is undertaken.
Determining what benefits should be attributed to a project is not always simple and the
required accounting can become rather involved.
In a benefit-cost analysis, economic conditions
should first be projected for a base case in which
no project is implemented and the benefits and
costs are estimated for that scenario. Then the
benefits and costs for each project are measured
as the incremental economic impacts that occur
in the economy over these baseline conditions
due to the project. The appropriate method for
benefit and cost estimation depends on whether
or not the market prices reflecting true social
values are available or if such prices can be
constructed.

Fig. 9.3 Demand function
defining how much water
Q will be purchased for a
specified unit price p. The
shaded area represents the
willingness to pay for a
quantity q*

383

Market Prices Equal Social Values
Consider the estimation of irrigation benefits in
the irrigation-recreation example discussed in the
previous section. Let X be the quantity of irrigation water supplied by the project each year. If
the prevailing market price pX reflects the marginal social value of water and if that price is not
affected by the project’s operation, the value of
the water is just pX X. However, it often happens
that large water projects have a major impact on
the prices of the commodities they supply. In
such cases, the value of water from our example
irrigation district would not be based on prices
before or after project implementation. Rather,
the total value of the water X to the users is the
total amount they would be willing to pay for it.
Let Q(p) be the amount of water the consumers would want to buy at a unit price p. For
any price p, consumers will continue to buy the
water until the value of another unit of water is
less than or equal to the price p. The function Q
(p) defines what is called the demand function.
As illustrated by Fig. 9.3, the lower the unit
price, the more water individuals are willing to
buy, i.e., the greater will be the demand.
The willingness to pay a given unit price p is
defined by the area under the demand curve. As
Fig. 9.3 suggests, there are some who would be
willing to pay a higher price for a given amount
of water than others. As the unit price decreases,
more individuals are willing to buy more water.
The total willingness to pay for a given amount

384

9

Performance Criteria

of water q* is the area under the demand curve
from Q = 0 to Q = q*.


Willingness to pay for q ¼

Z
0

q

Q1 ðqÞdq
ð9:3Þ

Consumer’s willingness to pay for a product is
an important concept in welfare economics.
Market Prices not Equal to Social Values
It frequently occurs that market prices do not
truly reflect the true social value of the various
goods and services supplied by a water resources
project. In such cases it is necessary for the
planner to estimate the appropriate values of the
quantities in question. There are several procedures that can be used depending on the situation. A rather simple technique that can reach
absurd conclusions if incorrectly applied is to
equate the benefits of a service to the cost or
supplying the service by the least expensive
alternative method. Thus the benefits from
hydroelectricity generation could be estimated as
the cost of generating that electricity by the
least-cost alternative method using solar, wind,
geothermal, coal-fired, natural gas, or nuclear
energy sources. Clearly, this approach to benefit
estimation is only valid if in the absence of the
project’s adoption, the service in question would
in fact be demanded at, and supplied by, the
least-cost alternative method. The pitfalls associated with this method of benefit and cost estimation can be avoided if one clearly identifies
reasonable with—and without project scenarios.
In other situations, simulated or imagined
markets can be used to derive the demand
function for a good or service and to estimate the
value of the amount of that good or service
generated or consumed by the project. The following hypothetical example illustrates how this
technique can be used to estimate the value of
outdoor recreation.
Assume a unique recreation area is to be
developed which will serve two population centers. Center A has a population of 10,000 and the
more distant Center B has a population or
30,000. From questionnaires it is estimated that if

Fig. 9.4 Estimated relationship between travel cost and
visits per capita

access to the recreation area is free, 20,000 visits
per year will be made from Center A at an
average round-trip travel cost of $1. Similarly
30,000 visits per year will come from Center B,
at an average round-trip cost of $2.
The benefits derived from the proposed
recreation area can be estimated from an imputed
demand curve. First, as illustrated in Fig. 9.4, a
graph of travel cost as a function of the average
number of visits per capita can be constructed.
Two points are available: an average of two visits
per capita (from center A) at a cost of $1/visit,
and an average of one visit per capita (from
Center B) at a cost of $2 visit. These travel cost
data are extrapolated to the ordinate and abscissa.
Even assuming that there are no plans to
charge admission at the site, if users respond to
an entrance fee as they respond to travel costs, it
is possible to estimate what the user response
might be if an entrance fee were to be charged.
This information will provide a demand curve for
recreation at the site.
Consider first a $1 admission price to be
added to the travel cost for recreation. The total
cost to users from population Center A would
then be $2 per visit. From Fig. 9.4 at $2 per visit,
one visit per capita is made; hence 10,000 visits
per year can be expected from Center A. The
resulting cost to users from Center B is $3 per
visit, and hence from Fig. 9.4 no visits would be
expected. Therefore, one point on the demand
curve (10,000 visits at $1) is obtained. Similarly,

9.4 Quantifying Performance Criteria

Fig. 9.5 Demand function for recreation facility

it can be inferred that at a $2 admission price
there will be no visits from either center. A final
point, corresponding to a zero admission price
(no added costs) is just the expected site attendance (20,000 + 30,000 = 50,000 visits). The
resulting demand curve is shown in Fig. 9.5.
Recreation total willingness to pay benefits are
equal to the area under the demand curve or
$35,000. Assuming no entrance fee is charged,
this amounts to $0.70 per visitor-day based on
the expected 50,000 visits.
Obviously, this example is illustrative only.
The average cost of travel time (which differs for
each population center) and the availability of
alternative sites must be included in a more
detailed analysis.
No Market Processes
In the absence of any market-like process (real or
simulated) it is difficult to associate specific
monetary values with benefits. The benefits
associated with esthetics and with many aspects
of environmental and ecosystem quality have
long been considered difficult to quantify in
monetary terms. Although attempts (some by
highly respected economists) have been made to
express environmental benefits in monetary
terms, the results have had limited success. In
most regions in the world, water resources
management guidelines, where they exist, do not
encourage the assignment of monetary values to
these criteria. Rather the approach is to establish
environmental and ecological requirements or

385

regulations. These constraints are to be met perhaps while maximizing other economic benefits.
Their shadow prices or dual variables (indicating
the marginal cost associated with a unit change in
the regulation or requirement) is likely to be as
close to monetizing such non-monetary impacts
as one can get, yet recognizing the actual monetary benefits could be greater. Legislative and
administrative processes rarely if ever determine
the explicit benefits derived from meeting these
environmental and ecological requirements or
regulations.
To a certain extent, environmental (including
esthetic) objectives, if quantified, can be incorporated into a multiobjective decision-making
process. However, this falls short of the assignment of monetary benefits that is often possible
for the first three categories of benefits.

9.4.1.2 A Note Concerning Costs
To be consistent in the estimation of net benefits
from water resources projects, cost estimates
should reflect opportunity costs, the value of
resources in their most productive alternative
uses. This principle is much easier stated than
implemented, and as a result true opportunity
costs are seldom included in a benefit-cost analysis. For example, if land must be purchased for
a flood control project, is the purchase price
(which would typically be used in a benefit-cost
analysis) the land’s opportunity cost? Suppose
that the land is currently a natural area and its
alternative use is as a nature preserve and
camping area. The land’s purchase price may be
low, but this price unlikely reflects the land’s true
value to society. Furthermore, assume that individuals who would otherwise be unemployed are
hired to work on the land. The opportunity cost
for such labor is the marginal value of leisure
forgone, since there is no alternative productive
use of those individuals. Yet, the labors must be
paid and their wages must be included in the
budget for the project.
The results of rigorous benefit-cost analyses
seldom dictate which of competing water
resources projects and plans should be implemented. This is in part because of the multiobjective nature of the decisions. One must

386

consider environmental impacts, income redistribution effects, and a host of other local, , and
national goals.
Other important considerations are the financial, technical, and political feasibilities of alternative plans. Particularly important when plan
are proposed by government agencies is the relative political and legal clout of those who support the plans and those who oppose them. Still,
a plan’s economic efficiency is an important
measure of a plan’s value to society and often
serves as an indicator of whether it should be
considered at all.

9.4.1.3 Long- and Short-Run Benefit
Functions
When planning the capacities and target values
associated with water resource development
projects, it is often convenient to think of two
types of benefit functions: the long-run benefits
and the short-run benefits. In long-run planning,
the capacities of proposed facilities and the target allocations of flows to alternative uses or
target storage volumes in reservoirs are unknown
decision variables. The values of these variables
are to be determined in a way that achieves the
most beneficial use of available resources, even
when the available resources vary in magnitude
over time. In short-run planning, the capacity of
facilities and any associated targets are assumed
known. The problem is one of managing or
operating a given or proposed system under
varying supply and demand conditions.
For example, if a water-using firm is interested in building a factory requiring water from a
river, of interest to those designing the factory is
the amount of water the factory can expect to get.
This in part may dictate the capacity of that
factory, the number of employees hired, and the
amount of product produced, etc. On the other
hand, if the factory already exists, the likely issue
is how to manage or operate the factory when the
water supply varies from the target levels that
were (and perhaps still are) expected.
Long-run benefits are those benefits obtained
if all target allocations are met. Whatever the
target, if it is satisfied, long-run benefits result,
Short-run benefits are the benefits one actually

9

Performance Criteria

can obtain by operating a system having fixed
capacities and target values. If the water resources available in the short run are those that can
meet all the targets that were established when
long-run decisions were made, then estimated
long-run benefits can be achieved. Otherwise the
benefits actually obtained may differ from those
expected. The goal is to determine the values of
the long-run decision variables in a way that
maximizes the present value of all the short run
benefits obtained given the varying water supply
and demand conditions.
The distinction between long-run and
short-run benefits can be illustrated by considering again a potential water user at a particular
site. Assume that the long-run net benefits associated with various target allocations of water to
that use can be estimated. These long-run net
benefits are those that will be obtained if the
actual allocation Q equals the target allocation
QT. This long-run net benefit function can be
denoted as B(QT). Next assume that for various
fixed values of the target QT the actual net benefits derived from various allocations Q can be
estimated. These short-run benefit functions b(Q|
QT) of allocations Q given a target allocation QT
are dependent on the target QT and obviously on
the actual allocation Q. The relationship between
the long-run net benefits B(T) and the short-run
net benefits b(Q|QT) for a particular target QT is
illustrated in Fig. 9.6.
The long-run function B(QT) in Fig. 9.6
reflects the benefits users receive when they have
adjusted their plans in anticipation of receiving
an allocation equal to the target QT and actually
receive it. The short-run benefits function specifies the benefits users actually receive when a
particular allocation is less (e.g., Q1) or more
(e.g., Q2) than the anticipated allocation, QT, and
they cannot completely adjust their plans to the
resulting deficit or surplus.
Clearly the short-run benefits associated with
any allocation cannot be greater than the
long-run benefits obtainable had the firm planned
or targeted for that allocation. The short-run
benefit function is always going to be under, or
tangent to, the long-run benefit function, as
shown in Fig. 9.6. In other words the short-run

9.4 Quantifying Performance Criteria

387

Fig. 9.6 Long-run and
short-run benefit functions,
together with the loss, L
(Q1|QT), associated with a
deficit allocation Q1 and
target allocation QT

benefits b(Q|QT) will never exceed the long-run
benefits B(Q) that could be obtained if the target
QT were equal to the allocation Q. When the
target QT equals the allocation Q, the values of
both functions are equal.
Flipping the short-run benefit function upside
down along a horizontal axis running through the
long-run benefit function at the target QT defines
the short-run loss function, as illustrated in
Fig. 9.7. The short-run loss of any actual allocation Q equals the long-run benefit of the target allocation QT minus the short-run benefit of
the actual allocation, Q.








L QjQT ¼ B QT  b QjQT

ð9:4Þ

This function defines the losses that occur when
the target allocation cannot be met. When the
actual allocation equals the target allocation, the
short-run loss is zero. It is possible there might be
short-run gains or benefits if there is a surplus
allocation over the target allocation, possibly for
a limited range of excess allocations.
The short-run benefit function, or its corresponding loss function, usually depends on the
value of the target allocation. However, if the
short-run losses associated with any deficit allocation (QT − Q1) or surplus allocation (Q2 − QT)
are relatively constant over a reasonable range of

targets, it may not be necessary to define the loss
as a function of the target QT. In this case the loss
can be defined as a function of the deficit
D and/or as a function of the surplus (excess)
E. Both the deficit D or excess E can be defined
by the constraint
Q ¼ QT D þ E

ð9:5Þ

Denote the loss function for a deficit allocation as
LD(D) and for a surplus allocation, LE(E). As
indicated above, the later may be a negative loss,
i.e., a gain, at least for some range of E, as shown
in Fig. 9.7.
The costs of the capacity of many components
or multipurpose projects are not easily expressed
as functions of the targets associated with each
use. For example the capacity, K, of a multipurpose reservoir is not usually equal to, or even
a function of, its recreation level target or its
active or flood storage capacities. The costs of its
total capacity, C(K), are best defined as functions
of that total capacity. If expressed as an annual
cost it would include the annual amortized capital costs as well as the annual operation, maintenance, and repair costs.
Assuming that the benefit and loss functions
reflect annual benefits and losses, the annual net
benefits, NB, from all projects j is the sum of

388

9

Performance Criteria

Fig. 9.7 Short-run loss
function, L(Q|QT),
associated with an
allocation Q and
target allocation QT

each project’s long-run benefits Bj(Tj) that are
functions of their targets, Tj, less short-run losses
Lj(Qj|Tj) and capacity costs Cj(Kj).
NB ¼

X






Bj Tj  Lj Qj jTj  Cj Kj
ð9:6Þ
j

The monetary net benefits accrued by each group
of water users can also be determined so that the
income redistribution effects of a project can be
evaluated.
The formulation of benefit functions as either
long- or short-run is, of course, a simplification
of reality. In reality, planning takes place on
many time scales and for each time scale one
could construct a benefit function. Consider the
planning problems of farmers. In the very long
run, they decide whether or not to own farms,
and if so how big they are to be in a particular
area. On a shorter time scale, farmers allocate
their resources to different activities depending
on what products they are producing and on the
processes used to produce them. Different activities, of course, require capital investments in
farm machinery, storage facilities, pipes, pumps,
etc., some of which cannot easily be transferred
to other uses. At least on an annual basis, most
farmers reappraise these resource allocations in
light of the projected market prices of the generated commodities and the availability and cost
of water, energy, labor and other required inputs.
Farmers can then make marginal adjustments in
the amounts of land devoted to different crops,
animals, and related activities within the bounds

allowed by available resources, including land,
water, capital and labor. At times during any
growing season some changes can be made in
response to changes in prices and the actual
availability of water.
If the farmers frequently find that insufficient
water is available in the short run to meet livestock and crop requirements, then they will
reassess and perhaps change their long-run plans.
They may shift to less water-intensive activities,
seek additional water from other sources (such as
deep wells), or sell their farms (and possibly
water rights) and engage in other activities.
For the purposes of modeling, however, this
planning hierarchy can generally be described by
two levels, denoted as long-run and short-run.
The appropriate decisions included in each
category will depend on the time scale of a
model.
These long-run and short-run benefit and loss
functions are applicable to some water users, but
not all. They may apply in situations where
benefits or losses can be attributed to particular
allocations in each of the time periods being
modeled. They do not apply in situations where
the benefits or losses result only at the end of a
series of time periods, each involving an allocation decision. Consider irrigation, for example. If
each growing season is divided into multiple
periods, then the benefits derived from each
period’s water allocation cannot be defined
independent of any other period’s allocation, at
least very easily. The benefits from irrigation
come only when the crops are harvested, e.g., at

9.4 Quantifying Performance Criteria

the end of the last period of each growing season.
In this case some mechanism is needed to
determine the benefits obtained from a series of
allocations over time, as will be presented in the
next chapter.

9.4.2 Environmental Criteria
Environmental criteria for water resource projects can include water quantity and quality
conditions. These conditions are usually expressed in terms of targets or constraints for flows,
depths, hydroperiods (duration of flooding),
storage volumes, flow or depth regimes and
water quality concentrations that are considered
desirable for esthetic or public health reasons or
for various ecosystem habitats. These constraints
or targets could specify desired minimum or
maximum acceptable ranges, or rates of changes,
of these values, either for various times within
each year or over an n-year period.
Water quality constituent concentrations are
usually expressed in terms of some maximum or
minimum acceptable concentration, depending
on the particular constituents themselves and the
intended uses of the applicable water body. For
example, phosphorus would normally have a
maximum permissible concentration and dissolved oxygen would normally have a minimum
acceptable concentration. These limiting concentrations and their specified reliabilities are
often based on standards established by governmental or international environmental or health
organizations. As standards they are viewed as
constraints. These standards could also be considered as targets and the maximum or average
adverse deviations from these standards or targets could be a system performance measure.
Few water quality criteria may be expressed in
qualitative terms. Qualitative quality criteria can
provide a ‘fuzzy’ limit on the concentration of
some constituent in the water. Such criteria might
be expressed as, for example, “the surface water
shall be virtually free from floating petroleumderived oils and non-petroleum oils of vegetable
or animal origin.” Stakeholder membership
functions can define what is considered virtually

389

free, and these can be included as objectives or
constraints in models (as discussed in Chap. 5).
Environmental performance criteria can vary
depending on the specific sites and on the
intended uses of water at that site. They should
be designed to assess, or define, the risks of
adverse impacts on the health of humans and
aquatic life from exposure to pollutants.
Environmental performance criteria of concern to water resource planners and managers can
also relate to recreational and land use activities.
These typically address hydrologic conditions,
such as streamflows or lake or reservoir storage
volume elevations during specified times or land
use activities on watersheds. For example, to
increase the safety of boaters and individuals
fishing downstream of hydropower reservoirs,
release rules may have to be altered to reduce the
rate of flow increase that occurs during peak
power production. Performance criteria applicable to the adverse environmental impacts of
sediment loads, say caused by logging or construction activities, or to the impact of nutrient
loads in the runoff, perhaps from agricultural and
urban areas, are other examples.

9.4.3 Ecological Criteria
Criteria that apply to aquatic ecosystems involve
both water quantity and quality and are often
compatible with environmental criteria. It is the
time-varying regimes of water quantities and
qualities, not minimum or maximum values that
benefit and impact ecosystems. It is not possible
to manage water and its constituent concentrations in a way that maximizes the health or well
being, however, measured, of all living matter in
an ecosystem. (Like people, it’s hard to satisfy
everyone all the time.) If one species feeds on
another, it is hard to imagine how to maximize
the health of both. The conditions that favor one
species group may not favor another. Hence
variation in habitat conditions is important for the
sustainability of both, and indeed for achieving
resilient biodiverse ecosystems.
While ecosystem habitats exhibit more
diversity when hydrologic conditions vary, as in

390

nature, than when they are relatively constant,
hydrologic variation is often not desired by many
human users of water resource systems. Reducing hydrologic variation and increasing the reliability of water resource systems has often been
the motivating factor in the design and operation
of hydraulic engineering works.
The state of ecological habitats is in part
functions of how water is managed. One way to
develop performance indicators of ecosystems is
to model the individual species making up the
ecosystem, or at least a subset of important indicator species. This is often difficult. Alternatively
one can define habitat suitability indices for these
important indicator species. This requires (1) selecting the indicator species representative of
each particular ecosystem, (2) identifying the
hydrological attributes that affect the wellbeing of
those indicator species during various stages of
their life cycles, and (3) quantifying the functional relationships between the wellbeing of
those species and values of the applicable
hydrological attributes, usually on a scale from 0
to 1. A habitat suitability value of 1 is considered
an ideal condition. A value of 0 is considered to
be very unfavorable.
Examples of hydrological attributes that
impact ecosystems could include flow depths,
velocities, constituent concentrations and temperatures, their durations or the rate of change in
any of those values over space and/or time at
particular times of the year. In wetlands the
hydrologic attributes could include the duration
of inundation (hydroperiod), time since last
drawdown below some threshold depth, the
duration of time below or above some threshold
depth, and time rates of change in depth. The
applicable attributes themselves, or perhaps just
their functional relationships, can vary depending
on the time of year and on the stage of species
development.
Figure 9.8 illustrates three proposed habitat
suitability indices for periphyton (algae) and fish
in parts of the Everglades region of southern
Florida in the US. All are functions of

9

Performance Criteria

hydrological attributes that can be managed.
Shown in this figure is the impact of hydroperiod duration on the habitat of three different
species of periphyton located in different parts
of the Everglades, and the impact of the duration of the hydroperiod as well as the number of
years since the last dry period on a species of fish.
There are other functions that would influence
the growth of periphyton and fish, such as the
concentrations of phosphorus or other nutrients
in the water. These are not shown. Figure 9.8
merely illustrates the construction of habitat
suitability indices.
There are situations where it may be much
easier and more realistic to define a range of
some hydrological attribute values as being ideal.
Consider fish living in streams or rivers for
example. Fish desire a variety of depths,
depending on their feeding and spawning activities. Ideal trout habitat, for example, requires
both deeper pools and shallower riffles in
streams. There is no one ideal depth or velocity
or even a range of depths or velocities above or
below some threshold value. In such cases it is
possible to divide the hydrologic attribute
scale into discrete ranges and identify the ideal
fraction of the entire stream or river reach or
wetland that ideally would be within each discrete range. This exercise will result in a function
resembling a probability density function. The
total area under the function is 1. (The first and
last segments of the function can represent anything less or greater than some discrete value,
where appropriate, and if so the applicable segments are understood to cover those ranges.)
Such a function is shown in Fig. 9.10. Figure 9.9
happens to be a discrete distribution, but it
could have been a continuous one as well.
Any predicted distribution of attribute values
resulting from a simulation of a water management policy can be compared to this ideal
distribution, as is shown in Fig. 9.10. The fraction of overlapping areas of both distributions
is an indication of the suitability of that habitat
for that indicator species.

9.4 Quantifying Performance Criteria

391

Fig. 9.8 Some proposed habitat suitability indices (SI) for three types of periphyton (algae) and a species of fish in
portions of the Everglades region in southern Florida of the US

392

9

Fig. 9.9 Ideal range of values of a hydrologic attribute
for a particular component of an ecosystem

Performance Criteria

some agreement or consensus. This just points to
the complexity of ecosystems and the nontrival
task of trying to simplify it to define habitat
suitability performance criteria. However once
identified, these habitat suitability performance
criteria can give water resource planners and
managers an admittedly incomplete but at least
relative indication of the ecosystem impacts of
alternative water management policies or
practices.
The use of these habitat suitability functions
along with other performance criteria in optimization and simulation models will be discussed later in this chapter.

9.4.4 Social Criteria

Fig. 9.10 Predicted (simulated indicated in blue) and
ideal (indicated in red) distributions of attribute values for
some ecosystem indicator species. The fraction of the area
common to both distributions is a measure of habitat
suitability

Habitat suitability ¼Fraction of area under
the ideal and simulated
distributions ofattribute values:

ð9:7Þ
In Fig. 9.10, this is the red shaded area under the
blue curve.
To identify a representative set of indicator
species of any ecosystem, the hydrologic attributes or ‘stressors’ that impact those indicator
species, and finally the specific functional relationships between the hydrologic attributes and
the habitat suitability performance indicators, is
not a trivial task. The greater the number of
experienced individuals involved in such an
exercise, the more difficult it may be to reach

Social performance criteria are often not easily
defined as direct functions of hydrological attributes. Most social objectives are only indirectly
related to hydrological attributes or other measures of water resource system performance.
Economic, environmental, and ecological
impacts resulting from water management policies directly affect people. One social performance criterion that has been considered in some
water resources development projects, especially
in developing regions, has been employment.
Where employment is considered important,
alternatives that provide more jobs may be preferred to those that use more heavy machinery in
place of labor, for example.
Another social performance criterion is
human settlement displacement. The number of
families that must move from their homes
because of, for example, flood plain restoration
or reservoir construction, is always of concern.
These impacts can be expressed as a function of
the extent of flood plain restoration or reservoir
storage capacity, respectively. Often the people
most affected are in the lower-income groups,
and this raises legitimate issues of social justice
and equity. Human resettlement impacts have
both social and economic dimensions.
Social objectives are often the more fundamental objectives discussed earlier in this chapter. Asking ‘why’ identifies them. Why improve

9.4 Quantifying Performance Criteria

water quality? Why prevent flood damage? Why,
or why not, build a reservoir? Why restore a
flood plain or wetland? Conversely, if social
objectives are first identified, by asking and then
answering ‘how’ they can be achieved usually
results in the identification of economic, environmental and ecological objectives more
directly related to water management.
The extent of press coverage or of public
interest and participation in the planning and
evaluation processes can also be an indicator of
social satisfaction with water management. In
times of social stress due to, for example, floods,
droughts, or disease caused by waterborne bacteria, viruses, and pollutants, press coverage and
public involvement often increases. (Public
interest also increases when there is a lot of
money to be spent but this is often a result of
substantial public interest as well.) It is a continuing challenge to actively engage an often
disinterested public in water management planning at times when there are no critical water
management impacts being felt and not a lot of
money is being spent. Yet this is just the time
such planning for more stressful conditions
should take place.

9.5

Multicriteria Analyses

Given multiple performance criteria measured in
multiple ways, how can one determine the best
decision, i.e., the best way to develop and manage water? Just what is best, or as some put it,
rational? The answer to these questions will often
differ depending on who is being asked. There is
rarely an alternative that makes every interest
group or impacted stakeholder the happiest.
When agreement is not universal and when some
objectives conflict with others, we can identify
the efficient tradeoffs among the objective values
each stakeholder would like to have. In this
section some ways of identifying efficient tradeoffs are reviewed. These methods of multicriteria
or multiobjective analyses are not designed to
identify the best solution, but only to provide

393

information on the tradeoffs among conflicting
quantitative performance criteria. Again, any
final decision will be based on qualitative as well
as this quantitative information in a political and
social process, not by or in a computer.
Even if the same units of measure, e.g.,
monetary ones, can be used for each performance
measure or objective, it is not always appropriate
to simply sum them together into a single measure or objective that can be maximized or
minimized. Consider for example a water
resources development project to be designed to
maximize net economic benefits. In the US this is
sometimes designated the national economic
development (NED) objective. Another objective
may be to distribute the costs and benefits of the
project in an equitable way. Both objectives are
measured in the same monetary units. While
everyone may agree that the biggest pie (i.e., the
maximum net benefits) should be obtained,
subject to various environmental and ecological
constraints perhaps, not everyone will likely
agree as to how that pie should be divided up
among all the stakeholders. It also matters who
pays and who benefits, and by how much. Again,
issues of equity and social justice involve judgments, and the challenge of water resource
planners and managers, and elected politicians, is
to make good judgments. The result is often a
plan or policy that does not maximize net economic benefits. Requiring all producers of
wastewater effluent to treat their wastes to the
best practical level before discharging the
remaining effluent, regardless of the quality or
assimilative capacity of the receiving bodies of
water, is one example of this compromise
between economic, environmental, and social
criteria.
The irrigation-recreation example presented
earlier in this chapter illustrates some basic
concepts in multiobjective planning. As indicated
in Fig. 9.1, one of the functions of multiobjective
planning is the identification of plans that are
technologically efficient. These are plans that
define the production-possibility frontier. Feasible plans that are not on this frontier are inferior

394

9

in the sense that it is always possible to identity
alternatives that will improve one or more
objective values without making others worse.
Although the identification of feasible and
efficient plans is seldom a trivial matter, it is
conceptually straightforward. Deciding which of
these efficient plans is the best is quite another
matter. One needs some way to compare them.
Social welfare functions that could provide a
basis for comparison is impossible to construct,
and the reduction of multiple objectives to a
single criterion (as in Fig. 9.2), especially if they
are conflicting, is seldom acceptable in practice.
When the various objectives of a water
resources planning project cannot be combined
into a single scalar objective function, a vector
optimization representation of the problem may
be applicable. Let the vector X represent the set
of unknown decision variables whose values are
to be determined and let Zj(X) be a performance
criterion or objective that is to be maximized.
Each performance criterion or objective j is a
function of these unknown decision variable
values. Assuming that all objectives Zj(X) are to
be maximized, the model can be written


maximize Z1 ðXÞ; Z2 ðXÞ; . . .; Zj ðXÞ; . . .; ZJ ðXÞ
subject to:
gi ðXÞ ¼ bi

i ¼ 1; 2; . . .; m

ð9:8Þ

The objective in Eq. 9.8 is a vector consisting of
J separate objectives. The m constraints gi(X) =
bi define the feasible region of solutions. Again,
the vector X represents all the unknown decision
variables whose values are to be determined by
solving the model.
The vector optimization model is a concise
way of representing a multiobjective problem but
it is not very useful when trying to solve it. In
reality, a vector can be maximized or minimized
only if it can be reduced to a scalar. Thus the
multiobjective planning problem defined by
Model 9.8 cannot, in general, be solved without
additional information. Various ways of solving
this multiobjective model are discussed in the
following subsections.

Performance Criteria

The goal of multiobjective modeling is the
generation of a set of technologically feasible and
efficient plans. Recall that an efficient plan is one
that is not dominated.

9.5.1 Dominance
A plan X dominates all others if it results in an
equal or superior value for all objectives, and at
least one objective value is strictly superior to
those of each other plan. In symbols, assuming
that all objectives j are to be maximized, plan
alternative i, Xi, dominates if
Zj ðX i ÞZj ðXk Þ for all objectives j and plans k
ð9:9Þ
and for each plan k ≠ i there is at least one
objective j* such that
Zj ðX i Þ [ Zj ðX k Þ

ð9:10Þ

Not very often does one plan dominate all others.
If it does, pick it! More often different plans will
dominate all plans for different objectives. However, if there exists two plans k and h such that the
values of all objectives j for plan k are never less
than that for plan h (Zj(Xk) ≥ Zj(Xh)), and for some
objective j*, plan k provides a higher value than
does plan h, (Zj*(Xk) > Zj*(Xh)), then plan
k dominates plan h and plan Xh can be dropped
from further consideration. This assumes of
course that all objectives are being considered. If
some objectives are not included in the analysis,
perhaps because they cannot be quantified, inferior plans with respect to those objectives that are
included in the analysis should not be rejected
from eventual consideration just based on this
quantitative analysis. To work, dominance analysis must consider all objectives. In practice this
condition is often impossible to meet.
Dominance analysis requires that participants
in the planning and management process specify
all the objectives that are to be considered. It
does not require the assessment of the relative
importance of each objective. Non-inferior, efficient, or nondominated solutions are often called

9.5 Multicriteria Analyses

Pareto optimal or Pareto efficient because they
satisfy the conditions proposed by the Italian
economist and social theorist Vilfredo Pareto
(1848–1923). A set of objective values is efficient if in order to improve the value of any
single objective, one must accept a diminishment
of at least one other objective.
Consider for example three alternatives A, B,
and C. Assume, as shown in Fig. 9.11, that plan
C is inferior to plan A with respect to objective
Z1(X) and also inferior to plan B with respect to
objective Z2(X). Plan C might still be considered
the best with respect to both objectives Z1(X) and
Z2(X). While plan C could have been inferior to
both A and B, as is plan D in Fig. 9.11, it should
not necessarily be eliminated from consideration
just based on a pair-wise comparison. In fact plan
D, even though inferior with respect to both
objectives Z1(X) and Z2(X), might be the preferred plan if another objective were included.
Two common approaches for identifying
nondominated plans that together identify the
efficient tradeoffs among all the objectives
Zj(X) in the Model Eq. 9.8 are the weighting and
constraint methods. Both methods require
numerous solutions of a single objective management model to generate points on the objective functions’ production-possibility frontier
(the blue line in Fig. 9.11).

Fig. 9.11 Four discrete
plans along with a
continuous efficiency
frontier associated with two
objectives, Z1 and Z2.
A pair-wise comparison of
plans or objectives may not
identify all the
nondominated plans. All
objectives should be
considered before declaring
a plan inferior

395

9.5.2 The Weighting Method
The weighting approach involves assigning a
relative weight to each objective to convert the
objective vector (in Eq. 9.8) to a scalar. This
scalar is the weighted sum of the separate
objective functions. The multiobjective Model
9.8 becomes
maximize Z ¼ ½w1 Z1 ðXÞ þ w2 Z2 ðXÞ. . .
þ wj Zj ðXÞ. . . þ wJ ZJ ðXÞ
subject to:
gi ðXÞ ¼ bi

i ¼ 1; 2; . . .; m

ð9:11Þ

where the nonnegative weights wj are specified
constants. The values of these weights wj are
varied systematically, and the model is solved
for each combination of weight values to generate a set of technically efficient (or non-inferior)
plans.
The foremost attribute of the weighting
approach is that the tradeoffs or marginal rate of
substitution of one objective for another at each
identified point on the objective functions’
production-possibility frontier is explicitly specified by the relative weights. The marginal rate of
substitution between any two objectives Zj and
Zk, at a specified constant value of X, is

396

9

Performance Criteria

Fig. 9.12 The efficiency
frontier between two
objectives, Z1(X) and
Z2(X), showing the
reduction in one objective,
say Z1(X), as the relative
weight, w2, associated with
the other objective,
increases



 dZj =dZk jx¼constant ¼ wk =wj

ð9:12Þ

This applies when each of the objectives is
continuously differentiable at the point X in
question. This is illustrated for a two-objective
maximization problem in Fig. 9.12.
These relative weights can be varied over
reasonable ranges to generate a wide range of
plans that reflect different priorities. Alternatively, specific values of the weights can be
selected to reflect preconceived ideas of the relative importance of each objective. It is clear that
the prior selection of weights requires value
judgments. If each objective value is divided by
its maximum possible value, then the weights
can range from 0 to 1 and sum to 1, to reflect the
relative importance given to each objective.
For many projects within developing countries, these weights are often estimated by the
agencies financing the development projects. The
weights specified by these agencies can, and
often do, differ from those implied by national or
regional policy. But regardless of who does it,
the estimation of appropriate weights requires a
study of the impacts on the economy, society,
and development priorities involved.
Fortunately here we are not concerned with
finding the best set of weights, but merely using
these weights to identify the efficient tradeoffs

among conflicting objectives. After a decision is
made, the weights that produce that solution
might be considered the best, at least under the
circumstances and at the time when the decision
was made. They will unlikely be the weights that
will apply in other places in other circumstances
at other times.
A principal disadvantage of the weighting
approach is that it cannot generate the complete set
of efficient plans unless the efficiency frontier is
strictly concave (decreasing slopes) for maximization, as it is in Figs. 9.1 and 9.12. If the
frontier, or any portion of it, is convex, only the
endpoints of the convex region will be identified
using the weighting method, as illustrated in
Fig. 9.13.

9.5.3 The Constraint Method
The constraint method for multiobjective planning can produce the entire set of efficient plans
for any shape of efficiency frontier, including that
shown in Fig. 9.13, assuming there are tradeoffs
among the objectives. In this method one
objective, say Zk(X) is maximized subject to
lower limits Lj, on the other objectives, j ≠ k. The
solution of the model, corresponding to any set
of feasible lower limits Lj, produces an efficient

9.5 Multicriteria Analyses

397

Fig. 9.13 An efficiency
frontier that cannot be
completely identified
within its convex region
using the weighting method
when objectives are being
maximized. Similarly for
concave regions when
objectives are being
minimized

alternative if the lower bounds on the other
objective values are binding.
In its general form, the constraint model is
maximize Zk ðXÞ

ð9:13Þ

subject to
gi ðXÞ ¼ bi

i ¼ 1; 2; . . .; m

Zj ðXÞ  Lj

8j 6¼ k:

ð9:14Þ
ð9:15Þ

Note that the dual variables associated with the
right-hand-side values Lj are the marginal rates of
substitution or rate or change of Zk(X) per unit
change in Lj (or Zj(X) if binding).
Figure 9.14 illustrates the constraint method
for a two-objective problem.
An efficiency frontier identifying the tradeoffs
among conflicting objectives can be defined by
solving the model many times for many values of
the lower bounds. Just as with the weighting
method, this can be a big job if there are many
objectives. If there are more than two or three
objectives the results cannot be plotted. Pair-wise
tradeoffs that can easily be plotted do not always
clearly identify nondominated alternatives, as
previously demonstrated.

The number of solutions to a weighting or
constraint method model can be reduced considerably if the participants in the planning and
management process can identify the acceptable
weights or lower limits. However this is not the
language of decision-makers. Decision-makers
who count on the support of each stakeholder
interest group are not happy in assigning weights
that imply the relative importance of those various stakeholder interests. In addition, decisionmakers should not be expected to know what
they may want until they know what they can
get, and at what cost (often politically as much as
economically). However, there are ways of
modifying the weighting or constraint methods to
reduce the amount of effort in identifying these
tradeoffs as well as the amount of information
generated that is of no interest to those making
decisions. This can be done using interactive
methods that will be discussed shortly.
The weighting and constraint methods are
among many methods available for generating
efficient or non-inferior solutions (see, for example, Steuer 1986). The use of methods that generate many solutions, even just efficient ones,
assumes that once all the non-inferior alternatives
have been identified, the participants in the
planning and management process will be able to

398

9

Performance Criteria

Fig. 9.14 The constraint
method for identifying the
efficiency frontier by
maximizing Z1(X) while
constraining Z2(X) to be no
less than L2

select the best compromise alternative from
among them. In some situations this has worked.
Undoubtedly, there will be planning activities in
the future where the use of these non-inferior
solution generation techniques alone will continue to be of value. However, in many other
planning situations, they alone will not be sufficient. Often, the number of feasible non-inferior
alternatives is simply too large. Participants in the
planning and management process will not have
the time or patience to examine, evaluate and
compare each alternative plan. Planners or managers may also need help in identifying which
alternatives they prefer. If they are willing to
work with analysts, these analysts can help them
identify what alternatives they prefer without
generating and comparing all the other plans.
There are a number of methods available for
assisting in selecting the most desirable nondominated plan. Some of the more common and
simpler ones are described next.

each objective that is to be maximized. Those
alternatives that do not meet these minimum
performance values are eliminated from further
consideration. Those that remain can again be
screened if the minimal acceptable values of one
or more objectives are increased. When used in
an iterative fashion, the number of non-inferior
alternatives can be reduced down to a single best
compromise or a set of plans which the participants in the planning and management process
are essentially indifferent. This process is illustrated in Fig. 9.15.
Of course, sometimes the participants in the
planning process will be unwilling or unable to
sufficiently narrow down the set of available
non-inferior plans with the iterative satisfying
method. Then it may be necessary to examine in
more detail the possible tradeoffs among the
competing alternatives.

9.5.5 Lexicography
9.5.4 Satisficing
One method of further reducing the number of
alternatives is called satisficing. It requires that
the participants in the planning and management
process specify a minimum acceptable value for

Another simple approach is called lexicography.
To use this approach, the participants in the
planning process must rank the objectives in
order of priority. This ranking process takes
place without considering the particular values of
these multiple objectives. Then, from among the

9.5 Multicriteria Analyses

399

Fig. 9.15 Two successive
iterations of the satisficing
method in which minimum
levels of each objective are
set thereby eliminating
those alternatives that do
not meet these minimum
levels

non-inferior plans that satisfy minimum levels of
each objective, the plan that is the best with
respect to the highest priority objective will be
the one selected as superior.
If there is more than one plan that has the
same value of the highest priority objective, then
among this set of preferred plans the one that
achieves the highest value of the second priority
objective is selected. If here too there are multiple such plans, the process can continue until
there is a unique plan selected.
This method assumes such a ranking of the
objectives is possible. Often the relative values
of the objectives of each alternative plan are of
more importance to those involved in the
decision-making process. Consider, for example, the problem of purchasing apples and
oranges. Assuming you like both types of fruit,
which type of fruit should you buy if you have
only enough money to buy one type? If you
know you already have lots of apples, but no
oranges, perhaps you would buy oranges, and
vice versa. Hence the ranking of objectives can
depend on the current state and needs of those
who will be impacted by the plan.

9.5.6 Indifference Analysis
Another method of selecting the best plan is
called indifference analysis. To illustrate the
possible application of indifference analysis to
plan selection, consider a simple situation in
which there are only two alternative plans (A and
B) and two planning objectives (1 and 2) being
considered. Let ZA1 and ZA2 be the values of the
two respective objectives for plan A and let ZB1
and ZB2 be the values of the two respective
objectives for plan B. Comparing both plans
when a different objective is better for each plan
can be difficult. Indifference analysis can reduce
the problem to one of comparing the values of
only one objective.
Indifference analysis first requires the selection of an arbitrary value for one of the objectives, say Z*2 for objective 2 in this two-objective
example. It is usually a value within the range of
the values ZA2 and ZB2 , or in a more general case
between the maximum and minimum of all
objective 2 values.
Next, a value of objective 1, say Z1 must be
selected such that the participants involved are

400

9

indifferent (equally happy or satisfied) between
the hypothetical plan that would have as its objective values (Z1, Z*2) and plan A that has as its
objective values (ZA1 , ZA2 ). In other words, Z1 must
be determined such that (Z1, Z*2) is as desirable as
or equivalent to (ZA1 , ZA2 ).






Z1 ; Z2  Z1A ; Z2A :

ð9:16Þ

Next another value of the first objective, say Z′1,
must be selected such that the participants are
indifferent between a hypothetical plan (Z′1, Z*2)
and the objective values (ZB1 , ZB2 ) of plan B.






Z10 ; Z2  Z1B ; Z2B

ð9:17Þ

These comparisons yield hypothetical but
equally desirable plans for each actual plan.
These hypothetical plans differ only in the value
of objective 1 and hence they are easily compared. If both objectives are to be maximized and
Z1 is larger than Z′1, then the first hypothetical
plan yielding Z1 is preferred to the second
hypothetical plan yielding Z′1. Since the two
hypothetical plans are equivalent to plans A and
B, respectively, plan A must be preferred to plan
B. Conversely, if Z′1 is larger than Z1 then plan
B is preferred to plan A.
This process can be extended to a larger
number of objectives and plans, all of which may
be ranked by a common objective. For example,
assume that there are three objectives Zi1, Zi2, Zi3,
and n alternative plans i. A reference value Z*3 for
objective 3 can be chosen and a value zi1 estimated for each alternative plan i such that one is
indifferent between (Zil, Zi2, Z*3) and (Zil, Zi2, Zi3).
The second objective value remains the same as
in the actual alternative in each of the hypothetical alternatives. Thus the focus is on the tradeoff
between the values of objectives 1 and 3.
Assuming that each objective is to be maximized,
if Z*3 is selected so that Z*3 < Zi3, then zil will no
doubt be greater than Zi3. Conversely, if Z*3 > Zi3,
then zil will be less than Zi3.
Next, a new hypothetical plan containing a
reference value Z*2 and Z*3 can be created. The

Performance Criteria

focus now is on the tradeoff between the values of
objectives 1 and 2 given the same Z*3. A value zzi′1
must be selected such that the participants are
indifferent between (zi1, Zi2, Z*3) and (zzil, Z*2, Z*3).
Hence for all plans i, the participants are indifferent between two hypothetical plans and the
actual one. The last hypothetical plans differ only
by the value of the first objective. The plan that
has the largest value for objective 1 will be the
best plan. This was achieved by pair-wise comparisons only.
In the first step objective 2 remained constant
and only objectives 1 and 3 were compared to get

i i i
Zl ; Z2 ; Z3 :  ðzil ; Z2i ; Z3 Þ
ð9:18Þ
In the next step involving the hypothetical
plans just defined objective 3 remained constant
and only objectives 1 and 2 were compared to get



ðzi1 ; Z2i ; Z3 Þ  zzi1 ; Z2 ; Z3

ð9:19Þ

Hence

i i i



Z1 ; Z2 ; Z3  ðzi1 ; Z2i ; Z3 Þ  zzi1 ; Z2 ; Z3
ð9:20Þ
Having done this for all n plans, there are now
n hypothetical plans (zzil, Z*2, Z*3) that differ only
in the value of zzil. All n plans can be ranked just
based on the value of this single objective.
Each of these plan selection techniques
requires the prior identification of discrete alternative plans.

9.5.7 Goal Attainment
The goal attainment method combines some of
the advantages of both the weighting and constraint plan generation methods already discussed. If the participants are unable to specify
these weights, the analyst must select them and
then later change them on the basis of their
reactions to the generated plans.
The goal attainment method identifies the
plans that minimize the maximum weighted

9.5 Multicriteria Analyses

401

deviation of any objective value, Zj(X), from its
specified target, Tj. The problem is to
minimize D

ð9:21Þ

subject to
gi ðXÞ ¼ bi

i ¼ 1; 2; . . .; m



wj Tj Zj ðXÞ  D

j ¼ 1; 2; . . .; J

ð9:22Þ
ð9:23Þ

Constraints 9.22 contain the relationships among
the decision variables in the vector X. They
define the feasible region of decision variable
values.
This method of multicriteria analysis can
generate efficient or non-inferior plans by
adjusting the weights and targets. It is illustrated
for a two-objective problem in Fig. 9.16.
If the weights are equal, then the deviations
will be equal and the resulting feasible solution
will be the closest to the ideal but infeasible one.
Unless Tj ≥ Zj(X) some plans generated from a
goal attainment method may be inferior with
respect to the objectives being considered.

Fig. 9.16 The goal
attainment method of
generating points on the
efficiency frontier using
different values of the
weights w1 and w2 for fixed
objective target values T1
and T2

9.5.8 Goal Programming
Goal programming methods also require specified target values along with relative losses or
penalties associated with deviations from these
target values. The objective is to find the plan
that minimizes the sum of all such losses or
penalties. Assuming for this illustration that all
such losses can be expressed as functions of
deviations from target values, and again assuming each objective is to be maximized, the general goal programming problem is to
X



Lj Dj

ð9:24Þ

i ¼ 1; 2; . . .; m

ð9:25Þ

minimize

j

subject to:
gi ðXÞ ¼ bi
Tj Zj ðXÞ  Dj

j ¼ 1; 2; . . .; J

ð9:26Þ

If the loss functions are linear or piecewise linear
the model can be solved using linear programming methods. Again, the target and loss values
can be changed to generate alternative plans X.

402

9.5.9 Interactive Methods
Interactive methods allow participants in the
planning process to explore the range of possible
decisions without having to generate all of them,
especially those of little interest to the
participants.
One such iterative approach, called the step
method, requires, at each iteration preference,
information from the participants in the process.
This information identifies constraints on various
objective values. The weighting method is used
to get an initial solution on the efficiency frontier.
The weights, wj, are calculated based on the
relative range of values each objective j can
assume, and on whether or not the participants
have indicated satisfaction regarding a particular
objective value obtained from a previous solution. If they are satisfied with the value of, say,
an objective Zj(X), they must indicate how much
of that value they would be willing to give up to
obtain unspecified improvements in objectives
whose values they consider unsatisfactory. This
defines a lower bound on Zj(X). Then the weight
wj for that objective is set to 0, and the weights of
all remaining objectives are recalculated. The
problem is again solved. The process is repeated
until some best compromise plan is identified.
This step method guides the participants in the
planning and management process among
non-inferior alternatives toward the plan or
solution the participants consider best without
requiring an exhaustive generation of all
non-inferior alternatives. Even if the best compromise solution is not identified or agreed upon,
the method provides a way for participants to
learn what the tradeoffs are in the region of
solutions of interest to them. However, the participants must be willing to indicate how much of
some objective value can be reduced to obtain
some unknown improvement of other objective
values. This is not as easy as indicating how
much more is desired of any or all objectives
whose values are unsatisfactory.
To overcome this objection to the step
method, other interactive methods have been
developed. These begin with an obviously inferior solution. Based on a series of questions

9

Performance Criteria

concerning how much more important it is to
obtain various improvements of each objective,
the methods proceed from that inferior solution
to more improved solutions. The end result is
either a solution everyone agrees is best, or an
efficient one where no more improvements can
be made in one objective without decreasing the
value of another.
These iterative interactive approaches are
illustrated in Fig. 9.17. To work, they require the
participation of the participants in the planning
and management process.

9.5.10 Plan Simulation
and Evaluation
The methods outlined above provide a brief
introduction to some of the simpler approaches
available for plan identification and selection.
Details on these and other potentially useful
techniques can be found in many books, some of
which are devoted solely to this subject of multiobjective planning (Cohon 1978; Steuer 1986).
Most have been formulated in an optimization
framework. This section describes ways of
evaluating alternative water management plans
or policies based on the time series of performance criteria values derived from simulation
models.
Simulation models of water resource systems
yield time series of output variable values. These
values in turn impact multiple system performance criteria, each pertaining to a specific
interest and measured in its appropriate units.
A process for evaluating alternative water
resource management plans or policies based on
these simulation model results includes the following steps:
1. Identify system performance indicators that
are impacted by one or more hydrologic
attributes whose values will vary depending
on the management policy or plan being
simulated. For example, navigation benefits,
measured in monetary units, might depend on
water depths and velocities. Hydropower
production is affected by water heads and

9.5 Multicriteria Analyses
Fig. 9.17 Two interactive
iterative multiple criteria
approaches for identifying
the tradeoffs of interest and
possibly the best decision

Fig. 9.18 Performance
indicators expressed as
functions of simulated
hydrological attributes

403

404

9

Performance Criteria

Fig. 9.20 Combining multiple time series of values of a
specified performance indictor into a single time series of
values of that performance indicator

Fig. 9.19 Mapping a time series of hydrological attribute values into a corresponding time series of performance indicator values

discharges through the power plant. Water
quality might be expressed as the average or
maximum concentration of various potential
pollutants over a fixed time period at certain
locations, and will depend in part on the
flows. Ecological habitats may be impacted
by flows, water depths, water quality, flooding frequency or duration, and/or rates of
changes in these attributes.
2. Define the functional relationships between
these performance indicators and the

hydrologic attributes. Figure 9.18 illustrates
such functions. The units on each axis may
differ for each such function.
3. Simulate to obtain time-series of hydrologic
attribute values and map them into a time
series of performance indicator values using
the functional relationships defined in step 2.
This step is illustrated in Fig. 9.19.
4. Combine multiple time series values for the
same performance criterion, as applicable, as
shown in Fig. 9.20. This can be done using
maximum or minimum values, or arithmetic
or geometric means, as appropriate. For
example, flow velocities, depths, and algal
biomass concentrations may impact recreational boating. The three sets of time series
of recreational boating benefits or suitability
can be combined into one time series, and
statistics of this overall time series can be
compared to similar statistics of other

9.5 Multicriteria Analyses

performance indicators. This step gives the
modeler an opportunity to calibrate the
resulting
single
system
performance
indicator.
5. Develop and compare system performance
exceedance distributions, or divide the range
of performance values into color-coded ranges and display on maps or in scorecards, as
illustrated in Fig. 9.21.
The area under each exceedance curve is the
mean. Different exceedance functions will result
from different water management policies, as
illustrated in Fig. 9.22.
One can establish thresholds to identify discrete zones of performance indicator values and
assign a color to each zone. Measures of reliability, resilience and vulnerability can then be
calculated and displayed as well.
Scorecards can show the mean values of each
indicator for any set of sites. The best value for
each indicator can be colored green; the worst
value for each indicator can be colored red. The
water management alternative having the most
number of green boxes will stand out and will
probably be considered more seriously than the
alternative having the most number of red (worst
value) boxes.
This five-step process has been used in a study
of improved ways of managing lake levels and
flows in Lake Ontario and its discharges into the
St. Lawrence River in North America. Performance criteria were defined for domestic and
industrial water supplies, navigation depths,
shore bank erosion, recreational boating, hydropower production, flooding, water quality, ecological habitats. The performance measures for
each of these interests were identified and
expressed as functions of one or more hydrologic
variable values related to flow and lake level
management. Models designed to simulate
alternative lake level and flow management
policies were used to generate sets of time series
for each system performance criterion. These in
turn were combined, summarized and compared.
The same five-step process has been implemented in the Everglades restoration project in
southern Florida. The Everglades is a long very

405

wide and extremely flat ‘river of grass’ flowing
generally south into, eventually, the Atlantic
Ocean and Gulf of Mexico. This ecosystem
restoration project has involved numerous local,
state and federal agencies. The project impacts a
large population and agricultural industry that
want secure and reliable water supplies and flood
protection. Its current estimated cost over some
three decades is about $8 billon. Hence it
involves politics. But its goal is primarily
focused on restoring a unique ecosystem that is
increasingly degraded due to extensive alterations in its hydrology over the past half-century.
The motto of the Everglades restoration project in south Florida is ‘to get the water right.’
Those who manage the region’s water are
attempting to restore the ecosystem by restoring
the hydrologic regime, i.e., the flows, depths,
hydroperiods, and water qualities, throughout
this region to what they think existed some
60 years ago. The trick is to accomplish this and
still meet water supply, flood protection, and land
development needs of those who live in the
region. Clearly achieving a hydrologic condition
that existed before people began populating that
region in significant numbers will not be possible. Hence the question: what if water managers
are not able to ‘get the water right?’ What if they
can only get the water right on 90 % of the area,
or what if they can only get 90 % of the water
right on all the area? In either case what will be
its likely impact on the ecosystem? Are there
opportunities for changing hydrology to improve
ecology? Where?
To address questions such as these in the
Everglades, at least in a preliminary way prior to
when more detailed ecological models will
become available, this five-step approach outlined above is being applied. It is being used to
extend their simulated hydrological predictions
to produce relative values of ecological habitat
suitability indicators for selected indicator species, as illustrated in Fig. 9.8, and topographic
characteristics.
This five-step procedure does not find an
‘optimal’ water management policy. It can however contribute useful information to the political
debate that must take place in the search of that

406

Fig. 9.21 Ways of summarizing and displaying time
series performance indicator data involving exceedance
distributions, and color-coded maps and scorecards.
Color-coded map displays on computers can be dynamic,

9

Performance Criteria

showing changes over time. Green and red colored
scorecard entries indicate best and worst plan or strategy,
respectively, for associated performance indicator

9.5 Multicriteria Analyses

407

Fig. 9.22 Two
exceedance functions
showing decrease in mean
performance value if policy
2 is followed compared to
Policy 1

optimum. Each step of the approach can and
should include and involve the various stakeholders and publics in the basin. These individuals are sources of important inputs in this
evaluation process. Stakeholders who will be
influencing or making water management decisions need to understand just how this multiobjective evaluation process works if they are to
accept and benefit from its results. Stakeholder
involvement in this process can help lead to a
common understanding (or ‘shared vision’) of
how their system works and the tradeoffs that
exist among conflicting objectives. The extent to
which all stakeholders understand this evaluation
approach or procedure and how it is applied in
their basin will largely determine their ability to
participate effectively in the political process of
selecting the best water management policy or
practice.

9.6

Statistical Summaries
of Performance Criteria

There are numerous ways of summarizing time
series data in addition to the methods just mentioned above. Weighted arithmetic mean values
or geometric mean values are two ways of summarizing multiple time series data. The overall
mean itself generally provides too little information about a dynamic process. Multiple time series plots themselves are often hard to compare.

Another way to summarize and compare time
series data is to calculate and compare the variance of the data.
Consider a time series of T values Xt whose
mean is X. For example, suppose the time series
consisted of 8, 5, 4, 9, 2, 1, 1, 3, 6, and 7. The
mean of these 10 values is 4.6. The variance is
XT
h

t

ðXt XÞ2 =T ¼

i
ð84:6Þ2 þ ð54:6Þ2 þ . . . þ ð74:6Þ2 =10

¼ 7:44

ð9:27Þ
A plot of these values and their mean is shown in
Fig. 9.23.The mean and variance for the time
series shown in Fig. 9.23 however are the same
for its upside down image, as shown in Fig. 9.24.
They do not even depend on the order of the time
series data.
Consider these two sets of time series shown
again in Fig. 9.25, each having the same mean
and variance. Assume that any value equal or
less than the dashed line (just above 2) is considered unsatisfactory. This value is called a
threshold value, dividing the time series data into
satisfactory and unsatisfactory values.
It is clear from Fig. 9.25 that the impact of
these two time series could differ. The original
time series shown in a red line remained in an
unsatisfactory condition for a longer time than
did the time series shown in blue. However, the

408
Fig. 9.23 Plot of time
series data having a mean
value of 4.6 and a variance
of 7.44

Fig. 9.24 A plot of two
different time series having
the same mean and
variance

Fig. 9.25 Threshold value
distinguishing values considered satisfactory, and
those
considered
unsatisfactory

9

Performance Criteria

9.6 Statistical Summaries of Performance Criteria

maximum extent of failure when it, the red series,
failed was less than the blue time series. These
characteristics can be captured by the measures
of reliability, resilience and vulnerability
(Hashimoto et al. 1982).

9.6.1 Reliability
The reliability of any time series can be defined
as the number of data in a satisfactory state
divided by the total number of data in the time
series. Assuming satisfactory values in the time
series Xt containing n values are those equal to or
greater than some threshold XT, then
Reliabilty[X ¼ ½number of time periods t
such that Xt  X T =n

ð9:28Þ

The reliability of the red time series is 0.7. It
failed three times in 10. The reliability of the blue
time series is also 0.7, failing three times in 10.
Is a more reliable system better than a less
reliable system? Not necessarily. Reliability
measures tell one nothing about how quickly a
system recovers and returns to a satisfactory
value, nor does it indicate how bad an unsatisfactory value might be should one occur. It may
well be that a system that fails relatively often,
but by insignificant amounts and for short durations will be preferred to one whose reliability is
much higher, but when a failure does occur, it is
likely to be much more severe. Resilience and
vulnerability measures can provide measures of
these system characteristics.

9.6.2 Resilience
Resilience can be expressed as the probability
that if in an unsatisfactory state, the next state
will be satisfactory. It is the probability of having
a satisfactory value in time period t + 1 given an
unsatisfactory value in any time period t. It can
be calculated as

409

Resilience[X ¼ ½number of times a satisfactory
value follows an unsatisfactory value=½number
of times an unsatisfactory value occurred
ð9:29Þ
Resilience is not defined if no unsatisfactory
values occur in the time series. For the time
series shown in red, the resilience is 1/3, again
assuming the value of 2 or less is considered a
failure. For the time series shown in blue the
resilience is 2/2 = 1. We cannot judge the resilience of the blue time series based on the last
failure in period 10 because we do not have an
observation in period 11.

9.6.3 Vulnerability
Vulnerability is a measure of the extent of the
differences between the threshold value and
the unsatisfactory time series values. Clearly this is
a probabilistic measure. Some use expected values,
some use maximum observed values, and others
may assign a probability of exceedance to their
vulnerability measures. Assuming an expected
value measure of vulnerability is to be used




Vulnerability½X ¼ sum of positive values of X T  Xt
=½number of times an unsatisfactory value occurred]

ð9:30Þ
The expected vulnerability of the original red
time series is [(2 − 2) + (2 − 1) + (2 − 1)]/
3 = 0.67. The expected vulnerability of the time
series shown by the blue line in Fig. 9.25 is
[(2 − 1.2) + (2 − 0.2)]/2 = 1.3.
So, while in this example the reliability of red
time series equals that of the blue time series, the
resilience of the blue time series is better than
that of the red time series. Yet the expected
vulnerability of the red time series is less than
that of the rotated blue time series. This shows
the typical tradeoffs one can observe using these
three measures of system performance.

410

9.7

9

Conclusions

Many theoretical and practical approaches have
been proposed in the literature for identifying
and quantifying objectives and for considering
multiple criteria or objectives in water resources
planning. The discussion and techniques presented in this chapter serve merely as an introduction to this subject. These tools, including
their modifications and extensions, are designed
to provide information that can be of value to the
planning and decision-making process.
Water resource systems planners and managers and the numerous other participants typically involved in decision-making face a
challenge when they are required to select one of
many alternatives, each characterized by different
values among multiple performance criteria. It
requires a balancing of the goals and values of
the various individuals and groups concerned
with the project. There is virtually no way in
which the plan selection step can be a normative
process or procedure; there can be no standard
set of criteria or methods which will identify the
preferred project. At best, an iterative procedure
in which those using tools similar to those
described in this chapter together with all the
interested stakeholders may reach some shared
vision of what is best to do, at least until conditions change or new knowledge or new goals
or new requirements emerge. This may be the
only way to identify a plan that is politically as
well as technically, socially, financially, and
institutionally feasible.
To many participants in the planning process
some of these approaches for objective quantification and multiobjective planning may seem
theoretical or academic. Many may be reluctant to
learn quantitative policy analysis techniques or to
spend time answering seemingly irrelevant questions that might lead eventually to a “compromise”
plan. Reluctance to engage in quantification of
tradeoffs among particular objectives of alternative plans, and by implication tradeoffs among the
interests of multiple stakeholder groups, may stem
from the support decision-makers desire from all
of these conflicting interest groups. In such situations it is obviously to their advantage not to be

Performance Criteria

too explicit in quantifying political values. They
might prefer that the “analyst” make these tradeoffs and just not discuss them. (We writers have
participated in such situations.) Planners, engineers, or analysts are often very willing to make
these tradeoffs because they pertain to subject
areas in which they often consider themselves
expert. However when political tradeoffs are at
issue, no one is an expert. No one has the ‘optimal’
answer, but professionals should be engaged in
and informing and facilitating the process of
coming to an acceptable, and often compromise,
decision.
Through further development and use of
practical analytical multiobjective planning
techniques, analysts can begin to interact with all
participants in the planning and management
process and can enlighten any who would argue
that water resources policy evaluation and analyses should not be political. Analysts, managers
and planners have to work in a political environment. They need to understand the process of
decision-making, what information is most useful to that process, and how it can best be presented. Knowledge of these facts in a particular
planning situation might dictate substantially the
particular approach to objective identification
and quantification and to plan selection that is
most appropriate.
The method deemed most appropriate for a
particular situation will depend not only on the
physical scale of the situation itself but also on the
decision-makers, the decision-making process,
and the responsibilities accepted by the analysts,
the participants, and the decision-makers.
Finally, a reminder that the decisions being
made at the current time are only those in a
sequence of decisions that will continue to be
made on into the future. No one can predict with
certainty what future generations will consider as
being important or what they will want to do, but
spending some time trying to guess is not an idle
exercise. It pays to plan ahead, as best one can,
and ask ourselves if the decisions being considered today will be those we think our descendants would have wanted us to make. This kind
of thinking gets us into issues of adaptive management and sustainability (ASCE 1999).

References

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stability definitions for strategic analysis of generic
water resources conflicts. Water Resources Management, 25(8), 1949–1977.
Steuer, R. E. (1986). Multiple criteria optimization: Theory,
computation, and application. New York: Wiley.

Additional References
(Further Reading)
Chen, H. W., & Chang, N.-B. (1998). Water pollution
control in the river basin by fuzzy genetic
algorithm-based multiobjective programming modeling. Water Science and Technology, 37(8), 55–63.
Cieniawski, S. E., Eheart, J. W., & Ranjithan, S. (1995).
Using genetic algorithms to solve a multiobjective
groundwater monitoring problem. Water Resources
Research, 31(2), 399–409.
Eschenauer, H., Koski, J., & Osyczka, A. (1990).
Multicriteria
design
optimization.
Berlin:
Springer-Verlag.
Gupta, H. V., Sorroshian, S., & Yapo, P. O. (1998).
Towards improved calibration of hydrologic models:
Multiple and non-commensurable measures of information. Water Resources Research, 34(4), 751–763.
Haimes, Y. Y., Hall, W. A., & Freedman, H. T. (1975).
Multiobjective optimization in water resources systems. Amsterdam: Elsevier Scientific.
Halhal, D., Walters, G. A., Ouazar, D., & Savic, D. A.
(1997). Multi-objective improvement of water

411
distribution systems using a structure messy genetic
algorithm approach. Journal of Water Resources
Planning and Management ASCE, 123(3), 137–146.
Hashimoto, T., Stedinger, J. R., & Loucks, D. P. (1982a).
Reliability resiliency and vulnerability criteria for
water resource system performance evaluation. Water
Resources Research, 18(1), 14–20.
Hashimoto, T., Loucks, D. P., & Stedinger, J. R. (1982b).
Robustness of water resource systems. Water
Resources Research, 18(1), 21–26.
Keeney, R. L., & Raiffa, H. (1976). Decisions with
multiple objectives: Preferences and value tradeoffs.
New York: Wiley.
Loucks, D. P., Stedinger, J. R., & Haith, D. A. (1981).
Water resource systems planning and analysis. Englewood Cliffs, NJ: Prentice Hall.
Maas, A., et al. (1962). Design of water resource systems.
Cambridge: Harvard University Press.
Reed, P. M., Minsker, B. S., & Goldberg, D. E. (2001).
A multiobjective approach to cost effective long-term
groundwater monitoring using an elitist nondominated
sorted genetic algorithm with historical data. Journal
of Hydroinformatics, 3(2), 71–89. April.
Ritzel, B. J., Eheart, J. W., & Ranjithan, S. (1994). Using
genetic algorithms to solve a multiple objective
groundwater pollution containment problem. Water
Resources Research, 30(5), 1589–1603. May.
Sawaragi, Y., Nakayama, H., & Tanino, T. (1985).
Theory of multiobjective optimization. Orlando, Florida: Academic Press Inc.
Statnikov, R. B., & Matusov, J. B. (1995). Multicriteria
optimization and engineering. New York: Chapman
and Hall.
Yeh, C.-H., & Labadie, J. W. (1997). Multiobjective
watershed-level planning of storm-water detention
systems. Journal of Water Resources Planning and
Management, 123(6), 336–343. November/December.
Zeleny, M. (1982). Multiple criteria decision making.
New York: McGraw-Hill.

Exercises
9:1 Distinguish between multiple purposes and
multiple objectives and give some examples
of complementary and conflicting purposes
and objectives of water resources projects.
9:2 Assume that farmers’ demand for water q is
a linear function a − b(p) of the price p,
where a, b > 0. Calculate the farmers’
willingness to pay for a quantity of water
q. If the cost of delivering a quantity of
water q is cq, c > 0, how much water
should a public agency supply to maximize
willingness to pay minus total cost? If the

412

9

local water district is owned and operated
by a private firm whose objective is to
maximize profit, how much water would
they supply and how much would they
earn? The farmers’ consumer surplus is
their willingness to pay minus what they
must pay for the resource. Compare the
farmers’ consumer surplus in two cases. Do
the farmers loose more than the private firm
gains by moving from the social optimum
to the point that maximizes the firm’s
profit? Illustrate these relationships with a
graph showing the demand curve and the
unit cost c of water. Which areas on the
graph represent the firm’s profits and the
farmers’ consumer surplus?
9:3 Consider the water allocation problem used
in the earlier chapters of this book. The
returns, Bi(Xi) from allocating Xi amount of
water to each of three uses i are as follows,
along with the optimal allocations from the
point of view of each use.
B1 ðX1 Þ ¼ 6X1  X12

!
and

B2 ðX2 Þ ¼ 7X2  1:5X22

!
and

X2opt ¼ 7=3



¼ B2 X2opt ¼ 147=18
Bmax
2

B3 ðX3 Þ ¼ 8X3  0:5X32

!

X3opt ¼ 8

and

year. The maximum irrigation demand
(capacity) is 4 × 106 m3. Let X1 be the
allocation of water to irrigation and X2 the
allocation for downstream flow augmentation. Assume that there are two objectives,
expressed as
Z1 ¼ 4X1 X2
Z2 ¼ 2X1 þ 6X2

(a) Write the multiobjective planning
model using a weighing approach and a
constraint approach.
(b) Define the efficient frontier. This
requires a plot of the feasible combinations of X1 and X2.
(c) Assume that various values are
assigned to a weight W1 for Z1 whereas
weight W2 for Z2 is constant and equal
to 1, verify the following solutions to
the weighing model.

X1opt ¼ 3




¼ B1 X1opt ¼ 9
Bmax
1




¼ B3 X3opt ¼ 32
Bmax
3

Consider this a multiobjective problem.
Instead of finding the best overall allocation that
maximizes the total return assume the objectives
are to maximize the returns from each user.
Show how the weighting, constraint, goal
attainment, and goal programming methods can
be used to identify the tradeoffs among each of
the three objectives for any limiting total amount
of water, for example, 6.
9:4 Under what circumstances will the weighting and constraint methods fail to identify
efficient solutions?
9:5 A reservoir is planned for irrigation and low
flow augmentation for water quality control.
A storage volume of 6 × 106 m3 will be
available for those two conflicting uses each

Performance Criteria

W1

X1

X2

Z1

Z2

>6

4

0

16

−8

6

4

0 to 12

16 to 14

−8 to 4

<6
to >1.6

4

2

14

4

1.6

4 to
0

2 to 6

14 to −6

4 to 36

<1.6

0

6

−6

36

9:6 Show that the following benefit, loss, and
cost functions can be included in a linear
optimization problem for finding the active
storage volume target Ts, annual release
target TR and the actual storage releases Rt
in each within-year period t, and the reservoir capacity K. The objective is to maximize annual net benefits from the
construction and operation of the reservoir.
Assume that the inflows are known in each
of 12 within-year periods t. Note that the
loss function associated with reservoir
recreation is independent of the value of Ts,

Exercises

413

variables used that are not defined below.
Let δtTR be the known release target in
period t.

unlike the loss function associated with
reservoir releases. Structure the complete
linear programming model. Define all

St = initial storage
volume
K = reservoir

Annual
recreation
benefits

1
3
20

Recreation
Loss in
periods t=

8

-10

6 through 9
(0 otherwise)

5

TS

-6
3

10
Storage target T S

5
7
Storage Volume St

LRBF
6

SRBF

3

Release
benefit
in period t
0

10

8

10

15

-20

δ tT R

Reservoir
release Rt

Annual
Cost of
reservoir

(if K > 0) 20
(if K = 0) 0

8
3
5

20

Reservoir capacity K

30

2

414

9

9:7 For the river basin shown, potential reservoirs exist at sites i = 1, 2, and 4 and a
diversion can be constructed between sites 1
and 2. The cost Ci(Ki) of each reservoir i is
a function of its active storage capacity Ki.
The cost of the diversion canal is Ci(Qi)
where Q is the flow capacity of the canal.
The cost of diverting a flow Qijt from site
i to site j is Cij(Qijt). The two users at sites 3
and 5 have known target allocations (demands) Tit in each period t. The return flow
from use 3 is 40 % of that allocated to use
3. Construct a model for finding the least
cost of meeting various percentages of the
target demands. Assume that the natural
streamflows Qit at each site i in each period
t, are known.
3 T3t
1
4

2

5 T5t

9:8 Suppose that there exist two polluters, A
and B, who can provide additional treatment, XA and XB, at a cost of CA(XA) and
CB(XB), respectively. Let WA and WB be the
waste produced at sites A and B, and
WA(1 − XA) and WB(1 − XB) be the resulting waste discharges at site A and B. These
discharges must be no greater than the
effluent standards Emax
and Emax
A
B . The
resulting
pollution
concentration
aAj(WA(1 − XA)) + aBj(WB(1 − XB)) + qj at
various sites j must not exceed the stream
standards Smax
. Assume that total cost and
j
cost inequity [i.e., CA(XA) + CB(XB) and|
CA(XA) − CB(XB)] are management objectives to be determined
(a) Discuss how you would model this
multiobjective problem using the

Performance Criteria

weighting and constraint (or target)
approaches.
(b) Discuss how you would use the model
to identify efficient, non-inferior
(Pareto-optimal) solutions.
(c) Effluent standards at sites A and B and
ambient stream standards at sites j could
be replaced by other planning objectives
(e.g., the minimization of waste discharged into the stream). What would
these objectives be, and how could they
be included in the multiobjective model?
9:9 (a) What conditions must apply if the goal
attainment method is to produce only
non-inferior alternatives for each
assumed target Tk and weight wk?
(b) Convert the goal programming objective
deviation components wi (zi  zi ðxÞ) to
a form suitable for solution by linear
programming.
9:10 Water quality objectives are sometimes
difficult to quantify. Various attempts
have been made to include the many
aspects of water quality in single water
quality indices. One such index was
proposed by Dinius (Social Accounting
Systems for Evaluating Water Resources, Water Resources Research, Vol. 8,
1972. pp. 1159–1177). Water quality, Q,
measured in percent is given by
Q¼

w1 Q1 þ w2 Q2 þ . . . þ wn Qn
w1 þ w2 þ . . . þ wn

where Qi is the ith quality constituent
(dissolved oxygen, chlorides, etc.) and
wi is the weight or relative importance of
the ith quality constituent. Write a
critique on the use of such an index in
multiobjective water resources planning.
9:11 Let objective Z1(X) = 5X1−2X2 and
objective Z2(X) = −X1 + 4 X2. Both are
to be maximized. Assume that the constraints on variables X1 and X2 are:

Exercises

1.
2.
3.
4.
5.

−X1 + X2 ≤ 3
X1 ≤ 6
X1 + X2 ≤ 8
X2 ≤ 4
X1, X2 ≥ 0

(a) Graph the Pareto-optimal or non-inferior
solutions in decision space.
(b) Graph the efficient combination of Z1 and Z2
in objective space.
(c) Reformulate the problem to illustrate the
weighting method for defining all efficient
solutions of part (a) and illustrate this method
in decision and objective space.
(d) Reformulate the problem to illustrate the
constraint method of defining all efficient
solutions of part (a) and illustrate this method
in decision and objective space.
(e) Solve for the compromise set of solutions using compromise programming as
defined by


a


a 1=a
Minimize w1 Z1  Z1 þ w2 Z2  Z2

Open Access This chapter is distributed under the terms
of the Creative Commons Attribution-NonCommercial
4.0 International License (http://creativecommons.org/
licenses/by-nc/4.0/), which permits any noncommercial
use, duplication, adaptation, distribution and reproduction
in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a
link to the Creative Commons license and indicate if
changes were made.

415

where Zi* represents the best value of objective
i with all weights w equal to 1 and α equal to 1, 2,
and ∞.
9:12 Illustrate the procedure for selecting
among three plans, each having three
objectives, using indifference analysis. Let
Zji represent the value of objective i for
plan j. The values of each objective for
each plan are given below. Assume that
each objective is to be maximized. Assume
that an identical indifference function for
all trade-offs between pairs of objectives,
namely one that implies you are willing to
give up twice as many units of your higher
(larger) objective value to gain one unit of
your lower (smaller) objective value. [For
example, you would be indifferent to two
plans having as their three objective values
(30, 5, 10) and (20, 5, 15).] Rank these
three plans in order of preference.
Plan 1 : ð5; 25; 15Þ;
Plan 2 : ð10; 20; 10Þ;
Plan 3 : ð15; 10; 15Þ

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reproduce the material.

Water Quality Modeling
and Prediction

The most fundamental human needs for water are
for drinking, cooking, and personal hygiene. The
quality of the water used to meet these needs
must pose no risk to human health. The quality of
the water in nature also impacts the condition of
all living organisms found in aquatic ecosystems
that we depend upon for our own wellbeing. At
the same time watersheds and their water bodies
serve as convenient sinks for domestic, industrial, and agricultural wastes. Runoff from agricultural and urban lands containing excess
nutrients, oils, and solid wastes together with
direct point source discharges of wastewaters
into water bodies degrades the quality of those
water bodies. Water resources management
involves the monitoring and management of
water quality as much as the monitoring and
management of water quantity. Various models
can assist in predicting the water quality impacts
of alternative land and water management policies and practices. This chapter introduces some
approaches to water quality modeling, leaving
descriptions of more advanced methods to textbooks devoted solely to this subject.

10.1

Introduction

Water quality management is a critical component of overall integrated water resources management. Most users of water depend on
adequate levels of water quality. When these
levels are not met, water users must then either
pay for water treatment or incur increased risks

10

of using lower quality water. As populations and
economies grow, more wastewater pollutants are
generated. Many of these are discharged into
surface and ground water bodies. Increasingly
the major efforts and costs involved in water
management are aimed at water quality protection and management. Conflicts among various
users of water are increasingly over issues
involving water quality.
Natural water bodies are able to serve many
uses. One of them is the transport and assimilation of many waterborne wastes. As natural water
bodies transport and assimilate wastes, their
quality changes. If the quality of water drops to
the extent that other uses are adversely impacted,
the assimilative capacities of those water bodies
have been exceeded with respect to those
impacted uses. Water management measures are
actions taken to ensure that the total pollutant
loads discharged into receiving water bodies do
not exceed the waste assimilative capacity of
those water bodies and that the quality meets the
quality standards set for those waters.
What uses depend on water quality? Almost
all one can identify. As everyone knows, all living organisms require water of sufficient quantity
and quality to survive. Different aquatic species
can tolerate different levels of pollutant concentrations that impact water quality. In much of the
developed world it is no longer ‘‘safe’’ to drink
natural surface or ground waters. Treatment is
usually required before these waters are safe for
humans to drink. Treatment is not a practical
option for improving the quality of water found in

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_10

417

418

nature yet this is the water that impacts the health
of fish and shellfish and other organisms in natural
aquatic ecosystems. Hence the focus in practice is
on the use of wastewater treatment facilities to
improve the quality of effluents being discharged
into natural water bodies.
Standards specifying minimum acceptable
levels of quality are commonly set for most
ambient waters. Various uses may have their own
quality requirements as well. Irrigation water
must not be too saline nor contain toxic substances that can be absorbed by the plants or
destroy the microorganisms in the soil. Water
quality standards for industry can be very
demanding, depending on course of the requirements of particular industrial processes.
Domestic wasteloads can contain high concentrations of bacteria, viruses, and other
organisms that impact human health. High
organic loadings can reduce dissolved oxygen
(DO) to levels that can kill parts of the aquatic
ecosystem and cause obnoxious odors. Nutrient
loadings from both urban and agricultural land
runoff can cause excessive algae growth that in
turn may degrade the water aesthetically, inhibit
boating and swimming, and upon death cause
low DO levels. Toxic heavy metals and other
micropollutants can accumulate in the bodies of
aquatic organisms, including fish, making them
unfit for human consumption even if they
themselves survive.
Pollutant discharges originate from point to
non-point sources. A common approach to controlling point source discharges, such as from
stormwater outfalls, municipal wastewater treatment plants or industries, is to impose standards
specifying maximum allowable pollutant loads or
concentrations in their effluents. This is often
done in ways that are not economically efficient
or even environmentally effective. Effluent standards typically do not take into account the
particular assimilative capacities of the receiving
water body. Nevertheless they are relatively easy
to monitor and control.
Non-point sources such as agricultural runoff
or atmospheric deposition are not as easily controlled and hence it is difficult to apply effluent

10

Water Quality Modeling and Prediction

standards to non-point source pollutants. Pollutant loadings from non-point sources can be
much higher than point source loadings. Management of non-point water quality impacts
requires a more ambient-focused water quality
management program.
The goal of an ambient water quality management program is to establish appropriate
standards for water quality in water bodies
receiving pollutant loads and then to ensure that
these standards are met. Realistic standard setting
takes into account the basin’s hydrologic, ecological, and land use conditions, the potential
uses of the receiving water bodies, and the
institutional capacity to set and enforce water
quality standards.
Ambient-based water quality prediction and
management involves considerable uncertainty.
No one can predict what pollutant loadings will
be in the future, especially from area-wide
non-point sources. In addition to uncertainties
inherent in measuring water quality, there are
uncertainties in models used to predict the
effectiveness of actions taken to meet water
quality standards. The models available to help
managers predict water quality impacts are relatively simple compared to the complexities of
actual water systems. If water quality models are
being used to inform those setting standards and
permissible waste loadings, these limitations and
uncertainties should be understood and
addressed.

10.2

Establishing Ambient Water
Quality Standards

A first step in setting water quality standards for
a water body is to identify the intended uses of
that water body, whether a lake, a section of a
stream, or areas of an estuary. The most restrictive (in terms of water quality) of the specific
desired uses of a water body is termed a designated use. Barriers to achieving the designated
use are the presence of pollutants or hydrologic
and geomorphic changes that impact the quality
of the water body.

10.2

Establishing Ambient Water Quality Standards

The designated use dictates the appropriate
type of water quality standard. For example, the
standards of a water body whose designated use
involves human contact recreation should protect
humans from exposure to microbial pathogens
while swimming, wading, or boating. Other uses
might require standards to protect humans and
aquatic life including fish, shellfish, and other
wildlife from consuming harmful substances.
Standards set upstream may impact the uses of
water downstream. For example, the water
quality of small headwater streams may affect the
ability of a downstream area to achieve a particular designated use such as being “fishable” or
“swimmable.” In this case, the designated use for
the smaller upstream water body may be defined
in terms of the achievement of the designated use
of the larger downstream water body.
In many areas human activities have sufficiently altered the landscape and aquatic
ecosystems to the point where they cannot be
restored to their pre-disturbance condition. For
example, a reproducing trout fishery in downtown Paris, Philadelphia, Phnom Penh, or Prague
may be desired by some, but may not be attainable because of the development history of the
areas or the altered hydrologic regimes of the
rivers flowing through them. Similarly, health
considerations would preclude designating an
area for shellfish harvesting near the outfall of a
Fig. 10.1 Factors
considered when
determining designated use
and associated water
quality standards

419

sewage treatment plant. Ambient water quality
standards must be realistic.
Decisions regarding the appropriate use for
water bodies can be informed by the use of water
quality prediction models. However, the final
standard selection should reflect a social consensus made in consideration of the current
condition of the watershed, its pre-disturbance
condition, the advantages derived from a certain
designated use, and the costs of achieving the
designated use.

10.2.1

Water Use Criteria

The designated use is a qualitative description of
a desired condition of a water body. A criterion is
a measurable indicator surrogate for use attainment. The criterion may be positioned at any
point in the causal chain of boxes shown in
Fig. 10.1.
Box 1 of Fig. 10.1 contains information about
the pollutant discharges, e.g., from a treatment
plant or in runoff (e.g., biological oxygen
demand, ammonia (NH3), pathogens, and suspended sediments). Effluent standards specifying
maximum permissible loadings may apply to
these pollutant loadings. Criteria in Boxes 2 and
3 are possible measures of ambient water quality
conditions. Box 2 includes measures of a water

420

quality parameter such as DO, pH, nitrogen
concentration, suspended sediment, or temperature. Criteria closer to the designated use (e.g.,
Box 3) include more combined or comprehensive
measures of the biological community as a
whole, such as the condition of the algal community (chlorophyll a) or a measure of contaminant concentration in fish tissue. Box 4
represents criteria that are associated with sources of pollution other than pollutants. These
criteria might include measures such as flow
timing and pattern (a hydrologic criterion),
abundance of nonindigenous taxa, some quantification of channel modification (e.g., decrease
in sinuosity), etc. (NRC 2001).
The more precise the statement of the designated use, the more accurate the criterion will be
as an indicator of that use. For example, the
criterion of fecal coliform count may be a suitable criterion for water contact recreation. The
maximum allowable count itself may differ
among water bodies that have water contact as
their designated use, however.
Surrogate indicators are often selected for use
as criteria because they are easy to measure and
in some cases are politically appealing. Although
a surrogate indicator may have appealing attributes, its usefulness can be limited unless it can
be logically related to a designated use.
As with setting designated uses, the connections among water bodies and segments must be
considered when determining criteria. For
example, where a segment of a water body is
designated as a mixing zone for a pollutant discharge, the criterion adopted should assure that
the mixing zone use will not adversely affect the
surrounding water body uses. Similarly, as previously discussed, the desired condition of a
small headwater stream may need to be chosen as
it relates to other water bodies downstream.
Thus, an ambient nutrient criterion may be set in
a small headwater stream to ensure a designated
use downstream, even if there are no local
adverse impacts resulting from the nutrients in
the small headwater stream. Conversely, a high
fecal coliform criterion may be permitted
upstream of a recreational area if the fecal load
dissipates before the flow reaches that area.

10

10.3

Water Quality Modeling and Prediction

Water Quality Model Use

Monitoring data are the preferred form of information for identifying impaired waters. Model
predictions might be used in addition to or
instead of monitoring data for several reasons
1. Modeling could be feasible in some situations
where monitoring is not.
2. Integrated monitoring and modeling systems
could provide better information than monitoring or modeling alone for the same total
cost. For example, regression analyses that
correlate pollutant concentration with some
more easily measurable factor (e.g., streamflow) could be used to extend monitoring data
for preliminary planning purposes. Models
can also be used to determine preliminary
probability distributions of impairment that
can help direct monitoring efforts and reduce
the quantity of monitoring data needed for
making listing decisions at a given level of
reliability (see Chaps. 7 and 9).
3. Modeling can be used to assess (predict)
future water quality situations resulting from
different management strategies. For example, assessing the improvement in water
quality after a new wastewater treatment plant
begins operating, or the effect of increased
industrial growth and effluent discharges.
A simple, but useful, modeling approach that
may be used in the absence of monitoring data is
“dilution calculations.” In this approach the rate
of pollutant loading from point sources in a water
body is divided by the stream flow to give a set
of estimated pollutant concentrations that may be
compared to the standard. Simple dilution calculations assume conservative movement of
pollutants. Thus, the use of dilution calculations
will tend to be conservative and lead to higher
than actual concentrations for decaying pollutants. Of course one could include a best estimate
of the effects of decay processes in the dilution
model.
Combined runoff and water quality prediction
models link stressors (sources of pollutants and
pollution) to responses. Stressors include human

10.3

Water Quality Model Use

activities likely to cause impairment, such as the
presence of impervious surfaces in a watershed,
cultivation of fields close to the stream,
over-irrigation of crops with resulting polluted
return flows, the discharge of domestic and
industrial effluents into water bodies, installing
dams and other channelization works, introduction of nonindigenous taxa, and over-harvesting
of fishes. Indirect effects of humans include land
cover changes that alter the rates of delivery of
water, pollutants, and sediment to water bodies.
Direct and indirect environmental effects of
human activities can include
•
•
•
•

alterations in physical habitat,
modifications in the seasonal flow of water,
changes in the food base of the system,
changes in interactions within the stream
biota, and
• release of contaminants (conventional pollutants) (Karr 1990; NRC 1992, 2001).

Ideally, models designed to manage water
quality should consider all five types of alternative management measures. The broad-based
approach that considers these five features provides a more integrative approach to reduce the
cause or causes of degradation (NRC 1992).
Models that relate stressors to responses can
be of varying levels of complexity. Sometimes,
models are simple qualitative conceptual representations of the relationships among important
variables and indicators of those variables. More
quantitative models can be used to make predictions about the assimilative capacity of a
water body, the movement of a pollutant from
various point and non-point sources through a
watershed, or the effectiveness of certain best
management practices.

10.3.1

Model Selection Criteria

There was a time when if one needed a water
quality model, they had to build it. Today there
exist a wide range of water quality models for
various types of water bodies, and for various
contaminants, and hence it makes little sense to

421

build another one if an existing model will suffice. This section discusses criteria that can be
used to select a particular model.
Water quality predictive models can include
both mathematical expressions and expert scientific judgment. They may be process-based
(mechanistic) models or data-based (statistical)
models. Quality models used for planning and
management should link management options to
meaningful response variables (e.g., pollutant
sources and water quality standard parameters).
They should incorporate the entire “chain” from
stressors to responses. Process-based models
should be consistent with scientific theory.
Model prediction uncertainty should be reported.
This provides decision-makers with estimates of
the risks of alternative options. To do this
requires prediction error estimates (Chap. 6).
Water quality management models should be
appropriate to the complexity of the situation and
to the available data. Simple water quality
problems can be addressed with simple models.
Complex water quality problems may or may not
require the use of more complex models. Models
requiring large amounts of monitoring data
should not be used in situations where such data
are unavailable. Models should be flexible
enough to allow updates and improvements as
appropriate based on new research and monitoring data.
Stakeholders need to accept the models proposed for use in any water quality management
study. Given the increasing role of stakeholders
in water management decision processes, they
need to understand and accept the models being
used, at least to the extent they wish to. Finally,
the cost of maintaining and updating the model
during its use must be acceptable.
Although predictions are typically made using
mathematical models, there are certainly situations where expert judgment can be just as good.
Reliance on professional judgment and simpler
models is often acceptable, especially when
limited data exist.
Highly detailed models require more time and
are more expensive to develop and apply.
Complex modeling studies should be undertaken
only if warranted by the complexity of the

422

10

management problem. More complex modeling
will not necessarily assure that uncertainty is
reduced, and in fact added complexity can
compound problems of uncertainty analyses
(Chap. 8).
Placing a priority on process description
usually leads to complex mechanistic model development and use over simpler mechanistic or
empirical models. In some cases this may result
in unnecessarily costly analyses for effective
decision-making. In addition, physical, chemical,
and biological processes in terrestrial and aquatic
environments are far too complex to be fully
represented in even the most detailed models. For
water quality management, the primary purpose
of modeling should be to support decisionmaking. The inability to completely describe all
relevant processes can be accounted for by
quantifying the uncertainty in the model
predictions.

10.3.2

Model Chains

Many water quality management analyses
require the use of a sequence of models, one
feeding data into another. For example, consider
the sequence or chain of models required for the
prediction of fish and shellfish survival as a
function of nutrient loadings into an estuary. Of
interest to the stakeholders are the conditions of
the fish and shellfish. One way to maintain
healthy fish and shellfish stocks is to maintain
sufficient levels of oxygen in the estuary. The
way to do this is to control algae blooms. This in
turn requires limiting the nutrient loadings to the
estuary that can promote algae growth and
blooms, and subsequent DO deficits. The modeling challenge is to link nutrient loading to fish
and shellfish survival.
The negative effects of excessive nutrients
(e.g., nitrogen) in an estuary are shown in
Fig. 10.2. Nutrients stimulate the growth of
algae. Algae die and accumulate on the bottom
where bacteria consume them. Under calm wind
conditions density stratification occurs. Oxygen
is depleted in the bottom water. Fish and shellfish

Water Quality Modeling and Prediction

may die or become weakened and more vulnerable to disease.
A model consisting of a sequence of conditional probabilities can be defined to predict the
probability of shellfish and fish abundance based
on upstream nutrient loadings causing problems
with fish and shellfish populations into the estuary. These conditional probabilities can be
judgmental, mechanistic, and/or statistical. Each
conditional probability can be a separate submodel. Assuming each submodel can identify a
conditional probability distribution, the probability Pr{C|N} of a specified amount of carbon,
C, given some specified loading of a nutrient, say
nitrogen, N, equals the probability Pr{C|A} of
that given amount of carbon given a concentration of algae biomass, A, times the probability Pr
{A|N, R} of that concentration of algae biomass
given the nitrogen loading, N, and the river flow,
R, times the probability Pr{R} of the river
flow, R.
PrfCjN g ¼ PrfCjAgPrfAjN; RgPrfRg ð10:1Þ
An empirical process-based model of the type
to be presented later in this chapter could be used
to predict the concentration of algae and the
chlorophyll violations based on the river flow
and nitrogen loadings. Similarly to predict the
production of carbon based on algae biomass.
A seasonal statistical regression model might be
used to predict the likelihood of algae blooms
based on algal biomass. A cross system comparison may be made to predict sediment oxygen
demand. A relatively simple hydraulic model
could be used to predict the duration of stratification and the frequency of hypoxia given both
the stratification duration and sediment oxygen
demand. Expert judgment and fish survival
models could be used to predict the shellfish
abundance and fishkill and fish health
probabilities.
The biological endpoints “shell-fish survival”
and “number of fishkills,” are meaningful indicators to stakeholders and can easily be related to
designated water body use. Models and even
conditional probabilities assigned to each link of

10.3

Water Quality Model Use

423

Fig. 10.2 The negative impacts of excessive nutrients in an estuary (NRC 2001)

the network in Fig. 10.3 can reflect a combination of simple mechanisms, statistical (regression) fitting, and expert judgment.
Advances in mechanistic modeling of aquatic
ecosystems have resulted in our ability to include
greater process (especially trophic) detail and
complexity, as well as to perform dynamic simulations. Still, mechanistic ecosystem models
have not advanced to the point of being able to
predict community structure or biotic integrity.
In this chapter, only some of the simpler mechanistic models will be introduced. More detail
can be found in books solely devoted to water
quality modeling (Chapra and Reckhow 1983;
Chapra 1997; McCutcheon 1989; Thomann and
Mueller 1987; Orlob 1983; Schnoor 1996) as
well as the current literature.

10.3.3

Model Data

Data availability and accuracy is of concern in
the development and use of models for water
quality management. The complexity of models
used for water quality management should be
compatible with the quantity and quality of
available data. The use of complex mechanistic
models for water quality prediction in situations
with little useful water quality data does not
compensate for a lack of data. Model complexity
can give the impression of credibility but this is
not always true.
It is often preferable to begin with simple
models and then over time add additional complexity as justified based on the collection and
analysis of additional data. This strategy makes

424

10

Water Quality Modeling and Prediction

chlorophyll
violations
nutrient
inputs

river
flow

algal
density

harmfull
algal
blooms

carbon
production

sediment
oxygen
demand

duration
of
stratification

frequency
of
hypoxia

shellfish
abundance

E020730c

number
of
fishkills

fish
health

Fig. 10.3 Cause and effect diagram for estuary eutrophication due to excessive nutrient loadings (Borsuk et al. 2001)

efficient use of resources. It focuses efforts
toward obtaining information and models that
will reduce the uncertainty as the analysis proceeds. Models should be selected (simple vs.
complex) in part based on the data available to
support their use.

Water quality models of water bodies receiving pollutant discharges require those pollutant
loadings as input data. These pollutant discharges
can originate from point and non-point sources.
Point source discharges are much easier to
measure, monitor, and estimate than non-point

10.3

Water Quality Model Use

425

source inputs. Non-point discharge data often
come from rainfall-runoff models that attempt to
predict the quantity of runoff and its constituent
concentrations. The reliability of the predictions
from these models is not very good, especially if
short time periods (e.g., each day or week) are
being simulated. Their average values over
longer time periods (e.g., each month or year)
tend to be more reliable. This is mainly because
the short-term inputs to those models, such as
constituent loadings on the land and the rainfall
within an area, which can vary over space and
time within the area and time period being simulated, are typically not known with any precision. Chapter 12 reviews some of these loading
models and their limitations.

is both a science as well as an art. Each model
reflects the creativity of its developer, the particular water quality management problems and
issues being addressed, the available data for
model parameter calibration and verification, and
the time available for modeling and associated
uncertainty and other analyses. The fact that
most, if not all, water quality models cannot
accurately predict what actually happens does
not necessarily detract from their value. Even
relatively simple models can help managers
understand the real-world prototype and estimate
at least the relative if not actual change in water
quality associated with given changes in the
inputs resulting from water and land management policies or practices.

10.4

10.4.1

Models of Water Quality
Processes

Water quality models can be applied to many
different types of water systems including
streams, rivers, lakes, reservoirs, estuaries,
coastal waters, and oceans. The models describe
the main water quality processes and typically
require the hydrologic and constituent inputs (the
water flows or volumes and the pollutant loadings). These models include terms for dispersive
and/or advective transport depending on the
hydrologic and hydrodynamic characteristics of
the water body, and terms for the biological,
chemical and physical reactions among constituents. Advective transport dominates in
flowing rivers. Dispersion is the predominant
transport phenomenon in estuaries subject to
tidal action. Lake water quality prediction is
complicated by the influence of random wind
directions and velocities that often affect surface
mixing, currents, and stratification. For this and
other reasons, obtaining reliable lake quality
predictions is often more difficult than for
streams, rivers, and estuaries. In coastal waters
and oceans, larges scale flow patterns and tides
are the most important transport mechanisms.
As with water quantity modeling, the development and application of water quality models

Mass Balance Principles

The basis principle of water quality models is that
of mass balance. A water system can be divided
into different segments or volumes elements (e.g.,
stream or river reaches, lake layers, estuary segments), also called ‘‘computational cells.’’ For
each segment or cell, there must be mass balance
for each water quality constituent over time. Most
water quality simulation models simulate quality
over a consecutive series of discrete time period
durations, Δt. Time is divided into discrete
intervals and the flows are assumed constant
within each of those time period intervals. For
each segment and each time period, the mass
balance of a substance in a segment can be
administrated. Components of the mass balance
for a segment include: (1) changes by transport
(Tr) into and out of the segment, (2) changes by
physical or chemical processes (P) occurring
within the segment and (3) changes by
sources/discharges to or from the segment (S).
Mit þ D t ¼ Mit þ Dt




DMi
Dt





þ Dt
Tr

DMi
Dt


P

þ Dt




DMi
Dt S

ð10:2Þ
The mass
components:

balance

has

the

following

426

10

• the mass in computational cell i at the
beginning of a time step t: Mit
• the mass in computational cell i at the end of
a time step t: Mit þ Dt
• changes in computational cell i by transport;
DM 
Dt

i

Tr

• changes in computational cell i by physical,
 
i
(bio)chemical or biological processes: DM
Dt P
• changes in computational cell i by sources
 
i
(e.g., waste loads, river discharges): DM
Dt S
Transport includes both advective and dispersive transport. Advective transport is the
transport by flowing water. Dispersive transport
results from concentration differences. Dispersion in the vertical direction is important if the
water column is stratified, and dispersion in the
horizontal direction can be in one or two
dimensions. Dispersion, as defined here, includes
the physical concept of molecular diffusion as it
represents all transport that is not described by
the advective transport.
Processes include physical processes such as
reaeration and settling, (bio)chemical processes
such as adsorption, transformation, and denitrification and biological processes such as primary
production and predation on phytoplankton. Water
quality processes convert one substance to another.
Sources include the addition of mass by waste
loads and the extraction of mass by intakes. Mass
entering over the model boundaries can be considered a source as well. The water flowing into
or flowing out of the modeled segment or volume
element (the computational cell) is derived from
monitoring data or a water quantity (possibly
hydrodynamic) model.
To model the transport of substances over
space, a water system can be divided into small
segments or volume elements. The complete
ensemble of all the segments or elements is
called the ‘‘grid’’ or ‘‘schematization.’’ Each
computational cell is defined by its volume and
its dimensions in one, two, or three directions
(Δx, Δy, Δz) depending on the nature of the
schematization (1D, 2D, or 3D). Note that the

Water Quality Modeling and Prediction

values of Δx, Δy and Δz do not have to be equal.
The computational cell can have any rectangular
shape. A computational cell can share surface
areas with other computational cells, the atmosphere and the bottom sediment or coast line.
The following sections will look at the transport processes in more detail, defining parameters or variables and their units in terms of mass
M, length L, and time T.

10.4.1.1
AdvectiveTransport
The advective transport, Tx0 A (M T−1), of a constituent at a site x0 is the product of the average
water velocity, vx0 (L T−1), at that site, the surface
or cross-sectional area, A (L2), through which
advection takes place at that site, and the average
concentration, Cx0 (M L−3), of the constituent:
Tx0 A ¼ vx0 A  Cx0

ð10:3Þ

10.4.1.2
Dispersive Transport
The dispersive transport, TxD0 (M T−1) across a
surface area is assumed to be proportional to the


concentration gradient @C
@x x¼x0 at site x0 times the
surface area A. Letting Dx0 (L2 T−1), be the dispersion or diffusion coefficient at site x0:

@C
TxD0 ¼  Dx0 A  
ð10:4Þ
@x x¼x0
Dispersion is commonly based on Fick’s diffusion law. The minus sign reflects the fact that
dispersion causes net transport from higher to
lower concentrations, so in the opposite direction
of the concentration gradient. The concentration
gradient is the difference of concentrations per
unit length, over a very small distance across the
cross section
ð@C=@xÞjx ¼ limðDx ! 0Þ
½ðCx þ 0:5Dx  Cx0:5Dx Þ=Dx

ð10:5Þ

Dispersion coefficients should be calibrated or
be obtained from calculations with turbulence
models.

10.4

Models of Water Quality Processes

427

10.4.1.3

Mass Transport
by Advection
and Dispersion
If the advective and dispersive terms are added
and the terms at a second surface at site x0 + Δx
are included, the one-dimensional equation results

@C
@2C
@C
@2C
¼ Dx 2  vx
þ Dy 2
@t
@x
@x
@y
@C
@2C
@C
þ D z 2  vz
þ S þ fR ðC; tÞ
 vy
@y
@z
@z
ð10:10Þ



Mi t þ Dt ¼ Mit þ Dt  mx0 Cx0  mx0 þ Dx Cx0 þ Dx
!


@C 
@C 
Dx0  þ Dx0 þ Dx 
A
@x x0
@x x0 þ Dx

with dispersion coefficients Dj defined for each
direction j. If source terms ‘‘S’’ and ‘‘fR’’ are
added as shown in the equation above, the
so-called advection–diffusion-reaction equation
emerges. The additional terms represent

ð10:6Þ
or equivalently:



Mi t þ Dt ¼ Mit þ Dt  Qx0 Cx0  Qx0 þ Dx Cx0 þ Dx

!


@C
@C 
Dx0 Ax0  þ Dx0 þ Dx Ax0 þ Dx 
;
@x x0
@x x0 þ Dx

ð10:7Þ
where Qx0 (L3 T−1) is the flow at site x0.
If the previous equation is divided by the
volume and the time interval Δt, then the following equation results in one dimension.

@C 



 Dx0 @C
@x x0
x0 þ Dx

Cit þ Dt  Cit Dx0 þ Dx @x
¼
Dt
Dx
vx0 Cx0  vx0 þ Dx Cx0 þ Dx
þ
Dx

ð10:8Þ

Taking the asymptotic limit Δt → 0 and
Δx → 0, the advection–diffusion equation for
one dimension results



@C
@
@C
@
¼
D
 ðvCÞ
@t
@x
@x
@x

ð10:9Þ

The finite volume computational method for
transport can be used to solve the advection–diffusion equation. The accuracy of the method will
be related to the size of Δx, A (=Δy Δz) and Δt.
By adding terms for transport in the y and zdirection a three-dimensional model is obtained.
Taking the asymptotic limit again will lead to a
three-dimensional advection–diffusion equation

• Discharges or ‘‘waste loads’’ (S): these source
terms are additional inflows of water or mass.
As many source terms as required may be
added to Eq. 10.10. These could include
small rivers, discharges of industries, sewage
treatment plants, small waste load outfalls,
etc.
• Reaction terms or ‘‘processes’’ (fR).
Processes can be split into physical processes
and other processes. Examples of physical processes are
• settling of suspended particulate matter
• water movement not affecting substances, like
evaporation
• volatilization of the substance itself at the
water surface.
Examples of other processes are
• biochemical conversions like ammonia and
oxygen forming nitrite
• growth of algae (primary production)
• predation by other animals
• chemical reactions.
These processes are described in more detail
in the remaining parts of this Sect. 10.4.
Multiplying each term in Eq. 10.9 by the
cross-sectional area A (L2), the expression DA
(∂C/∂x) − vAC for a one-dimensional model, or
its equivalent in Eq. 10.10 for a threedimensional model, is termed the total flux
(M T−1). Flux due to dispersion, DA(∂C/∂x),
is assumed to be proportional to the concentration gradient over distance. Constituents are

428

10

transferred by dispersion from higher concentration zones to lower concentrations zones. The
coefficient of dispersion D (L2 T−1) depends on
the amplitude and frequency of the tide, if
applicable, as well as upon the turbulence of the
water body. It is common practice to include in
this dispersion parameter everything affecting the
distribution of C other than advection. The term
vAC is the advective flux caused by the movement of water containing the constituent concentration C (M L−3) at a velocity rate v (L T−1)
across a cross-sectional area A.
The relative importance of dispersion and
advection depends on how detailed the velocity
field is defined. A good spatial and temporal
description of the velocity field within which the
constituent is being distributed will reduce the
importance of the dispersion term. Less precise
descriptions of the velocity field, such as averaging across irregular cross sections or approximating transients by steady flows, may lead to a
dominance of the dispersion term.
Many of the reactions affecting the decrease or
increase of constituent concentrations are often
represented by first-order kinetics which assumes
that the reaction rates are proportional to the
constituent concentration. While higher order
kinetics may be more correct in certain situations,
predictions of constituent concentrations based on
first-order kinetics have often been considered
acceptable for natural aquatic systems.

10.4.2

Water Quality Modeling and Prediction

site x = 0, the concentration C at any distance
x will equal

ðWC =QmÞexp½ðv=2DÞð1 þ mÞx x  0
CðxÞ ¼
ðWC =QmÞexp½ðv=2DÞð1mÞx x  0
ð10:12Þ
where


1=2
m ¼ 1 þ 4kD=v2

ð10:13Þ

Note from Eq. 10.13 that the parameter m is
always equal or greater than 1. Hence the exponent of e in Eq. 10.12 is always negative. As the
distance x increases in magnitude, either in the
positive or negative direction, the concentration
C(x) will decrease. The maximum concentration
C occurs at x = 0 and is WC/Qm.
Cð0Þ ¼ WC =Qm

ð10:14Þ

These equations are plotted in Fig. 10.4.
In flowing rivers not under the influence of
tidal actions the dispersion is small. Assuming
the dispersion coefficient D is 0, the parameter
m defined by Eq. 10.13, is 1. When D = 0, the
maximum concentration at x = 0 is WC/Q.
Cð0Þ ¼ WC =Q

if D ¼ 0:

ð10:15Þ

Assuming D = 0 and v, Q and k > 0,
Eq. 10.12 becomes

0
x0
CðxÞ ¼
ð10:16Þ
ðWC =QÞexp½kx=v x  0

Steady-State Models

Steady state means no change in the concentrations over time. In this case the left-hand side of
Eq. 10.9 or 10.10, ∂C/∂t, equals 0. Assume the
only sink is the natural decay of the constituent
defined as kC where k, (T−1), is the decay rate
coefficient or constant. Now Eq. 10.9 becomes
0 ¼ D@ 2 C=@x2  v@C=@x  kC

ð10:11Þ

Equation 10.11 can be integrated since river
reach parameters A, D, k, v, and Q are assumed
constant. For a constant loading, WC (M T−1) at

The above equation for x > 0 can be derived
from Eqs. 10.12 and 10.13 by noting that the term
(1 − m) equals (1 − m)(1 + m)/(1 + m) = (1 −
m2)/2 when D is 0. Thus, when D is 0 the
expression (v/2D)(1 − m)x in Eq. 10.12 becomes
−kx/v. The term x/v is sometimes denoted as a
single variable representing the time of flow—the
time flow Q takes to travel from site x = 0 to some
other downstream site x > 0.
As rivers approach the sea, the dispersion
coefficient D increases and the net downstream
velocity v decreases. Because the flow Q equals
the cross-sectional area A times the velocity v,

10.4

Models of Water Quality Processes

429

Fig. 10.4 Constituent
concentration distributions
along a river or estuary
resulting from a constant
discharge of that
constituent at a single point
source in that river or
estuary. The top plot
represents Eq. 10.16, the
middle plot represents
Eq. 10.12, and the bottom
plot represents Eq. 10.17

Q = Av, and since the parameter m can be defined
as (v2 + 4kD)1/2/v, then as the velocity v
approaches 0, the term Qm = Av(v2 + 4kD)1/2/v
approaches 2A(kD)1/2. The exponent vx(1 ± m)/2D
in Eq. 10.12 approaches ±x(k/D)1/2.
Hence for small velocities, Eq. 10.4 becomes
h
i
8
< WC =2AðkDÞ1=2 exp þ xðk=DÞ1=2
h
i
CðXÞ ¼ 
: WC =2AðkDÞ1=2 exp xðk=DÞ1=2

x0
x0

ð10:17Þ
Here dispersion is much more important than
advective transport and the concentration profile
approaches a symmetric distribution, as shown in
Fig. 10.4, about the point of discharge at x = 0.
Water quality management models are often
used to assess the effect of pollutant loadings on
ambient waters and to compare the results with
specific water quality standards. The above
steady state equations can be used to construct
such a model for estimating the wastewater
removal efficiencies required at each wastewater
discharge site that will result in an ambient

stream quality that meets the standards along a
stream or river.
Figure 10.5 shows a schematic of a river into
which wastewater containing constituent C is
being discharged at four sites. Assume maximum
allowable concentrations of the constituent C are
specified at each of those discharge sites. To
estimate the needed reduction in these discharges, the river must be divided into approximately homogenous reaches. Each reach can be
characterized by constant values of the
cross-sectional area, A, dispersion coefficient, D,
constituent decay rate constant, k, and velocity, v,
associated with some ‘‘design’’ flow and temperature conditions. These parameter values and
the length, x, of each reach can differ, hence the
subscript index i will be used to denote the particular reach. These reaches are shown in
Fig. 10.5.
In Fig. 10.5 each variable Ci represents the
constituent concentration at the beginning of
reach i. The flows Q represent the design flow
conditions. For each reach i the product (Qi mi) is
represented by (Qm)i. The downstream (forward)

430

10

Water Quality Modeling and Prediction

Fig. 10.5 Optimization model for finding constituent removal efficiencies, Ri, at each discharge site i that result in
, at least total cost
meeting stream quality standards, Cmax
i

transfer coefficient, TFi, equals the applicable
part of Eq. 10.12,
TFi ¼ exp½ðv=2DÞð1mÞx

ð10:18Þ

as does the upstream (backward) transfer coefficient, TBi.
TBi ¼ exp½ðv=2DÞð1 þ mÞx

ð10:19Þ

The parameter m is defined by Eq. 10.13.
Finding the cost solution of a model such as
shown in Fig. 10.5 does not mean that the
least-cost wasteload allocation plan will be
implemented, but such information can help
identify the additional costs of other imposed
constraints, for example, to ensure equity, or
extra safety. Models like this can be used to
identify the cost-quality tradeoffs inherent in any
water quality management program. Non economic objectives can also be used to obtain other
tradeoffs.

The model in Fig. 10.5 incorporates both
advection and dispersion. If upstream dispersion
under design streamflow conditions is not significant in some reaches, then the upstream
(backward) transfer coefficients, TBi, for those
reaches i will equal 0.

10.4.3

Design Streamflows
for Setting
and Evaluating
Quality Standards

In streams and rivers, the water quality may vary
significantly depending on the stream or river
flow and its quality prior to wastewater discharges. If waste load discharges are fairly constant, a high flow of high quality serves to dilute
the waste concentration while contaminant concentrations of low flows may become undesirably high. It is therefore common practice to pick

10.4

Models of Water Quality Processes

a more critical low-flow condition for judging
whether or not ambient water quality standards
are being met. This can also be seen from
Eqs. 10.12, 10.14, 10.15, and 10.16. This often is
the basis for the assumption that the smaller (or
more critical) the design flow, the more likely it
is that the stream quality standards will be met in
practice. This is not always the case, however.
Different regions of the world use different
design low-flow conditions. One example of
such a design flow, that is used in parts of North
America, is the minimum 7-day average flow
expected to be lower only once in 10 years on
average. Each year the lowest 7-day average flow
is determined, as shown in Fig. 10.6. The sum of
each of the 365 sequences of seven average daily

Fig. 10.6 Portion of
annual flow time series
showing low flows and the
calculation of average 7
and 14-day flows

Fig. 10.7 Determining the
minimum 7-day annual
average flow expected once
in 10 years, designated
7Q10, from the cumulative
probability distribution of
annual minimum 7-day
average flows

431

flows is divided by 7 and the minimum value is
selected. This is the minimum annual average
7-day flow for the year.
These minimum 7-day average flows for each
year of record define a probability distribution,
whose cumulative probabilities can be plotted. As
illustrated in Fig. 10.7, the particular flow on the
cumulative distribution that has a 90% chance of
being exceeded is the design flow. It is the minimum annual average 7-day flow expected once in
10 years. This flow is commonly called the 7Q10
flow. Analyses have shown that this daily design
flow is exceeded about 99% of the time in regions
where it is used (NRC 2001). This means that there
is on average only about a one percent chance that
any daily flow will be less than this 7Q10 flow.

432

Consider now any one of the river reaches
shown in Fig. 10.5. Assume an initial loading of
constituent mass, M/Δt, exists at the beginning of
the reach. As the reach flow, Q, increases and the
mass loading stays the same, the initial concentration, M/Q, will decrease. However, the flow
velocity will increase, and thus the time, Δt, it
takes to transport the constituent mass to the end
of that reach will decrease. This means less time
for the decay of the constituent. Hence it is
possible that ambient water quality standards that
are met during low flow conditions may not be
met under higher flow conditions, conditions that
are observed much more frequently. Figure 10.8
illustrates how this might happen. This does not
suggest that low flows should not be considered
when allocating waste loads, but rather that a
simulation of water quality concentrations over
varying flow conditions may show that higher
flow conditions at some sites are even more
critical and more frequent than they are during
less frequent low-flow conditions.
Figure 10.8 shows that for a fixed mass of
pollutant at x = 0, under low flow conditions the
more restrictive (lower) maximum pollutant
concentration standard in the downstream portion
of the river is met, but that same standard is
violated under higher flow conditions.

Fig. 10.8 Increasing
streamflows decreases
initial concentrations but
may increase downstream
concentrations

10

10.4.4

Water Quality Modeling and Prediction

Temperature

Temperature impacts almost all water quality
processes taking place in water bodies. For this
reason modeling temperature may be important
when the temperature varies over the period of
interest, or when the discharge of heat into water
bodies is to be managed.
Temperature models are based on a heat balance in the water body. A heat balance takes into
account the sources and sinks of heat. The main
sources of heat in a water body are shortwave
solar radiation, longwave atmospheric radiation,
conduction of heat from the atmosphere to the
water and direct heat inputs. The main sinks of
heat are long wave radiation emitted by the
water, evaporation, and conduction from the
water to atmosphere. Unfortunately, a model
with all the sources and sinks of heat requires
measurements of a number of variables and
coefficients that are not always readily available.
One temperature predictor is the simplified
model that assumes an equilibrium temperature
Te (°C) will be reached under steady-state
meteorological conditions. The temperature
mass balance in a volume segment depends on
the water density ρ (g/cm3), the heat capacity of
water, cp (cal/g/°C), and the water depth h (cm).

10.4

Models of Water Quality Processes

433

Assuming the net heat input, KH(Te − T)
(cal/cm2/day), is proportional to the difference of
the actual temperature, T, and the equilibrium
temperature, Te (°C),
dT=dt ¼ KH ðTe  T Þ=qcp h

ð10:20Þ

The overall heat exchange coefficient, KH
(cal/cm2/day/°C), is determined in units of
W/m2/°C (1 cal/cm2/day °C = 0.4840 W/m2/°C)
from empirical relationships that include wind
velocity Uw (m/s), dew point temperature Td (°C),
and actual temperature T (°C) (Thomann and
Mueller 1987).
The equilibrium temperature, Te, is obtained
from another empirical relationship involving the
overall heat exchange coefficient, KH, the dew
point temperature, Td, and the shortwave solar
radiation Hs (cal/cm2/day),
Te ¼ Td þ ðHs =KH Þ

ð10:21Þ

This model simplifies the mathematical relationships of a complete heat balance and requires
less data.

10.4.5

Sources and Sinks

Sources and sinks of water quality constituents
include the physical and biochemical processes
that are represented by the term S in Eq. 10.10.
External inputs of each constituent would have
the form W/Q, where W (M T−1) is the loading
rate of the constituent and Q represents the flow
of water into which the mass of waste W is
discharged.

10.4.6

First-Order Constituents

The first-order models are commonly used to
predict water quality constituent decay or
growth. They can represent constituent reactions
such as decay or growth in situations where the
time rate of change (dC/dt) in the concentration
C of the constituent, say organic matter that
creates a biochemical oxygen demand (BOD), is

proportional to the concentration of either the
same or another constituent concentration. The
temperature-dependent proportionality constant
kc (1/day) is called a rate coefficient or constant.
In general, if the rate of change in some constituent concentration Cj is proportional to the
magnitude of concentration Ci, of constituent i,
then
ðT20Þ

dCj =dt ¼ aij ki hi

Ci ;

ð10:22Þ

where θi is temperature correction coefficient for
ki at 20 °C and T is the temperature in °C. The
parameter aij is the grams of Cj produced (aij > 0)
or consumed (aij < 0) per gram Ci. For the prediction of BOD concentration over time, Ci =
Cj = BOD and aij = aBOD = −1 in Eq. 10.22.
Conservative substances, such as salt, will have a
decay rate constant k of 0. The concentration of
conservative substances depends only on the
amount of water, i.e., dilution.
The typical values for the rate coefficients kc
and temperature coefficients θi of some constituents C are in Table 10.1. For bacteria, the
first-order decay rate (kB) can also be expressed
in terms of the time to reach 90% mortality (t90,
days). The relationship between these coefficients
is given by kB = 2.3/t90.

10.4.7

Dissolved Oxygen

DO concentration is a common indicator of the
health of the aquatic ecosystem. DO was originally modeled by Streeter and Phelps (1925).
Since then a number of modifications and
extensions of the model have been made
depending on the number of sinks and sources of
DO being considered and how processes
involving the nitrogen cycle and phytoplankton,
are being modeled, as illustrated in Fig. 10.9.
The sources of DO in a water body include
reaeration from the atmosphere, photosynthetic
oxygen production from aquatic plants, denitrification, and DO inputs. The sinks include oxidation of carbonaceous and nitrogenous material,
sediment oxygen demand and respiration by
aquatic plants.

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10

Water Quality Modeling and Prediction

Table 10.1 Typical values of the first-order decay rate, k, and the temperature correction factor, θ, for some
constituents

DO2 =Dt ¼ loads þ transport þ reaeration
þ net primary production þ denitrification
 mineralization  nitrificationSOD

The rate of reaeration is assumed to be proportional to the difference between the saturation
concentration, DOsat (mg/l), and the concentration of DO, DO (mg/l). The proportionality
coefficient is the reaeration rate kr (1/day), defined
at temperature T = 20 °C, which can be corrected
for any temperature T with the coefficient θ(T−20)
.
r
The value of this temperature correction coefficient, θ, depends on the mixing condition of the
water body. Values generally range from 1.005 to

1.030. In practice a value of 1.024 is often used
(Thomann and Mueller 1987). The reaeration rate
constant is a sensitive parameter. There have been
numerous equations developed to define this rate
constant. Table 10.2 lists some of them.
The saturation concentration, DOsat, of oxygen in water is a function of the water temperature and salinity [chloride concentration, Cl
(g/m3)], and can be approximated by
n
DOsat ¼ 14:652  0:41022T þ ð0:089392TÞ2
 ð0:042685TÞ3 gf1  ðCl=100000Þg ð10:23aÞ

10.4

Models of Water Quality Processes

435

Table 10.2 Some equations for defining the reaeration rate constant, kr (1/day)

Fig. 10.9 The dissolved oxygen interactions in a water
body, showing the decay (satisfaction) of carbonaceous,
nitrogenous and sediment oxygen demands and water

body reaeration or deaeration (if supersaturated occurs at
the air–water interface)

436

10

Water Quality Modeling and Prediction

Fig. 10.10 Fitted curve to
the saturation dissolved
oxygen concentration
(mg/l) as a function of
temperature (°C)

Elmore and Hayes (1960) derived an analytical expression for the DO saturation concentration, DOsat (mg/l), as a function of temperature
(T, °C):
DOsat ¼ 14:652  0:41022T þ 0:007991T 2
 0:000077774T 3
ð10:23bÞ
Fitting a second-order polynomial curve to the
data presented in Chapra (1997) results in
DOsat ¼ 14:407  0:3369T þ 0:0035T 2
ð10:23cÞ
as is shown in Fig. 10.10.
Because photosynthesis occurs during daylight
hours, photosynthetic oxygen production follows
a cyclic, diurnal, pattern in water. During the day,
oxygen concentrations in water are high and can
even become supersaturated, i.e., concentrations
exceeding the saturation concentration. At night,
the concentrations drop due to respiration and
other oxygen consuming processes.
The biochemical oxygen demand results from
carbonaceous organic matter (CBOD, mg/l) and
from nitrogenous organic matter (NBOD, mg/l)
in the water. There is also the oxygen demand
from carbonaceous and nitrogenous organic
matter in the sediments (SOD, mg/l/day). These

oxygen demands are typically modeled as
first-order decay reactions with decay rate constants kCBOD (1/day) for CBOD and kNBOD
(1/day) for NBOD. These rate constants vary
with temperature, hence they are typically
defined for 20 °C. The decay rates are corrected
for temperatures other than 20 °C using temperature coefficients θCBOD and θNBOD, respectively.
The sediment oxygen demand (SOD)
(mg/liter/day) is usually expressed as a zero-order
reaction, i.e., a constant demand. One important
feature in modeling NBOD is ensuring the
appropriate time lag between when it is discharged
into a water body and when the oxygen demand is
observed. This lag is in part a function of the level
of treatment in the wastewater treatment plant.
The DO model with CBOD, NBOD, and SOD
is
ðT20Þ

ðT20Þ

dDO=dt ¼ kCBOD hCBOD CBOD  kNBOD hNBOD NBOD
þ kr qrðT20Þ ðDOsat  DOÞ  SOD

ð10:24Þ
ðT20Þ

dCBOD=dt ¼ kCBOD hCBOD CBOD ð10:25Þ
ðT20Þ

dNBOD=dt ¼ kNBOD hNBOD NBOD ð10:26Þ
The mean and range values for coefficients
included in these DO models are in Table 10.3.

10.4

Models of Water Quality Processes

437

Table 10.3 Typical values of parameters used in the dissolved oxygen models

10.4.8

Nutrients
and Eutrophication

Eutrophication is the progressive process of
nutrient enrichment of water systems. An increase

in nutrients leads to an increase in the productivity of the water system that may result in an
excessive increase in the biomass of algae or
other primary producers such as macrophytes or
duck weed. When it is visible on the surface of the

438

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Water Quality Modeling and Prediction

water it is called an algae bloom. Excessive algal
biomass could affect the water quality, especially
if it causes anaerobic conditions and thus impairs
the drinking, recreational, and ecological uses.
The eutrophication component of the model
relates the concentration of nutrients and the algal
biomass. For example, as shown in Fig. 10.11,
consider the growth of algae A (mg/l—not to be
confused with area A used in previous equations),
depending on phosphate phosphorus, P (mg/l),
and nitrite/nitrate nitrogen, Nn (mg/l), as the limiting nutrients. There could be other limiting
nutrients or other conditions as well, but here
consider only these two. If either of these two
nutrients is absent, the algae cannot grow
regardless of the abundance of the other nutrient.

The uptake of the more abundant nutrient will not
occur.
To account for this, algal growth is commonly
modeled as a Michaelis–Menten multiplicative
effect, i.e., the nutrients have a synergistic effect.
Model parameters include a maximum algal
growth rate l (1/day) times the fraction of a day,
fd, that rate applies (Fig. 10.12), the half saturation constants KP and KN (mg/l) (shown as Kc in
Fig. 10.13) for phosphate and nitrate, respectively, and a combined algal respiration and
specific death rate constant e (1/day) that creates
an oxygen demand. The uptake of phosphate,
ammonia, and nitrite/nitrate by algae is assumed
to be in proportion to their contents in the algae

Fig. 10.11 The dissolved oxygen, nitrogen, and phosphorus cycles, and phytoplankton interactions in a water
body, showing the decay (satisfaction) of carbonaceous
and sediment oxygen demands, reaeration or deaeration of
oxygen at the air–water interface, ammonification of
organic nitrogen in the detritus, nitrification (oxidation) of

ammonium to nitrate–nitrogen and oxidation of organic
phosphorus in the sediment or bottom layer to phosphate
phosphorus, phytoplankton production from nitrate and
phosphate consumption, and phytoplankton respiration
and death contributing to the organic nitrogen and
phosphorus

10.4

Models of Water Quality Processes

439

Fig. 10.12 Calculation of
the fraction, fd, of the
maximum growth rate
constant, l, to use in the
algal growth equations. The
fraction fd is the ratio of
actual production
zone/potential production
zone

Fig. 10.13 Defining the
half saturation constant for
a Michaelis–Menten model
of algae. The actual growth
rate constant = lC/
(C + KC)

biomass. Define these proportions as aP, aA, and
aN, respectively.
In addition to the above parameters, one needs
to know the amounts of oxygen consumed in the
oxidation of organic phosphorus, Po, and the
amounts of oxygen produced by photosynthesis
and consumed by respiration. In the model
below, some average values have been assumed.
Also assumed are constant temperature correction factors for all processes pertaining to any
individual constituent. This reduces the number
of parameters needed, but is not necessarily
realistic. Clearly other processes as well as other
parameters could be added, but the purpose here
is to illustrate how these models are developed.

Users of water quality simulation programs will
appreciate the many different assumptions that
can be made and the large amount of parameters
associated with most of them.
The source and sink terms of the relatively
simple eutrophication model shown in Fig. 10.11
can be written as follows:
For algae biomass
ðT20Þ

dA=dt ¼ lfd hA
½P=ðP þ KP Þ½Nn =ðNn þ KN ÞA
ðT20Þ
 ehA
A
ð10:27Þ
For organic phosphorus

440

10

dPo =dt ¼ kop hðT20Þ
Po
op

ð10:28Þ

For phosphate phosphorus
ðT20Þ

dP=dt ¼ lfd hA

For DO
ðT20Þ

dDO=dt ¼ kCBOD hCBOD CBOD  4:57ka haðT20Þ Na

½P=ðP þ KP Þ½Nn =ðNn þ KN ÞaP A

ð10:29Þ

ðT20Þ

ðT20Þ
 2kop hop
Po þ ð1:5 lfd  2eÞhA

ðDOsat
þ kr hðT20Þ
r

dNo =dt ¼ kon hðT20Þ
No
on

ð10:30Þ

For ammonia–nitrogen
ðT20Þ

½P=ðP þ KP Þ½Nn =ðNn þ KN ÞaA A

þ kon hðonT20Þ No

 ka hðaT20Þ Na

ð10:31Þ
For nitrate–nitrogen
ðT20Þ

dNn =dt ¼ lfd hA

½P=ðP þ KP Þ½Nn =ðNn þ KN ÞaN A

þ ka haðT20Þ Na

A

 DOÞ  SOD

ð10:33Þ

For organic nitrogen

dNa =dt ¼ lfd hA

Water Quality Modeling and Prediction

ðT20Þ

 kn hA

Nn

ð10:32Þ

Representative values of the coefficients for
this model are in Table 10.4.
Because of the growth of phytoplankton
cannot occur without nutrients, the eutrophication modeling must be coupled with that of
nutrients. Nutrient modeling must include all the
different biochemical forms of the nutrients,
primarily nitrogen and phosphorus, as well as all
the interactions between the different forms. The
sum of all these interactions is referred to a
‘‘nutrient cycling’’.
The nitrogen cycle includes ammonium
(NH4–N) and nitrate/nitrite (represented as

Table 10.4 Typical values of coefficients in the eutrophication model

10.4

Models of Water Quality Processes

441

NO3–N) as the main forms of dissolved nitrogen
in water. Furthermore, nitrogen is present in
algae, as well as in detritus, resulting from algae
mortality, and in suspended (non-detritus)
organic nitrogen. Nitrogen can also be present in
different forms in the bottom sediment.
Two important reactions in the nitrogen
nutrient cycle are nitrification and denitrification,
which affect the flux of ammonium and nitrate in
the water column. Nitrification is the conversion
of ammonium to nitrite and finally nitrate,
requiring the presence of oxygen
NH4 þ þ 2O2 ! NO3  þ H2 O þ 2H þ ð10:34Þ
Denitrification is the process occurring during
the breakdown (oxidation) of organic matter by
which nitrate is transformed to nitrogen gas,
which is then usually lost from the water system.
Denitrification occurs in anaerobic systems
NO3  ! N2

ð10:35Þ

The phosphorus cycle is simpler than the
nitrogen cycle because there are fewer forms in
which phosphorus can be present. There is only
one form of dissolved phosphorus, orthophosphorus (also called orthophosphate, PO4–P).
Similar to nitrogen, phosphorus also exists in
algae, in detritus, and other organic material as

Fig. 10.14 Schematic of
the adsorption/desorption
and decay processes of
various toxic chemicals in
water bodies and bottom
sediments

well as in the bottom sediment. Unlike nitrogen,
there can also be inorganic phosphorus in the
particulate phase.

10.4.9

Toxic Chemicals

Toxic chemicals, also referred to as ‘‘micropollutants,’’ are substances that at low concentrations can impair the reproduction and growth of
organisms including fish and human beings.
These substances include heavy metals, many
synthetic organic compounds (organic micropollutants), and radioactive substances.

10.4.9.1

Adsorbed and Dissolved
Pollutants
An important characteristic of many of these
substances is their affinity with the surface areas
of suspended or bottom sediments. Many chemicals preferentially sorb onto particulate matter
rather than remaining dissolved in water. To
model the transport and fate of these substances,
the adsorption–desorption process, estimations of
the suspended sediment concentration, resuspension from the bottom, and settling are required.
Figure 10.14 depicts the adsorption–desorption and first-order decay processes for toxic
chemicals and their interaction in water and
sediment. This applies to the water and sediment

442

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Water Quality Modeling and Prediction

Fig. 10.15 Defining the partition coefficient Kp
(liters/kg) as the slope of the fixed ratio between
concentrations of a constituent in the water and sediment

phases of either a water body or bottom sediments.
Different constituents have different partition coefficients,
when they apply

phases in both the water body and in the bottom
sediments.
The adsorption–desorption model assumes
(conveniently but not always precisely) that an
equilibrium exists between the dissolved (in
water) and absorbed (on sediments) concentrations of a toxic constituent such as a heavy metal
or organic contaminant. This equilibrium follows
a linear relationship. The slope of that linear
relation is the partition coefficient Kp (l/kg). This
is shown in Fig. 10.15.
Each partition coefficient Kp (liters per kilogram or l/kg) is defined as the ratio of the particulate concentration Cp′ of a micropollutant
(mg/kg C) divided by the dissolved concentration
Cd′ of a micropollutant (mg/l water).

the model in most cases. It may be necessary to
include dissolved organic matter as well.
The adsorbed fractions in the water column
are subject to settling. The fractions in the sediment are subject to resuspension. The adsorbed
fractions in the sediment can also be removed
from the modeled part of the water system by
burial.
The rates of settling and resuspension of
micropollutants are proportional to the rates for
particulate matter. An additional process called
bioturbation leads to redistribution of the
micropollutant among sediment layers. Bioturbation is caused by physical activity of organisms, and affects both the particulate and
dissolved phases but with different rates. Bioturbation is taken into account by means of dispersion coefficients.
For modeling purposes, it is important to
know how much of a toxic chemical is present as
a dissolved constituent as opposed to adsorbed.
Assuming partition coefficients apply to a particular toxic constituent, the concentration, Cw, of
that constituent in the water body is divided into
a dissolved fraction (fdw) and an absorbed fraction (faw).

Kp ¼ Cp0 =Cd0

ð10:36Þ

Representative values of partition coefficients
Kp are in Table 10.5.
The presence of a micropollutant in water is
described by the total concentration (sum of
dissolved and particulate concentrations), the
total particulate concentration, and the total dissolved concentration for each water and sediment
compartment. The particulate and dissolved
concentrations are derived from the total concentration and the respective fractions.
Because the fate of most micropollutants is
largely determined by adsorption to particulate
matter, suspended inorganic and organic matter
(including phytoplankton) have to be included in

C w ¼ ðfdw þ faw ÞCw

ð10:37Þ

In turn, the adsorbed fraction of a micropollutant is composed of the fractions adsorbed to
inorganic particulate matter, fim, dead particulate
organic matter, fpoc, and algae, falg. The total

10.4

Models of Water Quality Processes

443

Table 10.5 Typical values of partition coefficients in toxic chemical model from Thomann and Mueller (1987)

micropollutant concentration, Cw (mg/m3) is the
sum of all these fractions.


Cw ¼ fdw þ fim ; þ fpoc þ falg Cw

ð10:38Þ

Considering the simple division into dissolved
and adsorbed fractions (fdw and faw), these fractions depend on the partition coefficient, Kp, and
on the suspended sediment concentration, SS
(mg/l). The proportions of the total constituent
concentration in the water body, Cw, dissolved in
the water, DCw (mg/l), and adsorbed to the suspended sediments, ACw (mg/l) are defined as
DCw ¼ fdw Cw

ð10:39Þ

ACw ¼ faw Cw ;

ð10:40Þ

where the fractions


fdw ¼ 1= 1 þ Kp SS


faw ¼ 1  fdw ¼ Kp SS= 1 þ Kp SS

ð10:41Þ
ð10:42Þ

Similarly in the bottom sediments, the dissolved concentration DCs (mg/l) and adsorbed
concentration ACs (mg/l) are fractions, fds and fas,
of the total concentration Cs (mg/l).

DCs ¼ fds Cs

ð10:43Þ

ACs ¼ fas Cs

ð10:44Þ

These fractions are dependent on the sediment
porosity, ϕ, and density, ρs (kg/l).
fds ¼ 1=½/ þ qs ð1  /ÞKp 

ð10:45Þ

fas ¼ 1  fds
¼ ð½/ þ qs ð1  /ÞKp 1Þ=½/ þ qs ð1  /ÞKp 
ð10:46Þ
First-order decay occurs in the water and
sediment phases only in the dissolved fraction
with decay rate constants kw and ks (1/day),
respectively. Thus
dCw =dt ¼ kw hðT20Þ
fdw Cw  faw CwS þ fas Cs r
w
ð10:47Þ
fds Cs þ faw CwS  fas Cs r
dCs =dt ¼ ks hðT20Þ
s
ð10:48Þ

444

In the above two equations the parameter
s represents the mass of settling sediments
(mg/day), r the mass of resuspension sediments
(mg/day), and θ the temperature correction
coefficient of the constituent at temperature
T = 20 °C. If data are not available to distinguish
between the values of the decay rate constants
k in water and on sediments, they may be
assumed to be the same. Similarly for the values
of the temperature correction coefficients θ.
Suspended solids settling and resuspension can
be determined at each day from a sediment
model.

10.4.9.2
Heavy Metals
The behavior of heavy metals in the environment
depends on their inherent chemical properties.
Heavy metals can be divided into different categories depending on their dissolved form and
redox (reduction oxidation) status. Some metals,
including copper, cadmium lead, mercury,
nickel, tin and zinc form free or complexed
cations when dissolved in water (e.g., Cu2+ or
CuCl−). The soluble complexes are formed with
negatively charged ions such as chlorine, oxygen, or dissolved organic compounds. These
heavy metals also tend to form poorly soluble
sulfhides under chemically reducing conditions.
These sulfhides generally settle in bottom sediments and are essentially ecologically unavailable. Other metals such as arsenic and vanadium
are present as anions in dissolved form. The
differences between groups of metals have
important consequences for the partitioning of
the metals among several dissolved and particulate phases.
Metals are non-decaying substances. The fate
of heavy metals in a water system is determined
primarily by partitioning to water and particulate
matter (including phytoplankton), and by transport. The partitioning divides the total amount of
a pollutant into a ‘dissolved’ fraction and several
‘adsorbed’ fractions (as described in Eqs. 10.39–
10.42. The fractions of a metal that are adsorbed
onto particulate matter are influenced by all the

10

Water Quality Modeling and Prediction

processes that affect particulate matter, such as
settling and resuspension.
Partitioning is described in general by sorption to particulates, precipitation in minerals, and
complexation in solution. Complexation with
inorganic and organic ligands can be considered
explicitly in connection with the other processes.
Sorption can be modeled as an equilibrium process (equilibrium partitioning) or as the resultant
of slow adsorption and desorption reactions (kinetic formulations). In the latter case, partitioning
is assumed to proceed at a finite rate proportional
to the difference between the actual state and the
equilibrium state.
To describe the fate of certain heavy metals in
reducing environments, such as sediment layers,
the formation of metal sulfhides or hydroxides
can be modeled. The soluble metal concentration
is determined on the basis of the relevant solubility product. The excess metal is stored in a
precipitated metal fraction.
Sorption and precipitation affect the dissolved
metal concentration in different ways. Both the
adsorbed and dissolved fractions increase at
increasing total concentration as long as no solubility product is exceeded. When it is, precipitation occurs.

10.4.9.3
Organic Mircropollutants
Organic micropollutants generally are biocides
(such as pesticides or herbicides), solvents or
combustion products and include substances
such as hexachlorohexane, hexachlorobenzene,
PCB’s or polychlorobiphenyls, benzo-a-pyrene
and fluoranthene (PAH’s or polycyclic aromatic
hydrocarbons), diuron and linuron, atrazine and
simazine, mevinfos and dichlorvos, and dinoseb.
The short-term fate of organic micropollutants in a water system is determined primarily
by partitioning to water and organic particulate
matter (including phytoplankton), and by
transport. Additional processes such as
volatilization and degradation influence organic
micropollutant concentrations (this is in contrast
to heavy metals which do not decay). Many

10.4

Models of Water Quality Processes

445

toxic organic compounds have decay (or
“daughter”) products that are equally, if not
more, toxic than the original compound. The
rates of these processes are concentration and
temperature dependent.
Organic micropollutants are generally poorly
soluble in water and prefer to absorb to particulate matter in the water, especially particulate
organic matter and algae. Therefore, the fractions
of a micropollutant adsorbed to inorganic matter,
fim, dead particulate organic matter, fpoc, the
dissolved fraction of a micropollutant, fd, and
algae, falg, add up to the total micropollutant
concentration, C (mg/m3).
C ¼ ðfd þ fim þ fpoc þ falg ÞC

ð10:49Þ

The fractions are functions of the partition
coefficients Kp [for algae (m3/g C), for inorganic
matter (m3 g DW−1) and for dead particulate
organic matter (m3/g C)], the individual concentrations C [for algae biomass (g C/m3), for
dissolved (in water) inorganic matter (g/m3) and
for dead particulate organic matter (g C/m3)], and
the porosity ϕ (m3 water/m3 bulk). In surface
water the value for porosity is 1.
fd ¼ /=½/ þ Kpalg Calg þ Kpim Cim þ Kppoc Cpoc 
ð10:50Þ
fim ¼ ð1  fd ÞKpim Cim =½Kpalg Calg
þ Kpim Cim þ Kppoc Cpoc 
fpoc ¼ ð1  fd ÞKppoc Cpoc =½Kpalg Calg
þ Kpim Cim þ Kppoc Cpoc 
falg ¼ ð1  fd  fim  fpoc Þ

ð10:51Þ

ð10:52Þ

ð10:53Þ

In terms of bulk measures, each partition
coefficient Kp (see Eq. 10.36) also equals the
porosity ϕ times the bulk particulate concentration Cp (mg/m3 bulk) divided by the product of
the dissolved (mg/l bulk) and particulate (mg/m3

bulk) bulk concentrations, Cd Cs, all times
106 mg/kg.
Kp ¼ 106 / Cp =ðCd Cs Þ

ð10:54Þ

Partitioning can be simulated based on the
above equilibrium approach or according to slow
sorption kinetics. For the latter, the rate, dCp/dt,
of adsorption or desorption (mg/m3/day) depends
on a first-order kinetic constant ksorp (1/day) for
adsorption and desorption times the difference
between equilibrium particulate concentration
Cpe of a micropollutant (mg/m3 bulk) and the
actual particulate concentration Cp (mg/m3 bulk)
of a micropollutant.
dCp =dt ¼ ksorp ðCpe  Cp Þ

ð10:55Þ

The kinetic constant for sorption is not temperature dependent. All other kinetic constants
for micropollutants are temperature dependent.
Mass balance equations are similar for all
micropollutants except for the loss processes.
Metals are conservative substances. They
can be transformed into various species
either through complexation, adsorption, or precipitation. Organic micropollutants are lost
by volatilization, biodegradation, photolysis,
hydrolysis, and overall degradation. Most of
these processes are usually modeled as first-order
processes, with associated rate constants.
The volatization rate, dCd/dt (mg/m3/day) of
dissolved micropollutant concentrations, Cd
(mg/m3 water) in water depends on an overall
transfer coefficient, kvol (m/day), for volatilization and the depth of the water column, H (m).
dCd =dt ¼ kvol Cd =H

ð10:56Þ

The numerator (kvol Cd) is the volatilization
mass flux (mg/m2/day).
This equation is only valid when the atmospheric concentration is negligibly small, which
is the normal situation.
All other loss rates such as biodegradation,
photolysis, hydrolysis, or overall degradation
(mg/m3/day) are usually modeled as

446

10

dC=dt ¼ kC;

ð10:57Þ

where C is the total concentration of a micropollutant (mg/m3), and k is a (pseudo) temperature
dependent first-order kinetic rate constant for
biodegradation, photolysis, hydrolysis or overall
degradation (1/day). This is similar to Eq. 10.22.

10.4.9.4
Radioactive Substances
The fate of most radionuclides such as isotopes
of iodine (131I) and cesium (137Cs) in water is
determined primarily by partitioning to water and
particulate matter (including phytoplankton), by
transport, and by decay. Cesium (Ce+) adsorbs to
particulate inorganic matter, to dead particulate
organic material, and to phytoplankton, both
reversibly and irreversibly. The irreversible
fraction increases over time as the reversible
fraction gradually transforms into the irreversible
fraction. Radioactive decay proceeds equally for
all fractions. Precipitation of cesium does not
occur at low concentrations in natural water
systems.
Iodine is only present in soluble form as an
anion (IO3−) and does not adsorb to particulate
matter. Consequently, the transport iodine is only
subject to advection and dispersion.
Concentrations of radionuclides, CR (mg/m3)
are essentially conservative in a chemical sense,
but they decay by falling apart in other nuclides
and various types of radiation. The radioactive
decay rate (mg/m3/day) is usually modeled as a
first-order process involving a kinetic radioactive
decay constant, kdec (1/day). This kinetic constant
is derived from the half-life time of the
radionuclide. The initial concentration may be
expressed as radioactivity, in order to simulate
the activity instead of the concentration. These
state variables can be converted into each other
using
Ac ¼ 103 NA kdec CR =½86400 Mw; ð10:58Þ
whereAc = activity of the radionuclide (Bq/m3/s)
NA = Avogadro’s number (6.02 × 1023 mol)
Mw = molecular weight of the radionuclide
(g/mole)

10.4.10

Water Quality Modeling and Prediction

Sediments

Sediments in water play an important role in the
transport and fate of chemical pollutants in water.
Natural waters can contain a mixture of particles
ranging from gravel (2–20 mm) or sand (0.07–
2 mm) down to very small particles classified as
silt or clay (smaller than 0.07 mm). The very fine
fractions can be carried as colloidal suspension
for which electrochemical forces play a predominant role. Considering the large adsorbing
capacities, the fine fraction is characterized as
cohesive sediment. Cohesive sediment can
include silt and clay particles as well as particulate organic matter such as detritus and other
forms of organic carbon, diatoms and other
algae. Since flocculation and adsorbing capacities are of minor importance for larger particles,
they are classified as non-cohesive sediment.
The behavior of this fine-grained suspended
matter impacts water quality. First, turbidity and
its effect on the underwater light is an important
environmental condition for algae growth. The
presence of suspended sediment increases the
attenuation of light in the water column that leads
to an inhibition of photosynthetic activity and
hence, a reduction in primary production. Second, the fate of contaminants in waters is closely
related to suspended solids due to their large
adsorbing capacities. Like dissolved matter,
sediment is transported by advection and by
turbulent motion. In addition, the fate of the
suspended cohesive sediment is determined by
settling and deposition, as well as by bed processes of consolidation, bioturbation, and
resuspension.

10.4.11

Processes in Lakes
and Reservoirs

The water quality modeling principles discussed
above are applicable to different types of water
systems such as streams, rivers, lakes, estuaries,
and even coastal or ocean waters. This section
presents some of the unique aspects of water
quality modeling in lakes. The physical character

10.4

Models of Water Quality Processes

and water quality of rivers draining into lakes
and reservoirs are governed in part by the
velocity and the volume of river water. The
characteristics of the river water typically
undergo significant changes as the water enters
the lake or reservoir, primarily because its
velocity reduces. Portions of the sediment and
other material carried in the faster flowing water
settle out in the basin.
The structure of the biological communities
also changes from organisms suited to living in
flowing waters to those that thrive in standing or
pooled waters. Greater opportunities for the
growth of algae (phytoplankton) and the development of eutrophication are present.
Reservoirs typically receive larger inputs of
water, as well as soil and other materials carried
in rivers than lakes. As a result, reservoirs may
receive larger pollutant loads than lakes. However, because of greater water inflows flushing
rates are more rapid than in lakes. Thus, although
reservoirs may receive greater pollutant loads
than lakes, they have the potential to flush out the
pollutants more rapidly than do lakes. Reservoirs
may therefore exhibit fewer or less severe negative water quality or biological impacts than
lakes for the same pollutant load.
The water quality of lakes and reservoirs is
defined by
• water clarity or transparency (greater water
clarity usually indicates better water quality),
• concentration of nutrients (lower concentrations indicate better water quality),
• quantity of algae (lower levels indicate better
water quality),
• oxygen concentration (higher concentrations
are preferred for fisheries),
• concentration of dissolved minerals (lower
values indicate better water quality), and
• acidity (a neutral pH of 7 is preferred).
Many lakes and reservoirs receive discharges
of waste chemical compounds from industry,
some with toxic or deleterious effects on humans
and/or other water-dependent organisms and
products. Some of these pollutants can kill

447

aquatic organisms and damage irrigated crops.
Inadequate water purification resulting in the
discharge of bacteria, viruses, and other organisms into natural waters can be a primary cause
of waterborne disease. Although dangerous to
human health worldwide, such problems are
particularly severe in developing countries.
There can be major differences between deep
and shallow lakes or reservoirs. Deep lakes,
particularly in nontropical regions, usually have
poorer water quality in lower layers, due to
stratification (see Sect. 4.11.3). Shallow lakes do
not exhibit this depth differentiation in quality.
Their more shallow, shoreline areas have relatively poorer water quality because those sites are
where pollutants are discharged and have a
greater potential for disturbance of bottom muds,
etc. The water quality of a natural lake usually
improves as one moves from the shoreline to the
deeper central part.
In contrast, the deepest end of a reservoir is
usually immediately upstream of the dam. Water
quality usually improves along the length of a
reservoir, from the shallow inflow end to the
deeper, “lake-like” end near the dam, as shown in
Fig. 10.16.
Reservoirs, particularly the deeper ones, are
also distinguished from lakes by the presence of
a longitudinal gradient in physical, chemical, and
biological water quality characteristics from the
upstream river end to the downstream dam end.
Because of this, reservoirs have been characterized as comprising three major zones: an
upstream riverine zone, a downstream lake-like
zone at the dam end, and a transitional zone
separating these two zones (Fig. 10.16). The
relative size and volume of the three zones can
vary greatly in a given reservoir.

10.4.11.1

Water Quality Changes
and Impacts
Dams can produce changes in the downstream
river channels below them. These are quite unlike
downstream changes from lakes. Because reservoirs act as sediment and nutrient traps, the water at
the downstream dam end of a reservoir is typically
of higher quality than water entering the reservoir.

448

10

Water Quality Modeling and Prediction

Fig. 10.16 Longitudinal zonation of water quality and other variables in reservoirs

This higher quality water subsequently flows into
the downstream river channel below the dam. This
is sometimes a problem in that the smaller the
quantity of sediments and other materials transported in the discharged water, the greater the
quantity that can be picked up and transported as it
moves downstream. Because it contains less sediment, the discharged ‘‘hungry’’ water can scour
and erode the streambed and banks, picking up
new sediment as it continues downstream. This
scouring effect can negatively impact the flora,
fauna, and biological community structure in the
downstream river channel. The removal of sediments from a river by reservoirs also has important
biological effects, particularly on floodplains.

Many reservoirs, especially those used for
drinking water supplies, have water release or
discharge structures located at different vertical
levels in their dams (Fig. 10.17). This allows for
the withdrawal or discharge of water from different layers within the reservoir, so-called “selective withdrawal.” Depending on the quality of
the water discharged, selective withdrawal can
significantly affect water quality within the
reservoir itself, as well as the chemical composition and temperature of the downstream river.
Being able to regulate both quantities and qualities of the downstream hydrological regimes can
impact both flora and fauna and possibly even
the geomorphology of the stream or river.

10.4

Models of Water Quality Processes

449

Fig. 10.17 A multiple
outlet reservoir can be
better used to regulate the
temperature and water
quality downstream and
possibly the sediment in the
reservoir

Constructing a reservoir may have significant
social and economic implications, including the
potential for stimulating urban and agricultural
development adjacent to, and below, the reservoir. These activities can have both positive and
negative impacts on downstream water quality,
depending on the nature and size of development.
Agricultural and urban runoff is often the
leading source of pollution in lakes. Healthy lake
ecosystems contain nutrients in small quantities
from natural sources, but extra inputs of nutrients
(primarily nitrogen and phosphorus) adversely
impact lake ecosystems. When temperature and
light conditions are favorable, excessive nutrients
stimulate population explosions of undesirable
algae and aquatic weeds. After they die the algae
sink to the lake bottom, where bacteria consume
DO as they decompose the algae. Fish kills and
foul odors may result if dissolved is depleted.
Heavy metals are another major cause of lake
quality impairment. Heavy metals accumulate in
fish tissue. Since it is difficult to measure heavy
metals (e.g., mercury) in ambient water and since
they accumulate in fish tissue, fish samples are
commonly used to monitor heavy metal contamination. Common sources of heavy metals are
‘‘smoke-stack’’ industries, including power
plants, whose airborne discharges of mercury
eventually end up in our water supplies.
In addition to nutrients and metals siltation,
enrichment by organic wastes that deplete

oxygen and noxious aquatic plants impact lakes
and reservoirs. Often, several pollutants and
processes impact a single lake. For example, a
process such as removal of shoreline vegetation
may accelerate erosion of sediment and nutrients
into a lake. Extreme acidity (low pH) resulting
from acid rain can eliminate fish in isolated lakes.
Urban runoff and storm sewers, municipal sewage treatment plants, and hydrologic modifications are also sources of lake pollutants.

10.4.11.2 Lake Quality Models
The prediction of water quality in surface water
impoundments is based on mass balance relationships similar to those used to predict water quality
concentrations in streams and estuaries. There are
also significant problems in predicting the water
quality of lakes or reservoirs compared to those of
river and estuarine systems. One is the increased
importance of wind-induced mixing processes and
thermal stratification. Another for reservoirs is the
impact of various reservoir-operating policies.
Perhaps the simplest way to begin modeling
lakes is to consider shallow well-mixed
constant-volume lakes subject to a constant pollutant loading. The flux of any constituent concentration, C, in the lake equals the mass input of
the constituent less the mass output less losses
due to decay or sedimentation, if any, all divided
by the lake volume V (m3). Given a constant
constituent input rate WC (g/day) of a constituent

450

10

having a net decay and sedimentation rate constant kC (1/day) into a lake having a volume
V (m3) and inflow and outflow rate of
Q (m3/day), then the rate of change in the concentration C (g/m3/day) is
dC=dt ¼ ð1=VÞðWC  QC  kC CV Þ ð10:59Þ
Integrating this equation yields a predictive
expression of the concentration C(t) of the constituent at the end of any time period t based in
part on what the concentration, C(t − 1), was at
the end of that previous time period, t − 1. For a
period duration of Δt days,
CðtÞ ¼ ½WC =ðQ þ kC V Þ½1  expfDtððQ=V Þ þ kC Þg
þ Cðt  1ÞexpfDtððQ=V Þ þ kC Þg

ð10:60Þ
The equilibrium concentration, Ce, can be
obtained by setting the rate, dC/dt, in Eq. 10.59
to 0. The net result is
Ce ¼ WC =ðQ þ kC V Þ

ð10:61Þ

The time, tα, since the introduction of a mass
input WC that is required to reach a given fraction
α of the equilibrium concentration (i.e.,
C(t)/Ce = α) is
ta ¼ V½lnð1  aÞ=ðQ þ KC V Þ

ð10:62Þ

Similar equations can be developed to estimate the concentrations and times associated
with a decrease in a pollutant concentration. For
the perfectly mixed lake having an initial constituent concentration C(0), say after an accidental spill, and no further additions, the change
in concentration with respect to time is
dC=dt ¼ CðQ þ kC V Þ=V

ð10:63Þ

Integrating this equation, the concentration C
(t) is
CðtÞ ¼ Cð0ÞexpftððQ=VÞ þ KC Þ

ð10:64Þ

In this case one can solve for the time tα
required for the constituent to reach a fraction

Water Quality Modeling and Prediction

(1 − α) of the initial concentration C(0) (i.e.,
C(t)/Ce = 1 − α). The result is Eq. 10.62.
Equation 10.60 can be used to form an optimization model for determining the wasteload
inputs to this well-mixed lake that meet water
quality standards. Assuming that the total of all
natural wasteloads WC(t), inflows and outflows
Q(t), and the maximum allowable constituent
concentrations in the lake, C(t)max, may vary
among different within-year periods t, the minimum fraction, X, of total waste removal required
can be found by solving the following linear
optimization model:
Minimize X

ð10:65Þ

The following mass balance and constituent
concentration constraints apply for each period t :
CðtÞ ¼ ½WC ðtÞð1  XÞ=ðQðtÞ þ kC V Þ
½1  expfDtððQðtÞ=VÞ þ kC Þg
þ Cðt  1ÞexpfDtððQðtÞ=VÞ þ kC Þg
ð10:66Þ
CðtÞ  CðtÞmax

ð10:67Þ

If each period t is a within-year period, and if
the waste loadings and flows in each year are the
same, then no initial concentrations need be
assumed and a steady-state solution can be found.
This solution will indicate, for the loadings WC(t),
the fraction X of waste removal that meet the
quality standards, C(t)max, throughout the year.

10.4.11.3 Stratified Impoundments
Many deep reservoirs and lakes become stratified
during particular times of the year. Vertical
temperature gradients arise that imply vertical
density gradients. The depth-dependent density
gradients effectively prevent complete vertical
mixing. Particularly in the summer season, two
zones may form, an upper volume of warm water
called the epilimnion and a lower colder volume
called the hypolimnion. The transition zone or
boundary between the two zones is called the
thermocline (Fig. 10.17).
Because of stratification many models divide
the depth of water into layers, each of which is

10.4

Models of Water Quality Processes

assumed to be fully mixed. To illustrate this
approach without getting into too much detail,
consider a simple two-layer lake in the summer
that becomes a one-layer lake in the winter. This
is illustrated in Fig. 10.18.
Discharges of a mass WC of constituent C in a
flow Qin(t) into the lake in period t have concentrations of WC/Qin(t). The concentration in the
outflows from the summer epilimnion is Ce(t) for
each period t in the summer season. The concentration of the outflows from the winter lake as
a whole is C(t) for each period t in the winter
season. The summer time rates of change in the
epilimnion constituent concentrations Ce(t) and
hypolimnion concentrations Ch(t) depend on the

451

mass inflow, WC(t), and outflow, Ce(t)Qout(t), the
net vertical transfer across the thermocline, (v/DT)
[Ch(t)Vh(t) − Ce(t)Ve(t)], the settling on sediment
interface, sHh(t) Ch(t), and the decay, kCe(t):
dCe ðtÞ=dt ¼ ð1=Ve ðtÞÞfðWC ðtÞ  Ce ðtÞQout ðtÞÞ
þ ðv=DT Þ½Ch ðtÞVh ðtÞ  Ce ðtÞVe ðtÞg
 kCe ðtÞ
ð10:68Þ
dCh ðtÞ=dt ¼ kCh ðtÞ
 ðv=DT Þ½Ch ðtÞ  Ce ðtÞVe ðtÞ=Vh ðtÞ
 sHh ðtÞCh ðtÞ

Fig. 10.18 Lake stratification during summer and complete mixing during winter season

ð10:69Þ

452

10

In the above two equations, Ve and Vh (m3) are
the time-dependent volumes of the epilimnion
and hypolimnion, respectively, k (1/day) is the
temperature corrected decay rate constant,
v (m/day) is the net vertical exchange velocity
that includes effects of vertical dispersion, erosion of hypolimnion, and other processes that
transfer materials across the thermocline of
thickness DT (m), s is the settling rate velocity
(m/day) and Hh(t) is the average depth of the
hypolimnion (m).
In the winter season the lake is assumed to be
fully mixed. Thus for all periods t in the winter
season the initial concentration of a constituent is
CðtÞ ¼ Ce ðtÞVe ðtÞ þ Ch ðtÞVh ðtÞ=½Ve þ Vh 
ð10:70Þ
dCðtÞ=dt ¼ ð1=VðtÞÞfðWC ðtÞ  CðtÞQout ðtÞÞ
 decay kCe ðtÞ  sHðtÞCðtÞ
ð10:71Þ
At the beginning of the summer season, each
epilimnion and hypolimnion concentration will
be the same.

10.5

Ce ðtÞ ¼ CðtÞ

ð10:72Þ

Ch ðtÞ ¼ CðtÞ

ð10:73Þ

Simulation Methods

Most who will be using water quality models will
be using simulation models that are commonly
available from governmental agencies (e.g., US
EPA), universities, or private consulting and
research institutions such as the Danish Hydraulics Institute, Wallilngford software or Deltares
(Ambrose et al. 1996; Brown and Barnwell 1987;
Cerco and Cole 1995; DeMarchi et al. 1999;
Ivanov et al. 1996; Reichert 1994; USEPA 2001;
WL|Delft Hydraulics 2003).
These simulation models are typically based on
numerical methods incorporating a combination

Water Quality Modeling and Prediction

of plug flow and continuously stirred reactor
approaches to pollutant transport. Users must
divide streams, rivers, and lakes and reservoirs
into a series of well-mixed segments or volume
elements. A hydrologic or hydrodynamic model
calculates the flow of water between all the segments and volume elements. In each simulation
time step plug flow enters these segments or volume elements from upstream segments or elements. Flow also exits these segments or volume
elements to downstream segments or elements.
During this time the constituents can decay or
grow, as appropriate, depending the conditions in
those segments or volume elements. At the end of
each time step the volumes and their constituents
within each segment or element are fully mixed.
The length of each segment or the volume in each
element reflects the extent of dispersion in the
system.

10.5.1

Numerical Accuracy

As presented in Sect. 10.4, equations describing
water quality processes typically include time
rate of change terms such as dC/dt. While it is
possible to solve analytically some of these differential equations, most water quality simulation
models use numerical methods. The purpose of
this section is not to explain how this can be
done, but rather to point to some of the restrictions placed on the modeler because of these
numerical methods.
Consider first the relationship between the
stream, river, or lake segments and the duration
of time steps, Δt. The basic first-order decay flux,
dC/dt (g/m3/day), for a constituent concentration,
C, that is dependent on a rate constant, k (1/day),
is
dC=dt ¼ kC

ð10:74Þ

The finite difference approximation of this
equation can be written
Cðt þ DtÞ  CðtÞ ¼ CðtÞkDt
or

ð10:75Þ

10.5

Simulation Methods

Cðt þ DtÞ ¼ CðtÞð1  kDtÞ

453

ð10:76Þ

This equation can be used to illustrate the
restriction placed on the term kΔt. That term
cannot exceed a value of 1 or else C(t + Δt) will
be negative.
Figure 10.19 is a plot of various values of
C(t + Δt)/C(t) versus kΔt. This plot is compared
with the analytical solution resulting from the
integration of Eq. 10.110, namely
Cðt þ DtÞ ¼ CðtÞexpfkDtg

ð10:77Þ

Reducing the value of Δt will increase the
accuracy of the numerical solution. Any value of
Δt can be divided by a positive integer n to
become 1/nth of its original value. In this case
the predicted concentration C(t + Δt) will equal
Cðt þ DtÞ ¼ CðtÞð1  kDt=nÞn

Cðt þ DtÞ ¼ CðtÞð1 þ 1=mÞmðkDtÞ

ð10:80Þ

As n approaches infinity so does the variable
m, and hence the expression (1 + 1/m)m becomes
the natural logarithm base e = 2.718282. Thus as
n approaches infinity, Eq. 10.80 becomes
Eq. 10.77, the exact solution to Eq. 10.74.

10.5.2

Traditional Approach

ð10:78Þ

For example if kΔt = 1, and n = 2, the final
concentration ratio will equal
Cðt þ DtÞ=CðtÞ ¼ ð1  1=2Þ2 ¼ 0:25 ð10:79Þ
Compare this 0.25–0.37, the exact solution,
and to 0.0, the approximate solution when n is 1.
Having n = 2 brings a big improvement. If
n = 3, the concentration ratio will be 0.30, an
even greater improvement compared to 0. However no matter what value of n is selected, the
predicted concentration will always less than the
actual value based on Eq. 10.77, and hence the

Fig. 10.19 Plot of
numerical approximation
(red line) based on
Eq. 10.77 compared to the
true analytical (blue line)
value obtained from
Eq. 10.76

error is cumulative. Whenever Δt > n/k the predicted concentrations will alternate between
positive and negative values, either diverging,
converging or just repeating the cycle, depending
on how much Δt exceeds n/k. In any event, the
predicted concentrations are not very useful.
Letting m = −n/kΔt, Eq. 10.78 can be written
as

Most water quality simulation models simulate
quality over a period of time. Time is divided
into discrete intervals and the water and
wastewater flows are assumed constant within
each of those time period intervals. Each water
body is divided into segments or volume elements and these ‘‘computational cells’’ are considered to be in steady-state conditions within
each simulation time period. Advection or plug
flow (i.e., no mixing or dispersion) is assumed
during each time period. At the end of each
period mixing occurs within each segment or
volume element to obtain the concentrations in
the segment or volume element at the beginning

454

of the next time step. The larger the computational cell the greater is the dispersion.
This method is illustrated in Fig. 10.20. The
indices i − 1, i and i + 1 refer to stream or river
reach segments. The indices t and t + 1 refer to
two successive time periods, respectively. At the
beginning of time period t, each segment is
completely mixed. During the time interval Δt of
period t the water quality model predicts the
concentrations assuming plug flow in the direction of flow from segment i toward segment
i + 1. The time interval Δt is such that the flow
from any segment i does not pass through any
following segment i + 1. Hence at the end of
each time period, each segment has some of its
original water that was there at the beginning of
the period, and its end-of-period concentrations

10

Water Quality Modeling and Prediction

of constituents, plus some of the immediately
upstream segment’s water and its end-of-period
concentrations of constituents. These two volumes of water and their respective constituent
concentrations are then mixed to achieve a constant concentration within the entire segment.
This is done for all segments in each time
step. Included in this plug flow and then mixing
process are the inputs to the reach from point and
non-point sources of constituents.
In Fig. 10.20, a mass of waste enters reach i at
a rate of Wit . The volume in each reach segment
is denoted by V and the flows from one segment
to the next are denoted by Q. The drawing shown
on the left represents a portion of a stream or
river divided into well-mixed segments. During a
period t waste constituents enter reach segment

Fig. 10.20 Water quality modeling approach showing a water system schematized into reach segments or
‘computational cells’

10.5

Simulation Methods

455

i from the immediate upstream reach i − 1 and
from the point waste source. In this illustration,
the mass of each of these wastes is assumed to
decay during each time period, independent of
other wastes in the water. Depending on the
types of wastes, the decay, or even growth,
processes that take place may be more complex
than those assumed in this illustration. At the end
of each time period, these altered wastes are
mixed together to create an average concentration for the entire reach segment. This illustration
applies for each reach segment i and for each
time period t.
The length, Δxi, of each completely mixed
segment or volume element depends on the
extent of dispersion. Reducing the length of each
reach segment or size of each volume element
reduces the dispersion within the entire stream or
river. Reducing segment lengths, together with
increasing flow velocities, also reduces the
allowable duration of each time period t. The
duration of each simulation time step Δt must be
such that flow from any segment or element
enters only the adjacent downstream segment or
element during that time step. Stated formally,
the restriction is
Dt  Ti

ð10:81Þ

where Ti is the residence time in reach segment
or volume element i. For a 1-dimensional stream
or river system consisting of a series of segments
i of length Δxi, cross-section area Ai and average
flow Qit, the restriction is
Dt  minfDxi Ai =Qit ; 8i; tg

ð10:82Þ

If time steps are chosen which violate this
condition, then numerical solutions will be in
error. The restriction defined by Eq. 10.82 is
often termed the ‘‘courant condition.’’ It limits the
maximum time step value. Since the flows being
simulated are not always known, this leads to the
selection of very small time steps, especially in
water bodies having very little dispersion. While
smaller simulation time steps increase the accuracy of the model they also increase the

computational times. Thus, the balance between
computational speed and numerical accuracy
restricts the model efficiency in the traditional
approach to simulating water quality.

10.5.3

Backtracking Approach

An alternative backtracking approach to water
quality simulation eliminates the need to consider
the simulation time step duration restriction
indicated by Eq. 10.82 (Manson and Wallis
2000; Yin 2002). The backtracking approach
permits any simulation time step duration to be
used along with any segmenting scheme. Unlike
the traditional approach, water can travel through
any number of successive segments or volume
elements in each simulation time step.
This approach differs from the traditional one
in that instead of following the water in a segment or volume element downstream, the system
tracks back upstream to find the source concentrations of the contaminants at time t that will be
in the control volume or segment i + 1 at the
beginning of time period t + 1.
The backtracking process works from
upstream to downstream. It starts from the segment of interest, i, and finds all the upstream
sources of contaminants that flow into segment
i during time period t having a time interval Δt.
The contaminants could come from segments in
the same river reach or storage site, or from
upstream river reach or storage volume segments.
They could also come from incremental flows into
upstream segments. Flows between the source site
and the segment i + 1 transport the contaminants
from their source sites to segment i during the time
interval Δt, as shown in Fig. 10.21.
The simulation process for each segment and
for each time period involves three steps. To
compute the concentration of each constituent in
segment i at the end of time period t, as shown in
Fig. 10.21, the approach first backtracks
upstream to locate all the contaminant particles at
the beginning of period t that will be in the
segment i at the end of period t. This is achieved
by finding the most upstream and downstream

456

10

Water Quality Modeling and Prediction

Fig. 10.21 The backtracking approach for computing the concentrations of constituents in each reach segment or
volume element i during time step duration of Δt

positions of all reach intervals that will be at the
corresponding boundaries of segment i at the end
of time period t. This requires computing the
velocities through each of the intermediate segments or volume elements. Second, the changes
in the amounts of the modeled quality constituents, e.g., temperature, organics, nutrients
and toxics, are calculated assuming plug flow
during the time interval, Δt, using the appropriate
differential equations and numerical methods for
solving them. Finally, all the multiple incoming
blocks of water with their end-of-period constituent concentrations are completely mixed in
the segment i to obtain initial concentrations in
that segment for the next time step, t + 1. This is

done for each segment i in each time period t,
proceeding in the downstream direction.
If no dispersion is assumed, the backtracking
process can be simplified to consider only the
end points of each reach. Backtracking can take
place to each end-of-reach location whose time
of travel to the point of interest is just equal or
greater than Δt. Then using interpolation between
end-of-period constituent concentrations at those
upstream sites, plus all loadings between those
sites and the downstream site of interest, the
constituent concentrations at the end of the time
period t at the downstream ends of each reach
can be computed. This process, like the one
involving fully mixed reach segments, must take

10.5

Simulation Methods

457

into account the possibility of multiple paths
from each pollutant source to the site of interest,
and the different values of rate constants, temperatures, and other water quality parameters in
each reach along those paths.
Figure 10.21 illustrates an example of backtracking involving simple first-order decay processes. Assume contaminants that end up in reach
segment i at time the beginning of period t + 1
come from J sources with initial concentrations
Ct1 ; Ct2 ; Ct3 ; . . .; CtJ at the beginning of time period
t. Decay of mass from each source j during time
Δt in each segment or volume element is determined by the following differential equation:
ðT20Þ

dCjt =dt ¼ kj hj

Cjt

ð10:83Þ

The decay rate constant kj, temperature correction coefficient θj and water temperature T are
all temporally and spatially varied variables. Their
values depend on the particular river reaches and
storage volume sites through which water travels
during the period t from sites j to segment i.
Integrating Eq. 10.83 yields:
ðT20Þ

Cjt þ 1 ¼ Cjt expfkj hj

Dtg

ð10:84Þ

Since Δt is the time it takes water having an
initial concentration Cjt to travel to reach i, the
values Cjt þ 1 can be denoted as Cijt þ 1 .
ðTij20Þ

Cijt þ 1 ¼ Cjt expfkij hj

Dtg

ð10:85Þ

In Eq. 10.85 the values of the parameters are
the appropriate ones for the stream or river
between the source segments j and the destination segment i. These concentrations times their
respective volumes, Vjt , can then be mixed

together to define the initial concentration, Cit þ 1 ,
in segment i at the beginning of the next time
period t + 1.

10.5.4

Model Uncertainty

There are two significant sources of uncertainty
in water quality management models. One stems

from incomplete knowledge or lack of sufficient
data needed to estimate the probabilities of various events that might happen. Sometimes it is
difficult to even identify possible future events.
This type of uncertainty [sometimes called epistemic (Stewart 2000)] stems from our incomplete
conceptual understanding of the systems under
study, by models that are necessarily simplified
representations of the complexity of the natural
and socioeconomic systems, as well as by limited
data for testing hypotheses and/or simulating the
systems.
Limited conceptual understanding leads to
parameter uncertainty. For example, there is an
ongoing debate about the parameters that can
best represent the fate and transfer of pollutants
through watersheds and water bodies. Arguably
more complete data and more work on model
development can reduce this uncertainty. Thus, a
goal of water quality management should be to
increase the availability of data, improve their
reliabilities, and advance our modeling
capabilities.
However, even if it were possible to eliminate
knowledge uncertainty, complete model prediction certainty in support of water quality management decisions will likely never be achieved
until we can predict the variability of natural
processes. This is the other significant source of
uncertainty in water quality management models. This type of uncertainty arises in systems
characterized by randomness. Assuming past
observations are indicative of what might happen in the future and with the same frequency,
i.e., assuming stationary stochastic processes, we
can estimate from these past observations the
possible future events or outcomes that could
occur and their probabilities. Even if we think
we can estimate how likely any possible type of
event may be in the future we cannot predict
precisely when or to what extent that event will
occur.
For ecosystems, we cannot be certain we
know even what events may occur in the future,
let alone their probabilities. Ecosystems are
open systems in which it is not possible to know
in advance what all the possible biological
outcomes will be. Surprises are not only

458

10

possible, but likely. Hence both types of
uncertainty, knowledge uncertainty, and unpredictable variability or randomness, cannot be
eliminated.
Thus, uncertainty is a reality of water quantity
and quality management. This must be recognized when considering the results of water
quality management models that relate actions
taken to meet the desired water quality criteria
and designated uses of water bodies. Chapters 6,
7, and 8 suggests some ways of characterizing
this uncertainty.

10.6

Conclusions—Implementing
a Water Quality Management
Policy

This chapter provides only a brief introduction to
some of the relationships contained in water
quality models. As can be said for other chapters
as well, entire texts, and very good ones, have
been written on this subject (see, for example,
Chapra and Reckhow 1983; Chapra 1997;
McCutcheon 1989; Orlob 1983; Schnoor 1996;
Thomann and Mueller 1987). Water quality
modeling and management require skill and data.
Skill comes with experience.
If accompanied by field data and uncertainty
analysis, many existing models can be used to
assist those responsible for developing water
quality management plans in an adaptive
implementation or management framework.
Adaptive implementation or management will
allow for both model and data improvements
over time. Adaptive approaches strive toward
achieving water quality standards while relying
on monitoring and experimentation to reduce
uncertainty. This is often a way one can proceed
given the complexity of the real world compared
to the predictive models and data and time
usually available at the time a water quality
analysis is needed. Starting with simple analyses
and iteratively expanding data collection and
modeling as the need arises is a reasonable
approach.

Water Quality Modeling and Prediction

An adaptive management process begins with
initial actions that have reasonable chances of
succeeding. Future actions must be based on
continued monitoring of the water body to
determine how it responds to the actions taken.
Plans for future regulatory rules and public
spending should be subject to revision as stakeholders learn more about how the system
responds to actions taken. Monitoring is an
essential aspect of adaptive water quality management and modeling.
Regardless of what immediate actions are
taken, there may not be an immediate measurable
response. For example there may be significant
time lags between when actions are taken to
reduce nutrient loads and the resulting changes in
nutrient concentrations. This is especially likely
if nutrients from past activities are bound to
sediments or if nutrient-contaminated groundwater has a long residence time before its release
to surface water. For many reasons, lags between
actions taken and responses must be expected.
Water bodies should be monitored to establish
whether the “trajectories” of the measured water
quality criteria point toward attainment of the
designated use.
Waste load allocations will inevitably be
required if quality standards are not being met.
These allocations involve costs. Different allocations will have a different total costs and different distributions of those costs; hence they will
have different perceived levels of fairness.
A minimum cost policy may result in a cost
distribution that places most of the burden on just
some of the stakeholders. But until such a policy
is identified one will not know this. An alternative may be to reduce loads from all sources by
the same proportion. Such a policy has prevailed
in the US over the past several decades. Even
though not very cost effective from the point of
view of water quality management, the ease of
administration and the fulfillment of other
objectives must have made such a policy politically acceptable, even though expensive. However, more than these types of waste load
allocations policies will be needed for many of

10.6

Conclusions—Implementing a Water Quality Management Policy

the ecosystem restoration efforts that are
increasingly being made. Restoration activities
are motivated in part by a recognition of the
services ecosystems provide for water quality
management.
Our capabilities of including ecosystem
components within water quantity and quality
management models are at a fairly elementary
level. Given the uncertainty, especially with
respect to the prediction of how ecosystems will
respond to water management actions, together
with the need to take actions now, the popular
call is for adaptive management. The trial and
error aspects of adaptive management based on
monitoring and imperfect models may not satisfy
those who seek more definitive direction from
water quality analysts and their predictive models. Stakeholders and responsible agencies seeking assurances that the actions taken will always
work, as predicted, may be disappointed. Even
the best predictive capabilities of science cannot
assure that an action leading to attainment of
designated uses will be initially identified.
Adaptive management is a reasonable option in
most cases for allowing water quality management programs to move forward in the face of
considerable uncertainties.

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10

Water Quality Modeling and Prediction

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Exercises

10:2 Based on the integrated differential equations in Exercise 10.1
(a) Derive the equation for the distance Xc
downstream from a single point
source of BOD that for a given
streamflow will have the lowest dissolved oxygen concentration.
(b) Determine the relative sensitivity of
the deoxygenation rate constant Kd and
the reaeration rate constant Ka on the
critical distance Xc and on the corresponding critical deficit Dc. For initial
conditions, assume that the reach has a
velocity of 2 m/s (172.8 km/day), a Kd
of 0.30 per day, and a Ka of 0.4 per
day. Assume that the DO saturation
concentration is 8 mg/l, the initial
deficit is 1.0 mg/l, and the BOD concentration at the beginning of the reach
(including that discharged into the
reach at that point) is 15 mg/l.
10:3 To account for settling of BOD, in proportion to the BOD concentration, and for
a constant rate of BOD addition R due to
runoff and scour, and oxygen production
(A > 0) or reduction (A < 0) due to plants
and benthal deposits, the following differential equations have been proposed:
ðaÞ

1. The common version of the Streeter–
Phelps equations for predicting biochemical oxygen demand BOD and DO deficit
D concentrations are based on the following two differential equations
ðaÞ

dðBODÞ=dt ¼ K d ðBODÞ

ðbÞ

dD=dt ¼ Kd ðBODÞ  Ka D;

where Kd is the deoxygenating rate constant
(T−1), Ka is the reaeration-rate constant (T−1),
and τ is the time of flow along a uniform reach of
stream in which dispersion is not significant.
Show the integrated forms of (a) and (b).

Water Quality Modeling and Prediction

dðBODÞ=ds ¼ ðK d þ K s Þ ðBODÞ þ R

ð10:86Þ
ðbÞ

dD=ds ¼ Kd ðBODÞ  Ka D  A;
ð10:87Þ

where Ks is the settling rate constant (T−1) and s
is the time of flow. Integrating these two equations results in the following deficit equation:
Ds ¼

Kd
Ka  ðKd þ Ks Þ




BOD0 

R
Kd þ KS



fexp½Kd þ KS   expðKa sÞg




Kd
R
A

½1  expðKa sÞ
þ
Kd þ Ks Kd
Ka
þ D0 expðKa sÞ

ð10:88Þ

Exercises

463

where BODo and Do are the BOD and DO
deficit concentrations at s ¼ 0.

dT=ds ¼ kðTE  T Þ=qcD
(a) Assuming that k, D, and TE are constant over interval of time of flow
t2 − t1, integrate the equation above
to derive the temperature T1 at
locations X1.
(b) Develop a model for predicting the
temperature at a point in a nondispersive stream downstream from multiple
point sources (discharges) of heat.

(a) Compare this equation with that found
in Exercise 10.1 if Ks, R, and A are 0
(b) Integrate Eq. (10.86) to predict the
BODτ at any flow time τ.
10:4 Develop finite difference equations for
predicting the steady-state nitrogen component and DO deficit concentrations D in
a multi-section one-dimensional estuary.
Define every parameter or variable used.
10:5 Using Michaelis–Menten kinetics develop
equations for
(a) Predicting the time rate of change of a
nutrient concentration N ðdN=dtÞ as a
function of the concentration of bacterial biomass B;
(b) Predicting the time rate of change in
the bacterial biomass BðdB=dtÞ as a
function of its maximum growth rate
lmax
B , temperature T, B, N, and the
specific-loss rate of bacteria qB ; and
(c) Predicting the time rate of change in
DO deficit ðdD=dtÞ also as a function
of N; B; qB ; and the reaeration-rate
constant Ka ðT 1 Þ.
How would these three equations be altered by
the inclusion of protozoa P that feed on bacteria,
and in turn require oxygen? Also write the differential equations for the time rate of change in
the concentration of protozoa PðdP=dtÞ:
10:6 Many equations for predicting stream
temperature use Eulerian coordinates. The
actual behavior of the stream temperature
is more easily demonstrated if Lagrangian
coordinates (i.e., time of flow t rather than
distance X) are used. Assuming insignificant dispersion, the “time-of-flow” rate of
temperature change of a water parcel as it
moves downstream is

10:7 Consider three well-mixed bodies of water
that have the following constant volumes
and freshwater inflows
Water
body

Volume
(m3)

Flow
(m3/s)

Displacement
time

1

3 × 1012

3 × 103

3.17 years

2

3 × 10

3 × 102

11.6 days

3

3 × 10

3 × 10

2.8 h

8
4

4

The first body is representative of the Great
Lakes in North America, the second is characteristic in size to the upper New York harbor with
the summer flow of the Hudson River, and the
third is typical of a small bay or cove. Compute
the time required to achieve 99% of the equilibrium concentration, and that concentration, of
a substance having an initial concentration, and
that concentration, of a substance having an initial concentration of 0 (at time = 0) and an input
of N (M T−1) for each of the three water bodies.
Assume that the decay rate constant K is 0, 0.01,
0.05, 0.25, 1.0, and 5.0 days−1 and compare the
results.
10:8 Consider the water pollution problem as
shown in Figure below. There are two
sources of nitrogen, 200 mg/l at site 1 and
100 mg/l at site 2, going into the river,
whereas the nitrogen concentration in the
river just upstream of site 1 is 32 mg/l.

464

10

Water Quality Modeling and Prediction

200 mg/l
2
32 mg/l

3

1
100 mg/l

The unknown variables are the fraction of
nitrogen removal at each of those sites that would
achieve concentrations no greater than 20 and
25 mg/l just upstream of site 2 and at site 3,
respectively, at a total minimum cost. Let those
nitrogen removal fractions be X1 and X2. Assuming unit costs of removal as $30 and $20 at site 1
and site 2, respectively, the model can be written as
Minimize
Subject to :

30X1 þ 20X2
200ð1  X1 Þ0:25 þ 8  20
200ð1  X1 Þ0:15 þ 100ð1X2 Þ0:60 þ 5  25
X1  0:9; X2  0:9

Another way to write the two quality constraints of this model is to define variables Si
(i = 1, 2, 3) as the concentration of nitrogen just
upstream of site i. Beginning with a concentration
of 32 mg/l just upstream of site 1, the concentration of nitrogen just upstream of site 2 will be
½32 þ 200ð1  X1 Þ0:25 ¼ S2 and S2  20:
The concentration of nitrogen at site 3 will be
½S2 þ 100ð1  X2 Þ0:60 ¼ S3 and S3  25:
This makes the problem easier to solve using
discrete dynamic programming. The nodes or
states of the network can be discrete values of Si,
the concentration of nitrogen in the river at sites
i (just upstream of sites 1 and 2 and at site 3). The
links represent the decision variable values, Xi
that will result in the next discrete concentration,
Si+1 given Si. The stages i are the different source
sites or river reaches. A section of the network in
stage 1 (reach from site 1 to site 2) will look like

32

[32 + 200(1 - X1)]0.25 = S 2

S2

So if S2 is 20, X1 will be 0.76; if S2 is 15, X1 will
be 0.86. For S2 values of 10 or less X1 must exceed
0.90 and these values are infeasible. The cost
associated with the link or decision will be 30 X1.
Setup the dynamic programming network. It
begins with a single node representing the state
(concentration) of 32 mg/l just upstream of
site 1. It will end with a single node representing the state (concentration) 25 mg/l. The
maximum possible state (concentration value
just upstream of site 2 must be no greater than
20 mg/l. You can use discrete concentration
values in increments of 5 mg/l. This will be a
very simple network. Find the least-cost solution using both forward and backward moving
dynamic programming procedures. Please show
your work.
10:9 Identify three alternative sets (feasible
solutions) of storage lagoon volume
capacities V and corresponding land
application areas A and irrigation volumes Q2t in each month t within a year
that satisfy a 10 mg/l maximum NO3–N
content in the drainage water of a land
disposal system. In addition to the data
listed below, assume that the influent
nitrogen n1t is 50 mg/l each month, with
10% (α = 0.1) of the nitrogen in organic
form. Also assume that the soil is a
well-drained silt loam containing
4500 kg/ha of organic nitrogen in the soil
above the drains. The soil has a monthly
drainage capacity d of 60 cm and has a

Exercises

465

field capacity moisture content M of
10 cm. Maximum plant nitrogen uptake
values Ntmax are 35 kg/ha during April till
October, and 70 kg/ha during May till
September. Finally, assume that because
of cold temperatures, no wastewater irrigation is permitted during November till
March. December, January, and February’s precipitation is in the form of snow
and will melt and be added to the soil
moisture inventory in March.
Reach no.

Design BOD load (mg/l)

removal is required at each discharge site,
solve for the least-cost solution given the
data in the accompanying table. Can you
identify more than one type of model to
solve this problem? How would this
model be expanded to specifically
include both carbonaceous BOD and
nitrogenous BOD and non-point waste
discharges?

Present % removal load

Annual costs of various design BOD removal
60%

75%

85%

90%

1

248

67

0

22,100

77,500

1,20,600

2

408

30

6,30,000

7,80,000

9,87,000

11,70,000

3

240

30

2,10,000

2,77,500

3,23,000

3,78,000

4

1440

30

4,13,000

5,23,000

6,26,000

6,98,000

6

2180

30

5,00,000

6,38,000

7,90,000

9,00,000

7

279

30

8,40,000

10,72,000

12,32,500

13,50,000

Reach
no.

Time
of
flow
(days)

1

0.235

2

1.330

3

1.087

4

2.067

Wastewater
discharge
(103
m3/day)

DO
saturation
conc.
(mg/l)

Maximum
allowable
DO deficit
(mg/l)

DO deficit
of
wastewater
(mg/l)

DO Conc
at
beginning
of reach
(mg/l)

BOD Conc
at
beginning
of reach
(mg/l)

Av. deoxgn
rate constant
for reach
(days−1)

Reaeration
rate
constant
days−1)

5148

10.2

3.2

1

9.5

1.66

0.31

1.02

5023

9.95

2.45

1

8

0.68

0.41

0.6

9

2

1

?

?

0.36

0.63

9.7

3.75

1

9.54

1

0.35

0.09

Entering
reach
flow (103
m3/day)

Total
reach
flow
(103
m3/day)

19

5129

140

4883

30

10,171

10,201

53

1120

1173

5

0.306

0

11,374

11,374

9

2.5

–

?

?

0.34

0.72

6

1.050

98

11,374

11,472

8.35

2.35

1

–

–

0.35

0.14

7

6.130

155

11,472

11,627

8.17

4.17

1

–

–

0.3

0.02

10:10 Consider the problem of estimating the
minimum total cost of waste treatment in
order to satisfy quality standards within a
stream. Let the stream contain seven
homogenous reaches r, reach r = 1 being
at the upstream end and reach r = 7 at the
downstream end. Reaches r = 2 and 4 are
tributaries entering the mainstream at the
beginning of 1, 3, 5, 6, and 7. Point
sources of BOD enter the stream at the
beginning of reaches 1, 2, 3, 4, 6, and 7.
Assuming that at least 60% BOD

10:11 Discuss what would be required to analyze flow augmentation alternatives in
Exercise 12.8. How would the costs of
flow augmentation be defined and how
would you modify water quality models
to
include
flow
augmentation
alternatives?
10:12 Develop a dynamic programming model
to estimate the least-cost number, capacity, and location of artificial aerators to
ensure meeting minimum allowable DO
standards where they would otherwise be

466

10

violated during an extreme low-flow
design condition in a nonbranching section of a stream. Show how wastewater
treatment alternatives, and their costs,
could also be included in the dynamic
programming model.
10:13 Using the data provided, find the
steady-state concentrations Ct of a constituent in a well-mixed lake of constant
volume 30 × 106 m3. The production Nti
of the constituent occurs at three sites i,
and is constant in each of four seasons in
the year. The required fractions of constituent removal Pi at each site i are to be
set so that they are equal at all sites i and
the maximum concentration in the lake
in each period t must not exceed
20 mg/l.
Flow, Qt
(103
m3/day)

Constituent decay
rate, constant, Kt
(days−1)

Period,
t

Days
in
period

1

100

90

0.02

2

80

150

0.03

3

90

200

0.05

4

95

120

0.04

Water Quality Modeling and Prediction

particular period t were 1.1, 0.1, and
0.8 mg/l, respectively. Assume that all
other nutrients required for algal growth
are in abundance. The algal species of
concern are three in number and are
denoted by j = 1, 2, 3. The data required
to estimate the probable maximum algal
bloom biomass concentration are given in
the accompanying table. Compute this
bloom potential for all ki and k equal to 0,
0.8, and 1.0.
Parameter (algae species
index j)

Parameter value
1

2

3

a1j = mg N/mg dry wt of
algae j

0.04

0.01

0.20

a2j = mg P/mg dry wt of
algae j

0.06

0.02

0.10

a3j = mg Si/mg dry wt of
algae j

0.08

0.01

0.03

Dj = morality and grazing
rate constant (days−1)

0.6

0.4

0.20

dj = morality rate constant,
(days−1)

0.3

0.1

0.10

v = extinction reduction
rate constant for dead algae,
(days−1)

0.07

0.07

0.07

Constituent discharge
site, i

Constituent production
(kg/day)

hmax
= max. extinction
j
coef. (m−1)

0.07

0.07

0.10

1

38,000

0.03

0.03

25,000

gmin
j Z = min. extinction
coef. (m−1)

0.01

2
3

47,000

gj = increase in extinction
coef. per unit increase in
mg/l (g/m3) of dry wt of
species j (m2/g)

0.05

0.164

0.04

Nutrient index i

1

2

3

10:14 Suppose that the solution of a model such
as that used in Exercise 10.13, or measured data, indicated that for a wellmixed portion of a saltwater lake, the
concentrations of nitrogen (i = 1), phosphorus (i = 2), and silicon (i = 3) in a

Nutrient

N

P

Si

li = mineralization rate
constant, (days−1)

0.02

0.69

0.62

Exercises

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467

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River Basin Modeling

Multipurpose river basin development typically
involves the identification and use of both
structural and nonstructural measures designed to
increase the reliability and decrease the cost of
municipal, industrial, and agriculture water supplies, to protect against droughts and floods, to
improve quality, to provide for commercial
navigation and recreation, to enhance aquatic
ecosystems, and to produce hydropower, as
appropriate for the particular river basin. Structural measures may include diversion canals,
reservoirs, hydropower plants, levees, flood
proofing, irrigation delivery and drainage systems, navigation locks, recreational facilities,
groundwater wells, and water and treatment
treatment plants along with their distribution and
collection systems. Nonstructural measures may
include land-use controls and zoning, flood
warning and evacuation measures, and economic
incentives that affect human behavior with regard
to water and watershed use. Planning the development and management of water resource systems involves identifying just what and when and
where structural or nonstructural measures are
needed, the extent to which they are needed, and
their combined economic, environmental, ecological, and social impacts. This chapter introduces some modeling approaches for doing this.
Having just reviewed some water quality modeling approaches in the previous chapter, this
chapter focuses on quantity management.

11.1

11

Introduction

Various types of models can be used to assist
those responsible for planning and managing
various components of river systems. These
components include streams, rivers, lakes, reservoirs, and wetlands, and diversions to demand
sites that could be within or outside the basin
boundaries. Each of these components can be
impacted by water management policies and
practices. The management of any single component can impact the performance of other
components in the basin. Hence, for the overall
management of the water in river basin systems, a
systems view is usually taken. This systems view
requires the modeling of multiple interacting and
interdependent components. These multicomponent models are useful for analyzing alternative
designs and management policies for improving
the performance of integrated river basin systems.
The discussion in this chapter is limited to
water quantity management. Clearly the regimes
of flows, velocities, volumes, and other properties
of water quantity will impact the quality of that
water as well. However, unless water allocations
allocations and uses are based on requirements for
water quality, such as for the dilution of pollution,
water quality does not normally affect water
quantity. For this reason among others, it is
common to separate discussions of water quantity

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_11

469

470

11

management from water quality management
(Chap. 10). However, when attempting to predict
the impacts of any management policy on both
water quantity and quality, both water quantity
and quality models are needed.
This chapter begins with a discussion of
selecting appropriate model time periods that will
depend on the issues being addressed as well as
on the variability of the water supplies and
demands. Discussed next are methods for estimating streamflows at various sites of interest
throughout a basin based on gage (measured)
flows at other sites. Following these discussions
several methods are reviewed and compared for
estimating reservoir storage requirements for
water supplies. Model components are defined
for withdrawals and diversions, and for reservoir
storage. Reservoir storage can serve the needs
over time for water supply, flood control, recreation, and hydroelectric power generation. Next
flood control structures, such as levees and
channel flow capacity improvements at potential
flood damage in a river basin are introduced.
These components are then combined into a
multiple purpose multi-objective planning model
for a hypothetical river basin. The chapter concludes with an introduction to some dynamic
models for assisting in the scheduling and time
sequencing of multiple projects within a river
basin.

11.2

Model Time Periods

When analyzing and evaluating various water
management plans designed to distribute the
natural unregulated flows over time and space, it
is usually sufficient to consider average conditions within discrete time periods. In optimization
models, weekly, monthly, or seasonal flows are
commonly used as opposed to daily flows. The
shortest time period duration usually considered
in optimization models developed for identifying
and evaluating alternative water management
plans and operating policies is one that is no less
than the time water takes to flow from the upper

River Basin Modeling

end of the applicable river basin to the lower end
of the basin. In this case stream and river flows
can be defined by simple mass balance or continuity equations. For shorter duration time periods
flow routing may be required.
The actual length or duration of each
within-year period defined in a model may vary
from period to period. Modeled time period time
periods need not be equal. Generally what is
important is to capture in the model the needed
capacities of infrastructure that are determined in
large measure by the variation in supplies and
demands. These variations should be captured in
the model by appropriately selecting the number
and duration of time periods. If say over a
three-month period there is little variation in both
water supplies and demands or for the purposes
water serves, such as flood control, hydropower,
or recreation, there is no need to divide that
three-month period into multiple time periods.
Another important factor to consider in making a decision regarding the number and duration
of time periods to include in any model is the
purpose for which the model is to be used. Some
analyses are concerned only with identifying
designs designs and operating policies of various
engineering projects for managing water resources at some fixed time (say a typical year) in the
future. Multiple years of hydrological records are
used, usually in simulation models, to obtain an
estimate of just how well a system might perform, at least in a statistical sense, in that future
time period. The within-year period durations
can have an impact on those performance indicator values as well as on the estimate of
over-year as well as within-year storage that may
be needed to meet various goals. These static
analyses are not concerned with investment
project scheduling or sequencing.
Dynamic planning models are used to estimate the impacts of changing conditions over
time. These changes could include hydrological
inputs, economic, environmental and other
objectives, water demands, and design and
operating parameters. As a result, dynamic
models generally span many more years than do

11.2

Model Time Periods

static models, but they may have fewer
within-year periods.

11.3

Streamflow Estimation

Water resource managers need estimates of
streamflows at each site, where management
decisions are being considered. These streamflow
estimates can be based on the results of
rainfall-runoff models or on measured historical
flows at gage sites. For modeling alternative
management policies, these gage-based flows at
the sites of interest should be those that would
have occurred under natural conditions. These are
called naturalized flows that have been derived
from measured flows or rainfall–runoff models
and then adjusted to take into account any
upstream regulation and diversions. Many gage
flow values reflect actions such as diversions and
reservoir releases that occurred upstream that
altered the downstream flows, unless such
upstream water management and use policies are
to continue, these measured gage flows should be
converted to unregulated or natural flows prior to
their use in management models.
Assuming that unregulated streamflow data
are available at gage sites, these data can be used
to estimate the unregulated flows at sites where
they are needed. These sites would include any
place where diversions might occur or where
reservoirs for regulating flows might be built.
Consider, for example, the simple river basin
illustrated in Fig. 11.1. Assume streamflows
have been recorded over a number of years at

Fig. 11.1 River basin
gage sites where
streamflows are measured
and recorded

471

gage sites 1 and 9. Knowledge of the flows Qst in
each period t at gage sites 1 and 9 permits the
estimation of flows at any other site in the basin
as well as the incremental flows between those
sites in each period t.
The method used to estimate flows at ungaged
sites will depend on the characteristics of the
watershed or river basin. In humid regions where
streamflows increase in the downstream direction
due to rainfall runoff, and the spatial distribution
of average monthly or seasonal rainfall is more or
less the same from one part of the river basin to
another, the runoff per unit land area is typically
assumed constant. In these situations, estimated
flows, qst , at any site s can be based on the
watershed areas, As, contributing flow to those
sites, and the corresponding streamflows and
watershed areas above the nearest or most representative gage sites.
For each gage site, the runoff per unit land area
can be calculated by dividing the gage flow Qst by
the upstream drainage area, As. This can be done
for each gage site in the basin. Thus for any gage
site g, the runoff per unit drainage area in month
or season t is Qgt divided by Ag. This runoff per
unit land area times the drainage area upstream of
any site s of interest will be the estimated
streamflow in that period at that site s. If there are
multiple gage sites, such as illustrated in
Fig. 11.1, the estimated streamflow at some
ungaged site s can be a weighted combination of
those unit area runoffs times the area contributing
to the flow at site s. The nonnegative weights, wg,
that sum to 1, reflect the relative significance of
each gage site with respect to site s. Their values

472

11

will be based on the judgments of those who are
familiar with the basin’s hydrology.
(
Qst

¼

X
g

)
wg Qgt =Ag

As

ð11:1Þ

In all the models developed and discussed
below, the variable Qst will refer to the mean
natural (unregulated) flow (L3/T) at a site s in a
period t.
The difference between the natural streamflows at any two sites is called the incremental
flow. Using Eq. 11.1 to estimate streamflows will
result in positive incremental flows. The downstream flow will be greater than the upstream
flow. In arid regions runoff is not constant over
the region. Incremental flows may not exist and
hence due to losses, the flows may be decreasing
in the downstream direction. In these cases there
is a net loss in flow in the downstream direction.
This might be the case when a stream originates
in a wet area and flows into a region that receives
less rainfall. In such arid areas the runoff is often
less than the evapotranspiration and infiltration
into the ground along the stream channel.
For stream channels where there exist relatively uniform conditions affecting loss loss,
where there are no known sites where the stream
abruptly enters or exits the ground, as can occur
in karst conditions, the average streamflow for a
particular period t at site s can be based on the
nearest or most representative gage flow, Qgt , and
a loss rate per unit length of the stream or river,
Lgs between gage site g and an ungaged site s. If
there are at least two gage sites along the portion
of the stream or river that is in the dry region, one
can compute the loss of flow per unit stream
length, and apply this loss rate to various sites
along the stream or river. This loss rate per unit
length may not be constant over the entire length
between the gage stations, or even for all flow
rates, however. Losses will likely increase with
increasing flows simply because more water
surface is exposed to evaporation and seepage. In
these cases one can define a loss rate per unit

River Basin Modeling

length of stream or river as a function of the
magnitude of flow.
In watersheds characterized by significant
elevation changes and consequently varying
rainfall and runoff runoff, other methods may be
required for estimating average streamflows at
ungaged sites. The selection of the most appropriate method to use, as well as the most
appropriate gage sites to use for estimating the
streamflow, Qst , at a particular site s can be a
matter of judgment. The best gage site need not
necessarily be the nearest gage to site s, but
rather the site most similar with respect to
important hydrologic variables.
The natural incremental flow between any two
sites is simply the difference between their
respective natural flows.

11.4

Streamflow Routing

If the duration of a within-year period is less than
the time of flow throughout the stream or river
system being modeled, and the flows vary within
the system, some type of streamflow routing
must be used to keep track of where the varying
amounts of water are in each time period. There
are many proposed routing methods (as described in any hydrology text or handbook, e.g.,
Maidment 1993). Many of these more traditional
methods can be approximated with sufficient
accuracy using relatively simple methods. Two
such methods are described in the following
paragraphs.
The outflow, Ot, from a reach of stream or
river during a time period t is a function of the
amount of water in that reach, i.e., its initial
storage, St, and its inflow, It. Because of bank
storage, that outflow is often dependent on whether the quantity of water in the reach is increasing or decreasing. If bank inflows and outflows
are explicitly modeled, or if bank storage is
not that significant, the outflow from a reach
in any period t can be expressed as a simple
two-parameter power function of the form
a(St + It)b. Mass balance equations, that may take

11.4

Streamflow Routing

473

into account losses, update the initial storage
volumes in each succeeding time period. The
reach–dependent parameters a and b can be
determined through calibration procedures such
as genetic algorithms (Chap. 5) using a time series
of reach inflows and outflows. The resulting
outflow function is typically concave (the
parameter b will be less than 1), and thus the
minimum value of St + It must be at least 1. If due
to evaporation or other losses the reach volume
drops below this or any preselected higher
amount, the outflow can be assumed to be 0.
Alternatively one can adopt a three- or
four-parameter routing approach that fits a wider
range of conditions. Each stream or river reach
can be divided into a number of segments. That
number n is one of the parameters to be determined. Each segment s can be modeled as a
storage unit, having an initial storage volume, Sst,
and an inflow, Ist. The three-parameter approach
assumes the outflow, Ost, is a linear function of
the initial storage volume and inflow:
Ost ¼ a Sst þ bIst

ð11:2Þ

Equation 11.2 applies for all time periods
t and for all reach segments s in a particular
reach. Different reaches will likely have different
values of the parameters n, α, and β. The calibrated values of α and β are nonnegative and no
greater than 1. Again a mass balance equation
updates each segment’s initial storage volume in
the following time period. The outflow from each
reach segment is the inflow into the succeeding
reach segment.
The four-parameter approach assumes that the
outflow, Ost, is a nonlinear function of the initial
storage volume and inflow
Ost ¼ ða Sst þ bIst Þc

ð11:3Þ

The parameter γ is greater than 0 and no
greater than 1. In practice γ is very close to 1.
Again the values of these parameters, including
the number of reaches n, can be found using
nonlinear optimization methods, such as genetic
algorithms, together with a time series of
observed reach inflows and outflows.

Note the flexibility available when using the
three- or four-parameter routing approach. Even
blocks of flow can be routed a specified distance
downstream over a specified time, regardless of
the actual flow. This can be done by setting α and
γ to 1, β to 0, and the number of segments n to
the number of time periods it takes to travel that
distance. This may not be very realistic, but there
exist some river basin reaches where managers
believe this particular routing applies.

11.5

Lakes and Reservoirs

Lakes and reservoirs are sites in a basin where
surface water storage needs to be modeled. Thus,
variables defining the water volumes at those
sites must be defined. Let Sst be the initial storage
volume of a lake or reservoir at site s in period
t. Omitting the site index s for the moment, the
final storage volume in period t, St+1, (which is
the same as the initial storage in the following
period t + 1) will equal the initial volume, St,
plus the net surface and groundwater inflows, Qt,
less the release or discharge, Rt, and evaporation
and seepage losses, Lt. All models of lakes and
reservoirs include this mass balance equation for
each period t being modeled.
St þ Qt Rt Lt ¼ St þ 1

ð11:4Þ

The release from a natural lake is a function of
its surrounding topography and its water surface
elevation. It is determined by nature, and unless
it is made into a reservoir its discharge or release
is not controlled or managed. The release from a
reservoir is controllable, and, as discussed in
Chaps. 4 and 8, is usually a function of the
reservoir storage volume and time of year.
Reservoirs also have fixed storage capacities,
K. In each period t, reservoir storage volumes, St,
cannot exceed their storage capacities, K.
St  K

for each period t:

ð11:5Þ

Equations 11.4 and 11.5 are the two fundamental equations required when modeling water
supply reservoirs. They apply for each period t.

474

The primary purpose of all reservoirs is to
provide a means of regulating downstream surface water flows over time and space. Other
purposes may include storage volume management for recreation and flood control, and storage
and release management for hydropower production. Reservoirs are built to alter the natural
spatial and temporal distribution of the streamflows. The capacity of a reservoir together with
its release (or operating) policy determine the
extent to which surface water flows can be stored
for later release.
The use of reservoirs for temporarily storing
streamflows often results in a net loss of total
streamflow due to increased evaporation and
seepage. Reservoirs also bring with them changes in the ecology of a watershed and river
system. They may also displace humans and
human settlements. When considering new
reservoirs, any benefits derived from regulation
of water supplies, from flood damage reduction,
from hydroelectric power, and from any navigational and recreational activities should be
compared to any ecological and social losses and
costs. The benefits of reservoirs can be substantial, but so may the costs. Such comparisons of
benefits and costs are always challenging
because of the difficulty of expressing all such
benefits and costs in a common metric (Chap. 9).
Reservoir storage capacity can be divided
among the three major uses: (1) the active storage used for downstream flow regulation and for
Fig. 11.2 Total reservoir
storage volume capacity
consisting of the sum of
dead storage, active
storage, and flood control
storage capacities

11

River Basin Modeling

water supply, recreational development or
hydropower production; (2) the the dead storage
used for sediment collection; and (3) the flood
storage capacity reserved to reduce potential
downstream flood damage during flood events.
These separate storage capacities are illustrated
in Fig. 11.2. The distribution of active and flood
control storage capacities may change over the
year. For example there is no need for flood
control storage in seasons that are not likely to
experience floods.
The next several sections of this chapter
address how these capacities may be determined.

11.5.1

Estimating Active Storage
Capacity

11.5.1.1
Mass diagram Analyses
Perhaps one of the earliest methods used to calculate the active storage capacity required to
meet a specified reservoir release, Rt, in a
sequence of periods t, was developed by Rippl
(1883). His mass diagram analysis is still used
today by many planners. It involves finding the
maximum positive cumulative difference
between a sequence of prespecified (desired)
reservoir releases Rt and known inflows Qt. One
can visualize this as starting with a full reservoir,
and going through a sequence of simulations in
which the inflows and releases are added and
subtracted from that initial storage volume value.

11.5

Lakes and Reservoirs

Doing this over two cycles of the record of
inflows will identify the maximum deficit volume associated with that release. This is the
required reservoir storage. Having this initial
storage volume, the reservoir would always have
enough water to meet the desired releases.
However, this only works if the sum of all the
desired releases does not exceed the sum of all
the inflows over the same sequence of time
periods. Reservoirs cannot make water.
Equation 11.6 represents this process. The
active storage capacity, Ka, will equal the maximum accumulated storage deficit one can find
Fig. 11.3 The Rippl or
mass diagram method for
identifying reservoir active
storage capacity
requirements. The releases
Rt are assumed constant for
each period t

Fig. 11.4 Alternative plot
for identifying reservoir
active storage capacity
requirements

475

over some interval of time within two successive
record periods, T.
"
Ka ¼ maximum

j
X

#
ðRt Qt Þ ;

ð11:6Þ

t¼i

where 1 ≤ i ≤ j ≤ 2T.
Equation 11.6 is the analytical equivalent of
graphical procedures proposed by Rippl for
finding the active storage requirements. Two of
these graphical procedures are illustrated in
Figs. 11.3 and 11.4 for a 9-period inflow record
of 1, 3, 3, 5, 8, 6, 7, 2, and 1. Rippl’s original

476

approach, shown in Fig. 11.3, involves plotting
Pt
the cumulative inflow
t¼1 Qs versus time
t. Assuming a constant reservoir release, R, in
each period t, a line with slope R is placed so that
it is tangent to the cumulative inflow curve. To
the right of these points of tangency the release
R exceeds the inflow Qt. The maximum vertical
distance between the cumulative inflow curve
and the release line of slope R equals the maximum water deficit, and hence the required active
storage capacity. Clearly, if the average release
R is greater than the mean inflow, a reservoir will
not be able to meet the demand no matter what its
active storage capacity.
An alternative way to identify the required
reservoir storage capacity is to plot the cumulaP
tive nonnegative deviations, ts ðRs  Qs Þ, and
note the biggest total deviation, as shown in
Fig. 11.4.
These graphical approaches do not account
for losses. Furthermore, the method shown in
Fig. 11.3 is awkward if the desired releases in
each period t are not the same. The equivalent
method shown in Fig. 11.4 is called the sequent
peak method. If the sum of the desired releases
does not exceed the sum of the inflows, calculations over at most two successive hydrologic
records of flows are needed to identify the largest
cumulative deficit inflow. After that the procedure will produce repetitive results. It is much
easier to consider changing release values when
determining the maximum deficit by the sequent
peak method.

11.5.1.2
Sequent peak analyses
The sequent peak procedure is illustrated in
Table 11.1. Let Kt be the maximum total storage
requirement needed for periods 1 up through
period t. As before, let Rt be the required release
in period t, and Qt be the inflow in that period.
Setting K0 equal to 0, the procedure involves
calculating Kt using Eq. 11.7 consecutively for
up to twice the total length of record. This
assumes that the record repeats itself to take care
of the case when the critical sequence of flows
occurs at the end of the streamflow record, as

11

River Basin Modeling

indeed it does in the example 9-period record of
1, 3, 3, 5, 8, 6, 7, 2, and 1.
Kt ¼ Rt Qt þ Kt1
¼0

if positive;
otherwise

ð11:7Þ

The maximum of all Kt is the required storage
capacity for the specified releases Rt and inflows,
Qt. Table 11.1 illustrates this sequent peak procedure for computing the active capacity Ka, i.e.,
the maximum of all Kt, required to achieve a
release Rt = 3.5 in each period given the series of
9 streamflows. Note this method does not require
all releases to be the same.

11.5.2

Reservoir Storage-Yield
Functions

Reservoir storage-yield functions define the
minimum active active storage capacity required
to insure a given constant release rate for a
specified sequence of reservoir inflows. Mass
diagrams, sequent peak analyses, and linear
optimization (Chap. 4) are three methods that can
be used to define these functions. Given the same
sequence of known inflows and specified releases, each method will provide identical results.
Using optimization models, it is possible to
obtain such functions from multiple reservoirs
and to account for losses based on storage volume surface areas, as will be discussed later.
There are two ways of defining a linear optimization (linear programming) model to estimate
the active storage capacity requirements. One
approach is to minimize the active storage
capacity, Ka, subject to minimum required constant releases, Y, the yield. This minimum active
storage capacity is the maximum storage volume,
St, required given the sequence of known inflows
Qt, and the specified yield, Y, in each period
t. The problem is to find the storage volumes, St,
and releases, Rt that
Minimize Ka

ð11:8Þ

11.5

Lakes and Reservoirs

477

Table 11.1 Illustration of the sequent peak procedure for computing active storage requirements

subject to
mass balance constraints
St þ Qt Rt ¼ St þ 1
T þ1 ¼ 1

t ¼ 1; 2; . . .; T;

ð11:9Þ

Alternatively one can maximize the constant
release yield, Y, for various values of active
storage capacity, Ka, subject to the same constraint Eqs. 11.9–11.11.
Maximize Y

ð11:12Þ

capacity constraints
St  Ka

t ¼ 1:2; . . .; T

ð11:10Þ

minimum release constraints
Rt  Y

t ¼ 1:2; . . .; T

for various values of the yield, Y.

ð11:11Þ

Constraints 11.9 and 11.11 can be combined
to reduce the model size by T constraints.
St þ Qt Y  St þ 1
T þ1 ¼ 1

t ¼ 1; 2; . . .; T;

ð11:13Þ

The solutions of these two linear programming
models, using the 9-period flow sequence referred

478

11

River Basin Modeling

Fig. 11.5 Storage-yield
for the sequence of flows 1,
3, 3, 5, 8, 6, 7, 2, and 1

to above and solved for various values of yield or
capacity, respectively, are plotted in Fig. 11.5.
The results are the same as could be found using
the mass diagram or sequent peak methods.
There is a probability that the storage-yield
function just defined will fail. A record of only 9
flows, for example, is not very long and hence
will not give one much confidence that they will
define the critical low-flow period of the future.
One rough way to estimate the reliability of a
storage-yield function is to rearrange and rank
the inflows in order of their magnitudes. If there
are n ranked inflows there will be n + 1 intervals
separating them. Assuming there is an equal
probability that any future flow could occur in
any interval between these ranked flows, there is
a probability of 1/(n + 1) that a future flow will
be less than the lowest recorded flow. If that
record low flow occurs during a critical low-flow
period, more storage may be required than indicated in the function.
Hence for a record of only 9 flows that are
considered representative of the future, one can
be only about 90% confident that the resulting
storage-yield function will apply in the future.
One can be only 90% sure of the predicted yield
Y associated with any storage volume K. A much
more confident estimate of the reliability of any
derived sstorage-yield function can be obtained
by synthetic flows to supplement any measured
streamflow record, taking parameter uncertainty

into account (as discussed in Chaps. 6 and 8).
This will provide alternative sequences as well as
more intervals between ranked flows.
While the mass diagram and sequent peak
procedures are relatively simple, they are not
readily adaptable to reservoirs where evaporation
losses and/or lake level regulation are important
considerations, or to problems involving more
than one reservoir. Mathematical programming
(optimization) methods provide this capability.
These optimization methods are based on mass
balance equations for routing flows through each
reservoir. The mass balance or continuity equations explicitly define storage volumes (and
hence storage areas from which evaporation
occurs) at the beginning of each period t.

11.5.3

Evaporation Losses

Evaporation losses, Lt, from lakes and reservoirs,
if any, take place on their surface areas. Hence to
compute these losses their surface areas must be
estimated in each period t. Storage surface areas
are functions of the storage volumes, St. These
functions are typically concave, as shown in
Fig. 11.6.
In addition to the storage area-volume function, seasonal surface water evaporation loss
depths, Emax
, must be assumed, perhaps based on
t
measured evaporation losses over time.

11.5

Lakes and Reservoirs

479

Fig. 11.6 Storage surface
area as a function of
reservoir storage volume
along with its linear
approximation. The slope
parameter a is the assumed
increase in surface area
associated with a unit
increase in the storage
volume

Multiplying the average surface area, At, based
on the initial and final storage volumes, St and St
max
, yields the volume of
+1, by the loss depth, Et
evaporation loss, Lt, in the period t. The linear
approximation of that loss is
Lt ¼ ½ao þ aðSt þ St þ 1 Þ=2Etmax

ð11:14Þ

Letting
at ¼ 0:5 a Etmax

ð11:15Þ

the mass balance equation for storage volumes
that include evaporation losses in each period
t can be approximated as
ð1  at ÞSt þ Qt Rt ao Etmax ¼ ð1 þ at ÞSt þ 1

An alternative way to estimate evaporation
loss that does not require a surface area—storage
volume relationship, such as shown in Fig. 11.6,
is to define the storage elevation-storage volume
function. Subtracting the evaporation loss depth
from the initial surface elevation associated with
the initial storage volume will result in an
adjusted storage elevation which in turn defines
the initial storage volume after evaporation losses
have been deducted. This adjusted initial volume
can be used in continuity Eqs. 11.9 or 11.13.
This procedure assumes that evaporation is only
a function of the initial storage volume in each
time period t. For relatively large volumes and
short time periods such an assumption is usually
satisfactory.

ð11:16Þ

11.5.4
If Eq. 11.16 are used in optimization models
for identifying preliminary designsof a proposed
reservoir and if the active storage storage capacity
turns out to be essentially zero, or just that required
to provide for the fixed evaporation loss, ao Emax
,
t
then clearly any reservoir at the site is not justified.
These mass balance equations together with any
reservoir storage capacity constraints should be
removed from the model before resolving it again.
This procedure is simpler than introducing 0,1
integer variables that will remove the terms ao
Emax
in Eq. 11.16 if the active storage volume is 0
t
(using methods discussed in Chap. 4).

Over- and Within-Year
Reservoir Storage
and Yields

An alternative approach to modeling reservoirs is
to separate out over-year storage and within-year
storage, and to focus not on total reservoir
releases, but on parts of the total releases that can
be assigned specific reliabilities. These release
components we call yields. To define these yields
and the corresponding reservoir rules, we divide
this section into four parts. The first outlines a
method for estimating the reliabilities of various
constant annual minimum flows or yields. The

480

second discusses a modeling approach for estimating over-year and within-year active storage
requirements to deliver a specified annual and
within-year period yields having a specified reliability. The third and fourth parts expand this
modeling approach to include multiple yields
having different reliabilities, evaporation losses,
and the construction of reservoir operation rule
curves using these flow release yields.
It will be convenient to illustrate the yield
models and their solutions using a simple
example consisting of a single reservoir and two
within-year periods per year. This example will
be sufficient to illustrate the method that can be
applied to models having more within-year
periods. Table 11.2 lists the nine years of

11

River Basin Modeling

available streamflow data for each within-year
season at a potential reservoir site. These
streamflows are used to solve and compare the
solutions of various yield models as well as to
illustrate the concept of yield reliability.

11.5.4.1
Reliability of Annual yields
The maximum flow that can be made available at
a specific site by the regulation of the historic
streamflows from a reservoir of a given size is
often referred to as the “firm yield” or “safe
yield.” These terms imply that the firm or safe
yield is that yield which the reservoir will always
be able to provide. Of course, this may not be
true. If historical flows are used to determine this
yield, then the resulting yield may be better

Table 11.2 Recorded unregulated historical streamflows at a reservoir site

11.5

Lakes and Reservoirs

called an ‘‘historical yield.’’ Historical and firm
yield are often used synonymously.
A minimum flow yield is 100% reliable only
if the sequence of flows in future years will never
sum to a smaller amount than those that have
occurred in the historic record. Usually one
cannot guarantee this condition. Hence associated with any historic yield is the uncertainty,
i.e., a probability, that it might not always be
available in the future. There are some ways of
estimating this probability.
Referring to the nine-year streamflow record
listed in Table 11.2, if no reservoir is built to
increase the yields downstream of the reservoir
site, the historic firm yield is the lowest flow on
record, namely 1.0 that occurred in year 5. The
reliability of this annual yield is the probability
that the streamflow in any year is greater than or
equal to this value. In other words, it is the
probability that this flow will be equaled or
exceeded. The expected value of the exceedance
probability of the lowest flow in an n-year record
is approximately n/(n + 1), which for the
n = 9 year flow record is 9/(9 + 1), or 0.90. This
is based on the assumption that any future flow
has an equal probability of being in any of the
intervals formed by ordering the record of 9
flows from the lowest to the highest value, and
that the lowest value has a rank of 9.
Ranking the n flows of record from the highest
to the lowest and assigning the rank m of 1 to the
highest flow, and n to the lowest flow, the
expected probability p that any flow of rank m will
be equaled or exceeded in any year is approximately m/(n + 1). An annual yield having a
probability p of exceedance will be denoted as Yp.
For independent events, the expected number
of years until a flow of rank m is equaled or
exceeded is the reciprocal of its probability of
exceedance p, namely 1/p = (n + 1)/m. The recurrence time or expected time until a failure (a
flow less than that of rank m) is the reciprocal of
the probability of failure in any year. Thus, the
expected recurrence time Tp associated with a
flow having an expected probability p of exceedance is 1/(1 − p).

481

11.5.4.2

Estimation of Active
Reservoir Storage
Capacities
for Specified Yields
A reservoir with active over-year storage capacity provides a means of increasing the magnitude
and/or the reliabilities of various annual yields.
For example, the sequent peak algorithm defined
by Eq. 11.7 provides a means of estimating the
reservoir storage volume capacity required to
meet various “firm” yields Y0.9, associated with
the nine annual flows presented in Table 11.1.
The same yields can be obtained from a linear
optimization model that minimizes active
over-year storage capacity, Koa,
Minimize Kao

ð11:17Þ

This active over-year storage capacity must
satisfy the following storage continuity and
capacity constraint equations involving only
annual storage volumes, Sy, inflows, Qy, yields,
Yp, and excess releases, Ry. For each year y:
Sy þ Qy  Yp  Ry ¼ Sy þ 1
Sy  Kao

ð11:18Þ
ð11:19Þ

Once again, if the year index y = n, the last
year of record, then year y + 1 is assumed to
equal 1. For annual yieldsof 3 and 4, the
over-year storage requirements are 3 and 8,
respectively, as can be determined just by
examining the right-hand column of annual flows
in Table 11.2.
The over-year model, Eqs. 11.17–11.19,
identifies only annual or over-year storage
requirements based on specified (known) annual
flows, Qy, and specified constant annual yields,
Yp. Within-year periods t requiring constant
yields ypt that sum to the annual yield Yp may
also be considered in the estimation of the
required over-year and within-year or total active
storage capacity, Ka. Any distribution of the
over-year yield within the year that differs from
the distribution of the within-year inflows may

482

11

require additional active reservoir storage
capacity. This additional capacity is called the
within-year storage capacity.
The sequent peak method, Eq. 11.7, can be
used to obtain the total over-year and within-year
active storage capacity requirements for specified
within-year period yields, ypt. Alternatively a
linear programming model can be developed to
obtain the same information along with associated reservoir storage volumes. The objective is
to find the minimum total active storage capacity,
Ka, subject to storage volume continuity and
capacity constraints for every within-year period
of every year. This model is defined as
minimize Ka

ð11:20Þ

subject to
Sty þ Qty  ypt  Rty ¼ St þ 1;y
Sty  Ka

8t; y

8t; y ð11:21Þ
ð11:22Þ

In Eq. 11.21, if t is the final period T in year y,
the next period T + 1 = 1 in year y + 1, or year 1
if y is the last year of record, n.
The within-year storage requirement, Kwa , is
the difference in the active capacities resulting
from these two models, Eqs. 11.17–11.19, and
Eqs. 11.20–11.22.
Table 11.3 shows some results from solving
both of the above models. The over-year storage
capacity requirements, Koa, are obtained from
Eqs. 11.17–11.19. The combined over-year and
within-year capacities, Ka, are obtained from
solving Eqs. 11.20–11.22. The difference between
the over-year storage capacity, Koa, required to
meet only the annual yields and the total capacity,
Ka, required to meet each specified within-year
yield distribution of those annual yields is the
within-year active storage capacity Kwa .
Clearly, the number of continuity and reservoir capacity constraints in the combined
over-year and within-year model (Eqs. 11.20–
11.22) can become very large when the number
of years n and within-year periods T are large.
Each reservoir site in the river system will

River Basin Modeling

require 2nT continuity and capacity constraints.
Not all these constraints are necessary, however.
It is only a subset of the sequence of flows within
the total record of flows that generally determines
the required active storage capacity Ka of a
reservoir. This is called the critical period. This
critical period is often used in engineering studies
to estimate the historical yield of any particular
reservoir or system of reservoirs.
Even though the severity of future droughts is
unknown, many planners accept the traditional
practice of using the historical critical drought
period for reservoir design and operation studies
on the assumption that having observed such an
event in the past, it is certainly possible to
experience similar conditions in the future. In
some parts of the world, notably those countries
in the lower portions of the southern hemisphere,
historical records are continually proven to be
unreliable indicators of future hydrological conditions. In these regions especially, synthetically
generated flows based on statistical methods
(Chap. 6) are more acceptable as a basis for yield
estimation.
Over and within-year storage Capacity
To begin the development of a smaller, but more
approximate, model, consider each combined
over-year and within-year storage reservoir to
consist of two separate reservoirs in series
(Fig. 11.7). The upper reservoir is the over-year
storage reservoir, whose capacity required for an
annual yield is determined by an over-year
model, e.g., Eqs. 11.17–11.19. The purpose of
the ‘‘downstream’’ within-year reservoir is to
distribute as desired in each within-year period
t a portion of the annual yield produced by the
‘‘upstream’’ over-year reservoir. Within-year
storage capacity would not be needed if the
distribution of the average inflows into the upper
over-year reservoir exactly coincided with the
desired distribution of within-year yields.
Otherwise within-year storage may be required.
The two separate reservoir capacities summed
together will be an approximation of the total
active reservoir storage requirement needed to
provide those desired within-year period yields.

11.5

Lakes and Reservoirs

483

Table 11.3 Active Storagerequirements for various within-year yields

Assume the annual yield produced and
released by the over-year reservoir is distributed
in each of the within-year periods in the same
ratio as the average within-year inflows divided
by the total average annual inflow. Let the ratio
of the average period t inflow divided by the total
annual inflow be βt. The general within-year
model is to find the minimum within-year storage
capacity, Kwa , subject to within-year storage volume continuity and capacity constraints.
Minimize Kaw

ð11:23Þ

subject to
st þ bt Yp  ypt ¼ st þ 1
Fig. 11.7 Approximating a combined over-year and
within-year reservoir as two separate reservoirs, one for
creating annual yields, the other for distributing them as
desired in the within-year periods

8t

T þ1 ¼ 1
ð11:24Þ

484

11

st  Kaw

8t

ð11:25Þ

Since the sum of βt over all within-year periods t is 1, the model guarantees that the sum of
the unknown within-year yields, ypt, equals the
annual yield, Yp.
The over-year model, Eqs. 11.17–11.19, and
within-year model, Eqs. 11.23–11.25, can be
combined into a single model for an n-year
sequence of flows
Minimize Ka

ð11:26Þ

subject to
Sy þ Qy  Yp  Ry ¼ Sy þ 1
if

8y

y ¼ n; y þ 1 ¼ 1
Sy  Kao

8y

st þ bt Yp  ypt ¼ st þ 1
if

ð11:28Þ
8t

t ¼ T; T þ 1 ¼ 1
st  Kaw
X
t

ð11:27Þ

ð11:29Þ

8t

ð11:30Þ

ypt ¼ Yp

ð11:31Þ

Ka  Kao þ Kaw

ð11:32Þ

Constraint 11.31 is not required due to
Eq. 11.29, but is included here to make it clear
that the sum of within-year yields will equal the
over-year yield. Such a constraint will be
required for each yield of reliability p if multiple
yields of different reliabilities are included in the
model. In addition, constraint Eq. 11.30 can be
combined with Eq. 11.32, saving a constraint. If
this is done, the combined model contains
2n + 2T + 1 constraints, compared to the more
accurate model, Eqs. 11.20–11.22, that contains
2nT constraints.
If the fractions βt are based on the ratios of the
average within-year inflow divided by average

River Basin Modeling

annual inflow in the two within-year periods
shown in Table 11.2, 0.25 of the total annual
yield flows into the fictitious within-year reservoir in period t = 1, and 0.75 of the total annual
yield flows into the reservoir in period t = 2.
Suppose the two desired within-year yields are to
be 3 and 0 for periods 1 and 2, respectively. The
total annual yield, Y0.9, is 3. Assuming the natural distribution of this annual yield of 3 in
period 1 is 0.25 Y0.9 = 0.75, and in period 2 it is
0.75 Y0.9 = 2.25, the within-year storage required
to redistribute these yields of 0.75 and 2.25 to
become 3 and 0, respectively, is Kwa = 2.25.
From Tables 11.2 or 11.3 we can see that an
annual yield of 3 requires an over-year storage
capacity of 3. Thus, the estimated total storage
capacity required to provide yields of 3 and 0 in
periods 1 and 2 is the over-year capacity of 3
plus the within-year capacity of 2.25 equaling
5.25. This compares with 3 plus 2.5 of actual
within-year capacity required for a total of 5.50,
as indicated in Table 11.3.
There are ways to reduce the number of
over-year constraints without changing the
solution of the over-year model. Sequences of
years whose annual inflow values equal or
exceed the desired annual yield can be combined
into one constraint. If the yield is an unknown
variable then the mean annual inflow can be used
as the desired annual yield since it is the upper
limit of the annual yield. For example in
Table 11.2 note that the last three years and the
first year have flows equal or greater than 4, the
mean annual inflow. Thus, these four successive
years can be combined into a single continuity
equation
S7 þ Q7 þ Q8 þ Q9 þ Q1  4Yp  R7 ¼ S2
ð11:33Þ
This saves a total of 3 over-year continuity
constraints and 3 over-year capacity constraints.
Note that the excess release, R7, represents the
excess release in all four periods. Furthermore,
not all reservoir capacity constraint Eq. 11.28 are
needed, since the initial storage volumes in the
years following low flows will probably be less
than the over-year capacity.

11.5

Lakes and Reservoirs

There are many ways to modify and extend
this yield model to include other objectives, fixed
ratios of the unknown annual yield for each
within-year period, and even multiple yields
having different exceedance probabilities p.
The number of over-year periods being modeled compared to the number of years of flow
records determines the highest exceedance
probability or reliability a yield can have; e.g.,
9/10 or 0.9 in the 9-year example used here. If
yields having lower reliabilities are desired, such
as a yield with a reliability of 0.80, then the yield
variable YP can be omitted from Eq. 11.27 in that
critical year that determines the required
over-year capacity for a 0.90 reliable yield.
(Since some outflow might be expected, even if it
is less than the 0.90 reliable yield, the outflow
could be forced to equal the inflow for that year.)
If a 0.70 reliable yield is desired, then the yield
variables in the two most critical years can be
omitted from Eq. 11.27, and so on.
The number of years of yield failure determines the estimated reliability of each yield. An
annual yield that fails in f years has an estimated
probability (n − f)/(n + 1) of being equaled or
exceeded in any future year. Once the desired
reliability of a yield is known, the problem is to
select the appropriate failure years and to specify
the permissible extent of failure in those f failure
years.
To consider different yield reliabilities p let
the parameter αpy be a specified value between 0
and 1 that indicates the extent of a failure in year
y associated with an annual yield having a reliability of p. When αpy is 1 there is no failure, and
when it is less than 1 there is a failure, but a
proportion of the yield Yp equal to αpy is released.
Its value is in part dependent on the consequences of failure and on the ability to forecast
when a failure may occur and to adjust the
reservoir operating policy accordingly.
Theover-year storage continuity constraints
for n years can now be written in a form appropriate for identifying any single annual yield Yp
having an exceedance probability p.

485

Sy þ Qy  apy Yp  Ry ¼ Sy þ 1
y ¼ n; y þ 1 ¼ 1

8y if

ð11:34Þ

When writing Eq. 11.34, the failure year or
years should be selected from among those in
which permitting a failure decreases the required
reservoir capacity Ka. If a failure year is selected
that has an excess release, no reduction in the
required active storage capacity will result, and
the reliability of the yield may be higher than
intended.
The critical year or years that determine the
required active storage volume capacity may be
dependent on the yield itself. Consider, for
example, the 7-year sequence of annual flows (4,
3, 3, 2, 8, 1, 7) whose mean is 4. If a yield of 2 is
desired in each of the 7 years, the critical year
requiring reservoir capacity is year 6. If a yield of
4 is desired (again assuming no losses), the
critical years are years 2–4. The streamflows and
yields in these critical years determine the
required over-year storage capacity. The failure
years, if any, must be selected from within the
critical low-flow periods for the desired yield.
When the magnitudes of the yields are
unknown, some trial and error solutions may be
necessary to ensure that any failure years are
within the critical period of years for the associated yields. To ensure a wider range of applicable
yield magnitudes, the year having the lowest flow
within the critical period should be selected as the
failure year if only one failure year is selected.
Even though the actual failure year may follow
that low-flow year, the resulting required reservoir storage volume capacity will be the same.
Multiple Yields and Evaporation Losses
The yield models developed so far define only
single annual and within-year yields. Incremental
secondary yields having lower reliabilities can
also be included in the model. Referring to the
9-year streamflow record in Table 11.3, assume
that two yields are desired, one 90% reliable and
the other 70% reliable. Let Y0.9 and Y0.7 represent
those annual yields having reliabilities of 0.9 and

486

11

0.7, respectively. The incremental secondary
yield Y0.7 represents the amount in addition to
Y0.9 that is only 70% reliable. Assume that the
problem is one of estimating the appropriate
values of Y0.9 and Y0.7, their respective
within-year allocations ypt and the total active
reservoir capacity Ka that maximizes some
function of these yield and capacity variables.
In this case the over-year andwithin-year
continuity constraints can be written
Sy þ Qy  Y0:9  a0:7
y Y0:7  Ry ¼ Sy þ 1
if

8y

y ¼ n; y þ 1 ¼ 1
ð11:35Þ

st þ bt ðY0:9 þ Y0:7 Þ  y0:9;t  y0:7;t ¼ st þ 1
if t ¼ T; T þ 1 ¼ 1

8t

ð11:36Þ
Now an additional constraint is needed to
insure that each within-year yield of a reliability
p adds up to the annual yield of the same reliability. Selecting the 90% reliable yield,
X
t

y0:9;t ¼ Y0:9

ð11:37Þ

Evaporation losses must be based on an
expected storage volume in each period and year
since the actual storage volumes are not identified using these yield models. The approximate
storage volume in any period t in year y can
be defined as the initial over-year volume Sy,
plus the estimated average within-year volume
(st + st+1)/2. Substituting this storage volume
into Eq. 11.14 (see also Fig. 11.6) results in an
estimated evaporation loss Lyt.


Lyt ¼ ao þ aðSy þ ðst þ st þ 1 Þ=2Þ Etmax
ð11:38Þ
Summing Lyt over all within-year periods t
defines the estimated annual evaporation loss, Ey.

Ey ¼

River Basin Modeling

X

ao þ aðSy þ ðst þ st þ 1 Þ=2Þ Etmax
t

ð11:39Þ
This annual evaporation loss applies, of
course, only when there is a nonzero active
storage capacity requirement. These annual
evaporation losses can be included in the
over-year continuity constraints, such as
Eq. 11.35. If they are, the assumption is being
made that their within-year distribution will be
defined by the fractions βt. This may not be
realistic. If it is not, an alternative would be to
include the average within-year period losses, Lt,
in the within-year constraints.
The average within-year period loss, Lt, can
be defined as the sum of each loss Lyt defined by
Eq. 11.38 over all years y divided by the total
number of years, n.
Lt ¼

n 
X
y


ao þ aðSy þ ðst þ st þ 1 Þ=2Þ Etmax =n
ð11:40Þ

This average within-year period loss, Lt, can
be added to the within-year’s highest reliability
yield, ypt, forcing greater total annual yields of all
reliabilities to meet corresponding total
within-year yield values. Hence, combining
Eq. 11.37 and 11.38, for p equal to 0.9 in the
example,
Yp ¼

X
t

(
ypt þ

n
X
y

)
½ao þ aðSy þ ðst þ 1 Þ=2ÞEtmax =n

ð11:41Þ
Since actual reservoir storage volumes in each
period t of each year y are not identified in this
model, system performance measures that are
functions of those storage volumes, such as
hydroelectric energy or reservoir recreation, are
only approximate. Thus, as with any of these
screening models, any set of solutions should be

11.5

Lakes and Reservoirs

evaluated and further improved using more precise simulation methods.
Simulation methods require reservoir operating rules. The information provided by the
solution of the yield model can aid in defining a
reservoir operating policy for such simulation
studies.
Reservoir Operation Rules
Reservoir operation rules are guides for those
responsible for reservoir operation. They apply to
reservoirs being operated in a steady-state condition (i.e., not filling up immediately after construction or being operated to meet a set of new
and temporary objectives). There are several
types of rules but each indicates the desired or
required reservoir release or storage volumes at
any particular time of year. Some rules identify
storage volume targets (rule curves) that the
operator is to maintain, if possible, and others
identify storage zones, each associated with a
particular release policy. This latter type of rule
can be developed from the solution of the yield
model.
To construct an operation rule that identifies
storage zones, each having a specific release
policy, the values of the dead and flood storage
capacities, KD and Kf are needed together with
the over-year storage capacity, Koa, and within year storage volumes st in each period t. Since
both Koa and all st derived from the yield model

Fig. 11.8 Reservoir
release rule showing the
identification of the most
reliable release zone
associated with the
within-year yields y0.9,t

487

are for all yields, Yp, being considered, it is
necessary to determine the over-year capacities
and within-year storage volumes required to
provide each separate within-year yield ypt.
Plotting the curves defined by the respective
over-year capacity plus the within-year storage
volume (Koa + st) in each within-year period t will
define a zone of storage whose yield releases
ypt from that zone should have a reliability of at
least p.
For example, assume again a 9-year flow
record and 10 within-year periods. Of interest are
the within-year yields, y0.9,t and y0.7,t, having
reliabilities of 0.9 and 0.7. The first step is to
compute the over-year storage capacity requirement, Koa, and the within-year storage volumes,
st, for just the yields y0.9,t. The sum of these
values, Koa + st, in each period t can be plotted as
illustrated in Fig. 11.8.
The sum of the over-year capacity and
within-year volume Koa + st in each period t defines the zone of active storage volumes for each
period t required to supply the within-year yields
y0.9,t. If the storage volume is in this shaded zone
shown in Fig. 11.8, only the yields y0.9,t should
be released. The reliability of these yields, when
simulated, should be about 0.9. If at any time
t the actual reservoir storage volume is within
this zone, then reservoir releases should not
exceed those required to meet the yield y0.9,t if
the reliability of this yield is to be maintained.

488

The next step is to solve the yield model for
both yields Y0.9 and Y0.7. The resulting sum of
over-year storage capacity and within-year storage volumes can be plotted over the first zone, as
shown in Fig. 11.9.
If at any time t the actual storage volume is in
the second lighter shaded zone in Fig. 11.9, both
the release should be the sum of the most reliable
yield, y0.9,t and the incremental secondary yield
y0.7,t. If only these releases are made, the probability of being in that zone, when simulated,
should be about 0.7. If the actual storage volume
is greater than the total required over-year storage capacity Koa plus the within-year volume st,
the non-shaded zone in Fig. 11.23, then a release
Fig. 11.9 Reservoir
release rule showing the
identification of the second
most reliable release zone
associated with the total
within-year yields y0.9,
t + y0.7,t

Fig. 11.10 Reservoir
release rule defined by the
yield model

11

River Basin Modeling

can be made to satisfy any downstream demand.
Converting storage volume to elevation, this
release policy is summarized in Fig. 11.10.
These yield models focus only on the active
storage capacity requirements. They can be a part
of a model that includes flood storage requirements as well (as previously discussed in this
chapter). If the actual storage volume is within
the flood control zone in the flood season,
releases should be made to reduce the actual
storage to a volume no greater than the total
capacity less the flood storage capacity.
Once again, reservoir rules developed from
simplified models such as this yield model are
only guides, and once developed these rules

11.5

Lakes and Reservoirs

should be simulated, evaluated, and refined prior
to their actual adoption.

11.6
11.6.1

Drought and Flood Risk
Reduction
Drought Planning
and Management

Droughts are natural hazards that unlike floods,
tornadoes, and hurricanes, occur slowly and
gradually over a period of time. The absence of a
precise drought threshold introduces some
uncertainty about whether a drought exists and, if
it does, its degree of severity. The impacts of
drought are nonstructural and typically spread
over a larger geographical area than are damages
resulting from other natural hazards. All of these
drought characteristics have impacted the development of effective drought preparedness plans.
Droughts result from a deficiency of precipitation compared to normal (long-term average)
amounts that, when extended over a season or
especially over a longer period of time, is
insufficient to meet the demands of human
activities and the environment. All types of
drought results in water shortages for one or
more water-using activities.
Droughts differ from one another in three
essential characteristics: intensity, duration, and
spatial coverage. Moreover, many disciplinary
perspectives of drought exist. Because of these
numerous and diverse disciplinary views, confusion often exists over exactly what constitutes
a drought. Regardless of such disparate views,
the overriding feature of drought is its negative
impacts on people and the environment.

11.6.1.1
Drought Types
Droughts are normally distinguished by type:
meteorological, hydrological, agricultural, and
socioeconomic. Meteorological drought is
expressed solely on the basis of the degree of
dryness in comparison to some normal or average amount and the duration of the dry period.
Drought intensity and duration are the key
descriptors of this type of drought. Agricultural

489

drought links various characteristics of meteorological drought to agricultural impacts, focusing on precipitation shortages, differences
between actual and potential evapotranspiration,
and soil water deficits.
Hydrological droughts are described based on
the effects of low precipitation on surface or
subsurface water supply (e.g., streamflow,
reservoir storage, lake levels, and groundwater )
rather than with precipitation shortfalls. Hydrological droughts usually lag the occurrence of
meteorological and agricultural droughts because
more time elapses before precipitation deficiencies are detected in rivers, reservoirs, groundwater aquifers, and other components of the
hydrologic system. As a result, hydrological
droughts are typically detected later than other
drought types. Water uses affected by drought
can include multiple purposes such as power
generation, flood control, irrigation, domestic
drinking water, industry, recreation, and ecosystem preservation.
Socioeconomic droughts are linked directly to
the supply of some economic good. Increases in
population can alter substantially the demand for
these economic goods over time. The incidence
of socioeconomic drought can increase because
of a change in the frequency of meteorological
drought, a change in societal vulnerability to
water shortages, or both. For example, poor land
use practices such as overgrazing can decrease
animal carrying capacity and increase soil erosion, which exacerbates the impacts of, and
vulnerability to, future droughts.

11.6.1.2
Drought Impacts
The impacts of drought are often widespread
through the economy. They can be direct and
indirect. Restrictions in water use resulting from
drought is a direct or first-order impact of
drought. However, the consequences of such
restrictions could result in loss of income, farm
and business foreclosures, and government relief
programs) are possible indirect second- or
third-order impacts.
The impacts of drought appear to be increasing in both developing and developed countries,
which in many cases reflects the persistence of

490

non-sustainable development and population
growth. Lessening the impacts of future drought
events typically requires the development of
drought risk policies that emphasize a wide range
of water conservation and early warning measures. Drought management techniques are often
conditional on the severity of the drought. Identifying the actions to take and the thresholds
indicating when to take them are best accomplished prior to a drought, as agreements among
stakeholders are easier to obtain when individuals are not having to deal with the impacts of an
ongoing drought.
Drought impacts can be economic, environmental, and social.
Economic impacts can include direct losses to
agricultural and industrial users, losses in recreation, transportation, and energy sectors. Other
indirect economic impacts can include resulting
unemployment and loss of tax revenue to local,
state, and federal governments.
Environmental losses include damages to
plant and animal species in natural habitats, and
reduced air and water quality; an increase in
forest and range fires; the degradation of landscape quality; and possible soil erosion. These
losses are difficult to quantify, but growing
public awareness and concern for environmental
quality has forced public officials to focus greater
attention on them.
Social impacts can involve public safety,
health, conflicts among water users, and inequities in the distribution of impacts and disaster
relief programs. As with all natural hazards, the
economic impacts of drought are highly variable
within and among economic sectors and geographic regions, producing a complex assortment
of winners and losers with the occurrence of each
disaster.

11.6.1.3

Drought Preparedness
and Mitigation
Droughts happen, and it makes no sense to wait
until realizing a drought is happening before
preparing plans and policies to mitigate the
adverse impacts from a drought. As evidenced by
the ongoing drought (at this writing) in California, and the even more severe drought those in

11

River Basin Modeling

southeastern Australia recently witnessed,
drought management has to involve the institutions that not only manage water supply systems,
but all those who use water, and all those who
make land-use decisions that impact water runoff. It can involve hydrologic modeling methods
discussed in Chap. 6, and reservoir modeling as
discussed in Chaps. 4 and 8. Appendix C of this
book (contained on a disk or downloadable from
the web) discusses drought management modeling methods and options in more detail.

11.6.2

Flood Protection
and Damage
Reduction

Next consider the other extreme—floods. Two
types of structural alternatives are often used for
flood risk reduction. One is the provision of flood
storage capacity in reservoirs designed to reduce
downstream peak flood flows. The other is channel
enhancement and/or flood-proofing structures that
are designed to contain peak flood flows and
reduce damage. This section introduces methods
of modeling both of these alternatives for inclusion
in either benefit–cost or cost-effectiveness analyses. The latter analyses apply to situations in which
a significant portion of the flood control benefits
cannot be expressed in monetary terms and the aim
is to provide a specified level of flood protection at
minimum cost.
The discussion will first focus on the estimation of flood storage capacity in a single reservoir
upstream of a potential flood damage site. This
analysis will then be expanded to include
downstream channel capacity improvements.
Each of the modeling methods discussed will be
appropriate for inclusion in multipurpose river
basin planning (optimization) models having
longer time step durations than those required to
predict flood peak flows.

11.6.2.1

Reservoir Flood Storage
Capacity
In addition to the active storage capacities in a
reservoir, some capacity may be allocated for the
temporary storage of flood flows during certain

Drought and Flood Risk Reduction

divided by 1000 or 10,000, respectively.
A one-hundred-year flood or greater has a
probability of 1/100 or 0.01 of occurring in any
given year. Assuming annual floods are independent, if a 100-year flood occurs this year, the
probability that a flood of that magnitude or
greater occurring next year remains 1/100 or
0.01.
If PQ is the random annual peak flood flow
and PQT is a particular peak flood flow having a
return period of T years, then by definition the
probability of an actual flood of PQ equaling or
exceeding PQT is 1/T.
Pr½PQ  PQT ¼ 1=T

ð11:42Þ

The higher the return period, i.e., the more
severe the flood, the lower the probability that a
flood of that magnitude or greater will occur.
Equation 11.42 is plotted in Fig. 11.11.
The exceedance probability
distribution
shown in Fig. 11.11 is simply 1 minus the
cumulative distribution function FPQ() of annual
peak flood flows. The area under the function is
the mean annual peak flood flow, E[PQ].
The expected annual flood damage at a
potential flood damage site can be estimated
from an exceedance probability distribution of
peak flood flows at that potential damage site
together with a flow or stage damage function.
The peak flow exceedance distribution at any
potential damage site will be a function of the

Pr [PQ > PQ T ]

periods in the flood season of the year, as shown
in Fig. 11.2. Flood flows usually occur over time
intervals lasting from a few hours up to a few
days or weeks. Computational limitations make it
impractical to include such short time durations
in many of our multipurpose planning models
that typically include time periods of a week, or
10 days, or months or seasons spanning several
months. If we modeled these short daily or
hourly durations, flood routing equations would
have to be included in the model; a simple mass
balance would not be sufficient. Nevertheless
there are ways of including unknown flood
storage variables within longer period optimization models .
Consider a potential flood damage site along
a river. A flood control reservoir can be built
upstream of that potential damage site. The
question is how much flood storage capacity, if
any, should the reservoir contain. For various
assumed capacities and operating policies, simulation models can be used to predict the impact
on the downstream flood peaks. These hydraulic
simulation models must include flood routing
procedures from the reservoir to the downstream
potential damage site and the flood control operating policy at the reservoir. For various
downstream flood peaks, water elevations and
associated economic flood damages on the
floodplain can be estimated. To calculate the
expected annual damages associated with any
upstream reservoir capacity , the probability of
various damage levels being exceeded in any
year needs to be calculated.
The likelihood of a flood peak of a given
magnitude or greater is often described by its
expected return period. How many years would
one expect to wait, on average, to observe
another flood of equal or greater than a flood of
some specified magnitude? This is the reciprocal
of the probability of observing such a flood or
greater in any given year. A T-year flood has a
probability of being equaled or exceeded in any
year of 1/T. This is the probability that could be
calculated by adding up the number of years an
annual flood of a given or greater magnitude
is observed, say in 1000 or 10,000 years,

491

1
1/T
0

E020121u

11.6

0

PQ T
peak flood flow

Fig. 11.11 Probability of annual peak flood flows being
exceeded

492

11

upstream reservoir flood storage capacity Kf
and the reservoir operating policy.
The probability that flood damage of FDT
associated with a flood of return period T will be
exceeded is precisely the same as the probability
that the peak flow PQT that causes the damage
will be exceeded. Letting FD be a random flood
damage variable, its probability of exceedance is
Pr½FD  FDT  ¼ 1=T

ð11:43Þ

The area under this exceedance probability
distribution is the expected annual flood damage
, E[FD].
Z1
E½FD ¼

Pr½FD  FDT dFDT

ð11:44Þ

0

Fig. 11.12 Calculation of the expected annual flood
damage shown as the shaded area in quadrant (d) derived
from the expected stage damage function (a), the expected

River Basin Modeling

This computational process is illustrated
graphically in Fig. 11.12. The analysis requires
three input functions that are shown in quadrants
(a), (b), and (c). The dashed-line rectangles
define point values on the three input functions in
quadrants (a), (b), and (c) and the corresponding
probabilities of exceeding a given level of damages in the lower right quadrant (d). The distribution in quadrant (d) is defined by the
intersections of these dashed-line rectangles. This
distribution defines the probability of equaling or
exceeding a specified damage in any given year.
The (shaded) area under the derived function is
the annual expected damage, E[FD].
The relationships between flood stage and
damage, and flood stage and peak flow, defined
in quadrants (a) and (b) of Fig. 11.12, must be
known. These do not depend on the flood storage

stage-flow relation (b), and the expected probability of
exceeding an annual peak flow (c)

11.6

Drought and Flood Risk Reduction

capacity in an upstream reservoir. The information in quadrant (c) is similar to that shown in
Fig. 11.11 defining the exceedance probabilities
of each peak flow. Unlike the other three functions, this distribution depends on the upstream
flood storage capacity and flood flow release
policy. This peak flow probability of exceedance
distribution is determined by simulating the
annual floods entering the upstream reservoir in
the years of record.
The difference between the expected annual
flood damage without any upstream flood storage
capacity and the expected annual flood damage
associated with a flood storage capacity of Kf is
the expected annual flood damage reduction.
This is illustrated in Fig. 11.13. Knowing the
expected annual damage reduction associated
with various flood storage capacities, Kf, permits
the definition of a flood damage reduction
function, Bf(Kf).

493

If the reservoir is a single purpose flood
control reservoir , the eventual tradeoff is
between the expected flood reduction benefits,
Bf(Kf), and the annual costs, C(Kf), of that
upstream reservoir capacity. The particular
reservoir flood storage capacity that maximizes
the net benefits, Bf(Kf) − C(Kf), may be appropriate from a national economic efficiency perspective but it may not be best from a local
perspective. Those occupying the potential
damage site may prefer a specified level of protection from that reservoir storage capacity,
rather than the protection that maximizes
expected annual net benefits, Bf(Kf) – C(Kf).
If the upstream reservoir is to serve multiple
purposes, say for water supply, hydropower, and
recreation, as well as for flood control, the
expected flood reduction benefit function just
derived could be a component in the overall
objective function for that reservoir.

(b)

(a)

(c)

(d)

Fig. 11.13 Calculation of expected annual flood damage reduction benefits, shown as the darkened portion of
quadrant (d), associated with a specified reservoir flood storagecapacity

494

11

Total reservoir capacity K will equal the sum
of dead storage capacity Kd, active storage
capacity Ka, and flood storage capacity Kf,
assuming they are the same in each period t. In
some cases they may vary over the year. If the
required active storage capacity can occupy the
flood storage zone when flood protection is not
needed, the total reservoir capacity K will be the
dead storage, Kd, plus the maximum of either
(1) the actual storage volume and flood storage
capacity in the flood season or (2) the actual
storage volume in non-flood season.
K  Kd þ St þ Kf

for all periods t in flood season
plus the following period that
represents the end of
the flood season

ð11:45Þ
K  Kd þ St

for all remaining periods t
ð11:46Þ

In the above equations the dead storage
capacity, Kd, is assumed known. It is included in
the capacity Eqs. 11.45 and 11.46 assuming that
the active storage capacity is greater than zero.
Clearly, if the active storage capacity were zero,
there would be no need for dead storage.

11.6.2.2
Channel Capacity
The unregulated natural peak flow of a particular
design flood at a potential flood damage site can be
reduced by upstream reservoir flood storage
capacity or it can be contained within the channel
at the potential damage site by levees and other
channel-capacity improvements. In this section,
the possibility of levees or dikes and other channel
capacity or flood-proofing improvements at a
downstream potential damage site will be considered. The approach used will provide a means of
estimating combinations of flood control storage
capacity in upstream reservoirs and downstream
channel capacity improvements that together will
provide a prespecified level of flood protection at
the downstream potential damage site.

River Basin Modeling

Let QNT be the unregulated natural peak flow
in the flood season having a return period of
T years. Assume that this peak flood flow is the
design flood for which protection is desired. To
protect from this design peak flow, a portion QS of
the peak flow may be reduced by upstream flood
storage capacity. The remainder of the peak flow
QR must be contained within the channel. Hence if
the potential damage site s is to be protected from
a peak flow of QNT, the peak flow reductions due
to upstream storage, QS, and channel improvements, QR, must at least equal to that peak flow.
QNT  QS þ QR

ð11:47Þ

The extent to which a specified upstream
reservoir flood storage capacity reduces the design peak flow at the downstream potential
damage site can be obtained by routing the
design flood through the reservoir and the
channel between the reservoir and the downstream site. Doing this for a number of reservoir
flood storage capacities permits the definition of
a peak flow reduction function, fT(Kf).
QS ¼ fT ðKf Þ

ð11:48Þ

This function is dependent on the relative
locations of the reservoir and the downstream
potential damage site, on the characteristics and
length of the channel between the reservoir and
downstream site, on the reservoir flood control
operating policy, and on the magnitude of the
peak flood flow.
An objective function for evaluating these two
structural flood control measures should include
the cost of reservoir flood storage capacity,
CostK(Kf), and the cost of channel capacity
improvements, CostR(QR), required to contain a
flood flow of QR. For a single purpose, single
damage site, single reservoir flood control problem, the minimum total cost required to protect
the potential damage site from a design flood peak
of QNT, may be obtained by solving the model:
 
minimize CostK Kf þ CostR ðQRÞ

ð11:49Þ

11.6

Drought and Flood Risk Reduction

495

Fig. 11.14 Tradeoff between minimum cost of flood
protection and flood risk, as expressed by the expected
return period

subject to
QNT  fT ðKf Þ þ QR

ð11:50Þ

Equations 11.49 and 11.50 assume that a
decision will be made to provide protection from
a design flood QNT of return period T; it is only a
question of how to provide the required protection, i.e., how much flood storage capacity and
how much levee protection.
Solving Eqs. 11.47 and 11.48 for peak flows
QNT of various return periods T will identify the
risk-cost tradeoff. This tradeoff function might
look like what is shown in Fig. 11.14.

11.6.2.3

Estimating Risk of Levee
Failures
Levees are built to reduce the likelihood of
flooding on the flood plain. Flood flows prevented from flowing over a floodplain due to a
levee will have relatively little effect on users of
the flood plain, unless of course the levee fails to
contain the flow. Levee failure can result from
flood events that exceed (overtop) the design
capacity of the levee. Failure can also result from
various types of geostructural weaknesses. If any
of the flow in the stream or river channel passes
over, through or under the levee and onto the
flood plain, the levee is said to fail. The probability of levee failure along a river reach is in part
a function of the levee height, the probability
distribution of flood flows in the stream or river
channel, and the probability of geo-structural

failure. The latter depends in part on how well
the levee and its foundation is constructed. Some
levees are purposely designed to ‘‘fail’’ at certain
sites at certain flood stages to reduce the likelihood of more substantial failures and flood
damages further downstream.
The probability of levee failure given the flood
stage (height) in the stream or river channel is
often modeled using two flood stages. The US
Army Corps of Engineers calls the lower stage the
probable non-failure point, PNP, and the higher
stage is called the probable failure point, PFP
(USACE 1991). At the PNP, the probability of
failure is assumed to be 15%. Similarly, the
probability of failure at the PFP is assumed to be
85%. A straight-line distribution between these
two points is also assumed, as shown in
Fig. 11.15. Of course these points and distributions are at best only guesses, as not many, if any,
data will exist to base them on at any given site.
To estimate the risk of a flood in the floodplain protected by a levee due to overtopping or
geo-structural levee failure, the relationships
between flood flows and flood stages in the
channel and on the floodplain must be defined,
just as it had to be to carry out the analyses
shown in Figs. 11.12 and 11.13.
Assuming no geo-structural levee failure, the
flood stage in the floodplain protected by a levee
is a function of the flow in the stream or river
channel, the cross sectional area of the channel
between the levees on either side, the channel
slope and roughness, and the levee height. If
floodwaters enter the floodplain, the resulting
water level or stage in the floodplain will depend
on the topological characteristics of the flood
plain. Figure 11.16 illustrates the relationship
between the flood stage in the channel and the
flood stage in the flood plain, assuming no
geo-structural failure of the levee. Obviously
once the flood begins overtopping the levee, the
flood stage in the flood plain begins to increase.
Once the flood flow is of sufficient magnitude
that its stage without the levee is the same as that
with the levee, the existence of a levee has only a
negligible impact on the flood stage.
Figure 11.17 illustrates the relationship
between flood flow and flood stage in a

496

11

River Basin Modeling

Fig. 11.15 Assumed cumulative probability distribution of levee failure expressed as function of flood stage in river
channel

Fig. 11.16 Influence of a levee on the flood stage in
floodplain compared to stream or river flood stage. The
channel flood stage where the curve is vertical is the stage

at which the levee fails due to overtopping or from
geo-structural causes

floodplain with and without flood levees, again
assuming no geo-structural levee failure.
Combining Figs. 11.16 and 11.17 defines the
relationship between reach flow and channel
stage. This is illustrated in the upper left quadrant
of Fig. 11.18.

Combining the relationship between flood
flow and flood stage in the channel (upper left
quadrant of Fig. 11.18) with the probability distribution of levee failure (Fig. 11.15) and the
probability distribution of annual peak flows
being equaled or exceeded (Fig. 11.11), provides

11.6

Drought and Flood Risk Reduction

497

Fig. 11.17 Relationship
between flood flow and
flood stage in a floodplain
with and without flood
levees, again assuming no
geo-structural levee failure

Fig. 11.18 Deriving the
relation (shown in the
upper left quadrant)
between flood flow and
flood stage in the channel

an estimate of the expected probability of levee
failure. Figure 11.19 illustrates this process of
finding, in the lower right quadrant, the shaded
area that equals the expected annual probability
of levee failure from overtopping and/or
geo-structural failure.
The channel flood-stage function, S(q), of
peak flow q shown in the upper left quadrant of
Fig. 11.19 is obtained from the upper left

quadrant of Fig. 11.18. The probability of levee
failure, PLF(S), a function of flood stage, S(q),
shown in the upper right quadrant is the same as
in Fig. 11.15. The annual peak flow exceedance
probability distribution, FQ(q), (or its inverse
Q(p)) in the lower left quadrant is the same as
Fig. 11.12 or that in the lower left quadrant (c) of
Fig. 11.13. The exceedance probability function
in the lower right quadrant of Fig. 11.19 is

498

11

River Basin Modeling

Fig. 11.19 Derivation of
the probability of
exceeding a given
probability of levee failure,
shown in lower right
quadrant. The shaded area
enclosed by this probability
distribution is the annual
expected probability of
levee failure

derived from each of the other three functions, as
indicated by the arrows, in the same manner as
described in Fig. 11.12.
In mathematical terms, the annual expected
probability of levee failure, E[PLF], found in the
lower right quadrant of Fig. 11.19, equals
Z1
E½PLF ¼

PLF½SðqÞf ðqÞdq
0

Z1
¼

Z1
PLF½SðQðpÞÞdp ¼

0

PLF 0 ðpÞdp;

exceeding a channel flow whose stage equals the
levee height, as defined in the lower left quadrant
of Fig. 11.19. This is shown in Fig. 11.20.
Referring to Fig. 11.20, if the levee height is
increased, the horizontal part of the curve in the
upper right quadrant would rise, as would the
horizontal part of the curve in the upper left
quadrant as it shifts to the left. Hence given the
same probability distribution as defined in the
lower left quadrant, the expected probability of
exceeding an increased levee capacity would
decrease, as it should.

0

ð11:51Þ
where PLF′(p) is the probability of levee failure
associated with a flood stage of S(q) having an
exceedance probability of p.
Note that if the failure of the levee was only
due to channel flood stages exceeding the levee
height (i.e., if the probability of geo-structural
failure were zero) the expected probability of
levee failure would be simply the probability of

11.6.2.4

Annual Expected Damage
from Levee Failure
A similar analysis can provide an estimate of the
expected annual flood plain damage for a stream
or river reach. Consider, for example, a parcel of
land on a flood plain at some location i. If an
economic efficiency objective were to guide the
development and use of this parcel, the owner
would want to maximize the net annual economic benefits derived from its use, Bi, less the

11.6

Drought and Flood Risk Reduction

499

Fig. 11.20 Calculation of
annual probability of
equaling or exceeding any
specified probability of
levee overtopping, shown
in the lower right quadrant.
The shaded area in this
quadrant is the expected
probability of levee
overtopping, assuming
there is no geo-structural
failure

annual (non-flood damage) costs, Ci, and the
expected annual flood damages, EADi. The issue
of concern here is the estimation of these
expected annual flood damages.
Damages at location i resulting from a flood
will depend in part on the depth of flooding at
that location and a host of other factors (flood
duration, velocity of and debris in flood flow,
time of year, etc.). Assume that the flood damage
at location i is a function of primarily the flood
stage, S, at that location. Denote this potential
damage function as Di(S). Such a function is
illustrated in Fig. 11.21.
Integrating the product of the annual exceedance probability of flood stage, Fs(S), and the
potential damages, Di(S), over all stages S will
yield the annual expected damages, E[Di], for
land parcel i.
Z
E½Di  ¼

Di ðSÞFs ðSÞdS

ð11:52Þ

Fig. 11.21 An example function defining the damages
that will occur given any flood stage S to a parcel of land i

The sum of these expected damage estimates
over all the parcels of land i on the floodplain is
the total expected damage that one can expect
each year, on average, on the floodplain.
EAD ¼

X
i

E½Di 

ð11:53Þ

500

11

Alternatively the annual expected flood damage could be based on a calculated probability of
exceeding a specified flood damage, as shown in
Fig. 11.12. For this method the potential flood
damages, Di(S), are determined for various stages
S and then summed over all land parcels i for
each of those stage values S to obtain the total
potential damage function, D(S), for the entire
floodplain, defined as a function of flood stage S.
DðSÞ ¼

X

Di ðSÞ

ð11:54Þ

i

This is the function shown in quadrant (a) in
Fig. 11.12.
Levee failure probabilities, PLF′(p), based on
the exceedance probability p of peak flows, or
stages, as defined in Fig. 11.31 and Eq. 11.49
can be included in calculations of expected
annual damages. Expressing the damage function, D(S), as a function, D′(p), of the stage
exceedance probability p and multiplying this
flood damage function D′(p) times the probability of levee failure, PLF′(p) defines the joint
exceedance probability of expected annual damages. Integrating over all values of p yields the
expected annual flood damage, EAD.
Z1
EAD ¼

D0 ðpÞ PLF 0 ðpÞ dp

ð11:55Þ

0

Note that the flood plain damages and probability of levee failure functions in Eq. 11.55
both increase with increasing flows or stages, but
as peak flows or stages increase, their exceedance
probabilities decrease. Hence with increasing
p the damage and levee failure probability
functions decrease. The effect of levees on the
expected annual flood damage, EAD, is shown in
Fig. 11.22. The “without levee” function in the
lower right quadrant of Fig. 11.18, is D′(p). The
“with levee” function is the product of D′(p) and
PLF′(p). If the probability of levee failure, PLF′
(p) function were as shown in Fig. 11.18, i.e., if
it were 1.0 for values of p below some overtopping stage associated with an exceedance probability p*, and 0 for values of p greater than p*,

River Basin Modeling

then the function would appear as shown “with
levee—overtop only” in Fig. 11.22.

11.6.2.5
Risk-Based Analyses
Risk-based analyses attempt to identify the
uncertainty associated with each of the inputs
used to define the appropriate capacities of various flood risk reduction measures. There are
numerous sources of uncertainty associated with
each of the functions shown in quadrants (a), (b),
and (c) in Fig. 11.25. This uncertainty translates
to uncertainty associated with estimates of flood
risk probabilities and expected annual flood
damage reductions obtained from reservoir flood
storage capacities and channel improvements.
Going to the substantial effort and cost of
quantifying these uncertainties, which themselves will be surely be uncertain, does however
provide additional information. The design of
any flood protection plan can be adjusted to
reflect attitudes of stakeholders toward the
uncertainty associated with specified flood peak
return periods or equivalently their probabilities
of occurring in any given year.
For example, assume a probability distribution capturing the uncertainty about the expected
probability of exceedance of the peak flows at the
potential damage site (as shown in Fig. 11.12) is
defined from a risk-based-analysis. Figure 11.23
shows that exceedance function together with its
90% confidence bands near the higher flood peak
return periods. To be, say, 90% sure that protection is provided for the T-year return period
flow, PQT, one may have to for an equivalent
expected T + Δ year return period flow, PQT+Δ,
i.e., the flow having a (1/T) − δ expected probability of being exceeded. Conversely, protection
from the expected T + Δ year peak flood flow
will provide 90% assurance of protection from
flows that will occur less than once in T years on
average.
If society wanted to eliminate flood damage it
could do it, but at a high cost. This would require
either costly flood control structures or eliminating economic activities on lands subject to
possible flooding. Both reduce expected economic returns from the floodplain. Hence such
actions are not likely to be taken. There will

11.6

Drought and Flood Risk Reduction

501

Fig. 11.22 Calculation of
expected annual flood
damage taking into account
probability of levee failure

Pr [PQ > PQT ]

always be a risk of flood damage. Analyses such
as those just presented help identify these risks.
Risks can be reduced and managed but not

1/T
1/T-δ
0
E020 130h

90% confidence
estimates

1

0

PQ T

eliminated. Finding the best levels of flood protection and flood risk, together with risk insurance or subsidies (illustrated in Fig. 11.24) is the
challenge for public and private agencies alike.
Floodplain management is as much concerned
with good things not happening on them as with
bad things—like floods—happening on them.

PQT+
peak flood flow

Fig. 11.23 Portion of peak flow probability of exceedance function showing contours containing 90% of the
uncertainty associated with this distribution. To be 90%
certain of protection from a peak flow of PQt, protection
is needed from the higher peak flow, PQt+Δ expected once
every T + Δ years, i.e., with an annual probability of 1/
(T + Δ) or (1/T) − δ of being equaled or exceeded

Fig. 11.24 Relationship between expected economic
return from flood plain use and risk of flooding. The
lowest flood risk does not always mean the best risk, and
what risk is acceptable may depend on the amount of
insurance or subsidy provided when flood damage occurs

502

11.7

11

Hydroelectric Power
Production

Hydropower plants, Fig. 11.25, convert the
energy from the flowing water to mechanical and
then electrical energy. These plants containing
turbines and generators are typically located
either in or adjacent to dams. Pipelines (penstocks) carry water under pressure from the
reservoir to the powerhouse. Power transmission
systems transport the produced electrical energy
from the powerhouse to where it is needed.
The principal advantages of using hydropower are the absence of polluting emissions
during operation, its capability to respond relatively quickly to changing utility load demands,
and its relatively low operating costs. Disadvantages can include high initial capital cost and
potential
site-specific
and
cumulative

Fig. 11.25 Hydropower system components

River Basin Modeling

environmental impacts. Potential environmental
impacts of hydropower projects include altered
flow regimes below storage reservoirs or within
diverted stream reaches, water quality degradation, mortality of fish that pass through turbines,
blockage of fish migration, and flooding of terrestrial ecosystems by impoundments. However,
in many cases, proper design and operation of
hydropower projects can mitigate some of these
impacts. Hydroelectric projects can also provide
additional benefits such as from recreation in
reservoirs or in tailwaters below dams.
Hydropower plants can be either conventional
or pumped storage. Conventional hydropower
plants use the available water from a river, stream,
canal system, or reservoir to produce electrical
energy. In conventional multipurpose reservoirs
and run-of-river systems, hydropower production
is just one of many competing purposes for which

11.7

Hydroelectric Power Production

water resources may be used. Competing water
uses may include irrigation, flood control, navigation, downstream flow dilution for quality
improvement, and municipal and industrial water
supply. Pumped storage plants pump the water,
usually through a reversible turbine that acts as a
pump, from a lower supply source to an upper
reservoir. While pumped storage facilities are net
energy consumers, they are income producers.
They are valued by a utility because they can be
brought online rapidly to operate in a peak power
production mode when energy prices are the
highest. The pumping to replenish the upper
reservoir is performed during off-peak hours
when electricity costs are low. Then they are
released through the power plant when the electricity prices are higher. The system makes
money even though it consumes more energy.
This process benefits the utility by increasing the
load factor and reducing the cycling of its base
load units. In most cases, pumped storage plants
run a full cycle every 24 h (DOE 2002).
Run-of-river projects use the natural flow of
the river and produce relatively little change in
the stream channel and stream flow. A peaking
project impounds and releases water when the
energy is needed. A storage project extensively
impounds and stores water during high-flow
periods to augment the water available during
low-flow periods, allowing the flow releases and
power production to be more constant. Many
projects combine the modes.
The power capacity of a hydropower plant is
primarily the function of the flow rate through
the turbines and the hydraulic head. The
hydraulic head is the elevation difference the
water falls (drops) in passing through the plant or
to the tailwater, which ever elevation difference
is less. Project design may concentrate on either
of these flow and head variables or both, and on
the hydropower plant installed designed capacity.
The production of hydroelectric energy during
any period at any particular reservoir site is
dependent on the installed plant capacity; the flow
through the turbines; the average effective productive storage head; the duration of the period; the
plant factor (the fraction of time energy is produced); and a constant for converting the product of

503

flow, head, and plant efficiency to electrical energy.
The kilowatt-hours of energy, KWHt, produced in
period t is proportional to the product of the plant
efficiency, e, the productive storage head Ht, and
the flow qt through the turbines.
A cubic meter of water, weighing l03 kg,
falling a distance of 1 m, acquires 9.81 × 103 J
(nm) of kinetic energy. The energy generated in
one second equals the watts (joules per second)
of power produced. Hence an average flow of qt
cubic meters per second falling a height of Ht
meters in period t yields 9.81 × 103qtHt watts or
9.81 qtHt kilowatts of power. Multiplying by the
number of hours in period t yields the
kilowatt-hours of energy produced given a head
of Ht and an average flow rate of qt. The total
kilowatt-hours of energy, KWHt, produced in
period t assuming 100% efficiency in conversion
of potential to electrical energy is
KWHt ¼ 9:81 qt Ht ðseconds in period tÞ=
ðseconds per hourÞ
¼ 9:81 qt Ht ðseconds in period tÞ=3600
ð11:56Þ
Since the total flow, QTt through the turbines
in period t, equals the average flow rate qt times
the number of seconds in the period, the total
kilowatt-hours of energy produced in period
t given a plant efficiency (fraction) of e equals
KWHt ¼ 9:81 QTt Ht e=3600
¼ 0:002725 QTt Ht e

ð11:57Þ

The energy required for pumped storage,
where instead of producing energy the turbines
are used to pump water up to a higher level, is
KWHt ¼ 0:002725 QTt Ht =e

ð11:58Þ

For Eqs. 11.57 and 11.58, QTt is expressed in
cubic meters and Ht is in meters. The storage
head, Ht, is the vertical distance between the
water surface elevation in the lake or reservoir
that is the source of the flow through the turbines
and the maximum of either the turbine elevation
or the downstream discharge elevation. In variable head reservoirs, storage heads are functions

504

11

of storage volumes (and possibly the reservoir
release if the tailwater elevation affects the head).
In optimization models for capacity planning,
these heads and the turbine flows are among the
unknown variables. The energy produced is
proportional to the product of these two
unknown variables. This results in non-separable
functions in model equations that must be written
at each hydroelectric site for each time period t.
A number of ways have been developed to
convert these non-separable energy production
functions to separable ones for use in linear
optimization models for estimating design and
operating policy variable values. These methods
inevitably increase the number of model variables
and constraints. For a preliminary screening of
hydropower capacities prior to a more detailed
analysis (e.g., using simulation or other nonlinear
or discrete dynamic programming methods) one
can (1) solve the model using both optimistic and
pessimistic assumed fixed head values, (2) compare the actual derived heads with the assumed
ones and adjust the assumed heads, (3) resolve the
model, and (4) compare the capacity values. From
this iterative process, one should be able to
identify the range of hydropower capacities that
can then be further refined using simulation.
Alternatively average heads, Hot , and flows,
o
Qt , can be used in a linear approximation of the
non-separable product terms, QTt Ht.
QTt Ht ¼ Hto QTt þ Qot Ht  Qot Hto

ð11:59Þ

Again, the model may need to be solved
several times in order to identify reasonably
accurate average flow and head estimates, Qot and
Hot , in each period t.
The amount of electrical energy produced is
limited by the installedkilowatts of plant capacity
P as well as on the plant factor pt. The plant
factor is a measure of hydroelectric power plant
use in each time period. Its value depends on the
characteristics of the power system and the
demand pattern for hydroelectric energy. The
plant factor is defined as the average power load
on the plant for the period divided by the
installed plant capacity. The plant factor accounts
for the variability in the demand for hydropower

River Basin Modeling

during each period t. This factor is usually
specified by those responsible for energy production and distribution. It may or may not vary
for different time periods.
The total energy produced cannot exceed the
product of the plant factor pt, the number of
hours, ht, in the period, and the plant capacity P,
measured in kilowatts.
KWHt  P ht pt

11.8

ð11:60Þ

Withdrawals and Diversions

Major demands for the withdrawal of water
include those for domestic or municipal uses,
industrial uses (including cooling water), and
agricultural uses including iirrigation. These uses
generally require the withdrawals of water from a
river system, from reservoirs, or from other surface or groundwater bodies. The water withdrawn
may be only partially consumed, and that which is
not consumed may be returned, perhaps at a different site, at a later time period, and containing
different concentrations of constituents.
Water can also be allocated to instream uses
that alter the distribution of flows in time and
space. Such uses include (1) reservoir storage,
possibly for recreational use as well as for water
supply; (2) for flow augmentation, possibly for
water quality control or for navigation or for
ecological benefits; and (3) for hydroelectric
power production. The instream uses may complement or compete with each other or with
various off-stream municipal, industrial, and
agricultural demands. One purpose of developing
management models of river basin systems is to
help derive policies that will best serve these
multiple uses, or at least identify the tradeoffs
among the multiple purposes and objectives.
The allocated flow qst to a particular use at site
s in period t must be no greater than the total flow
available, Qst , that site and in that period.
qst  Qst

ð11:61Þ

The quantity of water that any particular user
expects to receive in each particular period is

11.8

Withdrawals and Diversions

505

termed the target allocation. Given a
multi-period (e.g., annual) known or unknown
target allocation Ts at site s, some (usually
known) fraction, fst , of that target allocation will
be expected in period t. If the actual allocation,
qst , is less than the target allocation, fst Ts, there
will be a deficit, Dst . If the allocation is greater
than the target allocation, there will be an excess,
Est . Hence, to define those unknown variables the
following constraint equation can be written for
each applicable period t.
qst ¼ fts T s Dst þ Ets

ð11:62Þ

Even though allowed, one would not expect a
solution to contain nonzero values for both Dst
and Est .
Whether or not any deficit or excess allocation
should be allowed at any demand site s depends
on the quantity of water available and the losses
or penalties associated with deficit or excess allocations to that site. At sites where the benefits
derived in each period are independent of the
allocations in other periods, the losses associated
with deficits and the losses or benefits associated
with excesses can be defined in each period
t (Chap. 9). For example, the benefits derived
from the allocation of water for hydropower
production in period t in some cases will be
essentially independent of previous allocations.
For any use in which the benefits are dependent on a sequence of allocations, such as at
irrigation sites, the benefits may be based on the
annual (or growing season) target water allocations Ts and their within season distributions,
fst Ts. In these cases one can define the benefits
from those water uses as functions of the
unknown season or annual targets, Ts, where the
allocated flows qst must be no less than the
specified fraction of that unknown target.
qst  fts T s

for all relevant t

ð11:63Þ

If, for any reason, an allocation variable value
qst must be low, or even zero, due to other more
beneficial uses, then clearly from Eq. 11.63 the

annual or growing season target allocation Ts
would be low (or zero) and presumably so would
be the benefits associated with that target value.
Water stored in reservoirs can often be used to
augment downstream flows for instream uses
such as recreation, navigation, and water quality
control. During natural low-flow periods in the
dry season, it is not only the increased volume
but also the lower temperature of the augmented
flows that may provide the only means of
maintaining certain species of fish and other
aquatic life. Dilution of wastewater or runoff
from non-point sources may be another potential
benefit from flow augmentation. These and other
factors related to water quality management are
discussed in greater detail in Chap. 10.
The benefits derived from navigation on a
potentially navigable portion of a river system
can usually be expressed as a function of the
stage or depth of water in various periods.
Assuming known stream or river flow-stage
relationships at various sites in the river, a possible constraint might require at least a minimum
acceptable depth, and hence flow, for those sites.

11.9

Lake-Based Recreation

Recreation benefits derived from natural lakes as
well as reservoirs are usually dependent on their
storage levels. Where recreational facilities have
been built, recreational benefits will also be
dependent on recreational target lake levels as
well. If docks, boathouses, shelters, and other
recreational facilities were installed based on
some assumed (target) lake level, and the lake
levels deviate from the target value, there can be
reduced recreational benefits. These storage targets and any deviations can be modeled similar to
Eq. 11.62. The actual storage volume at the
beginning of a recreation period t equals the target
storage volume less any deficit or plus any excess.
Sst ¼ T s  Dst þ Ets

ð11:64Þ

506

11

The recreational benefits in any recreational
period t can be defined based on what they would
be if the target were met less the average of any
losses that may occur from initial and final
storage volume deviations, Dst or Est , from the
target storage volume in each period of the
recreation season (Chap. 9).

11.10

Model Synthesis

Each of the model components discussed above
can be combined, as applicable, into a model of a
river system. One such river system together with
some of its interested stakeholders is shown in
Fig. 11.26.
One of the first tasks in modeling this basin is
to identify the actual and potential system components and their interdependencies. This is

River Basin Modeling

facilitated by drawing a schematic of the system
at the level of detail that will address the issues
being discussed and of concern to these stakeholders. This schematic can be drawn over the
basin as in Fig. 11.27. The schematic without the
basin is illustrated in Fig. 11.28.
A site number must be assigned at each point
of interest. These sites are usually where some
decision must be made. Mass balance and other
constraints will need to be defined at each of
those sites.
As shown in the schematic in Fig. 11.28, this
river has one streamflow gage site, site 1, two
reservoirs, sites 3 and 5, two diversions, sites 2
and 3, one hydropower plant, site 5, and a levee
desired at site 4 to help protect against floods in
the urban area. The reservoir at site 5 is a
pumped storage facility. The upstream reservoir
at site 3 is used for recreation, water supply, and

Fig. 11.26 A multipurpose river system whose management is of concern to numerous stakeholders

11.10

Model Synthesis

507

Fig. 11.27 A schematic representation of the basin components and their interdependencies drawn over the map
image of the basin

Fig. 11.28 Schematic of
river system showing
components of interest at
designated sites

508

11

flood control. The downstream reservoir is
strictly for hydropower production.
Before developing a model of this river system, the number of time periods t to include in
the model and the length of each within-year
time period should be determined. If a river
system’s reservoirs are to contain storage for the
distribution of water among years, called
over-year storage, then a number of periods
encompassing multiple years of operation must
be included in the model. This will allow an
evaluation of the possible benefits of storing
excess water in wet years for release in dry years.
Many reservoir systems completely fill almost
every year, and in such cases one is concerned
only with the within-year operation of the system. This is the problem addressed here. To
model the within-year operation of the system, a
year is divided into a number of within-year
periods. The number of the periods and the
duration of each period will depend on the
variation in the hydrology, the demands, and on
the particular objectives, as previously discussed.
Once the number and duration of the time
steps to be modeled have been identified, the
variables and functions used at each site must be
named. It is convenient to use notation that can
be remembered when examining the model
solutions. The notation made up for this example
is shown in Table 11.4.
For this example assume we are interested in
maximizing a weighted combination of all the
net economic benefits derived from all the designated uses of water. There could, and no doubt
should, be other objective components defined as
well, as discussed in Chap. 9. Nevertheless these
economic objective components serve the purpose here of illustrating how a model of this river
system can be constructed:
The overall objective might be a weighted
combination of all net benefits, NBs, obtained at
each site s in the basin:
Maximize

X
s

ws NBs

ð11:65Þ

River Basin Modeling

This objective function does not identify how
much each stakeholder group would benefit and
how much they would pay. Who benefits and
who pays, and by how much, may matter. If it is
known how much of each of the net benefits
derived from each site are to be allocated to each
stakeholder group i, then these allocated fraction,
fi, of the total net benefits, NBs, can be included
in the overall objective:
Maximize

X
i

wi

X
s

fi NBs

ð11:66Þ

Using methods discussed in Chap. 9, solving
the model for various assumed values of these
weights can help identify the tradeoffs between
different conflicting objectives, Eq. 11.65, or
conflicting stakeholder interests, Eq. 11.66.
The next step in model development is to
define the constraints applicable at each site. It is
convenient to begin at the most upstream sites
and work downstream. As additional variables or
functions are needed, invent notation for them.
These constraints tie the decision variables
together and identify the interdependencies
among system components. In this example the
site index is shown as a superscript.
At site 1:
• No constraints are needed. It is the gage site.
At site 2:
• the diverted water, X2(t), cannot exceed the
streamflow, Q2(t), at that site.
Q2 ðtÞ  X 2 ðtÞ 8t in the irrigation season
ð11:67Þ
• the diversion flow, X2(t), cannot exceed the
diversion channel capacity, X2.
X 2  X 2 ðtÞ 8t in the irrigation season
ð11:68Þ

11.10

Model Synthesis

Table 11.4 Names associated with required variables and functions at each site in Fig. 11.28

The units of these variables and parameters, however defined, must be consistent

509

510

11

• the diversion flow, X2(t), must meet the irrigation target, δ2t T2
X 2 ðtÞ  d2t T 2

8t in the irrigation season
ð11:69Þ

• NB2 = benefit function associated with the
annual target irrigation allocation, T2, less the
annual cost function associated with the
diversion channel capacity, X2.

At site 4:
• define deficit diversion, D4(t), from site 3,
associated with target, δ4t T4, if any.
X 3 ðtÞ ¼ d4t T 4  D4 ðtÞ

Q4 ¼ PQ4T

• storage volume mass balances (continuity of
storage), assuming no losses.
S3 ðt þ 1Þ ¼ S3 ðtÞ þ Q3 ðtÞX 3 ðtÞR3 ðtÞ
8t; T þ 1 ¼ 1
ð11:70Þ
• define storage deficits, D3(t), and excesses,
E3(t), relative to recreation target, T3.
3

3

3

S ðtÞ ¼ T  D ðtÞ þ E ðtÞ
8t in recreation season plus

ð11:71Þ

following period:
• diverted water, X3(t), cannot exceed diversion
channel capacity, X3.
X 3 ðtÞ  X 3

8t

ð11:72Þ

• reservoir storage capacity constraints involving dead storage, K3D, and flood storage, K3F,
capacities.
S3 ðtÞ  K 3  KD3  KF3
8t in flood season plus following period:
S3 ðtÞ  K 3  KD3
for all other periods t:

ð11:73Þ
• NB3 = sum of annual benefit functions for T3
and K3F less annual costs of K3 and X3 less
annual recreation losses associated with all
D3(t) and E3(t).

8t

ð11:74Þ

• channel capacity, Q4, must equal peak flood
flow, PQ4t , associated with selected return
period, T.

At site 3:

3

River Basin Modeling

ð11:75Þ

• NB4 = sum of annual benefit functions for T4
less annual cost of Q4 less annual losses
associated with all D4(t).
At site 5:
• continuity of pumped storage volumes,
involving inflows, QI5(t), and outflows,
QO5(t), and assuming no losses.
S5 ðt þ 1Þ ¼ S5 ðtÞ þ QI 5 ðtÞ  QO5 ðtÞ 8t
ð11:76Þ
• active storage capacity involving dead storage, K5D.
S5 ðtÞ  K 5  KD5

8t

ð11:77Þ

• pumped inflows, QI5(t), cannot exceed the
amounts of water available at the intake. This
includes the release from the upstream reservoir, R3(t), and the incremental flow,
Q5(t) − Q3(t).
QI 5 ðtÞ  Q5 ðtÞ  Q3 ðtÞ þ R3 ðtÞ 8t
ð11:78Þ
• define the energy produced, EP5(t), given the
average storage head, H(t), flow through the
turbines, QO5(t), and efficiency, e.
EP5 ðtÞ ¼ ðconst:ÞðHðtÞÞðQO5 ðtÞÞe 8t
ð11:79Þ

11.10

Model Synthesis

511

• define the energy consumed, EC5(t), from
pumping given the amount pumped, QI5(t).

the functions in the objective function components and constraints. Models of this size and
complexity, even though this is a rather simple
EC 5 ðtÞ ¼ ðconst:ÞðHðtÞÞðQI 5 ðtÞÞ=e 8t
river system, are usually solved using linear
ð11:80Þ programming algorithms simply because other
nonlinear or dynamic programming (optimiza• Energy production, EP5(t), and consumption, tion) methods are more difficult to use.
The model just developed is for a typical
EC5(t), constraints given power plant capacsingle year. In some cases it may be more
ity, P5.
appropriate to incorporate over-year as well as
EP5 ðtÞ  P5 ðhours of energy production in tÞ within-year mass balance constraints, and yields
8t
with their respective reliabilities, within this
ð11:81Þ modeling framework. This can be done as outlined in Sect. 5.4 of this chapter.
The information derived from optimization
models of river systems such as this one should
EC 5 ðtÞ  P5 ðhours of pumping in tÞ 8t
ð11:82Þ not be considered as a final answer. Rather it is
an indication of the range of system design and
• Define the average storage head, H(t), based operation policies that should be further analyzed
using more detailed analyses. Optimization
on storage head functions, h(S5(t)).
models of the type just developed serve as ways
 5

5
to eliminate inferior alternatives from further
HðtÞ ¼ hðS ðt þ 1ÞÞ þ hðS ðtÞÞ =2 8t
consideration more than as ways of finding a
ð11:83Þ
solution all stakeholders will accept as the best.

• NB5 = Sum of benefits for the energy produced, EP5(t), less the costs of the energy
consumed, EC5(t), less the annual costs of
capacities K5 and P5.
Equations 11.67–11.83 together with objective Eq. 11.65 or 11.66 define the general
structure of this river system model. Before the
model can be solved, the actual functions must
be defined. Then they may have to be made
piecewise linear if linear programming is to be
the optimization procedure used to solve the
model. The process of defining functions may
add variables and constraints to the model, as
discussed in Chaps. 4 and 9.
For T within-year periods t, this static model
of a single year includes between 14T + 8 and
16T + 5 constraints, depending on the number of
periods in the irrigation and recreation seasons.
This number does not include the additional
constraints that surely will be needed to define

11.11

Project Scheduling

The river basin models discussed thus far in this
chapter deal with static planning situations in that
system components and their capacities once
determined are not assumed to change over time.
Project capacities, targets, and operating policies
take on fixed values and one examines “snapshot
steady-state” solutions for a particular time in the
future. These “snapshots” only allow for fluctuations caused by the variability of supplies and
demands. The non-hydrologic world is seldom
static, however. Targets and goals and policies
change in response to population growth, investment in agriculture and industry, and shifting
priorities for water use. In addition, financial
resources available for water resources investment
are limited and may vary from year to year.
Dynamic planning models can aid those
responsible for the long-run development and

512

expansion of water resources systems. Although
static models can identify target values and system configuration designs for a particular period
in the future, they are not well adapted to
long-run capacity expansion planning over a 10-,
20-, or 30-year period. But static models may
identify projects for implementation in early
years which in later year simulations do not
appear in the solutions (Chap. 4 contains an
example of this).
This is the common problem in capacity
expansion, where each project has a fixed construction and implementation cost as well as
variable operating, repair, and maintenance cost
component. If there are two mutually exclusive
competing projects, one may be preferred at a
site when the demand at that site is low, but the
other may be preferred if the demand is, as it is
later projected to be, much higher. Which of the
two projects should be selected now when the
demand is low, given the uncertainty of the
projected increase over time, especially assuming
it makes no economic sense to destroy and
replace a project already built?
Whereas static models consider how a water
resources system operates under a single set of
fixed conditions, dynamic expansion models
must consider the sequence of changing conditions that might occur over the planning period.
For this reason, dynamic expansion models are
potentially more complex and larger than are
their static counterparts. However, to keep the
size and cost of dynamic models within the
limitations of most studies, these models are
generally restricted to very simple descriptions of
the economic and hydrologic variables of concern. Most models use deterministic hydrology
and are constrained either to stay within predetermined investment budget constraints or to
meet predetermined future demand estimates.
Dynamic expansion models can be viewed as
network models for solution by linear or dynamic
programming methods. The challenge in river
system capacity expansion or projectscheduling
models is that each component’s performance, or
benefits, may depend on the design and operating
characteristics of other components in the system. River basin project impacts tend to be

11

River Basin Modeling

dependent on what else is happening in the basin,
i.e., what other projects are present and how they
are designed and operated.
Consider a situation in which n fixed-scale
discrete projects may be built during the planning
period. The scheduling problem is to determine
which of the projects to build and in what order.
The solution of this problem generally requires a
resolution of the timing problem. When should
each project be built, if at all?
For example, assume there are n = 3 discrete
projects that might be beneficial to implement
sometime over the next 20 years. Let this 20-year
period consists of four 5-year construction periods y. The actual benefits derived from any new
project may depend on the projects that already
exist. Let S be the set of projects existing at the
beginning of any construction period. Finally let
Nay(S) be the maximum present value of the total
net benefits derived in construction period y associated with the projects in the set S. Here
‘‘benefits’’ refer to any composite of system
performance measures.
These benefit values for various combinations
discrete projects could be obtained from static
river system models, solved for all combinations
of discrete projects for conditions existing at the
end of each of these four 5-year periods. It might
be possible to just do this for one or two of these
four periods and apply applicable discount rates
for the other periods. These static models can be
similar to those discussed in the previous section
of this chapter. Now the challenge is to find the
sequencing of these three projects over the periods y that meet budget constraints and that
maximize the total present value of benefits.
This problem can be visualized as a network.
As shown in Fig. 11.29, the nodes of this network represent the sets S of projects that exist at
the beginning of the construction period. For
these sets S we have the present value of their
benefits, NBy(S), in the next 5-years. The links
represent the project or projects implemented in
that construction period. Any set of new projects
that exceeds the construction funds available for
that period is not shown on the network. Those
links are infeasible. For the purposes of this
example, assume it is not financially feasible to

11.11

Project Scheduling

513

Fig. 11.29 Project scheduling options. Numbers in nodes represent existing projects. Links represent new projects, the
difference between the existing projects at both connecting nodes

add more than one project in any single construction period. Let Cky be the present value of
the cost of implementing project k in construction period y.

The optimal is to find the best (maximum
benefits less costs) paths through the network.
Each link represents a net benefit, NBy(S)
over the next 5-years obtained from the set of

514

11

projects, S, that exist less the cost of adding a
new project k.
Using linear programming, one can define a
continuous nonnegative unknown decision variable Xij for each link between node i and node j.
It will be an indicator of whether a link is on the
optimum path or not. If after solving the model
its value is 1, the link connecting nodes i and
j represents the decision to make in that construction period. Otherwise its value is 0 indicating the link is not on the optimal path. The
sequence of links having their Xij values equal to
1 will indicate the most beneficial sequence of
project implementations.
Let the net benefits associated with node i be
designated NBi (that equals the appropriate
NBy(S) value), and the cost, Cky, of the new
project k associated with that link. The objective
is to maximize the present value of net benefits less project implementation costs over all
periods y.
Maximize

XX
i

ðNBi  Cij ÞXij

ð11:84Þ

j

Subject to
Continuity at each node
X
h

Xhi ¼

X

Xij

j

for each node i in the network:

$y = the maximum project implementation
funds available in period y.
Fy(S) = the present value of the total benefits
over the remaining periods, y, y + 1, …, 4.
FY+1(S) = 0 for all sets of projects S following
the end of the last period.
The recursive equations for each construction
period, beginning with the last period, can be
written
Fy ðSÞ ¼ maximum fNBy ðSÞ 
for all S

s 62 S

X

X

Cky  $y

k2s

ð11:87Þ
Defining Fy′(S) as the present value of the
total benefits of all new projects in the set S implemented in all periods up to and including
period y, and the subset s of projects k being
considered in period y now belonging to the set
S of projects existing at the end of the period, the
recursive equations for a forward moving solution procedure beginning with the first period,
can be written

for all S

ð11:85Þ

ð11:86Þ

The sums in Eq. 11.86 are over nodes h having links to node i and over nodes j having links
from node i.
The optimal path through this network can
also be solved using dynamic programming.
(Refer to the capacity expansion problem illustrated in Chap. 4). For a backward moving
solution procedure, let
s = subset of projects k not contained in the
set S (s 62 S).

Cky þ Fy þ 1 ðS þ sÞg

k2s

Fy0 ðSÞ ¼ maximumf NBy ðS  sÞ 

Sum of all decision variable values on the
links in any one period y must be 1. For example
in period 1
X00 þ X01 þ X02 þ X03 ¼ 1

River Basin Modeling

s2S

X

X
k2s

0
Ck;y þ Fy1
ðSsÞg

Cky  $y

k2s

ð11:88Þ
where F0′(0) = 0.
Like the linear programming model, the
solutions of these dynamic programming models
identify the sequencing of projects recognizing
their interdependencies. Of interest, again, is
what to do in this first constriction period. The
only reason for looking into the future is to make
sure, as best as one can that the first period’s
decisions are not myopic. Models like these can
be developed and solved again with more updated estimates of future conditions when next
needed.
Additional constraints and variables might be
added to these scheduling models to enforce
requirements that some projects precede others or
that if one project is built another is infeasible.

11.11

Project Scheduling

These additional restrictions usually reduce the
size of a network of feasible nodes and links, as
shown in Fig. 11.29.
Another issue that these dynamic models can
address is the sizing or capacity expansion
problem. Frequently, the scale or capacity of a
reservoir, pipeline, pumping station, or irrigation
is variable and needs to be determined concurrently with the solution of the scheduling and
timing problems. To solve the sizing problem,
the costs and capacities in the scheduling model
become variables.
This project scheduling problem by its very
nature must deal with uncertainty. A relatively
recent contribution to this literature is the work of
Haasnoot et al. (2013), Walker et al. (2013).

11.12

Conclusions

This chapter on river basin planning models
introduces some ways of modeling river basin
components, separately and together within an
integrated model. Ignored during the development of these different model types were the
uncertainties associated with the results of these
models. As discussed in Chaps. 7 and 8, these
uncertainties may have a substantial effect on
model solution and the decision taken.
Most of this chapter has been focused on the
development of simplified screening models,
using simulation as well as optimization methods,
for identifying what and where and when infrastructure projects should be implemented, and of
what capacity. The solution of these screening
models, and any associated sensitivity and uncertainty analyses, can be of value prior to committing
to more costly design modeling exercises.
Preliminary screening of river basin systems,
especially given multiple objectives, is a challenge to accomplish in an efficient and effective
manner. The modeling methods and approaches
discussed in this chapter serve as an introduction to that art.

515

References
Haasnoot, M., Kwakkel, J. H., Walker, W. E., & ter Maat,
J. (2013). Dynamic adaptive policy pathways: A
method for crafting robust decisions for a deeply
uncertain world. Global environmental Change, 23,
485–498.
Maidment, D. R. (1993). Handbook of hydrology. NY:
McGraw-Hill Inc.
Rippl, W. (1883). The capacity of storage reservoirs for
water supply. In Proceedings of the Institute of Civil
Engineers (Brit.) (Vol. 71).
U.S. DOE. (2002, April). Hydropower program. http://
hydropower.inel.gov or http://hydropower.id.doe.gov/
USACE. (1991). Hydrology and hydraulics workshop on
Riverine Levee Freeboard, Monticello, Minnesota,
Report SP-24, Hydrologic Engineering Center, US
Army Corps of Engineers, 272 pp.
Walker, W. E., Haasnoot, M., & Kwakkel, J. H. (2013).
Adapt or Perish: A review of planning approaches for
adaptation under deep uncertainty. Sustainability, 5,
955–979.

Additional References
(Further Reading)
Becker, L., & Yeh, W. W. G. (1974). Timing and sizing
of complex water resource systems. Journal of the
Hydraulics Division, ASCE , 100(HYIO), 1457–1470.
Becker, L., & Yeh, W. W.-G. (1974a). Optimal timing,
sequencing, and sizing of multiple reservoir surface
water supply facilities. Water Resources Research, 10
(1), 57–62.
Bedient, P. B., & Huber, W. C. (1992). Hydrology and
floodplain analysis (2nd ed.). Reading: Addison
Westley.
Basson, M. S., Allen, R. B., Pegram, G. G. S., & van
Rooyen, J. A. (1994). Probabilistic management of
water resource and hydropower systems. Highlands
Ranch: Water Resources Publications.
Chin, D. A. (2000). Water-resources engineering. Upper
Saddle River: Prentice Hall.
Erlenkotter, D. (1973a). Sequencing of interdependent
hydroelectric projects. Water Resources Research, 9
(1), 21–27.
Erlenkotter, D. (1973b). Sequencing expansion projects.
Operations Research, 21(2), 542–553.
Erlenkotter, D. (1976). Coordinating scale and sequencing
decisions for water resources projects. Economic Modeling for Water Policy Evaluation, NorthHolland/TIMS
Studies in the Management Sciences, 3, 97–112.

516
Erlenkotter, D,. & Rogers, J. S. (1977). Sequencing
competitive expansion projects. Operations Research,
25(6), 937–951.
Hall, W. A., Tauxe, G. W., & Yeh, W. W.-G. (1969). An
alternative procedure for optimization of operations
for planning with multiple river, multiple purpose
systems. Water Resources Research, 5(6), 1367–1372.
Jacoby, H. D., & Loucks, D. P. (1972). Combined use of
optimization and simulation models in river basin
planning. Water Resources Research, 8(6), 1401–1414.
James, L. D., & Lee, R. R. (1971). Economics of water
resources planning. New York: McGraw-Hill Book
Company.
Linsley, R. K., Franzini, J. B., Freyberg, D. L., &
Tchobanoglous, G. (1992). Water-resources engineering. NY: McGraw-Hill Inc.
Loucks, D. P. (1976). Surface water quantity management, Chap. 5. In A. K. Biswas (Ed.), Systems
approach to water management. New York:
McGraw-Hill Book Company.
Loucks, D. P., Stedinger, J. R., & Haith, D. A. (1981).
Water resource systems planning and analysis. Englewood Cliffs: Prentice Hall Inc.
Major, D. C., & Lenton, R. L. (1979). Applied water
resources systems planning. Englewood Cliffs: Prentice Hall Inc.
Maass, A., Hufschmidt, M. M., Dorfman, R., Thomas, H. A.,
Jr., Marglin, S. A., & Fair, G. M. (1962). Design of water
resource systems. Cambridge: Harvard University Press.
Mays, L. W. (2001). Water resources engineering. NY:
Wiley.
Mays, L. W., & Tung, Y.-K. (1992). Hydrosystems
engineering and management. NY: McGraw-Hill Inc.
Morin, T. L. (1973). Optimal Sequencing of Capacity
Expansion Projects. Journal of the Hydraulics Division, ASCE, 99(HY9), 1605–1622.
Morin, T. L., & Esogbue, A. M. O. (1974). A useful theorem
in the dynamic programming solution of sequencing and
scheduling problems occurring in capital expenditure
planning. Water Resources Research, 10(1), 49–50.
National Research Council. (1999). New strategies for
America’s Watersheds. Washington, DC: National
Academy Press.
National Research Council. (2000). Risk analysis and
uncertainty in flood damage reduction studies. Washington, DC: National Academy Press.
O’Laoghaire, D. T., & Himmelblau, D. M. (1974).
Optimal expansion of a water resources system.
New York: Academic Press Inc.
ReVelle, C. (1999). Optimizing reservoir resources. NY:
Wiley.
Thomas, H. A., Jr., & Burden, R. P. (1963). Operations
research in water quality management. Cambridge:
Harvard Water Resources Group.
USACE. (1991a). Benefit determination involving existing levees. Policy Guidance Letter No. 26, Headquarters, U.S. Army Corps of Engineers, Washington, DC.
USACE. (1999). Risk analysis in geotechnical engineering for
support of planning studies, ETL 1110-2-556, Headquarters, U.S. Army Corps of Engineers, Washington, DC.

11

River Basin Modeling

Viessman, W., Jr., & Welty, C. (1985). Water management
technology and institutions. NY: Harper & Row, Pubs.
Wurbs, R. A., & James, W. P. (2002). Water resources
engineering. Upper Saddle River: Prentice Hall.

Exercises
11:1 Using the following information pertaining to the drainage area and discharge in
the Han River in South Korea, develop an
equation for predicting the natural unregulated flow at any site in the river, by
plotting average flow as a function of
catchment area. What does the slope of the
function equal?
Gage point

Catchment
area (km2)

Average flow
(106 m3/year)

First bridge of
the Han River

25,047

17,860

Pal Dang dam

23,713

16,916

So Yang dam

2703

1856

Chung Ju dam

6648

4428

10,319

7300

Hong Chun dam

1473

1094

Dal Chun dam

1348

1058

Kan Yun dam

1180

926

461

316

Yo Ju dam

Im Jae dam

11:2 In watersheds characterized by significant
elevation changes, one can often develop
reasonable predictive equations for average
annual runoff per hectare as a function of
elevation. Describe how one would use such
a function to estimate the natural average
annual flow at any gage in a watershed
which is marked by large elevation changes
and little loss of water from stream channels
due to evaporation or seepage.
11:3 Compute the storage-yield function for a
single reservoir system by the mass diagram and modified sequent peak methods
given the following sequences of annual
flows: (7, 3, 5, 1, 2, 5, 6, 3, 4). Next
assume that each year has two distinct

Exercises

517

hydrologic seasons, one wet and the other
dry, and that 80% of the annual inflow
occurs in season t = 1 and 80% of the
yield is desired in season t = 2. Using the
modified sequent peak method, show the
increase in storage capacity required for
the same annual yield resulting from
within-year redistribution requirements.
11:4 Write two different linear programming
models for estimating the maximum constant reservoir release or yield Y given a
fixed reservoir capacity K, and for estimating the minimum reservoir capacity
K required for a fixed yield Y. Assume that
there are T time periods of historical flows
available. How could these models be
used to define a storage capacity-yield
function indicating the yield Y available
from a given capacity K?
11:5(a) Construct an optimization model for estimating the least-cost combination of active
storage capacities, K1 and K2, of two
reservoirs located on a single stream, used
to produce a reliable constant annual flow
or yield (or greater) downstream of the two
reservoirs. Assume that the cost functions
Cs(Ks) at each reservoir site s are known
and there is no dead storage and no evaporation. (Do not linearize the cost functions; leave them in their functional form.)
Assume that 10 years of monthly unregulated flows are available at each site s.
(b) Describe the two-reservoir operating policy that you would incorporate into a
model model to check the solution
obtained from the optimization model.

Capacities K1

a

Annual
capacity
cost

Reservoir
capacity

Flood
stage for
flood of
return
period T
T=1

T=2

T=5

T = 10

T = 100

0

30

105

150

165

180

5

30

80

110

120

130

25

10

30

55

70

75

80

30

15

30

40

45

48

50

40

20

30

35

38

39

40

70

10a

10 is fixed cost if capacity > 0; otherwise, it is 0

Cost of flood damage

Flood stage
30

0

50

10

70

20

90

30

110

40

130

50

150

90

180

150

11:7 Develop a deterministic, static, within-year
model for evaluating the development
alternatives in the river basin shown in the
accompanying figures. Assume that there
are t = 1, 2, 3,…, n within-year periods
and that the objective is to maximize the
total annual net benefits in the basin. The
solution of the model should define the
reservoir capacities (active + flood storage
capacity), the annual allocation targets, the
levee capacity required to protect site 4
from a T-year flood, and the within-year

K2

1

11:6 Given the information in the accompanying tables, compute the reservoir capacity
that maximizes the net expected flood
damage reduction benefits less the annual
cost of reservoir capacity.

2

period allocations of water to the uses at
sites 3 and 7. Clearly define all variables
and functions used, and indicate how the
model would be solved to obtain the
maximum net benefit solution.

518

11

River Basin Modeling

1
3
2

4

5

7
6

Levee Potential flood
damage site

Gage
site
8

Irrigation
area

denote loss functions and C() will denote the cost
functions.

Site

Fraction of
gage flow

1

0

Potential reservoir for water
supply

2

0.3

Potential reservoir for water
supply, flood control

3

0.15

Diversion to a use, 60% of
allocation returned to river

4

–

Existing development,
possible flood protection from
levee

5

0.6

Potential reservoir for water
supply, recreation

6

–

Hydropower; plant
factor = 0.30

7

0.9

Potential diversion to an
irrigation district

8

1

Gage site

For simplicity, assume no evaporation losses
or dead storage requirements. Omitting the
appropriate subscripts t for time periods and s for
site, let T, K, D, E, and P be the target, reservoir
capacity, deficit, excess, and power plant capacity variables, respectively. Let Qt and Rt be the
natural streamflows and reservoir releases, and St
be the initial reservoir storage volumes in period
t. Kf will denote the flood storage capacity at site
2 that will contain a peak flow of QS and QR is
the downstream channel flood flow capacity. The
relationship between QS and Kf is defined by the
function κ(QS). The unregulated design flood
peak flow for which protection is required is QN.
KWH will be the kilowatt-hours of energy,
H will be net storage head, ht the hours in a
period t. The variable q will be the water supply
allocation. Benefit functions will be B(), L() will

11:8 List the potential difficulties involved
when attempting to structure models for
defining
(a) Water allocation policies for irrigation
during the growing season.
(b) Energy production and capacity of
hydroelectric plants.
(c) Dead storage volume requirements in
reservoirs.
(d) Active storage volume requirements in
reservoir.
(e) Flood storage capacities in reservoirs.
(f) Channel improvements for damage
reduction.
(g) Evaporation and seepage losses from
reservoirs.
(h) Water flow or storage targets using
long-run benefits and short- run loss
functions.
11:9 Assume that demand for water supply
capacity is expected to grow as t(60 − t),
for t in years. Determine the minimum
present value of construction cost of some
subset of water supply options described
below so as to always have sufficient
capacity to meet demand over the next
30 years. Assume that the water supply
network currently has no excess capacity
so that some project must be built immediately. In this problem, assume that project capacities are independent and thus
can be summed. Use a discount factor

Exercises

519

equal to exp(−0.07 t). Before you start,
what is your best guess at the optimal
solution?
Project number

Construction cost

Capacity

1

100

200

2

115

250

3

190

450

4

270

700

11:10 (a) Construct a flow diagram for a simulation model designed to define a
storage-yield function for a single
reservoir given known inflows in each
month t for n years. Indicate how you
would obtain a steady-state solution
not influenced by an arbitrary initial
storage volume in the reservoir at the
beginning of the first period. Assume
that evaporation rates (mm per month)
and the storage volume/surface area
functions are known.
(b) Write a flow diagram for a simulation
model to be used to estimate the probability that any specific reservoir capacity, K, will satisfy a series of known
release demands, rt, downstream given
unknown future inflows, it. You need
not discuss how to generate possible
future sequences of streamflows, only
how to use them to solve this problem.
11:11 (a) Develop an optimization model for
finding the cost-effective combination
of flood storage capacity at an
upstream reservoir and channel
improvements at a downstream
potential damage site that will protect
the downstream site from a prespecified design flood of return period
T. Define all variables and functions
used in the model
(b) How could this model be modified to
consider a number of design floods
T and the benefits from protecting the

potential damage site from those
design floods? Let BFT be the annual
expected flood protection benefits at
the damage site for a flood having
return periods of T.
(c) How could this model be further
modified to include water supply
requirements of At to be withdrawn
from the reservoir in each month t?
Assume known natural flows Qst at
each site s in the basin in each month t.
(d) How could the model be enlarged to
include recreation benefits or losses at
the reservoir site? Let Ts be the
unknown storage volume target and
Dst be the difference between the
storage volume Sst and the target Ts if
Sst – Ts > 0, and Ets be the difference if
Ts – Sst > 0. Assume that the annual
recreation benefits B(Ts) are a function
of the target storage volume Ts and the
losses LD ðDst Þ and LE ðEts Þ are associated with the deficit Dst and excess Ets
storage volumes.
11:12 Given the hydrologic and economic
data listed below, develop and solve a
linear programming model for estimating the reservoir capacity K, the
flood storage capacity Kf, and the
recreation storage volume target T that
maximize the annual expected flood
control benefits, Bf(Kf), plus the
annual recreation benefits, B(T), less
all losses LD ðDt Þ and LE ðEt Þ associated with deficits Dt or excesses Et in
the periods of the recreation season,
minus the annual cost C(K) of storage
reservoir capacity K. Assume that the
reservoir must also provide a constant
release or yield of Y = 30 in each
period t. The flood season begins at
the beginning of period 3 and lasts
through period 6. The recreation season begins at the beginning of period
4 and lasts though period 7.

520

11

Period

1

2

3

4

5

6

7

8

9

Inflows
to
reservoir

50

30

20

80

60

20

40

10

70

8
< 12Kf
Bf ðKf Þ ¼ 60 þ 8ðKf  5Þ
:
140 þ 4ðKf  15Þ

if
if
if

8
45 þ 10K
>
>
<
95 þ 6ðKf  5Þ
CðKf Þ ¼
185 þ 10ðKf  20Þ
>
>
:
385 þ 15ðK  40Þ

B(T) = 9T,

LD ðDt Þ = 4Dt,
LE ðEt Þ = 2 Et,

if
if
if
if

Kf  5
5  Kf  15
Kf  15

0  Kf  5
5  Kf  20
20  Kf  40
Kf  40

where T is a particular
unknown value of -reservoir
storage
where Dt is T − St if T ≥ St
where Et is St − T if St ≥ T

River Basin Modeling

11:13 The optimal operation of multiple reservoir systems for hydropower production
presents a very nonlinear and often difficult problem.
Use dynamic programming to determine the
operating policy that maximizes the total annual
hydropower production of a two-reservoir system, one downstream of the other. The releases
R1t from the upstream reservoir plus the unregulated incremental flow (Q2t − Q1t) constitute
the inflow to the downstream dam. The flows Q1t
into the upstream dam in each of the four seasons
along with the incremental flows (Q2t − Q1t) and
constraints on reservoir releases are given in the
accompanying two tables:
Upstream dam (flow in 106 m3/period)
Season
t

Inflow
Q1t

Minimum
release

Maximum
release through
turbines

1

60

20

90

2

40

30

90
(continued)

Y = 30

Qt = inflow

Rt = release
(excess)
St = initial storage
Kf = flood storage
K = total capacity

15
4

10

8
C(K)

Bf (Kf)
12

0

45
5

Kf

15

-15

6
12

5

20

K

40

Exercises

521

Upstream dam (flow in 106 m3/period)
Season
t

Inflow
Q1t

Minimum
release

Maximum
release through
turbines

3

80

20

90

4

120

20

90

storage might occur at each reservoir. Assume
that the conversion of potential energy equals to
the product RiHi to electric energy is 70% efficient independent of Ri and Hi. In calculating the
energy produced in any season t at reservoir i,
use the average head during the season
 i ¼ 1 ½Hi ðtÞ þ Hi ðt þ 1Þ
H
2

Downstream dam (flow in 106 m3/period)
Season
t

Incremental
flow,
(Q2t − Q1t)

Minimum
release

Maximum
release
through
turbines

1

50

30

140

2

30

40

140

3

60

30

140

4

90

30

140

Note that there is a limit on the quantity of water
that can be released through the turbines for energy
generation in any season due to the limited capacity
of the power plant and the desire to produce
hydropower during periods of peak demand.
Additional information that affects the operation of the two reservoirs are the limitations on
the fluctuations in the pool levels (head) and the
storage-head relationships:
Data

Upstream
dam

Downstream
dam

Maximum head,
Hmax (m)

70

90

Minimum head,
Hmin (m)

30

60

Maximum storage
volume, Smax (m3)

150 × 106

400 × 106

Storage-net head
relationship

H = Hmax(S/
Smax)0.64

H = Hmax(S/
Smax)0.62

In solving the problem, discretize the storage
levels in units of 10 × 106 m3. Do a preliminary
analysis to determine how large a variation in

Report your operating policy and the amount
of energy generated per year. Find another feasible policy and show that it generates less
energy than the optimal policy.
Show how you could use linear programming
to solve for the optimal operating policy by
 i by a linear
approximating the product term Ri H
expression.
11:14 You are responsible for planning a project that may involve the building of a
reservoir to provide water supply benefits
to a municipality, recreation benefits
associated with the water level in the lake
behind the dam, and flood damage
reduction benefits. First you need to
determine some design variable values,
and after doing that you need to determine the reservoir operating policy.
The design variables you need to determine
include
• the total reservoir storage capacity (K),
• the flood storage capacity (Kf) in the first
season that is the flood season,
• the particular storage level where recreation
facilities will be built, called the storage target
(ST) that will apply in seasons 3, 4, and 5—
the recreation seasons and finally,
• the dependable water supply or yield (Y) for
the municipality.

522

11

River Basin Modeling

ST
Kf

Assume you can determine these design
variable values based on average flows at the
reservoir site in six seasons of a year. These
average flows are 35, 42, 15, 3, 15, and 22 in the
seasons 1–6, respectively.
The objective is to design the system to maximize the total annual net benefits derived from
• flood control in season 1,
• recreation in seasons 3–5, and
• water supply in all seasons,
less the annual cost of the
• reservoir and
• any losses resulting from not meeting the
recreation storage targets in the recreation
seasons.
The flood benefits are estimated to be 2 Kf .
The recreation benefits for the entire recreation season are estimated to be 8 ST.
The water supply benefits for the entire year
are estimated to be 20 Y.
The annual reservoir cost is estimated to be
3 K1.2.
The recreation loss in each recreation season
depends on whether the actual storage volume is
lower or higher than the storage target. If it is
lower the losses are 12 per unit average deficit in
the season, and if they are higher the losses are 4
per unit average excess in the season. It is possible that a season could begin with a deficit and
end with an excess, or vice versa.
0.7

Develop and solve a nonlinear optimization
model for finding the values of each of the design
variables: K, Kf, ST, and Y and the maximum
annual net benefits. (There will be other variables
as well. Just define what you need and put it all
together in a model.)
Does the solution give you sufficient information that would allow you to simulate the
system using a sequence of inflows to the
reservoir that are different than the ones used to
get the design variable values? If not how would
you define a reservoir operating policy? After
determining the system’s design variable values
using optimization, and then determining the
reservoir operating policy, you would then simulate this system over many years to get a better
idea of how it might perform.
11:15 Suppose you have 19 years of monthly
flow data at a site where a reservoir could
be located. How could you construct a
model to estimate what the required
over-year and within year storage needed
to produce a specified annual yield Y that
is allocated to each month t by some
known fraction δt. What would be the
maximum reliability of those yields? If
you wanted to add to that an additional
secondary yield having only 80% reliability, how would the model change?
Make up 19 annual flows and assume that
the average monthly flows are specified
fractions of those annual flows. Just using
these annual flows and the average
monthly fractions, solve your model.

Exercises

523

Capacities K1

11:16

K2

1

(a) Develop an optimization model for
estimating the least-cost combination
of active storage K1 and K2 capacities
at two reservoir sites on a single
stream that are used to produce a
reliable flow or yield downstream of
the downstream reservoir. Assume
10 years of monthly flow data at each
reservoir site. Identify what other
data are needed.
(b) Describe the two-reservoir operating
policy that could be incorporated into
a simulation model to check the
solution obtained from the optimization model.
Define
C s ðK s Þ
Kds
Sst
Ls
R12
t
Yt
Qst
aso
as
est

cost of active storage capacity at site s;
where s = 1, 2
dead storage capacity of reservoir at
site s; Kds = 0
storage volume at beginning of period
t at site s
loss of water due to evaporation at site
s; Ls = 0
release from reservoir at site 1 to site 2
in period t
yield to downstream in period t
10 years of monthly natural flows
available at each site s
area associated with dead storage
volume at site s
area per unit storage volume at site s
evaporation depth in period t at site s

T periods t, and an operating policy that
defines the lagoon discharge as a function of the initial volume and inflow,
indicate how you would estimate the
probability distribution of lagoon storage volumes.
11:18 (a) Using the inflow data in the table
below, develop and solve a yield
model for estimating the storage capacity of a single reservoir required to
produce a yield of 1.5 that is 90% reliable in both of the two within-year periods t, and an additional yield of 1.0
that is 70% reliable in period t = 2.
(b) Construct a reservoir-operating rule that
defines reservoir release zones for these
yields.
(c) Using the operating rule, simulate the 18
periods of inflow data to evaluate the
adequacy of the reservoir capacity and
storage zones for delivering the required
yields and their reliabilities. (Note that
in this simulation of the historical record
the 90% reliable yield should be satisfied in all the 18 periods, and the
incremental 70% reliable yield should
fail only two times in the 9 years.)
(d) Compare the estimated reservoir capacity with that which is needed using the
sequent peak procedure.
Year
1
2

11:17 Given inflows to an effluent storage
lagoon that can be described by a simple
first-order Markov chain in each of

2

3

Period

Inflow

1

1.0

2

3.0

1

0.5

2

2.5

1

1.0

2

2.0
(continued)

524
Year
4
5
6
7
8
9

11
Period

Inflow

1

0.5

2

1.5

1

0.5

2

0.5

1

0.5

2

2.5

1

1.0

2

5.0

1

2.5

2

5.5

1

1.5

2

4.5

11:19 One possible modification of the yield
model of would permit the solution
algorithm to determine the appropriate
failure years associated with any desired
reliability instead of having to choose
these years prior to model solution. This
modification can provide an estimate of
the extent of yield failure in each failure
year and include the economic consequences of failures in the objective
function. It can also serve as a means of
estimating the optimal reliability with
respect to economic benefits and losses.
Letting Fy be the unknown yield reduction in a possible failure year y, then in
place of αpyYp in the over-year continuity
constraint, the term (Yp − Fy) can be
used. What additional constraints are
needed to ensure (1) that the average
shortage does not exceed (1 − αpy)Yp or
(2) that at most there are f failure years
and none of the shortages exceed
(1 − αpy)Yp.
11:20 In Indonesia there exists a wet season
followed by a dry season each year. In
one area of Indonesia all farmers within
an irrigation district plant and grow rice

River Basin Modeling

during the wet season. This crop brings
the farmer the largest income per hectare;
thus they would all prefer to continue
growing rice during the dry season.
However, there is insufficient water during the dry season for irrigating all
5000 ha of available irrigable land for
rice production. Assume an available
irrigation water supply of 32 × 106 m3 at
the beginning of each dry season, and a
minimum requirement of 7000 m3/ha for
rice and 1800 m3/ha for the second crop.
(a) What proportion of the 5000 ha
should the irrigation district manager
allocate for rice during the dry season
each year, provided that all available
hectares must be given sufficient
water for rice or the second crop?
(b) Suppose that crop production functions are available for the two crops,
indicating the increase in yield per
hectare per m3 of additional water,
up to 10,000 m3/ha for the second
crop. Develop a model in which the
water allocation per hectare, as well
as the hectares allocated to each crop,
is to be determined, assuming a
specified price or return per unit of
yield of each crop. Under what conditions would the solution of this
model be the same as in part (a)?
11:21 Along the Nile River in Egypt, irrigation
farming is practiced for the production of
cotton, maize, rice, sorghum, full and
short berseem for animal production,
wheat, barley, horsebeans, and winter
and summer tomatoes. Cattle and buffalo
are also produced, and together with the
crops that require labor, water. Fertilizer,
and land area (feddans). Farm types or
management practices are fairly uniform,
and hence in any analysis of irrigation

Exercises

525

policies in this region this distinction
need not be made. Given the accompanying data develop a model for determining the tons of crops and numbers of
animals to be grown that will maximize
(a) net economic benefits based on
Egyptian prices, and (b) net economic
benefits based on international prices.
Identify all variables used in the model.
Known parameters
Ci
PEi
PIi
v
g
fP
fN
Yi
α
β
rw
swh
sba
rs
lNi
lPi
lim
wim
him

miscellaneous cost of land preparation per
feddan
Egyptian price per 1000 tons of crop i
international price per 1000 tons of crop i
value of meat and dairy production per
animal
annual labor cost per worker
cost of P fertilizer per ton
cost of N fertilizer per ton
yield of crop i, tons/feddan
feddans serviced per animal
tons straw equivalent per ton of berseem
carryover from winter to summer
berseem requirements per animal in winter
straw yield from wheat, tons per feddan
straw yield from barley, tons per feddan
straw requirements per animal in summer
N fertilizer required per feddan of crop i
P fertilizer required per feddan of crop i
labor requirements per feddan in month m,
man-days
water requirements per feddan in
month m, 1000 m3
land requirements per month, fraction
(1 = full month)

Required Constraints. (Assume known
resource limitations for labor, water, and land)
(a) Summer and winter fodder
requirements for the animals.
(b) Monthly labor limitations.
(c) Monthly water limitations.
(d) Land availability each month.

(berseem)

(e) Minimum number of animals required for
cultivation.
(f) Upper bounds on summer and winter tomatoes (assume these are known).
(g) Lower bounds on cotton areas (assume this is
known).
Other possible constraints
(a)
(b)
(c)
(d)

Crop balances.
Fertilizer balances.
Labor balance.
Land balance.

11:22 In Algeria there are two distinct cropping
intensities, depending upon the availability of water. Consider a single crop
that can be grown under intensive rotation or extensive rotation on a total of
A hectares. Assume that the annual water
requirements for the intensive rotation
policy are 16,00 m3 per ha, and for the
extensive rotation policy they are
4000 m3 per ha. The annual net production returns are 4000 and 2000 dinars,
respectively. If the total water available is
320,000 m3, show that as the available
land area A increases, the rotation policy
that maximizes total net income changes
from one that is totally intensive to one
that is increasingly extensive.
Would the same conclusion hold if instead of
fixed net incomes of 4000 and 2000 dinars per
hectares of intensive and extensive rotation, the
net income depended on the quantity of crop
produced? Assuming that intensive rotation
produces twice as much produced by extensive
rotation, and that the net income per unit of crop
Y is defined by the simple linear function
5 − 0.05Y, develop and solve a linear programming model to determine the optimal rotation
policies if A equals 20, 50, and 80. Need this net
income or price function be linear to be included
in a linear programming model?

526

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11

River Basin Modeling

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Urban Water Systems

“Today, a simple turn of the tap provides clean
water—a precious resource. Engineering advances in managing this resource—with water
treatment, supply, and distribution systems—
changed urban life profoundly in the 20th century, virtually eliminating waterborne diseases in
developed nations, and providing clean and
abundant water for communities, farms, and
industries.” So states the US National Academy
of Engineering on its selection of water supply
systems to be among the top five greatest
achievements of engineering in the twentieth
century. But providing everyone with clean tap
water, especially in urban areas, has yet to be
achieved, even in developed nations. The world’s
population is growing by about 80 million people per year, and is predicted to approach
10 billion by 2050. Over 50% of people on our
planet today live in urban areas and that percentage will grow. As populations continually
move to cities for improved economic opportunities and a higher standard of living and as cities
merge to form megacities, the design and management of water becomes an increasingly
important part of integrated urban infrastructure
planning and management.

12.1

Introduction

Urban water management involves the planning,
design, and operation of infrastructure needed to
meet the demands for drinking water and sanitation, the control of infiltration and stormwater

12

runoff, and for recreational parks and the maintenance of urban ecosystems. As urban areas
grow, so do the demands for such services. In
addition there is an increasing need to make
urban water systems more resilient to climate
change. All this leads to the realization that urban
water management must be an integral part of
urban planning in general. Land use decisions
impact water supply and wastewater system
designs and operation, as well as measures needed for managing stormwater runoff. A functioning urban infrastructure system also requires
energy which in turn typically requires water.
Sustainable urban development must focus on
the relationships between water, energy, and land
use, and often on diversifying sources of water to
assure reliable supplies. Integrated urban water
management (IUWM) provides both a goal and a
framework for planning, designing, and managing
urban water systems. It is a flexible process that
responds to change and enables stakeholders to
participate in, and predict the impacts of, development decisions. It includes the environmental,
economic, social, technical, and political aspects
of urban water management. It enables better land
use planning and the management of its impacts on
fresh water supplies, treatment, and distribution;
wastewater collection, treatment, reuse and disposal; stormwater collection, use and disposal; and
solid waste collection, recycling, and disposal
systems. It makes urban development part of
integrated basin management oriented toward a
more economically, socially, and environmentally
sustainable mixed urban–rural landscape.

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_12

527

528

12

Urban Water Systems

Fig. 12.1 Schematic showing urban surface water source, water treatment prior to urban use, and some sources of
nonpoint urban drainage and runoff and its impacts

While recognizing the need for and the benefits derived from a systems approach to urban
planning and development, including its water
related components, this chapter will serve as an
introduction to each of these components separately, and not all together as a system. This
understanding of each component is needed if
indeed they eventually will be modeled,
designed, and managed as part of the overall
urban infrastructure system.
These urban water infrastructure components
typically include water collection and storage
facilities at source sites, water transport via
aqueducts (canals, tunnels, and/or pipelines)
from source sites to water treatment facilities;
water treatment, storage, and distribution systems; wastewater collection (sewer) systems and

treatment; and urban drainage works. This is
illustrated as a simple schematic in Fig. 12.1.
Generic data-driven simulation models of
components of urban water systems have been
developed and are commonly applied to study
specific component design and operation issues.
Increasingly optimization models are being used
to estimate cost-effective designs and operating
policies. Cost savings can be substantial, especially when applied to large complex urban
systems (Dandy and Engelhardt 2001; Savic and
Walters 1997).
Most urban water users desire, and many
require, high quality potable water. The quality of
natural surface and/or groundwater supplies,
called raw water, often cannot meet the quality
requirements of domestic and industrial users. In

12.1

Introduction

such situations water treatment prior to its use is
required. Once treated, water can be stored and
distributed within the urban area, usually through a
network of storage tanks and pipes. Pipe flows in
urban distribution systems should be under pressure to prevent contamination from groundwater
leakage and to meet fire protection and other user
requirements.
After use, the “wastewater” is collected in a
network of sewers, or in some cases ditches,
leading to a wastewater treatment plant or discharge site. Wastewater treatment plants remove
some of the impurities in the wastewater before
discharging it into receiving water bodies or on
land surfaces. Water bodies receiving effluents
from point sources such as wastewater treatment
plants may also receive runoff from the surrounding watershed area during storm events.
The discharge of point and nonpoint pollutants
into receiving water bodies can impact the quality
of the water in those receiving water bodies. The
fate and transport of these pollutants in water
bodies can be predicted using water quality
models similar to those discussed in Chap. 10.
This chapter briefly describes these urban
water system components and reviews some of
the general assumptions incorporated into optimization and simulation models used to plan and
manage urban water infrastructure systems. The
focus of urban water systems modeling is mainly
on the prediction and management of quantity
and quality of flows and pressure heads in water
distribution networks, wastewater flows in gravity sewer networks, and on the design efficiencies
of water and wastewater treatment plants. Other
models can be used for the real-time operation of
various components of urban systems.

Box 12.1 An urban drinking water
crisis
This ongoing (2016) case of urban water
management in Flint, Michigan (US) illustrates what can happen even in so-called
developed regions if decisions are made
without adequate analyses of possible
health impacts and the consequences of

529

poor follow-up decisions at various governmental levels.
After a change in April 2014 of the source
of the city’s drinking water, the city’s water
distribution system became contaminated
with lead. This has created a serious public
health danger. As a result, thousands of
residents have severely high levels of lead in
their blood and are experiencing a variety of
serious health problems. Local, state, and
federal authorities, and political leaders, did
not seem sufficiently concerned until
inhabitants of Flint, with the help of others
who performed water quality analyses and
obtained public health statistics from local
hospitals, made it a national issue. In
November, 2015, some of the residents filed
a federal class action lawsuit against the
Governor and 13 other city and state officials. Additional lawsuits have been filed
after that and resignations have occurred. In
January 2016, the Governor declared the
city to be in a state of emergency. Less than 2
weeks later the President of the US declared
the drinking water crisis in Flint as a federal
emergency authorizing additional help from
the federal government.

12.2

Water Treatment

Before water is to be used for human consumption
its harmful impurities need to be removed. Communities that do not have adequate water treatment
facilities, common in developing regions, often
have high incidences of disease and mortality due
to contaminated water supplies. A range of syndromes, including acute dehydrating diarrhea (cholera), prolonged febrile illness with abdominal
symptoms (typhoid fever), acute bloody diarrhea
(dysentery), and chronic diarrhea (Brainerd diarrhea) result in over 3 billion episodes of diarrhea
and an estimated 2 million deaths, mostly among
children, each year.
Contaminants in natural water supplies can
include microorganisms such as Cryptosporidium

530

and Giardia lamblia, inorganic and organic
cancer-causing chemicals (such as compounds
containing arsenic, chromium, copper, lead, and
mercury), and radioactive materials (such as
radium and uranium). As Box 12.1 illustrates, this
need to protect water supplies from such contaminants is not limited to developing regions.
To remove impurities and pathogens, a typical
municipal water purification system involves a
sequence of physical and chemical processes.
Physical and chemical removal processes include
initial and final filtering, coagulation, flocculation
sedimentation, and disinfection, as illustrated in
the schematic of Fig. 12.2.
As shown in Fig. 12.2, one of the first steps in
most water treatment plants involves passing raw
water through coarse filters to remove sand, grit,
and large solid objects. Next, a chemical such as
alum is added to the raw water to facilitate coagulation of remaining impurities. As the wastewater
is stirred the alum causes the formation of sticky
globs of small particles made up of bacteria, silt,
and other impurities. Once these globs of matter
are formed, the water is routed to a series of settling
tanks where the globs, or floc, sink to the bottom.
This settling process is called flocculation.

Fig. 12.2 Typical processes in water treatment plants

12

Urban Water Systems

After flocculation the water is pumped slowly
across another large settling basin. In this sedimentation or clarification process much of the
remaining floc and solid material accumulates at
the bottom of the basin. The clarified water is
then passed through layers of sand, coal, and
other granular material to remove microorganisms—including viruses, bacteria, and protozoa
such as Cryptosporidium—and any remaining
floc and silt. This stage of purification mimics the
natural filtration of water as it moves through the
ground.
The filtered water is then treated with chemical
disinfectants to kill any organisms that remain
after the filtration process. An effective disinfectant is chlorine but its use may cause potentially
dangerous substances such as trihalomethanes.
Alternatives to chlorine include ozone oxidation (Fig. 12.2). Unlike chlorine, ozone does not
stay in the water after it leaves the treatment plant,
so it offers no protection from bacteria that might
be in the storage tanks and water pipes of the water
distribution system. Water can also be treated with
ultraviolet light to kill microorganisms, but it has
the same limitation as oxidation. It is ineffective
outside the treatment plant.

12.2

Water Treatment

531

Fig. 12.3 A 6-million gallon per day water treatment plant at San Luis Obispo, located about halfway between Los
Angeles and San Francisco on the Central Coast of California

Figure 12.3 is an aerial view of a water
treatment plant serving a population of about
50,000.

12.3

Water Distribution

Water distribution systems include pumping
stations, distribution storage, and distribution
piping. The hydraulic performance of each
component in the distribution network depends
upon the performance of other components. Of
interest to designers are both the flows and their
pressures throughout the network.
The energy at any point within a network of
pipes is often represented in three parts: the
pressure head, p/γ, the elevation head, Z, and the
velocity head, V2/2g. (A more precise representation includes a kinetic energy correction factor,
but that factor is small and can be ignored.) For
open-channel flows, the elevation head is the
distance from some datum to the top of the water
surface. For pressure pipe flow, the elevation
head is the distance from some datum to the

center of the pipe. The parameter p is the pressure, e.g., newton per cubic meter (N/m3), γ is the
specific weight (N/m2) of water, Z is the elevation above some base elevation (m), V is the
velocity (m/s), and g is the gravitational acceleration (9.81 m/s2).
Energy can be added to the system such as by
a pump, and lost from the system such as by
friction. These changes in energy are referred to
as head gains and losses. Balancing the energy
across any two sites i and j in the system requires
that the total heads, including any head gains HG
and losses HL (m) are equal.


p=c þ Z þ V 2 =2g þ HG site i


¼ p=c þ Z þ V 2 =2g þ HL site j

ð12:1Þ

The hydraulic grade is the sum of the pressure
head and elevation head (p/γ + Z). For
open-channel flow the hydraulic grade is the
water surface slope, since the pressure head at its
surface is 0. For a pressure pipe, the hydraulic
head is the height to which a water column
would rise in a piezometer—a tube rising from

532

the pipe. When plotted in profile along the length
of the conveyance section, this is often referred
to as the hydraulic grade line, or HGL. The
hydraulic grade lines for open-channel and
pressure pipes are illustrated in Figs. 12.4 and
12.5.
The energy grade is the sum of the hydraulic
grade and the velocity head. This is the height to
which a column of water would rise in a Pitot
tube, but also accounting for fluid velocity. When

Fig. 12.4 The energy components along an open channel

Fig. 12.5 The energy components along a pressure pipe

12

Urban Water Systems

plotted in profile, as in Fig. 12.5, this is often
referred to as the energy grade line, or EGL. At a
lake or reservoir, where the velocity is essentially
zero, the EGL is equal to the HGL.
Specific energy, E, is the sum of the depth of
flow and the velocity head, V2/2 g. For
open-channel flow, the depth of flow, y, is the
elevation head minus the channel bottom elevation. For a given discharge, the specific energy is
solely a function of channel depth. There may be

12.3

Water Distribution

533

more than one depth having the same specific
energy. In one case the flow is subcritical (relatively higher depths, lower velocities) and in the
other case the flow is critical (relatively lower
depths and higher velocities). Whether or not the
flow is above or below the critical depth (the
depth that minimizes the specific energy) will
depend in part on the channel slope.
Friction is the main cause of head loss. There
are many equations that approximate friction loss
associated with fluid flow through a given section
of channel or pipe. These include Manning’s or
Strickler’s equation that is commonly used for
open-channel flow and Chezy’s or Kutter’s
equation, Hazen–Williams equation, and Darcy–
Weisbach or Colebrook–White equations that are
used for pressure pipe flow. They all define flow
velocity, V (m/s) as an empirical function of a
flow resistance factor, C, the hydraulic radius
(cross-sectional area divided by wetted perimeter), R, and the friction or energy slope, S = HL/
Length.
V ¼ kCRx Sy

ð12:2Þ

The terms k, x, and y of Eq. 12.2 are parameters.
The roughness of the flow channel usually
determines the flow resistance or roughness factor C. The value of C may also be a function of
the channel shape, depth, and fluid velocity.
Values of C for different types of pipes are listed
in hydraulics texts or handbooks (e.g., Mays
2000, 2001; Chin 2000).

12.3.1



V ¼ R2=3 S1=2 =n

ð12:3Þ

Q ¼ AV

ð12:4Þ

The values of various friction factors n can be
found in tables in hydraulics texts and
handbooks.
The energy balance between two ends of a
channel segment is defined in Eq. 12.5. For
open-channel flow the pressure heads are 0. Thus
for a channel containing water flowing from site
i to site j:


Z þ V 2 =2g


site i



¼ Z þ V 2 =2g þ HL site j
ð12:5Þ

The head loss HL is assumed to be primarily due
to friction.
The friction loss is computed based on the
average rate of friction loss along the segment
and the length of the segment. This is the difference in the energy grade line elevations
(EGL) between sites i and j.


HL ¼ EGLi  EGLj




¼ Z þ V 2 =2g site i  Z þ V 2 =2g site j
ð12:6Þ
The friction loss per unit distance along the
channel is the average of the friction slopes at the
two ends divided by the channel length. This
defines the energy grade line, EGL.

Open-Channel Networks
12.3.2

For open-channel flow Manning’s or Strickler’s
equation is commonly used to predict the average
velocity, V (m/s), and the flow, Q (m3/s) associated with a given cross-sectional area, A (m2).
The velocity depends on the hydraulic radius
R (m) and the slope S of the channel as well as a
friction factor n.

Pressure Pipe Networks

The Hazen–Williams equation is commonly used
to predict the flows or velocities in pressure
pipes. Flows and velocities are again dependent
on the slope, S, the hydraulic radius R (m) (that
equals the half the pipe radius, r), and the
cross-sectional area, A (m2).

534

12

V ¼ 0:849CR0:63 S0:54

ð12:7Þ

Q ¼ AV ¼ p r 2 V

ð12:8Þ

The head loss along a length L (m) of pipe of
diameter D (m) containing a flow of Q (m3/s) is
defined as
HL ¼ KQ1:85 ;

ð12:9Þ

where K is the pipe coefficient defined by
Eq. 12.10.


K ¼ ½10:66L= C1:85 D4:87

ð12:10Þ

Another pipe flow equation for head loss is the
Darcy–Weisbach equation based on a friction
factor f.
HL ¼ f LV 2 =D2g

ð12:11Þ

The friction factor is dependent on the Reynolds
number and the pipe roughness and diameter.
Given these equations it is possible to compute the distribution of flows and heads
throughout a network of open channels or pressure pipes. The two conditions are the continuity
of flows at each node, and the continuity of head
losses in loops for each time period t.
At each node i:

Fig. 12.6 An example of
a pipe network, showing
the values of K for
predicting head losses from
Eq. 12.10

Urban Water Systems

out
Storageit þ Qin
it  Qit ¼ Storagei;t þ 1 ð12:12Þ

In each section between nodes i and j:
HLit ¼ HLjt þ HLijt ;

ð12:13Þ

where the head loss between nodes i and j is
HLijt.
To compute the flows and head losses at each
node in Fig. 12.6 requires two sets of equations,
one for continuity of flows, and the other continuity of head losses. In this example, the direction of flow in two links, from A to C, and from
B to C are assumed unknown and hence each is
represented by two nonnegative flow variables.
Let Qij be the flow from site i to site j and Hi
be the head at site i. Continuity of flow in this
network requires:
0:5 ¼ QDA þ QDC

ð12:14Þ

0:1 ¼ QDA  QAC þ QCA  QAB

ð12:15Þ

0:25 ¼ QAB þ QCB  QBC

ð12:16Þ

0:15 ¼ QDC þ QAC  QCA þ QBC  QCB
ð12:17Þ
Continuity of heads at each node requires:

12.3

Water Distribution

535



HD ¼ HC þ 22  Q1:85
DC ;

ð12:18Þ



HD ¼ HA þ 11  Q1:85
DA ;

ð12:19Þ

HA ¼ HB þ 22  Q1:85
AB ;

ð12:20Þ



HC ¼ HA þ 25  ðQCA QAC Þ

1:85



between nodes A and C may not be economical, at
least for these flow conditions. Other flow conditions may prove otherwise. But even if they do
not, this pipe section increases the reliability of
the system, and reliability is an important consideration in water supply distribution networks.

12.3.3
;

ð12:21Þ



HC ¼ HB þ 11  ðQCB QBC Þ1:85

ð12:22Þ

Solving these Eqs. 12.14–12.22 simultaneously
for the five flow and four head variables yields
the flows Qij from nodes i to nodes j and heads Hi
at nodes i listed in Table 12.1.
This solution shown in Table 12.1 assumes
that the network is at a constant elevation, has no
storage capacity, and there are no minor losses.
Losses are usually expressed as a linear function
of the velocity head, due to hydraulic structures
(such as valves, restrictions, or meters) at each
node. This solution suggests that the pipe section

Water Quality

Many of the water quality models discussed in
Chap. 10 can be used to predict water quality
constituent concentrations in open channels and
in pressure pipes. The assumption of complete
mixing such as at junctions or in short segments
of pipe, is made. Reactions among constituents
can occur as water travels through the system at
predicted velocities. Water-resident times (the
ages of waters) in various parts of the network
are important variables for water quality prediction as constituent decay, transformation, and
growth processes take place over time.
Computer models typically use numerical
methods to find the hydraulic flow and head
relationships as well as the resulting water
quality concentrations. Most numerical models

Table 12.1 Flows and heads of the network shown in Fig. 12.6

536

12

assume combinations of plug flow (advection)
along pipe sections and complete mixing within
segments of the each pipe section at the end of
each simulation time step. Some models also use
Lagrangian approaches for tracking particles of
constituents within a network. (See Chap. 10).
Computer programs (e.g., EPANET) exist that
can perform simulations of the flows, heads, and
water quality behavior within pressurized networks of pipes, pipe junctions, pumps, valves,
and storage tanks or reservoirs. These programs
are designed to predict the movement and fate of
water constituents within distribution systems.
They can be used for many different kinds of
applications in distribution systems design,
hydraulic model calibration, chlorine residual
analysis, and consumer exposure assessment.
They can also be used to compare and evaluate
the performance of alternative management
strategies for improving water quality throughout
a system. These can include:
• altering the sources within multiple source
systems,
• altering pumping and tank filling/emptying
schedules,
• use of satellite treatment, such as rechlorination at storage tanks,
• targeted pipe cleaning and replacement.
Computer models that simulate the hydraulic
and water quality processes in water distribution
networks must be run long enough for the system
to reach equilibrium conditions, i.e., conditions
not influenced by initial boundary assumptions.
Equilibrium conditions within pipes are reached
relatively quickly compared to those in storage
tanks.

12.4
12.4.1

Wastewater Collection
Sewer Networks

Flows in urban sewers and their pollutant concentrations vary throughout a typical day, a typical
week, and over the seasons of a year. Flow conditions can range from free surface to surcharged

Urban Water Systems

flow, from steady to unsteady flow, and from
uniform to gradually or rapidly varying
non-uniform flow.
Urban drainage ditches normally have uniform cross sections along their lengths and uniform gradients. Because the dimensions of the
cross sections are typically one or two orders of
magnitude less than the lengths of the conduit,
unsteady free surface flow can be modeled using
one-dimensional flow equations.
When modeling the hydraulics of flow it is
important to distinguish between the speed of
propagation of the kinematic wave disturbance and
the speed of the bulk of the water. In general the
wave travels faster than the water particles. Thus if
water is injected with a tracer the tracer lags behind
the wave. The speed of the wave disturbance
depends on the depth, width, and velocity of the
flow.
Flood attenuation (or subsidence) is the
decrease in the peak of the wave as it propagates
downstream. Gravity tends to flatten, or spread
out, the wave along the channel. The magnitude of
the attenuation of a flood wave depends on the
peak discharge, the curvature of the wave profile at
the peak, and the width of flow. Flows can be
distorted (changed in shape) by the particular
channel characteristics.
Additional features of concern to hydraulic
modelers are the entrance and exit losses to the
conduit. Typically at each end of the conduit is a
manhole. Manholes are storage chambers that
provide access (for men and women) to the
conduits upstream and downstream. Manholes
induce some additional head loss.
Manholes usually cause a major part of the head
losses in sewer systems. A manhole head loss
represents a combination of the and contraction
losses. For pressure flow, the head loss HL due to
contraction can be written as a function of the
downstream velocity, VD, and the upstream and
downstream flow cross-sectional areas AU and AD.


HL ¼ K VD2 =2g ½1  ðAD =AU Þ2

ð12:23Þ

The coefficient K varies between 0.5 for sudden
contraction and about 0.1 for a well-designed
gradual contraction.

12.4

Wastewater Collection

An important parameter of a given
open-channel conduit is its capacity, that is, the
maximum flow that can occur without surcharging or flooding. Assuming the hydraulic
gradient is parallel to the bed of the conduit, each
conduit has an upper limit to the flow that it can
accept.
Pressurized flow is much more complex than
free surface flow. In marked contrast to the
propagation speed of disturbances under free
surface flow conditions, the propagation of disturbances under pressurized flow in a 1 m circular conduit 100 m long can be less than a
second. Some conduits can have the stable situation of free surface flow upstream and pressurized flow downstream.

12.5

Wastewater Treatment

The wastewater generated by residences, businesses, and industries in a community is largely
water. Wastewater often contains less than 10%
dissolved and suspended solid material. Its
cloudiness is caused by suspended particles whose
concentrations in untreated sewage range from
100 to 350 mg/l. One measure of the biodegradable constituents in the wastewater is its biochemical oxygen demand, or BOD5. BOD5 is the
amount of dissolved oxygen aquatic microorganisms consumed in 5 days as they metabolize
(eat) the organic material in the wastewater.
Untreated sewage typically has a BOD5 concentration ranging from 100 to 300 mg/l.
Pathogens or disease-causing organisms are
also present in sewage. Coliform bacteria are
used as an indicator of disease-causing organisms. Sewage also contains nutrients (such as
ammonia and phosphorus), minerals, and metals.
Ammonia can range from 12 to 50 mg/l and
phosphorus can range from 6 to 20 mg/l in
untreated sewage (Fig. 12.8).
As illustrated in Fig. 12.7, wastewater treatment is a multistage process. The goal is to
reduce or remove organic matter, solids, nutrients, disease-causing organisms, and other pollutants from wastewater before it is released into
a body of water, or on to the land, or is reused.

537

The first stage of treatment is called preliminary
treatment.
Preliminary treatment removes solid materials
(sticks, rags, large particles, sand, gravel, toys,
money, or anything people flush down toilets).
Equipment such as bar screens, and grit chambers are used to filter the wastewater as it enters a
treatment plant. The wastewater then passes on to
what is called primary treatment.
Clarifiers and septic tanks are usually used to
provide primary treatment. Primary treatment
separates suspended solids and greases from
wastewater. Wastewater is held in a tank for
several hours allowing the particles to settle to
the bottom and the greases to float to the top. The
solids drawn off the bottom and skimmed off the
top receive further treatment as sludge. The
clarified wastewater flows on to the next secondary stage of wastewater treatment.
Secondary treatment is typically a biological
treatment process designed to remove dissolved
organic matter from wastewater. Sewage
microorganisms cultivated and added to the
wastewater absorb organic matter from sewage as
their food supply. Three approaches are commonly
used to accomplish secondary treatment; fixed
film, suspended film, and lagoon systems.
Fixed film systems grow microorganisms on
substrates such as rocks, sand, or plastic. The
wastewater is spread over the substrate. As
organic matter and nutrients are absorbed from
the wastewater, the film of microorganisms
grows and thickens. Trickling filters, rotating
biological contactors, and sand filters are examples of fixed film systems.
Suspended film systems stir and suspend
microorganisms in wastewater. Activated sludge,
extended aeration, oxidation ditch, and sequential batch reactor systems are all examples of
suspended film systems. As the microorganisms
absorb organic matter and nutrients from the
wastewater they grow in size and number. After
the microorganisms have been suspended in the
wastewater for several hours, they are settled out
as sludge. Some of the sludge is pumped back
into the incoming wastewater to provide “seed”
microorganisms. The remainder is sent on to a
sludge treatment process.

538

12

Urban Water Systems

Fig. 12.7 A typical wastewater treatment plant showing the sequence of processes for removing impurities

Fig. 12.8 Wastewater treatment plant on the shores of
Western Lake Superior in Minnesota, USA. The Activated Sludge system is designed to treat 48 million

gallons of wastewater per day (mgd) with a peak
hydraulic capacity of 160 mgd. (Photo courtesy of
Western Lake Superior Sanitary District.)

12.5

Wastewater Treatment

Lagoons, where used, are shallow basins that
hold the wastewater for several months to allow for
the natural degradation of sewage. These systems
take advantage of natural aeration and microorganisms in the wastewater to renovate sewage.
Advanced treatment is necessary in some
treatment systems to remove nutrients from
wastewater. Chemicals are sometimes added
during the treatment process to help remove
phosphorus or nitrogen. Some examples of
nutrient removal systems include coagulant
addition for phosphorus removal and air stripping
for ammonia removal.
Final treatment focuses on removal of
disease-causing organisms from wastewater.
Treated wastewater can be disinfected by adding
chlorine or by exposing it to sufficient ultraviolet
light. High levels of chlorine may be harmful to
aquatic life in receiving streams. Treatment systems often add a chlorine-neutralizing chemical to
the treated wastewater before stream discharge.
Sludges are generated throughout the sewage
treatment process. This sludge needs to be treated
to reduce odors, remove some of the water and
reduce volume, decompose some of the organic
matter and reduce volume, and kill diseasecausing organisms. Following sludge treatment,
liquid and cake sludge free of toxic compounds
can be spread on fields, returning organic matter
and nutrients to the soil.

12.6

Urban Drainage Systems

Urban drainage involves a number of hydraulic
and biochemical processes. These typically
include:
•
•
•
•
•
•

Rainfall and surface runoff
Surface loading and washoff of pollutants
Stormwater sewer and pipe flow
Sediment transport
Separation of solids at structures
Outfalls

These components or processes are briefly
discussed in the following subsections.

539

12.6.1

Rainfall

Surface runoff of precipitation and the need to
collect urban wastewater are the primary reasons
for designing and maintaining urban drainage
systems. Storms are a major source of flow into
the system. Even sanitary sewer systems that are
designed to be completely separate from storm
drainage sewers are often influenced by rainfall
through illicit connections or even infiltration.
Rainfall varies over time and space. These
differences are normally small when considering
short time periods and small distances but they
increase as time and distance increase. The
ability to account for spatial differences in rainfall depends on the size of the catchment area and
on the number of functioning rainfall recording
points in the catchment. The use of radar permits
more precision in estimating precipitation patterns over space and time, as if more rain gauges
were used and as if they were monitored more
frequently. In practice spatial effects are not
measured at high resolution and therefore events
where significant spatial variations occur, such as
in summer thunderstorms, are usually not very
accurately represented.
There are two categories of rainfall records:
recorded (real) events and synthetic (not-real)
events. Synthetic rainfall comes in two forms: as
stochastically generated rainfall data and as design storms. These events are derived from
analyses of actual rainfall data and are used to
augment or replace those historical (real) data.
Design events are a synthesized set of rainfall
profiles that have been processed to produce
storms with specific return periods, i.e., how
often, on average, one can expect to observe
rainfall events of that magnitude or greater.
Design events are derived to reduce the number
of runs needed to analyze system performance
under design flow conditions.

12.6.1.1

Time Series Versus Design
Storms
Professionals can argue over whether infrastructure design capacities should be based on real
rainfall records or synthetic storm events. The

540

argument in favor of using synthetic storms is that
they are easy to use and require only a few events to
assess the system design performance. The argument in favor of a time series of real rainfall is that
these data typically include a wider range of conditions, and therefore are likely to contain the
conditions that are critical on each catchment.
The two methods are not contradictory. The use
of real rainfall involves some synthesis in choosing which storms to use in a time series, and in
adjusting them for use on a catchment other than
the one where they were measured. Time series of
rainfalls are generally used to look at aspects such
as overflow spill frequencies and volumes. On the
other hand, synthetic design storms can be generated for a wide range of conditions including the
same conditions as represented by real rainfall.
This is generally considered appropriate for
looking at pipe network performance.

12.6.1.2

Spatial–Temporal
Distributions
Rainfall varies in space as well as in time, and the
two effects are related. Short duration storms
typically come from small rain cells that have a
short life, or that move rapidly over the catchment. As these cells are small (of the order of a
kilometer in diameter) there is significant spatial
variation in rainfall intensity. Longer duration
storms tend to come from large rainfall cells
associated with large weather systems. These
have less spatial intensity variation.
Rainfall is generally measured at specific sites
using rain gauges. The recorded rainfall amount
and intensity will not be the same at each site. Thus
the use of recorded rainfall data requires some way
to account for this spatial and temporal variations.
The average rainfall over the catchment in any
period of time can be more or less than the measured values at one or more gauge sites. The runoff
from a portion of a catchment exposed to a high
intensity rainfall will more than the runoff from the
same amount of rainfall spread evenly over the
entire catchment.
12.6.1.3 Synthetic Rainfall
A convenient way of using rainfall data is to
analyze long rainfall records to define the

12

Urban Water Systems

statistical characteristics of the rainfall, and then
to use these statistics to produce synthetic rainstorms of various return periods and durations.
Three parameters are used to describe the
statistics of rainfall depth.
• The rainfall intensity or depth of rain in a
certain period
• The length of the period over which that
intensity occurs
• The frequency with which it is likely to occur,
or the probability of it occurring in any particular year.
In most of the work on urban drainage and
river modeling, the risks of occurrence are
expressed not by probabilities but by the inverse
of probability, the return period. An event that
has a probability of 0.2 of being equaled or
exceeded each year has an expected return period
of 1/0.2 or 5 years. An event having a probability
of 0.5 of being equaled or exceeded has an
expected return period of 1/0.5 = 2 years.
Rainfall data show an intensity–duration–frequency relationship. The intensity and duration
are inversely related. As the rainfall duration
increases the intensity reduces. The frequency
and intensity are inversely related so that as the
event becomes less frequent the intensity
increases.
An important part of this duration–intensity
relationship is the period of time over which the
intensity is averaged. It is not necessarily the
length of time for which it rained from start to
finish. In fact any period of rainfall can be analyzed for a large range of durations, and each
duration could be assigned a different return
period. The largest return period might be quoted
as the “return period of the storm”, but it is only
meaningful when quoted with its duration.
Intensity–duration–frequency relationships or
depth–duration–frequency
relationships,
as
shown in Fig. 12.9, are derived by analysis of a
long set of rainfall records. Intensity–duration–
frequency data is commonly available all over
the world and therefore it is important to be
aware of its method of derivation and ways it can
be used for simulation modeling.

12.6

Urban Drainage Systems

541

Fig. 12.9 Rainfall
intensity–duration–
frequency (return period)
curves

The depth of rainfall is the intensity times its
duration integrated over the total storm duration.

12.6.1.4 Design Rainfall
Design rainfall events (hyetographs) for use in
simulation models are derived from intensity–
duration–frequency data.
The rainfall intensity during an event is not
uniform in time, and its variation both in intensity and when the peak intensity occurs during
the storm can be characterized by the peakedness
of the storm and the skew of the storm
(Fig. 12.10).
A design storm is a synthetic storm that has an
appropriate peak intensity and storm profile.

12.6.2

Runoff

The runoff from rainfall involves a number of
processes and events, as illustrated in Fig. 12.11,
and can be modeled using various methods. Most
of these methods assume an initial loss, a continuing loss, and a remainder contributing to the
system runoff.

Most models assume that the first part of a
rainfall event goes to initial wetting of surfaces
and filling depression storage. The depth
assumed to be lost is usually related to the surface type and condition. Rain water can be
intercepted by vegetation or can be trapped in
depressions on the ground surface. It then either
infiltrates into the ground and/or evaporates.
Depression storage can occur on any surface,
paved, or otherwise.
Initial loss depths are defined as the minimum
quantity of rainfall causing overland runoff. The
initial loss depth of rainfall for catchment surfaces
can be estimated as the intercept on the rainfall axis
of plots of rainfall verses runoff (Fig. 12.12). The
runoff values shown in Fig. 12.12 were obtained
for various catchments in the UK (Price 2002).
As rainfall increases so does depression storage.
The relationship between depression storage and
surface slope S is assumed to be of the form aS −b,
where S is average slope of the subcatchment and
a and b are parameters between 0 and 1. The values
of a and b depend in part on the surface type.
Evaporation, another source of initial loss, is
generally considered to be relatively unimportant.

542

Fig. 12.10 Storm peak skewness profiles

Fig. 12.11 Schematic representation of urban rainfall–runoff processes

12

Urban Water Systems

12.6

Urban Drainage Systems

543

Fig. 12.12 Estimation of
depression storage based
on data from catchments in
the UK (Price 2002)

For example, considering a heavy summer storm
(25 mm rainfall depth) falling on hot-asphalt
(temperature say 60 °C falling to 20–30 °C as a
result of sensible heat-loss) a maximum evaporation loss of 1 mm is likely to occur.
Continuing losses are often separated into two
parts: evapotranspiration and infiltration. These
processes are usually assumed to continue
throughout and beyond the storm event as long
as water is available on the surface of the ground.
Losses due to vegetation transpiration and general evaporation are not particularly an issue for
single events, but can be during the interevent
periods where catchment drying takes place. This
is applicable to models where time-series data are
used and generated. Infiltration is usually
assumed to account for the remaining rainfall that
does not enter into the drainage system. The
proportion of this loss can range from 100 % for
very permeable surfaces to 0 % for completely
impermeable surfaces.
Many models try to account for the wetting of
the catchment and the increasing runoff that takes
place as wetting increases. The effect of this is
shown in Fig. 12.13.
It is impractical to take full account of the
variability in urban topography, and surface
condition. Impervious (paved) surfaces are often
dominant in an urban catchment and the loss of
rainfall prior to runoff is usually relatively small.

Runoff routing is the process of passing
rainfall across the surface to enter the drainage
network. This process results in attenuation and
delay. These are modeled using routing techniques that generally consider catchment area
size, ground slope, and rainfall intensity in
determining the flow rate into the network. The
topography and surface channels and even
upstream parts of the sewer system are usually
lumped together into this process and are not
explicitly described in a model. The runoff
routing process is often linked to catchment
surface type and empirical calibration factors are
used accordingly.
Various models for rainfall–runoff and routing are available and are used in different parts of
the world. Overland runoff on catchment surfaces
can be represented by the kinematic wave equation. However, direct solution of this equation in
combination with the continuity equation has not
been a practical approach when applied to basins
with a large number of contributing subcatchments. Simpler reservoir-based models, that are
less computationally and data demanding, represent the physical processes almost as accurately
as the more complex physically based approaches (Price 2002). In practice, models applied to
catchments typically assume an average or
combined behavior of a number of overland flow
planes, gutters, and feeder pipes. Therefore, the

544

12

Urban Water Systems

Fig. 12.13 Effect of
catchment wetness on
runoff Q over time t (Price
2002)

parameters of a physically based approach (for
example, the roughness value) as applied would
not relate directly to parameters representative of
individual surfaces and structures.
Many overland flow routing models are based
on a linear reservoir-routing concept. A single
reservoir model assumes that the outflow, Q
(t) (m3/s), at the catchment outlet is proportional
to the volume of stormwater, S(t) (m3), present
on the ground surface of that catchment including the nonexplicitly modeled network that
contributes stormwater to that outlet point of the
urban drainage system. To take into account the
effects of depression storage and other initial
losses, the first millimeter(s) of rainfall may not
contribute to the runoff.
The basic equation for runoff Q(t) (m3/s) at
time t is

QðtÞ ¼ SðtÞ=K;

ð12:24Þ

where K is a linear reservoir coefficient. This coefficientissometimesafunctionofthecatchmentslope,
area, length of longest sewer, and rainfall intensity.
For a two linear reservoir model, two reservoirs are
applied in series for each surface type with an equivalent storage–output relationship, as defined by
Eq. 12.24,foreachreservoir.
The simplest models rely on fixed runoff
coefficients K. They best apply to impervious
areas where antecedent soil moisture conditions
are not a factor.
Typical values for runoff coefficients are
given in Table 12.2 (Price 2002). Use of these
coefficients should be supported either by field
observations or by expert judgment.

12.6

Urban Drainage Systems

545

Table 12.2 Typical values of the runoff fraction (coefficient K)

12.6.2.1 The Horton Infiltration Model
The Horton model describes the increasing runoff
from permeable surfaces as a rainfall event occurs
by keeping track of decreasing infiltration as the
soil moisture content increases. The runoff from
paved surfaces is assumed to be constant while the
runoff from permeable surfaces is a function of the
conceptual wetting and infiltration processes.
Based on infiltrometer studies on small
catchments Horton defined the infiltration rate, f,
either on pervious surfaces or on semipervious
surfaces, as a function of time, t (hours), the
initial infiltration rate, fo (mm/h), the minimum
(limiting or critical) infiltration rate, fc (mm/h),
and an infiltration rate constant, k (1/h).
f ¼ fc þ ðfo fc Þekt

ð12:25Þ

The minimum or limiting infiltration rate, fc, is
commonly set to the saturated groundwater
hydraulic conductivity for the applicable soil
type.

The integration of Eq. 12.25 over time defines
the cumulative infiltration F(t).


FðtÞ ¼ fc t þ ðfo fc Þ 1ekt =k

ð12:26Þ

The Horton equation variant as defined in
Eq. 12.26 represents the potential infiltration
depth, F, as a function of time, t, assuming the
rainfall rate is not limiting, i.e., it is higher than
the potential infiltration rate. Expressed as a
function of time, it is not suited for use in a
continuous simulation model. The infiltration
capacity should be reduced in proportion to the
cumulative infiltration volume, F, rather than in
proportion to time. To do this Eq. 12.26 may be
solved iteratively to find the time it takes to cause
ponding, tp, as a function of F. That time tp is
used in Eq. 12.25 to establish the appropriate
infiltration rate for the next time interval (Bedient
and Huber 1992). This procedure is used, for
example, in the urban stormwater management
model (SWMM) (Huber and Dickinson 1988).

546

A flow chart of the calculations performed in a
simulation program in which the rainfall can vary
might be as shown in Fig. 12.14.
Various values for Horton’s infiltration model
are available in the published literature. Values of
fo and fc as determined by infiltrometer studies,

12

Urban Water Systems

Table 12.3, are highly variable even by an order of
magnitude on seemingly similar soil types. Furthermore, the direct transfer of values as measured
on rural catchments to urban catchments is not
advised due to the compaction and vegetation
differences associated with the latter surfaces.

Fig. 12.14 Flow chart of Horton model infiltration algorithm used in each time step of a simulation model

12.6

Urban Drainage Systems

547

Table 12.3 Values for Horton’s infiltration model for different soil groups as defined by the US Soil Conservation
Service

12.6.2.2

The US Soil Conservation
Method (SCS) Model
The SCS method is a widely used model for predicting runoff from rural catchments, especially in
the USA, France, Germany, Australia, and parts of
Africa. It has also been used for the permeable
component in a semi-urban environment. This
runoff model allows for variation in runoff
depending on catchment wetness. The model
relies on what are called curve numbers, CN.
The basis of the method is the continuity
equation. The total depth (mm) of rainfall, R,
either evaporates or is otherwise lost, Ia, infiltrates and is retained in the soil, F, or runs off the
land surface, Q
R ¼ Ia þ F þ Q

ð12:27Þ

The relationship between the depths (mm) of
rainfall, R, runoff, Q, the actual retention, F, and
the maximum potential retention storage, S (not
including Ia), is assumed to be
F=S ¼ Q=ðRIa Þ;

ð12:28Þ

when R > Ia. These equations combine to give
the SCS model
Q ¼ ðRIa Þ2 =ðRIa þ SÞ

ð12:29Þ

This model can be modified for use in continuous
simulation models.
Numerical representation of the derivative of
Eq. 12.29 can be written as Eq. 12.30 for

predicting the runoff, q (mm/Δt), over a time
interval Δt given the rainfall r (mm/Δt), in that
time interval.
q ¼ r ðRIa ÞðRIa þ 2SÞ=ðRIa þ SÞ2 ð12:30Þ
This equation is used incrementally enabling the
rainfall and runoff coefficients, r and q, to change
during the event.
The two parameters S and Ia are assumed to be
linearly related by
Ia ¼ k S;

ð12:31Þ

where 0 < k < 0.2.
The original SCS approach recommended
k = 0.2. However, other studies suggest that
k values between 0.05 and 0.1 may be more
appropriate.
The storage variable, S, itself is related to an
index known as the runoff curve number, CN,
representing the combined influence of soil type,
land management practices, vegetation cover,
urban development, and antecedent moisture
conditions on hydrological response. CN values
vary between 0 and 100, 0 representing no runoff
and 100 representing 100% runoff.
The storage parameter S is related to the curve
number CN by
S ¼ ð25400=CNÞ  254

ð12:32Þ

Curve number values depend on antecedent
moisture conditions (AMC) and hydrologic soil

548

group. The antecedent moisture conditions are
divided into three classes, as defined in
Table 12.4.
The four hydrologic soil groups are defined in
Table 12.5.
The CN value can either be defined globally
for the catchment model or can be associated
with specific surface types. CN values for

12

Urban Water Systems

different conditions are available from various
sources. Table 12.6 lists some of these relevant
to urban areas and antecedent moisture condition
class AMC II.
Figure 12.15 identifies the CN values for
antecedent moisture content (AMC) classes I and
III based on class II values.

Table 12.4 Antecedent moisture classes (AMC) for determining curve numbers CN

Table 12.5 SCS hydrologic soil groups used in Tables 12.3 and 12.6

12.6

Urban Drainage Systems

549

Table 12.6 Initial CN values for AMC II with various urban land use, cover quality, and hydrologic soil groups
(defined in Table 12.5)

12.6.2.3

The SWMM Rainfall–Runoff
Model
The rainfall–runoff element of the popular
SWMM model generally assumes 100% runoff
from impermeable surfaces and uses Horton or
the Green–Ampt model for permeable runoff.
The Green–Ampt model is similar to the Horton
model, in that it has a conceptual infiltration rate
that varies with time. It is therefore applicable to
pervious or semipervious catchments (Huber and
Dickinson 1988; Roesner et al. 1988).

12.6.3

Surface Pollutant Loading
and Washoff

The modeling of surface pollutant loading and
washoff into sewer systems is very imprecise.
Pollutants that build up on the surface of an
urban area originate from wind blown dust,
debris that is both natural and human-made,
including vehicular transport emissions. When
rainfall takes place some of this material, as
dissolved pollutants and fine solids, is washed

550

12

Urban Water Systems

Fig. 12.15 Runoff curve
numbers for AMC classes I
and III based on curve
numbers for AMC II

into the stormwater sewers or gullies. During
buildup time many of the pollutants degrade.
Deposition of this material is not homogeneous but rather is a function of climate, geography, land use, and human activity. The
mechanism of washoff is obviously a function of
location, land use, rainfall intensity, slope, flow
rate, vehicle disturbance, etc. None of these
factors are explicitly modeled in most washoff
and sewer flow models.
Measurements made of pollutant accumulation and washoff have been the basis of empirical
equations representing both loading and washoff
processes. In practice the level of information
available and the complexity of the processes
being represented make the models of pollutant
loading and washoff a tool whose outputs must
be viewed for what they are, merely guesses.
Modeling does not change this, it only has the
potential of making those guesses better.

12.6.3.1 Surface Loading
Pollutant loadings and accumulation on the surface of an urban catchment occur during dry
periods between rainstorms.

A common hypothesis for pollutant accumulation during dry periods is that the mass loading
rate, mP, (kg/ha/day) of pollutant P is constant.
This assumed constant loading rate on the surface
of the ground can vary over space and is related to
the land use of that catchment. In reality these
loadings on the land surface will not be the same,
neither over space nor over time. Hence to be
more statistically precise, a time series of loadings
may be created from one or more probability
distributions of observed loadings. (Just how this
may be done is discussed in Chaps. 6 and 7.)
Different probability distributions may apply
when, for example, weekend loadings differ from
workday loadings. However, given all the other
uncertain assumptions in any urban loading and
washoff model, the effort may not be justified.
As masses of pollutants accumulate over a dry
period they may degrade as well. The time rate of
degradation of a pollutant P is commonly
assumed to be proportional to its total accumulated mass MP (kg/ha). Assuming a proportionality constant (decay rate constant) of kP (1/day),
the rate of change in the accumulated mass MP
over time t is

12.6

Urban Drainage Systems

dMP =dt ¼ mP kP MP

551

ð12:33Þ

As the number of days during the dry period gets
very large the limiting accumulation of a mass
MP of pollutant P is mP/kP. If there is no decay,
then of course kP is 0 and the limiting accumulation is infinite.
Integrating Eq. 12.33 over the duration Δt
(days) of a dry period yields the mass, MP(Δt)
(kg/ha) of each pollutant available for washoff at
the beginning of a rainstorm.



MP ðDtÞ ¼ MP ð0Þ ekpDt þ mP 1ekpDt =kP ;
ð12:34Þ
where MP(0) is the initial mass of pollutant P on
the catchment surface at the beginning of the dry
period, i.e., at the end of the previous rainstorm.
Sediments (that become suspended solids in
the runoff) are among the pollutants accumulating
on the surface of urban catchments. They are
important by themselves, but also because some
of the other pollutants that accumulate become
attached to these sediments. Sediments are typically defined by their medium diameter size value
(d50). Normally a minimum of two sediment
fractions are modeled, one coarse high-density
material (grit) and one fine (organics).

The sediments of each diameter size class are
commonly assumed to have a fixed amount of
pollutants attached to them. The fraction of each
attached pollutant, sometimes referred to as the
potency factor of the pollutant, is expressed as kg
of pollutant per kg of sediment. Potency factors
are one method for defining pollutant inputs into
the system.

12.6.3.2 Surface Washoff
Pollutants in the washoff can be dissolved in
water, or they can be attached to the sediments.
Many models of the transport of dissolved and
particulate pollutants through a sewerage system
assume each pollutant is conservative—it does
not degrade with time. For practical purposes this
is a reasonable assumption when the time of flow
in the sewers is relative short. Otherwise it may
not be a good assumption, but at least it is a
conservative one.
Pollutants can enter the sewer system from a
number of sources. A major source is the washoff
of pollutants from the catchment surface during a
rainfall event. Their removal is caused by the
impact of rainfall and by erosion from runoff
flowing across the surface. Figure 12.16 shows
schematically some sources of pollution in the
washoff model.

Fig. 12.16 Some sources of pollutants in the washoff to sewer systems from land surfaces

552

12

The rate of pollutant washoff is dependent on
an erosion coefficient, αP, and the quantity, MP,
of available pollutant, P. As the storm event
proceeds and pollutants are removed from the
catchment, the quantities of available pollutants
decrease, hence the rate of pollutant washoff
decreases even with the same runoff.
When runoff occurs, a fraction of the accumulated load may be contained in that runoff.
This fraction will depend on the extent of runoff.
If a part of the surface loading of a pollutant is
attached to sediments, its runoff will depend on
the amount of sediment runoff, which in turn is
dependent on the amount of surface water runoff.
The fraction of total surface loading mass
contained in the runoff will depend on the runoff
intensity. The following approximate relation
may apply for the fraction, fRt , of pollutant P in
the runoff Rt in period t:
ftP ¼ aP Rt =ð1 þ aP Rt Þ

ð12:35Þ

The greater the runoff Rt the greater will be the
fraction fPt of the total remaining pollutant loading in that runoff. The values of the parameters
αP are indicators of the effectiveness of the runoff
in picking up and transporting the particular
pollutant mass. Their values are dependent on the
type of pollutant P and on the land cover and
topography of particular basin or drainage area.
They can be determined based on measured
pollutant mass surface loadings and on the mass
of pollutants contained in the rainfall and sediment runoff, preferably at the basin of interest.
Since such data are difficult, or at least expensive,
to obtain, they usually are based on experiments
in laboratories.
A mass balance of pollutant loadings can
define the total accumulated load, MPt+1 at the
end of each simulation time period t or equivalently at the beginning of each time period t + 1.
Assuming a daily simulation time step,


MPt þ 1 ¼ 1ftP MPt ekP þ mP

Urban Water Systems

part. The total mass of pollutant P in the runoff
must also include those attached fractions (potency factors), if any, of each sediment size class
being modeled as well.
As the sediments are routed through the system those from different sources are mixed
together. The concentrations of associated pollutants therefore change during the simulation as
different proportions of sediment from different
sources are mixed together. The results are given
as concentrations of sediment, concentrations of
dissolved pollutants, and concentrations of pollutants associated with each sediment fraction.

12.6.3.3

Stormwater Sewer and Pipe
Flow
Flows in pipes and sewers have been analyzed
extensively and their representation in models is
generally accurately defined. The hydraulic
characteristics of sewage are essentially the same
as clean water. Time-dependent effects are, in
part, a function of the change in storage in
manholes. Difficulties in obtaining convergence
occurs at pipes with steep to flat transitions, dry
pipes, etc., and therefore additional features and
checks are needed to achieve satisfactory model
results.
12.6.3.4 Sediment Transport
Pollutant transport modeling of both sediment
and dissolved fractions involves defining the
processes of erosion and deposition and advection and possibly dispersion. One-dimensional
models by their very nature cannot predict the
sediment gradient in the water column. In addition the concept of the sewer being a bioreactor is
not included in most simulation models. Most
models assume pollutants are conservative during the time in residence in the drainage system
before being discharged into a water body. All
these processes that take place in transient are
generally either ignored or approximated using a
range of assumptions.

ð12:36Þ

12.6.3.5
Of interest, of course, is the total pollutant mass
in the runoff. For each pollutant type P in each
period t these will be fPt (MPt) for the dissolved

Structures and Special Flow
Characteristics
Manholes, valves, pipes, pumping stations,
overflow weirs, etc., that impact the flows and

12.6

Urban Drainage Systems

head losses in sewers can be explicitly included
in deterministic simulation models. The impact
of some of these structures can only be predicted
using 2- or 3-dimensional models. However, the
ever-increasing power of computers is making
higher dimensional fluid dynamic analyses
increasingly available to practicing engineers.
The biggest limitation may be more related to
data and calibration than to computer models and
costs.

12.6.4

Water Quality Impacts

12.6.4.1 Slime
Slime can build up on the perimeter of sewers
that contain domestic sewage. The buildup of
slime may have a significant effect on roughness.
In a combined system the effect will be less as
the maximum daily flow of domestic sewage will
not usually be a significant part of pipe capacity.
The extent to which the roughness is
increased by sliming depends on the relation
between the sewage discharge and the pipe-full
capacity. Sliming will occur over the whole of
the perimeter below the water level that corresponds to the maximum daily flow. The slime
growth will be heaviest in the region of the
maximum water level. Over the lower part of the
perimeter, the surface will still be slimed, but to a
lesser extent than at the waterline. Above the
maximum waterline the sewer surface will tend
to be fairly free of slime.
12.6.4.2 Sediment
When sediment is present in the sewer the
roughness increases quite significantly. It is difficult to relate the roughness to the nature and
time history of the sediment deposits. Most
stormwater sewers contain some sediment
deposits, even if only temporarily. The only data
available suggest that the increase in head loss
can range from 30 to 300 mm, depending on the
configuration of the deposit and on the flow
conditions. The higher roughness value is more
appropriate when the sewer is flowing part full
and when considerable energy is lost as a result
of the generation of surface disturbances. In

553

practice the lower roughness values are used as
flow states of interest are usually extreme events
and therefore sewers are operating in surcharge.

12.6.4.3

Pollution Impact
on the Environment
The effects of combined sewer outflows (CSOs)
or discharges are particularly difficult to quantify
and regulate because of their intermittent and
varied nature. Their immediate impact can only
be measured during a spill event, and their
chronic effects are often difficult to isolate from
other pollution inputs. Yet CSOs are one of the
major causes of poor river water quality. Standards and performance criteria specifically for
intermittent discharges are therefore needed to
reduce the pollutants in CSOs.
Drainage discharges that affect water quality
include:
(1) oxygen-demanding substances. These can be
either organic, such as fecal matter, or inorganic. (Heated discharges, such as cooling
waters, reduce the saturated concentration of
dissolved oxygen),
(2) substances that physically hinder reoxygenation at the water surface, such as oils,
(3) discharges containing toxic compounds,
including ammonia, pesticides, and some
industrial effluents, and
(4) discharges that are high in suspended solids
and thus inhibit biological activity by
excluding light from water or by blanketing
the bed.
Problems arise when pollutant loads exceed
the self-purification capacity of the receiving
water, harming aquatic life, and restricting the
use of the water for consumption and many
industrial and recreational purposes. The assimilative capacity for many toxic substances is very
low. Water polluted by drainage discharges can
create nuisances such as unpleasant odors. It can
also be a direct hazard to health, particularly in
tropical regions where waterborne diseases such
as cholera and typhoid prevail.
The aim of good drainage design, with respect
to pollution, is to balance the effects of

554

12

continuous and intermittent discharges against
the assimilation capacity of the water, so as to
achieve in a cost and socially effective way the
desired quality of the receiving water.
Figure 12.17 shows the effect of a discharge
that contains suspended solids and organic matter. The important indicators showing the effect
of the discharge are the dissolved oxygen in
diagram “a” and the clean water fauna shown in
diagram “d”. The closeness with which the clean
water fauna follow the dissolved oxygen reflects

Urban Water Systems

the reliance of a diverse fauna population on
dissolved oxygen. These relationships are used
by biologists to argue for greater emphasis on
biological indicators of pollution as they respond
to intermittent discharges better than chemical
tests which, if not continuous, may miss the
pollution incident. There are a number of biological indexes in use in most countries in
Europe.
In Fig. 12.17 the BOD in diagram “a” rises or
stays constant after release despite some of the

(a)

(b)

(c)

(d)

Fig. 12.17 Pollution impact along a waterway downstream from its discharge

12.6

Urban Drainage Systems

pollutant being digested and depleting the dissolved oxygen. This is because there is a time lag
of up to several days while the bacteria, which
digest the pollutant, multiply. The suspended
solids (SS) settle relatively quickly and they can
then be a source of pollutants if the bed is disturbed by high flows. This can create a subsequent pollution incident especially if the
suspended solids contain quantities of toxic
heavy metals.
Diagram “b” of Fig. 12.17 shows ammonium
ions (NH4+) that are discharged as part of the
dissolved pollutants being oxidized to nitrates
(NO3−). The rise in the ammonium concentration
downstream of the discharge is relevant due to
the very low tolerance many aquatic organisms,
particularly fish, have to the chemical. The
ammonium concentration rises if the conditions
are anaerobic and will then decline once aerobic
conditions return and the ammonium ions are
oxidized to nitrates.
Diagrams “c” and “d” show the effect of
combined sewer overflows (CSOs) on flora and
fauna. The increased quantities of phosphate and
nitrate nutrients that they consume can lead to
eutrophication. The fauna show perhaps the
clearest pattern of response. The predictability of
this response has lead to the development of the
many biological indices of pollution. The rapid
succession of organisms illustrates the pattern of
dominance of only a few species in polluted
conditions. For example, tubificid worms can
exist in near anaerobic conditions, and having
few competitors, they can multiply prolifically.
As the oxygen levels increase these organisms
are succeeded by Chironomids (midge larvae)
and so on until in clean well-oxygenated water,
there is a wide diversity of species all competing
with each other.
In most circumstances, the concentration of
dissolved oxygen (DO) is the best indicator of
the “health” of a water source. A clean water
source with little or no biodegradable pollutants
will have an oxygen concentration of about 9–
10 mg/l when in equilibrium with air in a temperate environment. This maximum saturation

555

concentration is temperature dependent. The
hotter the water is the lower the DO saturation
concentration.
All higher forms of life in a river require
oxygen. In the absence of toxic impurities there
is a close correlation between DO and biodiversity. For example, most game fish die when the
DO concentration falls below about 4 mg/l.
Perhaps of more pragmatic significance is the
fact that oxygen is needed in the many natural
treatment processes performed by microorganisms that live in natural water bodies. The
quantity of oxygen required by these organisms
to breakdown a given quantity of organic waste
is, as previously discussed, the biochemical
oxygen demand (BOD). It is expressed as mg of
dissolved oxygen required by organisms to digest
and stabilize the waste in a liter of water. These
organisms take time to fully digest and stabilize
the waste. The rate at which they do so depends
on the temperature of, and the quantity of these
organisms available in, the water at the start.
Since the BOD test measures only the
biodegradable material it may not give an accurate assessment of the total quantity of oxidizable
material in a sample in all circumstances (e.g., in
the presence of substances toxic to the oxidizing
bacteria). In addition, measuring the BOD of a
sample of water typically takes a minimum of
5 days. The chemical oxygen demand (COD)
test is a quicker method and measures the total
oxygen demand. It is a measure of the total
amount of oxygen required to stabilize all the
waste. While the value of COD is never less than
the value of BOD, it is the faster reacting BOD
that impacts water quality. And this is what most
people care about. However, determining a
relationship between BOD and COD at any site
can provide guideline values for BOD based on
COD values.
Tables 12.7 and 12.8 provide some general
ranges of pollutant concentrations in CSOs from
urban catchments.
Water quality models for urban drainage are
similar, or simpler versions of water quality
models for other water bodies (as discussed in

556
Table 12.7 Typical quality of domestic sewage

Table 12.8 Pollutant concentrations (mg/l) in urban runoff

12

Urban Water Systems

12.6

Urban Drainage Systems

Chap. 10). Often just a simple mixing and dilution model will be sufficient to predict the concentration of pollutants at any point in a sewer
system. For example, such models may be sufficient for some toxic substances that are not
broken down. Once the flows in the CSO enter
the receiving water body, models discussed in
Chap. 10 can be used to estimate their fate as
they travel with the water in the receiving water
body.
A factor that makes predicting the impacts of
overflow discharges particularly difficult is the
noncontinuous nature of the discharges and their
pollutant concentrations. In the first sanitary or
foul flush the fine sediments deposited in the
pipes during dry periods are swept up and
washed out of the system. Most existing models,
termed constant concentration models, do not
account for this phenomenon. Since many of the
most significant pollution events occur when the
river has low flows, and hence low dilution factors, the quantity of spill in the first flush may be
very important in the overall pollution impact.

12.6.4.4

Bacteriological
and Pathogenic Factors
The modeling of pathogenic microorganisms is
particularly difficult since there are a very large
number of pathogenic organisms, each usually
with a unique testing procedure, many of which
are expensive. Also many pathogens may present
a significant risk to human health in very small
numbers. Incubation periods of over 24 h are not
uncommon and there are as yet no automatic
real-time monitoring techniques in commercial
use.
The detection of pathogens relies heavily on
indicator organisms that are present in feces in
far higher numbers than the pathogenic organisms. Escherichia Coliform (E. coli) bacteria is
the most common fecal indicator. This indicator
is commonly used throughout the world to test
water samples for fecal contamination.
Because of the problems in measuring
microbiological parameters involved in CSOs,
most sophisticated methods of determining the
quality of sewer water restrict themselves to
more easily measured determinants such as BOD,

557

COD, suspended solids, ammonia, nitrates, and
similar constituents.

12.6.4.5 Oil and Toxic Contaminants
Oils are typically discharged into sewers by
people, industries, or are picked up in the runoff
from roads and road accidents. Since oil floats on
water surfaces and disperses rapidly into a thin
layer, a small quantity of oil discharged into a
water body can prevent reoxygenation at the
surface and thus suffocate the organisms living
there. The dispersal rate changes with oil viscosity and the length of time the oil is a problem
will partly depend on the surface area of the
receiving water as well.
There are three main sources of toxic contaminants that may be discharged from CSOs:
• Industrial effluents. These could be anything
from heavy metals to herbicides.
• Surface washoff contaminants. These may be
contaminants washed off the surface in heavy
rainstorms that in agricultural and suburban
residential areas will probably include pesticides and herbicides. In many cases these
contaminants make a larger contribution to
the pollutant load than the domestic sanitary
flow.
• Substances produced naturally in the sewer.
Various poisonous gases are produced in
sewers. From the point of view of water
quality, ammonia is almost certainly the most
important though nitrogen sulfide can also be
significant.

12.6.4.6 Suspended Solids
Discharges high in suspended solids pose a
number of problems. They almost invariably
exert an oxygen demand. If they remain in suspension they can prevent light from penetrating
the water and thus inhibit photosynthesis. If
deposited they become a reservoir of oxygen
demanding particles that can form an anaerobic
layer on the bed, decreasing biodiversity. They
also can degrade the bed for fish (such as salmon)
spawning. If these suspended solids contain toxic
substances, such as heavy metals, the problems
can be more severe and complex.

558

12.6.5

12

Green Urban Infrastructure

Many urban areas are being recognized for successfully using natural ecosystems to reduce the
costs of providing clean drinking water and
managing stormwater. Moreover, this “green”
infrastructure provides many quality-of-life benefits, by improving air quality, increasing shading, contributing to higher property values, and
enhancing streetscapes. New York City in the
US, for example, estimates that such efforts have
saved ratepayers billions of dollars—by eliminating the need for construction of hard “gray”
infrastructure such as storm sewers and filtration
plants—while preserving large tracts of natural
areas. The department’s Green Infrastructure
Plan lays out how the city will improve the water
quality in New York Harbor by capturing and
retaining stormwater runoff before it enters the
sewer system using streetside swales, tree pits,
and blue and green rooftop detention techniques
to absorb and retain stormwater (Fig. 12.18).
This hybrid approach reduces combined sewer

Urban Water Systems

overflows by 12 billion gallons a year—over 2
billion gallons a year more than the current
all-gray strategy—while saving New Yorkers
$2.4 billion (NYCDEP 2012).

12.7

Urban Water System Modeling

Optimization and simulation models are becoming increasingly available and used to analyze a
variety of design and operation problems
involving urban water systems. Many are incorporated within graphics user interfaces that
facilitate the use of the models and the understanding and further analyses of their results.

12.7.1

Optimization

Methods for finding optimal solutions are
becoming increasingly effective in the design and
planning of urban infrastructure. Yet they are
challenged by the complexity and nonlinearity of
especially urban water distribution networks.

Fig. 12.18 Examples of “green” and “blue” (upper center) rooftop infrastructures for reducing urban storm runoff and
at the same time enhancing the attractiveness of the urban environment. Some go to extremes

12.7

Urban Water System Modeling

Numerous calibration procedures for water
distribution system models have been developed
since the 1970s. Trial and error approaches
(Rahal et al. 1980; Walski 1983) were replaced
with explicit type models (Ormsbee et al. 1986;
Boulos et al. 1990). More recently, calibration
problems have been formulated and solved as
optimization problems. Most of the approaches
used so far are either local or global search
methods. Local search gradient methods have
been used by Shamir (1974), Lansey et al. (1991),
Datta et al. (1994), Reddy et al. (1996), Pudar
et al. (1992), and Liggett et al. (1994) to solve
various steady-state and transient model calibration problems (Datta et al. 1994; Savic et al. 1995;
Greco et al. 1999; Vitkovsky et al. 2000).
Evolutionary search algorithms, discussed in
Chap. 5, are now commonly used for the design
and calibration of various highly nonlinear
hydraulic models of urban systems. They are
particularly suited for search in large and complex decision spaces (e.g., in water treatment,
storage and distribution networks). They do not
need complex mathematical matrix inversion
methods and they permit easy incorporation of
additional calibration parameters and constraints
into the optimization process (Savic et al. 1995;
Vitkovsky and Simpson 1997; Tucciarelli et al.
1999; Vitkovsky et al. 2000).
In addition to calibration, these evolutionary
search methods have been used extensively to find
least-cost designs of water distribution systems
(Simpson et al. 1994; Dandy et al. 1996; Savic and
Walters 1997). Other applications include the
development of optimal replacement strategies for
water mains (Dandy and Engelhardt 2001), finding the least expensive locations of water quality
monitoring stations (Al-Zahrani and Moied 2001),
minimizing the cost of operating water distribution
systems (Simpson et al. 1999), and identifying the
least-cost development sequence of new water
sources (Dandy and Connarty 1995).
These search methods are also finding a role
in developing master or capital improvement
plans for water authorities (Murphy et al. 1996;
Savic et al. 2000). In this role they have shown
their ability to identify low cost solutions for
highly complex water distribution systems

559

subject to a number of loading conditions and a
large number of constraints. Constraints on the
system include maximum and minimum pressures, maximum velocities in pipes, tank refill
conditions, and maximum and minimum tank
levels.
As part of any planning process, water authorities need to schedule the capital improvements to
their system over a specified planning period.
These capital improvements could include water
treatment plant upgrades, new water sources as
well as new, duplicate or replacement pipes, tanks,
pumps, and valves. This scheduling process
requires estimates of how water demands are
likely to grow over time in various parts of the
system. The output of a scheduling exercise is a
plan that identifies what facilities should be built,
installed, or replaced, to what capacity and when,
over the planning horizon.
The application of optimization to master
planning for complex urban water infrastructure
presents a significant challenge. Using optimization methods to find the minimum cost
design of a system of several thousand pipes for a
single demand at a single point in time is difficult
enough on its own. The development of
least-cost system designs over a number of time
periods experiencing multiple increasing
demands can be much more challenging.
Consider, for example, developing a master
plan for the next 20 years divided into four
5-year construction periods. The obvious way to
model this problem is to include the system
design variables for each of the next four 5-year
periods given the expected demands at those
times. The objective function for this optimization model might be to minimize the present
value of all construction, operation, and maintenance costs.
Dandy et al. (2002) have developed and
applied two alternative modeling approaches.
One approach is to find the optimal solution for
the system for only the final or “target” year. The
solution to this first optimization problem identifies those facilities that will need to be constructed sometime during the 20-year planning
period. A series of subproblems are then optimized, one for each intermediate planning stage,

560

to identify when each facility that is to be built
should be built. For these subproblems, the
decisions are either to build or not to build to a
predetermined capacity. If a component is to be
built, its capacity has already been determined in
the target year optimization.
For the second planning stage, all options
selected in the first planning stage are locked in
place and a choice is made from among the
remaining options. Therefore, the search space is
smaller for this case. A similar situation applies
for the third planning stage.
An alternative approach is to solve the first
optimization problem for just the first planning
stage. All options and all sizes are available. The
decisions chosen at this time are then fixed, and
all options are considered in the next planning
stage. These options include duplication of previously selected facilities. This pattern is repeated until the final “target” year is reached.
Each method has its advantages and disadvantages. For the first “Build-to-Target” method,
the optimum solution is found for the “target
year”. This is not necessarily the case for the
“Build-up” method. On the other hand, the latter
buildup method finds the optimal solution for the
first planning stage that the “Build-to-Target”
method does not necessarily do. As the demands
in the first planning stage are known more precisely than those for the “target” year, this may
be an advantage.
The buildup method allows small pipes to be
placed at some locations in the first time planning
stage, if warranted, and these can be duplicated at a
later time; the build-to-target method does not. This
allows greater flexibility, but may produce a solution that has a higher cost in present value terms.
The results obtained by these or any other
optimization methods will depend on the
assumed growth rate in demand, the durations of
the planning intervals, the economic discount
rate if present value of costs is being minimized,
and the physical configuration of the system
under consideration. Therefore, the use of both
methods is recommended. Their outputs, together with engineering judgment, can be the basis
of developing an adaptive master development
plan. Remember, it is only the current

12

Urban Water Systems

construction period’s solution that should be of
interest. Prior to the end of that period the
planning exercise with updated information can
be performed again to obtain a better estimate of
what the next period’s decisions should be.

12.7.2

Simulation

Dynamic simulation models are increasingly
replacing steady-state models for analyzing water
quantity, pressure and water quality in distribution
and collection networks. Dynamic models provide
estimates of the time-variant behavior of water
flows and their contaminants in distribution networks, even arising from flow reversals. The use
of long time-series analysis provides a continuous
representation of the variability of flow, pressure,
and quality variables throughout the system. It
also facilitates the understanding of transient
operational conditions that may influence, for
example, the way contaminants are transported
within the network. Dynamic simulation also
lends itself well to statistical analyses of exposure.
This methodology is practical for researchers and
practitioners using readily available hardware and
software (Harding and Walski 2002).
Models used to simulate a sequence of time
periods must be capable of simulating systems
that operate under highly variable conditions.
Urban water systems are driven by water use and
rainfall, which by its nature is stochastic. Changes
in water use, control responses and dispatch of
sources, and random storms over different parts of
the catchment all can affect flow quantities, the
flow direction and thus the spatial distribution of
contaminants. Because different water sources
often have different quality, changing water
sources can cause changes in the quality of water
within the system (see Box 12.1).
The simulation of water quantities and qualities in urban catchments serve three general
purposes:
• Planning/Design—These studies define system configurations, size or locate facilities, or
define long-term operating policies. They
adopt a long-term perspective but, under

12.7

Urban Water System Modeling

current practices, use short, hypothetical scenarios based on representative operating
conditions. In principle, the statistical distribution of system conditions should be an
important consideration, but in practice variability is considered only by analyses intended to represent worst-case conditions.
• Operations—These short-term studies analyze scenarios that are expected to occur in
the immediate future so as to inform immediate operational decisions. These are based
on current system conditions and expected
operating conditions. These analyses are often
driven by regulations.
• Forensics—These studies are used to link
presence of contaminants to the risk or actual
occurrence of disease. Depending on whether
the objective is cast in terms of acute or
chronic exposures, such studies may adopt
short- or long-term perspectives. Because
there are often dose/response relationships
and issues of latency in the etiology of disease, explicit consideration of the spatial
distribution, timing, frequency, duration and
level of contamination is important to these
studies (Rodenbeck and Maslia 1998; Aral
et al. 1996; Webler and Brown 1993).

12.8

561

of the waters into which urban runoff is discharged. In most urban areas in developed
regions, government regulations require designers
and operators of urban water systems to meet three
sets of standards. Pressures must be adequate for
fire protection, water quality must be adequate to
protect public health, and urban drainage of waste
and stormwaters must meet effluent and receiving
water body quality standards. This requires monitoring as well as the use of various models for
detecting leaks and for predicting the impacts of
alternative urban water treatment, distribution,
and collection system designs and operating,
maintenance and repair policies.
Modeling the water and wastewater flows,
pressure heads, and quality in urban water conveyance, treatment, distribution, and collection
systems is a challenging exercise, not only
because of its hydraulic complexity, but also
because of the stochastic inputs to and demands
on the system. This chapter has attempted to
provide an overview of some of the basic considerations used by modelers who develop
computer-based optimization and simulation
models for design and/or operation of parts
of such systems. These same considerations
should be in the minds of those who use such
models as well. Much more detail can be found in
many of the references listed at the end of this
chapter.

Conclusions

Urban water systems must include not only the
reservoirs, groundwater wells, and aqueducts that
are the sources of water supplies needed to meet
the varied demands in an urban area, but also the
water treatment plants, the water distribution
systems that transport that water, together with
the pressures required, to where the demands are
located. Once water is used, the now wastewater
needs to be collected and transported to where it
can be treated and either reused or discharged
back into the environment. Overlaying all of this
hydraulic infrastructure and plumbing is the
urban stormwater drainage system.
Well-designed and operated urban water systems are critically important for maintaining
public health as well as for controlling the quality

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Bedient, P. B., & Huber, W. C. (1992). Hydrology and
floodplain analysis, 2nd ed. Reading, MA:
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Boulos, P. F., & Wood, D. J. (1990). Explicit calculation
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Dandy, G. C., & Connarty, M. (1995). Use of genetic
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Dandy, G. C., & Engelhardt, M. (2001). The optimal
scheduling of water main replacement using genetic
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Dandy, G. C., Simpson, A. R., & Murphy, L. J. (1996).
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449–458.
Datta, R. S. N., & Sridharan, K. (1994). Parameter
estimation in water-distribution systems by least
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report_2013.pdf
Greco, M., & Del Guidice, G. (1999). New approach to
water distribution network calibration. Journal of
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Harding, B. L., & Walski, T. M. (2002). Long time-series
simulation of water quality in distribution systems.
Proceedings, ASCE Environmental and Water
Resources Planning Symposium, Roanoke, Va.
Huber, W. C., & Dickinson, R. E. (1988). Storm water
management model, Version 4, User’s Manual.
EPA/600/3-88/001a (NTIS PB88-236641/AS), U.S.
EPA, Athens, GA, 30605. 595 pp. Also see http://
www.epa.gov/ednnrmrl/swmm/.
Lansey, K. E., & Basnet, C. (1991). Parameter estimation
for water distribution networks. Journal of Water
Resources Planning and Management, ASCE, 117(1),
126–144.
Liggett, J. A., & Chen, L. C. (1994). Inverse transient
analysis in pipe networks. Journal of Hydraulic
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Mays, L. W. (Ed.). (2000). Water distribution systems
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Mays, L. W. (2001). Water resources engineering. New
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& Farrill, T. W. (1996). Genetic algorithm optimization of the Fort Collins-Loveland water distribution
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AWWA.
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network calibration. Journal of Water Resources
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Price, R. K. (2002). Urban drainage modeling, course
notes. Delft, NL: International Institute for Infrastructural, Hydraulic and Environmental Engineering.
Pudar, R. S., & Liggett, J. A. (1992). Leaks in pipe
networks. Journal of Hydraulic Engineering, ASCE,
118(7), 1031–1046.
Rahal, C. M., Sterling, M. J. H., & Coulbeck, B. (1980).
Parameter tuning for simulation models of water
distribution networks. Proceedings of Institution of
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Reddy, P. V. N., Sridharan, K., & Rao, P. V. (1996).
WLS method for parameter estimation in water
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Rodenbeck, S. E., & Maslia, M. L. (1998). Groundwater
modeling and GIS to determine exposure to TCE at
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Roesner, L. A., Aldrich, J. A., & Dickinson, R. E. (1988).
Storm Water Management Model User’s Manual,
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techniques for calibrating network models, Report
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Savic, D. A., & Walters, G. (1997). Genetic algorithms
for least-cost design of water distribution networks.
Journal of Water Resources Planning and Management, ASCE, 123(2), 67–77.
Savic, D. A., Walters, G. A., Randall Smith, M., &
Atkinson, R. M. (2000). Cost savings on large water
distribution systems: Design through GA optimisation.
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Additional References
(Further Reading)
Alegre, H., & Coelho, S. T. (2012). Infrastructure asset
management of urban water systems, chapter 3. In A.
Ostfeld (Ed.), Water supply system analysis—selected
topics. doi:10.5772/52377, ISBN 978-953-51-0889-4
Axworthy, D. H., & Karney, B. W. (1996). Modeling low
velocity/high dispersion flow in water distribution
systems. Journal of Water Resources Planning and
Management, ASCE, 122(3), 218–221.
Bahri, A. (2012). Integrated urban water management,
TEC background paper 16 global water partnership
SE-111 51. Stockholm, Sweden: Printed by Elanders.
Chaudry, M.H., & Islam, M. R. (1995). Water quality
modeling in pipe networks. In E. Cabreran & A.
F. Vela (Eds.), Improving efficiency and reliability in
water distribution systems. Dordrecht: Kluwer Academic Publishers.
Clark, R. M., Males, R., & Stevie, R. G. (1984). Water
supply simulation model volumes I, II and III., U.S.
EPA, Cincinnati, OH.
Clark, R. M., Grayman, W. M., Males, R. M., & Coyle,
J. M. (1986). Predicting water quality in distribution
systems”, proceedings, distribution system symposium, AWWA, September 7–10 (pp. 155–180). Minnesota: Minneapolis.
EPA. (1992). CEAM Systems Development Life Cycle
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Athens, GA, 30605.
Feyen, J., Shannon, K., & Neville, M. (2008). Water and
urban development paradigms: Towards an integration of engineering, design and management
approaches. Boca Raton: CRC Press.
Grayman, W. M., Clark, R. M., & Males, R. M. (1988).
Modeling distribution system water quality: Dynamic
approach. Journal of Water Resources Planning and
Management, ASCE, 114(3), 295–312.
Grayman, W. M., Loucks, D. P., & Saito, L. (Eds.).
(2012). Toward a sustainable water future: Visions for
2050. Reston, VA: ASCE Press.
Hering, J. G., Waite, T. D., Luthy, R. G., Drewes, J. E., &
Sedlak, D. L. (2013). A changing framework for urban

563
water systems environment. Science Technology, 47,
10721–10726. dx.doi.org/10.1021/es4007096
Hutton, C. J., Vamvakeridou-Lyroudia, L. S., Kapelan,
Z., & Savic, D. A. (UNEXE). (2011). Uncertainty
quantification and reduction in urban water systems
(UWS) modelling: Evaluation Report D3.6.1 prepared
2011.005, A framework for integrated modeling of
urban systems.
Islam, M. R., & Chaudry, M. H. (1998). Modeling of
constituent transport in unsteady flows in pipe
networks. Journal of Hydraulic Engineering, ASCE,
124(11), 1115–1124.
Medows, M. E., Walski, T. M., et al. (Eds.). (1997).
Computer applications in hydraulic engineering.
Waterbury, CT: Haestad Press, Haestad Methods Inc.
Morley, M. S., Atkinson, R. M., Savic, D. A., & Walters,
G. A. (2001). GAnet: Genetic algorithm platform for
pipe network optimisation. Advances in Engineering
Software, 32(6), 467–475.
Grimm, N. B., Grove, J. M., Pickett, S. T. A., & Redman,
C. L. (2000). Integrated approaches to long-term
studies of urban ecological systems. BioScience, 50
(7), 571–584.
Ormsbee, L. E. (1989). Implicit network calibration.
Journal of Water Resources Planning and Management, ASCE, 115(2), 243–257.
Ostfeld, A. (ed). (2012). Water supply system analysis—
selected topics, ISBN 978-953-51-0889-4, 158 pages,
Publisher: InTech, doi:10.5772/2882
Pickett, S. T. A., Cadenasso, M. L., Grove, J. M., Nilon, C. H.,
Pouyat, R. V., Zipperer, W. C., & Costanza, R. (2001).
Linking terrestrial ecological, physical, and socioeconomic components of metropolitan areas, urban ecological
systems. Annual Review of Ecology System, 32, 127–57.
Rossman, L. A., Boulos, P. F., & Altman, T. (1993).
Discrete volume-element method for network water
quality models. Journal of Water Resources Planning
and Management, ASCE, 119(5), 505–517.
Rossman, L. A., Clark, R. M., & Grayman, W. M. (1994).
Modeling chlorine residuals in drinking-water distribution systems. Journal of Environmental Engineering, ASCE, 120(4), 803–820.
Shamir, U., & Howard, C. D. D. (1968). Water distribution systems analysis. Journal of the Hydraulic
Division, ASCE, 94(1), 219–234.
Tang, K., Karney, B., Pendlebury, M., & Zhang, F.
(1999). Inverse transient calibration of water distribution systems using genetic algorithms. In D. A. Savic
& G. A. Walters (Eds.), Proceedings of Water
Industry Systems: Modelling and Optimisation Applications (Vol. 1, pp. 317–326). Exeter, UK.
Todini, E. (1999). Using a Kalman Filter approach for
looped water distribution network calibration. In D.
A. Savic & G. A. Walters (Eds.), Proceedings of
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Applications (Vol. 1, pp. 327–336), Exeter, UK.
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(2012). World Water Assessment Programme. Paris:
UNESCO.

564

12

Viessman, W. Jr., & Welty, C. (1985). Water Management Technology and Institutions. New York: Harper
& Row, Pub.

Exercises
12:1 Define the components of the infrastructure needed to bring water into your home
and then collect the wastewater and treat it
prior to discharging it back into a receiving water body. Draw a schematic of such
a system and show how it can be modeled
to determine the best design variable values. Define the data needed to model such
a system and then make up values of the
needed parameters and solve the model of
the system.
12:2 Compare the curve number approach to
the use of Manning’s equation to estimate
urban runoff quantities. Then define how
you would predict quality and sediment
runoff as well.
12:3 Develop a simple model for predicting the
runoff of water, sediment, and several
chemicals from a 10-ha urban watershed
in the northeastern United States during
August 1976. Recorded precipitation was
as follows:

Urban Water Systems

12:4 There exists a modest-sized urban subdivision of 100 ha containing 2000 people.
Land uses are 60% single-family residential, 10% commercial, and 30% undeveloped. An evaluation of the effects of
street-cleaning practices on nutrient losses in runoff is required for this catchment.
This evaluation is to be based on the 7-month
precipitation record given below. Present the
results of the simulations as 7-month PO4 and N
losses as functions of street-cleaning interval and
efficiency (i.e., show these losses for ranges of
intervals and efficiencies). Assume a runoff
threshold for washoff of Qo = 0.5 cm.
Precipitation (cm)
Day

A

M

1.1

0.4

4

0.1

1.5

5

0.1

0.9

6

0.1

1
2

J

J

0.6

A

S

O

1.4

0.7

1.9

0.5

3

0.1

7
S

1.4
1.4
0.1

0.1

9

0.1
0.6

1

6

7

8

9

10

13

14

15

26

29

11

R1(cm)

1.8

0.7

2.6

2.9

0.1

0.3

2.9

0.1

1.4

3.7

0.8

12
13

Solids (sediment) buildup on the watershed at
the rate of 50 kg/ha-day, and chemical concentrations in the solids are 100 mg/kg. Assume that
each runoff event washes the watershed surface
clean. Assume also that there is no initial sediment
buildup on August. The watershed is 30% impervious. For each storm use your model to compute:

14

(a) Runoff in cm and m3.
(b) Sediment loss (kg).
(c) Chemical loss (g), in dissolved and
solid-phase form for chemicals with three
different adsorption coefficients, k = 5,
100, 1000.

21

0.3

1.0
1.1

0.5

0.7

0.7
1.6

0.4

0.1

10
Day

1.9

0.1

1.5
0.1

02
0.1

02

0.2
02

02
1.5

0.5

3.5

15

0.8
1.0

16

4.3

17
18

0.5

0.7

0.8

0.8

1.9

0.4

0.1

0.8

0.1

0.4

0.9

03

2.3

19

0.4

20

0.7

0.5

2.8

0.3

22

0.1

23

2.0

24

3.2

25

0.1

0.1
0.1

0.4
02

0.6

4.7
2.8
(continued)

Exercises

565

Precipitation (cm)
26

3.0

1.6

27
28

0.3

29

02

30

0.1

31

0.1

1.1
0.6
0.2

12:5 Managing the quantity and quality of
stormwater runoff is a common urban
problem. Discuss the factors to be considered when planning storm sewer networks and detention basins, and how
might simulation and/or optimization
methods be used to help do this.
12:6 Multiple connected pipeline networks are
commonly make up urban water distribution systems. Define a simple pipeline
network and develop an optimization
model for finding the flows and heads in
the network needed to provide required
flow discharges and pressures at various
nodes or junctions of the network. Discuss
some of the complicating issues associated

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with the design of such networks that your
model may not be considering.
12:7 Consider a wastewater treatment plant and
associated effluent detention pond
designed to release treated and stored
effluent into a stream so as to adapt to
varying effluent concentration standards
associated with varying stream assimilative capacities as its flow and water temperature and existing pollutant loads vary
over time. Develop a model for estimating
the treatment plant efficiency for BOD
removal, and the size of the detention
basin, needed to meet the varying effluent
standards of the receiving stream, at a
minimum cost. Define the data you will
need to do this, and all model parameters.
Why might this proposed adaptive treatment scheme not be very practical?
12:8 Urban Infrastructural Asset Management
is increasingly becoming a key topic in the
move toward increased sustainability of
water supply and wastewater systems. List
some characteristics of sustainable infrastructural systems.

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permission from the license holder to duplicate, adapt or
reproduce the material.

Project Planning: Putting It All
Together

Water resources planning and management
issues are rarely simple. Projects focused on
addressing and finding solutions to these issues
are also rarely simple. These projects too need to
be planned and executed in ways that will maximize their likelihood of success, i.e., will lead to
useful results. When decision-makers and other
stakeholders disagree over what they want, and
what they consider useful and helpful, the challenge facing project planners and managers is
even more challenging. This chapter offers some
suggestions on project planning and management. These suggestions reflect years of experiences the writers and their institutions, have had
planning and participating in various water
resources development projects, at various
scales, in many river basins and watersheds
throughout much of the world.
Each water resources system is unique, and
the specific application of any planning and
analysis approach needs address the particular
issues of concern as well as adapt to the political
environment in which decisions are made. What
is important in all cases is that such planning and
analyses activities are comprehensive, systematic
and transparent, and are performed in full and
constant collaboration with the region’s planners,
decision-makers, and the interested and affected
public.

13.1

13

Water Management
Challenges

Managing water is important. The effectiveness
of strategies for dealing with water availability,
quality, and variability is a major determinant of
the survival of species, the functioning and
resilience of ecosystems, the vitality of societies,
and the strength of economies. Humans have
been managing water and adapting to surpluses
and shortfalls since the dawn of civilization, and
especially since the early origins of agriculture.
There is evidence across the globe of thousands
of years of dam building and canal construction
to direct water toward crops of various kinds.
Though the tools and infrastructure water managers can use today are dramatically more
sophisticated than those used in the past and the
scale on which water managers work is much
larger in almost all cases, the activities are still
very much the same: managing floods and
droughts through harvesting and storing water
above or underground, delivering water across
long distances through pipelines and canals,
treating, distributing water supplies to where they
are needed, collecting, and treating the resulting
wastewaters all designed to meet a variety of
economic, public health, environmental, and
social objectives.

© The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1_13

567

568

In regions witnessing increasing human populations demanding more energy and more food
together with a more uncertain climate has led to
a complicated dynamic interconnected web of
physical, economic, and social components with
many opportunities for intelligent adaptive
management interventions. These interventions
that change the distribution of water quantities
and qualities over time and space can result in
substantial economic, environmental, and social
benefits. They can also introduce unexpected
costs and risks. The constraints are physical (as
with the large inputs of energy required for
desalination), geographical (depending on the
available suitable locations for reservoirs),
financial (building, operating and maintaining
infrastructure required to manage water is
expensive), political (nobody wants to relinquish
rights to scarce water without compensation),
and ethical (what uses deserve to be prioritized,
and how they relate to the needs of the
environment).
Trade-offs are fundamental when allocating
water to various sectors of society. Water is
linked to the production of energy, food, industrial products and to human health and the condition of the broader environment. For many
kinds of water uses, allocating water to one use
usually means less water available for other uses.
Consumptive use for agriculture, industry, or
cities almost always involves trade-offs, as do
mandates for instream flows to protect ecosystems or fisheries. But even consumptive uses do
not diminish the total amount of global water.
Consumption shifts water to a different part of
the hydrological cycle: for example, from liquid
to vapor, from clean to contaminated, or from
fresh to salty.
Choices about managing water trade-offs
involve more than hydrology and economics.
They involve people’s values, ethics, and priorities that have evolved and been embedded in
societies over thousands of years. The juxtaposition of hydrology, economics, and values is at
the crux of the water–climate–food–energy–environmental and society (people) nexus. While it

13

Project Planning: Putting It All Together

is unreasonable to think that models of water
resource systems will or even should include
each component of this interconnected interdependent nexus of components, analysts must be
cognizant that the part of system that they model
is interacting with and being influenced by those
components assumed exogenous to the system.

13.2

13.2.1

Water Resources System
Components, Functions,
and Decisions
Components

For the purposes of planning and management
water
resource
systems
include
three
components:
• The natural resource system (NRS) component consists of the streams, rivers, lakes, and
their embankments and bottoms, and the
groundwater aquifers, and the water itself. This
includes the abiotic or physical, biological, and
chemical (“ABC”) components in and above
the soil. It also includes the infrastructure
needed to collect, store, treat, and transport
water such as canals, reservoirs, dams, weirs,
sluices, wells, pumping stations, pipes, sewers,
and water and wastewater treatment plants, and
the policies or rules for operating them.
• The socioeconomic system (SES) component
is the water using and water-related human
activities. This component can also include
the stakeholders, i.e., the interested and
affected public.
• The administrative and institutional system
(AIS) component are the institutions that are
responsible for the administration, legislation
and regulation of the supply (NRS) and the
demand (SES) components of the water
resource system (WRS). This component
includes those institutions that plan and build
and operate the infrastructure required to
insure that water is where and when and in the
condition needed in ways beneficial to society.

13.2

Water Resources System Components, Functions, and Decisions

13.2.2

Functions

Table 13.1 presents a framework of water
resource system functions. This framework distinguishes between tangible and intangible
functions. Tangible functions can be described
quantitatively. For example, hydropower generation or municipal water supply, may be assigned
a monetary value. Intangible functions are
activities such as nature conservation or preserving a beautiful view that are hard to quantify
in monetary terms. In between are environmental
functions, some of which may be given quantitative values and others valued only indirectly,
such as by using the opportunity cost associated
with a particular target. The self-purification
process of a river, for example, may be assigned
a value by comparing this “work done by nature”
with the costs of the least cost alternative that
accomplishes the same results, such as constructing, maintaining and operating a wastewater collection and treatment system.

Table 13.1 Functions of the water resources system

569

13.2.2.1 Subsistence Functions
Communities depend to a large extent on water
for household uses, and for irrigating home gardens and community outdoor green and recreation areas. They may also use streams, paddy
fields, ponds, and lakes for fishing. These uses
are often neglected in national economic
accounts, as they are not marketed or otherwise
assigned a monetary value. However, if the WRS
becomes unable to provide these products or
services, this may well be considered an economic loss.
13.2.2.2 Commercial Functions
Commercial uses of water resources are reflected
in national economic accounts because they are
marketed or otherwise given a monetary value,
e.g., the price to be paid for domestic water supplies. Catching fish for sale by individuals and
commercial enterprises is an example. These uses
have a commercial value and most are also consumptive in nature. The concept of

570

“nonconsumptive use” should be regarded with
certain reservations. Nonconsumptive water use
may alter the performance of the WRS in various
ways. For example, consider reservoirs built for
hydropower. Reduced sediment and fish passage
and increased evaporation losses may impact
downstream ecosystems and users. Second,
operation of the reservoirs for the production of
“peak power” may alter the flow regimes downstream, and this can adversely affect downstream
ecological habitats and users. Finally, water
quality problems resulting from reservoirs may
impact users and ecosystems. Another example of
partly nonconsumptive use is inland water transportation. Oil and chemical pollution caused by
water transport activities can affect other users
and the ecosystem that depend upon the water
resources. Moreover, inland water transportation
may involve a real consumptive demand for
water. If water depths are to be maintained at a
certain level for navigational purposes, releases
from reservoirs may be required which provide
no value to other water users. An example is the
Lower Nile system, where water is released from
Lake Nasser to enable navigation and energy
generation during the so-called winter closure.
This water could otherwise remain stored for
(consumptive) use by agriculture during the
growing season.

13.2.2.3 Environmental Functions
The drainage basin of a river fulfills a series of
environmental functions that require no human
intervention, and thus have no need of regulatory
systems. These functions include self-purification
of the water and recreational and tourism uses. It
is sometimes difficult to assign values to environmental functions. They may be assessed by
using opportunity costs, calculated as the costs of
providing similar functions in other ways, e.g.,
the cost of additional wastewater treatment.
Lower bounds on recreational and tourism values
may be estimated by assessing the economic
benefits accruing from the use of tourist facilities
including hotels, and/or the revenue obtained
from the sale of fishing licenses.

13

Project Planning: Putting It All Together

13.2.2.4 Ecological Functions
Rivers, streams, and lakes and their associated
wetlands, floodplains, and marshes offer an environment for aquatic species. Land–water ecotones
(transition areas between adjacent ecological
communities) are known to harbor a rich assemblage of species, and are also important for the
diversity of adjacent ecological communities.
These ecological entities have an intrinsic ecological value irrespective of actual or potential
human use. There are many concepts and expressions that describe this ecological value: “heritage
value,” “aesthetic value,” “nature value,” “option
value,” “existence value,” among others.
Box 13.1. Definitions
Policy goal: what do we want to
accomplish?
Strategy: how do we want to do it?
Decision: what are we going to do?
Scenario: the external economic, environmental, or political situation affecting
our strategy and decision.

13.2.3

Goals, Strategies, Decisions,
and Scenarios

In planning projects the terms goal, strategy,
decision, and scenario are frequently used. In
popular use their distinction is often confusing.
In this book we have used the following
meanings:
• A goal defines what is to be achieved or how
some target is to be met. Goals identify needs,
prioritize issues and define targets and constraints on the actions to be taken to meet the
targets. Goals may define preferred courses of
action. For example, the goal might be to
apply user-oriented demand management
measures rather than relying on large-scale
water supply infrastructure development.

13.2

Water Resources System Components, Functions, and Decisions

• A strategy is defined as a logical combination
of individual measures or decisions that
accomplishes the stated goals and satisfies the
constraints imposed on the WRS. For example, the construction of a reservoir plus the
widening of the canal downstream and the
increase of the intakes of the irrigation system
all in an effort to reduce the risk of damage to
the agriculture sector in a drought prone area
is one strategy. An alternative strategy might
be to implement a cropping pattern that uses
less water.
• A decision is the implementation of a particular strategy or course of action. A distinction can be made between:
– Technical (structural) measures: modifications of elements of the water resources
infrastructure such as canals, pumping
stations, reservoirs, and fish ladders.
Technical measures often include managerial measures such as better ways of
using the infrastructure.
– Ecological (nonstructural) measures to
improve the functioning of ecosystems,
for example, by introducing fish fry in
spawning areas, or large herbivores.
– Economical measures to induce water
consumers to alter their use of water by
changing the price of the resource use
(through charges, taxes, or subsidies).
– Regulatory measures to alter the use of
water (through land-use zoning, permits,
pollution control and other forms of
restrictive legislation).
– Institutional measures specifying which
governmental agencies are responsible for
which functions of the WRS, and specifying the necessary interactions between
the public and private sectors involved.
• A scenario is defined as the environment
exogenous to the water system under consideration that cannot be controlled. Examples of
scenario variables include rainfall and other
aspects of the climate, demographical trends
and changes, production functions (including
crop water requirements), and most economic

571

variables relating to benefits and costs. What
should be considered as a scenario and what as
a decision variable may depend on the system
boundaries that have been defined.

13.2.4

Systems Approaches to WRS
Planning and Decision
Making

Literature on the systems approach to planning
often emphasizes the mathematical techniques
used by practitioners of this approach. This book
is no exception. Most of it is devoted to modeling
water systems. The use of mathematical tools,
however, is only part of what constitutes a systems approach. The approach applied to complex
systems of many interdependent components,
involves:
– building predictive models to explain system
behavior,
– devising courses of action (strategies) that
combine observations with the use of models
and informed judgments,
– comparing the alternative courses of action
available to decision-makers,
– communicating
the
results
to
the
decision-makers in meaningful ways,
– recommending and making decisions based
on the information provided during these
exchanges between analysts, planners, and
decision-makers and stakeholders, and
– monitoring and evaluating the results of the
strategies implemented.
Systems analysis and policy analysis are often
considered as being the same. If a distinction is
to be made, one might define systems analysis as
being applicable to more than just policy issues
or problems. It can be applied to any system one
wants to analyze for whatever reason. System
diagrams or conceptual models identifying system components and their linkages are important
tools in systems analysis. A system diagram

572

13

Project Planning: Putting It All Together

Fig. 13.1 Identification of a water resources management (WRM) problem

represents cause–effect relations among the
components of the overall system. An example
of the use of system diagrams in analyzing water
resources problems is presented in Fig. 13.1.
As Fig. 13.1 shows, water using activities may
face two problems. First, the quantity demanded
may be greater than the supply; second, they
adversely impact the natural system (e.g., generate pollution or alter the water level). The perception of these problems can motivate analysis
and planning activities, which in turn can result in
management actions. The figure shows that the
problems can be addressed in two ways: either by
implementing demand-oriented measures (addressing the water use, i.e., SES), or by developing infrastructure that impacts the NRS).
Demand-oriented measures aim to reduce water
use and effluent discharge per unit of output.
Supply-oriented measures on the other hand are
aimed at increasing the water supply so that the
magnitude and frequency of shortages are
reduced or at increasing the assimilative capacity
of the receiving water bodies. Which measure or
combination of measures is most effective
depends on the criteria selected by the implementing authority.

13.3

Conceptual Description
of WRS

Water resources management aims to increase
the benefits to society from the existence and use
of water (NRS). Just how best to do it is society’s
(SES) choice, commonly made through its governing institutions (AIS). These three “entities”
are depicted in Fig. 13.2.
The management actions among the components of a WRS system are depicted by the
arrows shown in Fig. 13.2. The arrows represent only the actions, not the information flows.
There must be information feedbacks, otherwise
effective management would be impossible.
Each of the three systems is embedded within
its own environment. The NRS is bounded by
climate and physical conditions; the SES is
formed by the demographic, social, and economic conditions of the surrounding economies; and the AIS is formed and bounded by the
constitutional, legal, and political system it
operates within. Boundary conditions are usually considered fixed, but in some cases they
may not be. For example, climatic conditions

13.3

Conceptual Description of WRS

573

Fig. 13.2 Context for
water resources planning
involving the natural
resource, socioeconomic
and administrative–
institutional systems

may be considered to be changing due to global
warming. Similarly for laws and regulations.
Whether and, if so, when to consider the possibility of changes in this “external” environment should be decided at the start of any
planning project.
Consider, for example, regional economic .
This predicted growth is often treated as given. If
the water resources available cannot sustain this
projected growth (or only at very high costs), it
may be appropriate to reconsider this assumed
growth. By learning the consequences of unrestricted growth at the regional level, planners can
consider the desirability of other options that
might be considered at higher (usually national)
planning levels. This is represented in Fig. 13.2
by the border frame “socioeconomic development plans”. In fact, the arrow pointing inwards
to the SES is reversed in such a case: the analysis
provides information to a higher planning level
that can change the boundary conditions.

13.3.1

Characteristics
of the Natural
Resources System

The natural resources system (NRS) is defined by
its boundaries, its processes, and its control
measures.

13.3.1.1 System Boundaries
The study area of a planning project will often
coincide with an administrative boundary (state,
county, district, province, etc.). However, a WRS
is typically defined by its hydrologic boundary.
These political and hydrologic boundaries can
differ. Clearly, any planning project for a WRS
must include the larger of these boundaries, but
not necessarily everything within them depending on the purpose of the study and the particular
WRS. The consideration of problem sheds that
contain the components that impact water sheds
is often useful.

574

For the purposes of modeling it has often
proven useful to subdivide the NRS into smaller
units with suitable boundaries. Examples are
subdivisions into a groundwater and a surface
water system, subdivision of a surface water
system into catchments and sub-catchments, and
subdivision of a groundwater system into different aquifers or aquifer components. The definition of (sub) systems and their boundaries should
be done in such a way that the transport of water
across area boundaries can be reasonably determined and modeled.

13.3.1.2

Physical, Chemical,
and Biological
Characteristics
The physical processes in an NRS are transport and
storage within and between its subsystems. For the
surface water system, a distinction is usually made
between the infrastructure of rivers, canals, reservoirs, and regulating structures (the open channel
network) and the catchments draining to the open
channel network. The biological and chemical
characteristics define the biological and chemical
composition of groundwater and surface water and
the transport, degradation and adsorption processes that may influence this composition. The
level of detail to which these characteristics are
considered will depend on the requirements and
threats they impose on the water using and
water-based activities.
13.3.1.3 Control Measures
By adding or changing the values of system
parameters defining design and operating policy
options of NRS, water resources managers can
change the state of the system. An example is the
rule curve defining how much water to release
and when for different purposes. Another example is the flow capacity of feeder canals.
Increasing the capacity of these canals permits
greater allocations of water to farmers. An
example of nonphysical control that changes the
state of the biotic system is the release of
predator fish in reservoirs to reach a desired
balance of species in the ecosystem.

13

13.3.2

Project Planning: Putting It All Together

Characteristics
of the Socioeconomic
System

Like the NRS, the SES has its boundaries, processes, and control measures.

13.3.2.1 System Boundaries
The economic and social system generally does
not have a physical boundary like that of the
natural system. Economic and social activities in
a river basin, for example, are connected to the
world outside that basin through the exchange of
goods, people, and services. The factors that
determine the socioeconomic activities to include
in a project planning exercise will depend on the
context of the problems and development
opportunities being considered. Outside the
boundary of the socioeconomic system are factors or conditions that are beyond the control of
the WRS decision-makers.
13.3.2.2

System Elements
and Parameters
The socioeconomic part of the WRS can be
defined by identifying the main water using and
water-related activities, the expected changes and
developments in the study area, and the parameters whose values define these changes and
developments. Examples of activities or economic sectors that may be relevant and of the
type of information that has to be obtained to be
able to describe the SES include:
• Agriculture and fisheries: present practice,
location and area of irrigated agriculture,
desired and potential developments, water use
efficiency, and so on.
• Power production: existing and planned
reservoirs and power stations, operation and
capacity, future demands for electric energy.
• Public water supply: location of centers of
population and industrial activities, expected
growth, alternative resources.
• Recreation: nature and location, expected and
desired development, water quality conditions.

13.3

Conceptual Description of WRS

575

• Navigation: water depths in relevant parts of
the open channel system.
• Nature conservation: location of valuable and
vulnerable areas and their dependence on
water quality and quantity regimes.

13.3.3.1 System Elements
Administrative and institutional settings vary with
scale, and with the way governing institutions
exist and operate. In many countries, but certainly
not all, the institutional framework consists of:

Some examples of important system parameters of the SES are labor force and wage rates,
price levels in relation to national and international markets, subsidies, efficiency of production and water use, and income distribution.
When identifying and analyzing activities in
the study area, it is important to consider possible
discrepancies between the opinions of individual
actors or stakeholders and their representatives.
For example, individual farmers may have different interests than suggested by the official
agricultural organizations.

• the central government, divided into sectors
such as public works, irrigation, agriculture,
forestry, environment, housing, industry,
mining, and transport
• a coordinating body, for example, a national
water board, to coordinate actions by various
sectors of the national government
• regional bodies based upon the normal subdivisions of government, for example, provinces, districts, cities, and villages
• regional bodies based on a division according
to the physical characteristics of the area,
such as river basin authorities
• water user organizations, representing the
interests of directly involved stakeholders, for
example, in irrigation districts.

13.3.2.3 Control Measures
The functioning of the SES can be influenced by
legislative and regulatory measures, and the price
of water may be a particularly important factor in
deciding how much is demanded. This price can
be influenced by the water resources managers
and used as a control variable. When the cost of
water use represents only a small portion of the
total cost of an activity, however, an increase in
its price may have little if any impact on water
use. In some cases water use is a necessity of life
no matter how high the costs. In such cases, the
price of water (or taxation for waste water discharges) may not be an acceptable control variable (except perhaps to inform stakeholders on
the consequences of possible cost reduction
measures).

13.3.3

Characteristics
of the Administrative
and Institutional
System

The AIS, like the NRS and SES, has its boundaries (its authority or limits) and its processes
including its ways of reorganizing for improved
performance.

When initiating broad comprehensive water
planning projects knowing the following information is useful:
• the ministries and coordinating bodies having
authority and responsibilities related to water
resources management
• the agencies involved in the preparation of
water resources development plans
• existing national and regional water resources
development plans and the authorities
responsible for implementing these plans,
establishing and enforcing regulations, and
overseeing infrastructure construction and
operation
• the existing legislation (laws and regulations)
concerning water rights, allocation of water
resources, water quality control, and the financial aspects of water resources management.
Other often useful information includes the
policies and plans of various water-related sectors such as environment, agriculture, economy,
transportation, urban development and energy.

576

13

13.3.3.2 Control Measures
From a systems point of view, the decision or
control variables that can be changed in the AIS
are less clear than in the case of the NRS and
SES. Often measures can be taken to improve the
functioning of the system, for example, by
establishing coordinating bodies when these are
not present, shifting responsibilities toward lower
levels of government, privatization, and other
measures. If they cannot be changed, at least
possible beneficial changes can be identified and
presented to those responsible for making
decisions.

13.4

Framework for Analysis
and Implementation

A water resources planning study generally
comprises five general phases, as illustrated in
Fig. 13.3. Although we do not suggest the use of
any rigid framework, some distinct phases and
activities can be recognized and used to structure
the analysis as a logical sequence of steps. The
description of these phases, the activities in them
and the interactions among the activities in them,
is referred to as the analysis framework. A coherent set of models is typically used for the
quantitative analyses aimed at identifying and
evaluating alternative beneficial measures and
strategies.
A decision process is not a simple linear
sequence of steps as suggested in Fig. 13.3, but
involves feedbacks to earlier steps. Part of the
process is thus iterative. Feedback loops are
needed when:
• solutions fail to meet current criteria
• new insights change the perception of the
problem and its solutions
• essential system components and links have
been overlooked
• goals and objectives or the scope of study
change (e.g., due to changing political,
international, developments in society).
Communication and interaction with the
decision-makers are essential throughout the

Project Planning: Putting It All Together

duration of a planning project and the implementation of the selected development. To
ignore this increases the risk of generating plans
and policies that are no longer relevant or of
interest to the client. Regular reporting (inception and interim reports, etc.) helps in effective
communication, but a continuous dialogue is
important throughout all stages or phases of the
analysis.
Decision makers and stakeholders should be
involved in each of the five (idealized) stages of
this framework. Otherwise there is a risk of the
planning project producing results that those
potentially impacted will not support. Stakeholder involvement brings both knowledge and
preferences to the planning process—a process
that typically will need to find suitable compromises among all decision-makers and stakeholders if a consensus is to be reached.
The framework involves a series of decisions
at the end of each stage. The divergence–convergence process for involving stakeholders in
decision-making on the five analysis stages is
illustrated in the rhombus approach of Fig. 13.4.
The first inception stage of the process identifies the subject of the analysis (what is to be
analyzed and under what conditions), the objectives (the desired results of the analysis), and
constraints (its limitations). On the basis of this
analysis, during which intensive communication
with the decision-makers is essential, an agreement on the approach for the remainder of the
analysis needs to be achieved. The results of the
inception stage can be presented in an inception
report, which includes the work plan for the other
phases of the analysis.
In the situational analysis stage, the tools for
the analysis of the water resource system are
selected or developed. Major activities in this
phase typically include data collection and
modeling. The models will be used to quantify
the present and future problems in the system.
Scenarios will be developed to describe the
future boundary conditions for the system.
Identifying and screening of alternative decisions
can occur in this phase. If possible no regret
measures will be identified for immediate
implementation. A gradual improvement of the

13.4

Framework for Analysis and Implementation

577

Defining
Analysis

Stakeholder
Process
and Criteria

SocioEconomic
system

system

Base case
Analysis

Scenario
Analysis
Reference
Case
Measures /
Screening
Problem
No regrets

III – Strategy Building
/ Scorecard
Management
Analysis
Preferred
Strategy

Data
and
(Modelling)
Tools

Stakeholders and Decision-makers / Capacity Building

Natural
Resources
system

Investment Planning

Feasibility
and EIA

Design

Monitoring and

Fig. 13.3 Framework for analysis and implementation of water resources projects

understanding of various characteristics of the
WRS is often obtained as the study progresses
from limited data sets and simple tools to more
detailed data and models. Interaction with the
decision-makers will be greatly enhanced if they

or those they trust and communicate with are
involved as part of the analysis team. More formal interaction can be structured through presentations of results in meetings and in interim
progress reports.

13

III. Strategy
building

start

decision

decision

decision
analysis

Project Planning: Putting It All Together

decision

578

V. Implemenplanning

end
Convergence

level of knowledge

Divergence

Fig. 13.4 Divergence—convergence process in decision-making

In the strategy building stage alternative
strategies will be developed and discussed with
the decision makers/stakeholders. This will
include adaptive management elements to ensure
that the preferred strategy is sufficient robust and
flexible in case the future develops differently
than expected.
In the action planning stage the selected
strategy will be prepared for implementation. An
implementation plan will be developed that
describes what will be done, by who, how it will
be financed, etc. This stage often requires also
additional work on components of the strategy
(such as feasibility and design studies), and
environmental impact assessments (EIA). Promotion of the he selected strategy is needed to
“sell” the proposed measures to public. Finally,
institutional arrangements will have to be made
to ensure a smooth implementation.
Finally, in the implementation stage the actual
implementation will take place. Continuous
monitoring and evaluation is needed to adjust the

implementation plan when this appears to be
needed, for example, because the conditions
(e.g., finances, social pressures, political mood)
change.
Each stage or phase needs to provide the
information desired by those institutions who
will decide on what is best to do, and when, and
how. What those governing institutions need to
know to be better informed before making their
decisions will of course vary among different
planning projects. But whatever that information is the purpose of performing analyses is to
create and communicate it. The results of the
analyses performed in a planning project should
be of no surprise to those reading them in a final
project report. Again, communication between
the project and the requesting institutions, and
the affected public—the stakeholders—is
essential throughout the project. This communication may not guarantee a consensus but it
can certainly help the project team in their
efforts to find it.

13.4

Framework for Analysis and Implementation

13.4.1

Step I—Inception Phase

Water resources planning studies are often triggered by specific management problems such as
the need to increase power production or water
supply reliability, the occurrence of droughts or
floods, or the threat of water quality deterioration.
The need for water resources planning in relation
to other sector planning efforts may also be a
trigger. Which parts of the WRS are studied and
under what conditions follows primarily from the
objectives of the study (and from the available
budget, data, and time). The initiators of the study
generally have more or less concrete ideas about
the objectives and purpose of the analysis.
However, these can change during a study.
The client’s ideas about the problems and
issues to be addressed will usually be described
in a Project Formulation Document (PFD) or
Terms of Reference (ToR). The very first activity
of the project is to review and discuss the contents of these documents. If the subject (what
needs analyzing) and objectives (what is to be
accomplished) are adequately described in the
ToR, the next step of the study is to specify and
agree on the approach (how).
In many situations, however, the next task of
the project will be to assist the decision-makers
in further specifying the objectives and subject of

Fig. 13.5 Enabling
conditions (the “pillars”)
for IWRM

579

the analysis. For this activity, intensive communication is required with authorities involved in
water resources planning and the stakeholders.
They can provide information on the requirements of various interest groups related to water
and on expected problems. It is not uncommon to
have the stated objectives of a study differ from
the actual (often unstated) objectives of the client
(including just stalling for time hoping stakeholders will lose interest in a particular issue).
Furthermore, objectives can change over time.
As emphasized above, constant and effective
communication between analysts and their clients is absolutely essential to the success of any
planning project. We mention this often as it is
not always easy given busy time schedules and
often having to learn the differences in the
meanings of various words or expressions (jargon) used by all parties.

13.4.1.1 The Enabling Conditions
In order to successfully carry out a good planning
study certain conditions should be met. Most of
these conditions are external to the project
activities. This means that they should have been
set before the planning exercise starts. A generic
description of the enabling conditions for integrated planning is given in Background Paper no.
4 (GWP 2000) and is illustrated in Fig. 13.5.

580

• Enabling environment at national level:
– national water legislation and national
policies that guide the planning process
and enables enforcement.
• Institutional framework:
– existence of water institutions at national
and regional level with qualified staff;
– in case of river basin studies, existence of
some kind of river basin organization
(RBO) at river basin level.
• Management instruments:
– availability of data, information, and tools
that enables informed decision making.
In the Inception stage it should be determined
which conditions are relevant for the specific
planning exercise. This depends on the issues
involved. If needed, institutional measures can be
part of the planning project.

13.4.1.2

Setting Up the Stakeholder
Involvement Process
The very first step is to set up the stakeholder
involvement process. Which stakeholders to
involve and how will depend on the specific
basin and the issues to be addressed. In general
two categories of stakeholders can be identified:
• the people and organizations that will be
affected by the plan; and
• the people and organization that are needed to
implement the plan.
In some cases a stakeholder analysis might be
needed to determine the best stakeholder
involvement process. More detail on involving
stakeholders is given in Sect. 13.5.1.

13.4.1.3 Defining Analysis Conditions
In addition to the more legal and institutional
oriented conditions as described in Sect. 13.4.1.1
it is necessary to get agreement on the analysis
conditions for the planning study. This includes:
• The base year for the study:
– the most recent year for which basic data
on the present situation is available;

13

Project Planning: Putting It All Together

• The time horizon(s) for the study:
– this may include short term (e.g., 5 years),
medium term (e.g., 20 years) and long
term (>25 years);
• The discount rate to be applied in the economic analysis:
– taken as specified by (e.g.) the Ministry of
Finance or Economic Affairs, or by the
financier of the planned investments (e.g.,
ADB, World Bank and JICA);
• System boundaries of NRS, SES, and AIS—
the components and the level of detail that
will be included:
– e.g., will the coastal zone be included in a
river basin study?
– are the results to be presented at local
government unit level?
• Time periods based on within- and over-year
variability of systems processes and inputs
• Scenario assumptions concerning factors
external to the WRS, such as the growth of
population, food and energy consumption and
prices. See also Sect. 13.4.2.4.
• System assumptions. These concern factors
internal to the WRS, such as the response of
crop production to improved cultivation
practices, or the effectiveness of price incentives on per capita water consumption. These
system assumptions can be subject of additional (sensitivity) analysis.
• Data, time, and budget constraints. Studies
have to be executed within constraints of
available data, time, and budget.
The choice of the time horizon is often given
insufficient attention. Official planning horizons
(e.g., 5, 10, and 25 years) are typically used as
time horizons for elements of the analysis.
However, one should also consider the timescales of the system and the processes within it.
System components will have characteristic time
scales. For example:
• Economic activities have life cycles that are
usually determined by the amortization period
of the investments. Time horizons of planning
processes can be based on these conditions.

13.4

Framework for Analysis and Implementation

581

• Social institutions have time horizons that
depend on the pace of legal/institutional and
political decision making.
• Physical–chemical systems have time scales
that depend on the response or restoration
times of the systems. Restoration of polluted
rivers, for example, may be achieved within a
few months, while the restoration of a polluted groundwater aquifer may take decades.
• Ecosystems may have a time scale of a few
weeks (algae blooms) or tens of years
(degradation of mangrove forests), depending
on the type of process or intervention.

refer to national policy priorities and indicate the
contribution the plan will make to the various
development goals. Required information is
usually described in various national policy
documents. In addition to the national policy
documents any existing regional/provincial policy documents need to be taken into account.
Each plan need to have an agreed objective that
not only focuses on the main, but also expresses
the relation with above mentioned national and
other sector plans, as well as the contribution the
basin can make in realizing these higher level
plans.

To study the sustainability and ecological
integrity of the resource system, time horizons
should be tuned to the response times of the
system rather than to a planning horizon only.
Although more attention is now paid to sustainability, no operational procedure has been generally adopted to properly consider long-term
effects
in
the
evaluation
process.
Decision-makers tend to focus on short-range
decisions even if they impose possible risks in
the long term, because their political time horizons are often limited to (or renewable in) short
terms and hence they prefer short-term political
gains.

Operational objectives, criteria and targets
If needed, the general objectives as stated in the
national policy documents have to be translated
into operational objectives for the specific area
under consideration, e.g., a river basin. This
should be done by specifying them in socioeconomic terms, amongst others, which are meaningful to the decision makers and stakeholders.
For each objective evaluation criteria should be
defined as a measure of how far the defined
objectives have been achieved and, if possible,
clear targets should be specified. Monitoring will
indicate how far the objectives have actually
been achieved. This process, illustrated in
Fig. 13.6, is discussed in more detail in Chap. 9.
The evaluation criteria need to be comprehensive (i.e., sufficiently indicative of the degree
to which the objective is achieved) and measurable. The criteria do not all have to be expressed
in a single measurement scale. Criteria can be
expressed in monetary and nonmonetary terms.
It may be useful to incorporate sustainability
as an objective, and if so, it may also be useful to
relate them to the UN Sustainable Development
Goals (SDGs), the SDG targets and the indicators, that have been selected to monitor the
SDGs.

13.4.1.4 Objectives and Criteria
An essential activity in the inception phase is the
translation of general objectives, as described in
the ToR or in policy documents, into operational
objectives that can be quantified. Examples of
objectives and criteria are discussed throughout
this book and especially in Chap. 9. The objectives
and criteria used in a water resources management
study in West Java, Indonesia are presented as an
illustration at the end of this chapter.
National
and
regional
development
objectives
An essential component of an integrated plan is
the connection of the plan and its objective to
national development goals as well as to common international goals (e.g., the Sustainable
Development Goals—SDGs). The plan should

Illustrative river basin case
Table 13.2 presents a scorecard that summarizes
results of an analysis for a river basin case. The
results of the Inception step (i.e., the objectives
and criteria), for this river basin are given in the

582

13

Project Planning: Putting It All Together

• Objective 1: Provide safe water and sanitation
for the people;
– % people access to safe drinking water;
– % people access to sanitation facilities;
• Objective 2: Increase food production;
– Irrigation area (ha);
– Number of animal water points (#);
• Objective 3: Support economic sectors—industry and energy;
– Water supplied to mining (% of demand);
– Water supplied to industry (% of
demand);
– Hydropower generated (MWh);
• Objective 4: Protect the Environment;
– Protected watershed area (km2);
– Number of springs/sources protected (#);
– Average class water quality rivers (class A
to D);
• Objective 5: Decrease vulnerability to floods
and droughts;
– Vulnerability to floods—average damage
($/year);
– Vulnerability to droughts—average damage ($/year).
• In addition two implementation-related criteria were formulated to evaluate the strategies:

(e.g. food security)

(e.g. achieve self-sufficiency in rice)

(e.g. self-sufficiency index in %)
Target
(e.g. 100 %)

(e.g. 80 %)

Fig. 13.6 Making objectives operational

first two columns of the table. They show that for
this case five objectives were formulated. For
each objective 2 or 3 criteria were identified that
expresses in how far the objective is or will be
achieved:

Table 13.2 Example of a scorecard showing objective values associated with various strategies
Base
Objectives and criteria

Alternative (investment) strategies

Year
unit

2010

Targets
2020

Ref. case (no action)

2030

Perfect

2020

2030

Strategy 1
2020

2030

Strategy 2
2020

2030

Obj.1: Water and Sanitation
% people access to safe drinking water

%

50%

63%

73%

100%

63%

73%

63%

73%

63%

73%

% people access to sanitation facilities

%

30%

50%

70%

100%

50%

70%

50%

70%

50%

70%

Obj.2: Food production
Irrigation area

1000 ha

24

30

35

40

26

28

28

31

30

35

#

300

500

900

1000

400

700

500

900

500

900

Water supplied to mining

%

30%

80%

90%

100%

40%

50%

50%

70%

80%

90%

Water supplied to industry

%

70%

80%

90%

100%

70%

70%

80%

90%

80%

90%

MWh

34

80

120

120

34

34

70

110

80

120

# animal water points
Obj.3: Industry and Energy

Hydropower generated
Obj.4: Environment
Protected watershed area
Number of springs/sources protected
Average class water quality rivers

2

1200

2500

3500

3500

2000

2500

2500

3000

2500

3500

#

300

600

900

900

400

600

500

700

600

850

I-V

II

III

IV

V

II

III

III

III

III

IV

km

Obj.5: Vulnerability
Vulnerability to floods - average damage

m€/yr

120

< 78

< 50

0

100

80

100

80

78

50

Vulnerability to droughts - average damage

m€/yr

200

< 50

< 30

0

160

120

80

40

50

30

Implementation information
Required investments
B/C ratio economic categories (Obj.2, Obj.3)

m€

-

-

-

400

650

600

1200

-

> 1,3

> 1,2

-

1.3

1. 2

1.2

1.1

13.4

Framework for Analysis and Implementation

– Required investments ($);
– Benefit/Cost ratios of economic categories
(</>).

13.4.1.5

Work Plan
and Decision-Making
Once it is clear “what” will, as well as what will
not, be analyzed and “why”, analysts can specify
“how” this will be done. A description of the
system to be analyzed includes the conditions
and the assumptions under which the analysis
will be performed.
All required activities can be combined in a
work plan. It is often advantageous to develop a
critical path network of the various analysis
tasks. Critical path networks define the sequence
of various tasks required to complete an analysis,
or indeed the entire planning project, and their
start and finish times. This will guide the allocation of personnel and identify the time needed
to perform such tasks. These networks can be
updated as the project proceeds. Such networks
are useful for scheduling activities and personnel
involved in the project, and for ensuring (or at
least increasing the probability) that data and
personnel will be available for each activity when
needed and when decision-makers and stakeholders are to be involved in the analyses or in
workshops or meetings focused on improved
understanding of project progress and goals.
Data Availability
An important boundary condition for studies is
often the availability of data and other information required for the study. The availability of
data determines the level of detail and accuracy
that can be achieved in the analysis. If few data
are available, a more qualitative analysis may
have to be performed. The required level of detail
will primarily depend on the problems to be
addressed and the objectives to be satisfied.
Level of detail One of the main tasks of a
project leader is to motivate and manage the
experts from various disciplines. Not staying
focused on the appropriate level of detail is one
of the most common causes for project failure. If
the needed level of detail is underestimated at the

583

start of the project, the study will have to obtain
the additional detail needed fulfill the objectives
of the analyses. Sometimes the right level of
detail is chosen, but team members may be
tempted to spend too much time addressing more
detailed questions of interest to them and fail to
come up with the information desired within the
available time. Maintaining the proper level of
detail is one of the main reasons for feedback
loops in the analysis process.
Computational Requirements
An important element of the work plan will be
the determination of the computational resources
needed for the analysis. This includes mathematical models, databases, GIS, and the like.
Together these must be used in a way that
describes the system and permits an evaluation of
possible measures and strategies under different
scenarios at the level of detail desired. Often a
combination of simulation and optimization
models has proven useful.
For the purposes of analysis, the study area is
typically subdivided over space and time into
smaller units considered to be homogeneous with
respect to their characteristic parameters. Each
unit can be included in mathematical model(s).
The number of elements required for the analysis
depends on the issues being addressed, the
complexity of the study area, the measures to be
studied and the availability of data. It generally is
wise to start with a preliminary schematization
with the minimum number of elements. If more
spatial or temporal detail is required model elements can be subdivided. The assumptions and
conditions under which analyses are undertaken
should be specified in close cooperation with
those institutions overseeing and contributing to
the study.
Work Plan
The results of the inception phase are documented in an inception report. This report can
serve as a reference during the execution of the
study. An essential part of the report is the proposed work plan, in which time, budget and
human resource allocations to various activities

584

13

are specified. This work plan typically includes
bar charts (possibly derived from critical path
analyses) for activities and staffing, time schedules for deliverables, milestones, reporting procedures and similar features. The report should
include a communication plan that describes the
interaction between the decision-makers and
stakeholders and the analysis team.
Inception Report
An inception report is a specific and concrete
result of the inception phase. It contains the
findings of and decisions made during the inception phase. It should make clear what will be
studied, and why and how. In many cases it will
also specify what will not be studied and why. The
content of the inception report follows the subjects mentioned above. It is an important product
because it contains all that has been learned in this
first inception phase and that has been agreed
upon between the analyst and the “client” (the
decision-makers and the stakeholders).
A possibly even more important result of the
inception phase, however, is the interaction
between the analyst and the client that took place
during this phase. It should state the client’s
views about problems, objectives and other
aspects. Project analysts must understand the
client’s concerns, problems and objectives. Clients should feel they “own” the results of the
inception phase and view the inception report as
their own product, not merely a report of the
planners, analysts or consultants. To achieve
such ownership, frequent interaction must have
taken place among the analysts, the
decision-makers and stakeholders, to a much
greater extent than is indicated in Fig. 13.3. This
can be done in specific workshops, such as those
devoted to the problem statement or to the
specification of objectives and criteria.

13.4.2

Project Planning: Putting It All Together

will be collected and where necessary and possible the system components will be captioned in
models. As much as possible this should be done
in close collaboration with the stakeholders to
ensure that the analysts and stakeholders have the
same understanding of the system. Once these
models are available a structured analysis can be
carried out to quantify the present and future
problems and a start can be made with identifying measures to address these problems.

13.4.2.1

Understanding
and Describing
the Water Resources
System
A WRS comprises:
• Natural (Resources) System (NRS);
• Socioeconomic System (SES); and
• Administrative and Institutional System
(AIS).
Each of the three systems is embedded within
its own environment. The Natural Resources
System is bounded by climate and (geo)physical
conditions. The SES is formed by the demographic, social and economic conditions of the
surrounding economies. The AIS is formed and
bounded by the constitutional, legal and political
system. The interlinkages of the three systems
are illustrated in Fig. 13.7.
It is important that the plan includes a good
description of the integrated elements of the

Step II—Situation Analysis

In the situation analysis phase the study starts to
dig deeper in the water resource system. Its
various components will be studied in detail, data

Fig. 13.7 Systems components of a WRS

13.4

Framework for Analysis and Implementation

WRS. Most decision-makers and stakeholders
will be nontechnical or only know about a limited part of the overall system. To be able to
make balanced decisions they should understand
how the overall system functions and how
interventions in one part of the system will
impact other systems elements.
The situational analysis starts with an inventory of the characteristics of the WRS. This
requires the reduction of a complex reality into a
comprehensible description of system components and linkages. Choices have to be made
about what (the detail that) should be included
and what can be ignored. Such choices require
engineering and economic judgment in combination with an understanding of the problems and
possible measures that can be taken to improve
system performance. The next step will be an
inventory of the activities and ongoing developments that will determine how the system will
perform in the future and what kind of additional
activities can be expected. This can include
autonomous developments (such as population
and urban growth) as well as policy decisions
that have been or may be taken that could
influence the characteristics and performance of
the WRS. An inventory of policies and institutions is helpful for identifying who is involved in
the management and development of the system
(and hence who should be involved in the analyses) and their objectives and opinions. This
knowledge will contribute to the development of
scenarios for the analyses.
Analysis of the Natural Resources System
(NRS)
The NRS comprises the natural and engineered
infrastructure, including the hydrometeorological
boundary conditions. Models can be used to
simulate the processes of water distribution
through the infrastructure, taking into account the
storage of water and water withdrawals to satisfy
the demands of water-using activities. Such
models have been introduced in many of the
previous chapters of this book.
The results of the water quantity modeling
may be the inputs for water quality models. The
analysis of chemical components in the water

585

system is used to study the influence they have on
the user functions or the biological system. The
components and processes that are to be considered in the analysis should have been selected in
the inception phase. The analysis of the biological
system aims to determine the response of the
ecosystems to water resources management (see
Chap. 10). Since often there is too little exact
information on individual biotic components and
their behavior under different hydrologic and
chemical regimes, models of ecosystems typically
depend on habitat parameters.
Analysis of the Socioeconomic System
(SES)
Developments in the SES determine the way
demands on the NRS may change. Conversely, the
development of economic activities within the
study area may depend on the availability of water.
For example, good supplies of relatively cheap
surface water may stimulate the development of
irrigated agriculture, or attract industrial activities
that require large quantities of water for their
production processes. Another example is the
development of water-based recreation activities
adjacent to a reservoir. These SES developments
in turn increase the water demands. Economists or
planners may be able to estimate future levels of
the activities dependent on water discharges and
storage levels. These relations can be incorporated
into water resource planning models.
The starting point for an analysis of the SES is
an assessment of the present economic situation
with respect to the water-related activities and the
factors that determine these activities. Past trends
can help provide information on factors that have
been decisive in bringing about the present situation and that may give clues about the likely
impacts of future developments. One’s attention
should be on the most important factors that
determine relevant water-related activities rather
than on analyses of the total economy. However,
the difficulty in forecasting economic development is the uncertainty about which factors will
be decisive for this development.
Part of the data needed to develop planning
models is the relation between the economic

586

activities and their water use. Data are needed
that define the type and amount of water used by
various activities. Data are needed identify the
following with respect to each identified activity:
• the amounts of water (quantity and quality)
demanded and consumed during which periods of the year and at which locations
• the amounts of water discharged and the
pollution loads during which periods of the
year and at which locations
• the benefits to the user if these amounts are
made available
• the damage to the user if these amounts are
not available
• costs that can be recovered by having the user
pay for the water and its influence (both at the
intake and the discharge sites of his activity)
on the water use pattern.
All these data should be able to contribute to
the estimates of future water demands, consumption and wastewater discharges per unit of
activity. As well as the level of activities and the
resulting water demands, knowledge of the geographical location of water using activities (the
pattern of activities) is necessary. If the pattern of
activities is not expected to change, the analysis
can be focused on the present situation in the
study area. If new activities are expected to
develop within the study area and their water use
characteristics are unknown, it may be necessary
to study the water use characteristics of similar
activities in other regions.
The resulting water demand data need not
always be considered as “given.” Water-use
coefficients can be changed through measures
such as water pricing that aim at reaching a
socially preferred use pattern. Technological
developments may result in less water use and
pollution load per person or unit of product. If
supplies and demands are matched before the
effects of such incentives are analyzed then one
may over estimate needed capacities, because the
“given” demands may be lower if water users are
confronted with the costs as well as the benefits
of water use. This type of internal feedback
should be considered in the study.

13

Project Planning: Putting It All Together

Future water demands are often dependent on
future scenarios. A water demand scenario is a
logical but assumed combination of basic SES
parameters and their effects on water-related
activities, including the resulting water demands.
An understanding of the functioning of the SES
developed through the assessment of past and
present trends is often helpful when formulating
a limited number of consistent scenarios.
Box 13.2 is an example of one such scenario.
Box 13.2. Example demand scenario
The water demand in an agricultural area
depends largely on the availability of land
and the crops being irrigated. The demand
for agricultural products, however, will
develop in an autonomous way. If the
availability of water resources in a region is
limited, the autonomous development of
the agriculture sector will be limited as
well, and one would predict a small
increase in agricultural water demand. If the
demand for agricultural products increases
considerably and self-sufficiency in food
production is an objective, then the political
pressure for agricultural development to
meet this objective may be considerable.
The water demand corresponding to this
desired agricultural development could
show the need for further development of
the water resources in the region.

Analysis of the Administrative and Institutional System (AIS)
An analysis of the AIS is required to identify any
legal or regulatory or institutional constraints on
water resources management. Attention must be
given to the interaction between various authorities involved in water resources management
and to the effectiveness of the AIS. Arrangements made in the past concerning the use of
water (water rights) should be identified, since
these may significantly constrain the options for
water resources development.
Water resources management studies are often
limited to the preparation of policies for a certain

13.4

Framework for Analysis and Implementation

agency. In this situation, the analysis of the AIS
will mainly serve to identify measures that the
agency can implement effectively. The responsible agency should be aware of the possible role
they may have in solving the management
problems. Sometimes, the analysis of the AIS
may result in recommendations for institutional
and legal changes.

13.4.2.2 Data and Modeling
The result of the data collection and modeling
activities is a quantitative representation of the
WRS at an appropriate level of detail. The
framework is designed to assess the effects of
individual measures or combinations of measures, expressed in values for the evaluation
criteria chosen. If computer programs for running
models have to be developed or if existing
computer programs have to be adapted in a significant way, a considerable effort may be
required which may consume a large part of the
available budget and time. Careful selection of
the phenomena to be represented by the models,
tuned to the needs of the project, is important.
During the modeling activity, more information on the study area and the type of measures to
be considered may become available. This could
lead to changes in model structure. The models
should therefore be flexible and adaptable to new
information.
Model Integration
The various models and components developed
for the NRS and SES describe parts of the total
system. Some models may produce output that is
needed as input for another model. For example,
the output of a water quantity model may be the
input to a water quality model requiring different
spatial and temporal resolutions. Some models
may include links to various sub-models and run
interactively, others not. Depending on the models and the problem situation, single or multiple
linked models may be included within an interactive decision support system. In other cases, a
clear description of information flow from one
independent model to another may be sufficient.
Figure 13.8 provides an example in which
various simulation models are combined to

587

analyze a river basin under drought conditions.
The reservoirs in the system involve sedimentation and hydropower generation. The core of this
modeling framework is formed by the “core
models” block in the upper right corner of the
figure. In this block the demand for water is
determined, followed by a balancing of supply
through water allocation decisions. Links among
these core models are automatic. Other models
are linked through file transfer. This applies to
the required input on macroeconomic and
hydrometeorological conditions (generated by
scenarios) as well as the side analysis of the
sedimentation and water quality in the reservoirs.
The last parts of the computational framework
are the modules that determine the financial and
economic aspects (investments, operation and
maintenance, benefit–cost, etc.) and support a
multi-criteria analysis.
At various places in this modeling framework,
one can change the values of input parameters.
Scenarios can be analyzed by changing the
macroeconomic
and
hydrometeorological
conditions.
Figure 13.8 is just an example. Other problem
situations may require different modeling
frameworks. The goal in creating such model
frameworks is to make them as simple and
transparent as possible, and still adequately
address the problems to be solved. Sometimes
complexity is necessary. In any event it saves
time and money to start as simple as possible and
only add more detail when necessary to carry out
a proper analysis.
Collaborative modeling
Involving decision-makers and stakeholders in
the analysis process has till recently been limited
to the more general analysis about problems and
solutions. The quantitative information, e.g.
resulting from models, was provided by the analysts (e.g., consultants) as input for the discussions. More and more we see that stakeholders do
not accept this black box approach anymore. They
want to understand what went into the model, how
the models work and, preferably, they want to
“play” with the model themselves. This is a
promising development as this will increase the

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Project Planning: Putting It All Together

Fig. 13.8 Example of typical computational framework of simulation models

understanding of the stakeholders on how the
system works and let them see the opportunities
and constraints of that system. Having stakeholders involved in the development and running
of the models requires that these models are made
more accessible and intuitive, in particular their
input/output interfaces. It requires also a different
attitude of the modelers. Various approaches to

collaborative modeling are currently being
developed, sometimes under different names such
as Collaborative Modeling for Decision Support
(e.g., shared vision modeling), Mediated Modeling, Group Model Building, Companion Modeling,
Interactive
Modeling,
Networked
Environments for Stakeholder Participation or
Model-supported Collaborative Planning.

13.4

Framework for Analysis and Implementation

13.4.2.3

The Need for a Structured
Quantified Analysis
Process
Decision making on measures and strategies to
improve the performance of the WRS should be
based on quantified information about the present
problems (e.g., average flood damage) and the
impacts of proposed measures (e.g., the reduction
in flood damage) and the costs of these measures.
To be able to produce this quantified information
the following is needed:
• a structured analysis process (this section);
and
• a computational framework (see previous
section).
The analysis process starts with a quantified
problem description. The analysis of the present
situation is called the Base Case analysis. To be
able to predict possible future problems scenarios
should be defined on how this future might
develop. The computational framework will
calculate the impacts (the future problems) of
these possible external developments. This is
often called the Reference Case analysis.
Base case
The performance of the WRS is studied for the
infrastructure and water demands in the base
case. The base case is based on the base year,
which is the most recent year for which a complete set of data can be collected. The base case
describes thus the performance of the WRS in the
present situation. A comparison of the base case
with the criteria (and possible targets) specified
in the WRM objectives will result in a quantified
problem statement.
Scenario conditions
A good plan should also address the expected
water-related problems in the future. The analysis
for the future time horizon(s) should include
different scenario conditions. Possible scenario
conditions for WRM are socioeconomic developments (change in demand and pollution) and
climate change (including sea level rise). See the

589

next section on more information about developing scenarios.
Reference case
The reference case addresses the future situation
by considering the present infrastructure, to which
measures are added that have already been decided
or are being executed, together with selected scenario conditions. In the reference case an analysis
of the performance of the WRS is undertaken if
present policies and regulations are continued and
followed by the government and the water users.
Problem description—present and future
The problem description should be carried out
based on the results obtained from the base and
reference case analyses in combination with the
problems and issues perceived by the
decision-makers and stakeholders. A problem
analysis should be expressed as far as possible in
terms of the socioeconomic and environmental
impacts that have a meaning to the decision
makers and stakeholders. An integrated approach
is crucial for a solid understanding of the system
and its associated problems. The integrated
approach can only be achieved if the plan defines
the main problems and issues in the basin and its
interlinkages. For this, it is important that the
plan is aligned with other related plans such as
Watershed Plans (erosion), Flood Risk Management (FRM), and Integrated Coastal Zone Management (ICZM), amongst others.
Inventory of potential measures and selection of promising measures
Once the present and future problems are known
measures (including “no regrets” that can
immediately be implemented) can be identified
that will address these problems. An inventory
should be made of all the measures that the
stakeholders are planning or considering. Based
on the quantified problem analysis additional
measures might be formulated. The computational framework can be used to determine the
impacts of these measures. The most promising
measures will be kept for detailed analysis in the
next step: Strategy Building.

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13

base case analysis

reference case analysis

present
problems and issues

expected future
problems and issues

promising measures

Project Planning: Putting It All Together

scenario analysis

inventory of measures

No-regret
measures

…. to Strategy
Building phase

Planning phase

Fig. 13.9 Structured analysis process

The above described structured analysis process is illustrated in Fig. 13.9.

13.4.2.4 Scenario Analysis
A good plan should not only address the present
problems but should also prepare for problems
that might arise in future. To predict the future
scenario assumptions have to be made. Scenarios
are possible developments external to the WRS,
i.e., outside the control of the decision makers
involved in the project. The most usual scenario
components for water resources studies are
socioeconomic developments (e.g., growth of
population and economic activities) and climate
change (including sea-level rise). For the

economic evaluation of the plan it might be
needed to make assumption about the future
prices of energy and food. Changes in diet (e.g.,
the consumption of more meat) can also be
important.
The most used combination of scenario elements are presented in a quadrant of low and
high economic growth versus slow and fast climate change. Ideally the whole analysis should
be carried out for all kind of scenario combinations and the selection of the best strategy should
be based on the evaluation which strategy is able
to cope with all these possible future developments. In reality most analyses are carried out for
the most likely scenario based on a trend analysis

13.4

Framework for Analysis and Implementation

or Business-As-Usual (BAU). The strategy that
follows out of this is then analyzed in a “scenario
analysis,” to test that strategy on robustness and
flexibility for other possible futures. See also
Sect. 13.4.3.2 on adaptive management analysis.

13.4.2.5 Quantified Problem Analysis
A problem analysis should address and be
expressed in terms of the socioeconomic and
environmental or ecosystem impacts that are of
interest to the decision makers. Not all stakeholders may be able to relate to predicted changes
in flows, water levels, or pollutant concentrations.
Some may want to know how much money is
involved, the rate of shore line erosion, the relative change in fish population, or the number of
people affected by flooding. Expressing outcomes
in terms of socioeconomic impacts makes it easier
to relate the problems to the (socioeconomic)
development objectives that decision-makers
have formulated for the particular region or system under consideration.
A good problem analysis will also indicate the
measures that can be taken to eliminate, reduce
or alleviate the identified problems or to take
advantage of new beneficial opportunities. The
identification of measures not only helps to
clarify the problems and possible solutions; but
also helps in the design of the computational
framework and the data collection activities.
These activities should be designed in such a way
that the measures can be evaluated in the analysis
phases of the study.
On completion of the initial analysis, project
staff (and the decision makers/stakeholders)
should have a clear idea about what will be
studied in subsequent phases, for what purpose
and under what conditions.
13.4.2.6

Identification and Screening
of Potential Measures
Once the base and reference cases have been
defined, and the problems and bottlenecks identified, measures to address resource management
problems can be considered. Measures can be
divided into different categories. An inventory of

591

all possible kinds of actions that can be taken will
in general result in hundreds of discrete possibilities. In most cases it will not be practicable to
analyze all of them in detail. A screening process
is needed to select the most promising ones. This
can be done in several ways. As mentioned in
various chapters of this book, separate optimization models can be used to eliminate less
attractive or less promising alternatives. It can
also be done by using the modeling framework
developed for the project but limiting the analysis to a few criteria, such as economic or environmental ones. A third kind of screening
analysis is to apply judgment as to criteria
effectiveness, efficiency, legitimacy and sustainability. Box 13.3 describes these criteria.
Box 13.3. Criteria for screening
Effectiveness. Measures to be taken are
those which solve the most serious problems and have the highest impact on the
objectives. Measures to prevent problems
will be preferred to those that solve them.
Similarly, measures that solve problems
will be preferred to those that only control
them.
Efficiency. Measures to be taken should
not meet the explicit objectives at the
expense of other implicit objectives. The
cost–benefit analysis (at the national level)
is one indicator of efficiency. An example
is to create a law that forces industrial firms
to incur the full cost of end-of-pipe
wastewater treatment. In Egypt, this
would improve the Nile system water
quality, and thus improve health of those
who drink it and reduce environmental
damage. On the other hand it might impose
high costs to the firms, possibly resulting in
loss of employment. An efficient decision
may be to opt only for cost sharing rather
than full cost recovery.
Legitimacy. Measures to be included in
the strategy should not rely on uncertain
legal/institutional
changes.
Measures
should also be as fair as possible, thus

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13

13.4.3
reducing public opposition so that they will
be favored by as many stakeholders as
possible.
Sustainability. Measures to be taken are
those that improve (or at least do not
degrade) the present environmental and
socioeconomic conditions for future
generations.

The aim of the screening process is to identify
those measures that should be further analyzed.
The screening of measures is a cyclic process.
Assessing the measures will contribute to a better
understanding of their effectiveness and new ones
may be identified (comprehension loop). Combinations of measures may be considered for
specific parts of the WRS, for instance for solving
the water quality problems in a subbasin. The
result of the screening process is a set of promising measures that can be used for strategy design.
The whole process of base case and reference case
analysis and screening is depicted in Fig. 13.9.
No regrets
A special category of promising measures are the
“no regrets.” More realistic we should speak of
“likely no regrets” and “low-regret” measures.
These are measures on which there is a very large
agreement among the decision-makers and
stakeholders that these should absolutely be
implemented, preferably as soon as possible. It
should be ascertained that these measures will
not have negative impacts on other measures or
will prevent other possible promising measures
to be implemented. The reason to define such no
regret measures is that in quite some situations
there is a huge pressure to actual implement
measures and not to wait till (another) big integrated study has been completed and accepted in
its full extend. In particular in developing countries there is a big need for proposals for such
measures. These measures can proceed immediately to step IV on Action Planning.

Project Planning: Putting It All Together

Step III—Strategy Building

In the Strategy Building step, promising measures
are combined into strategies. The effects of various
strategies are assessed and a limited set of
promising ones is defined. For these promising
strategies, the effects are assessed in more detail.
The sensitivity of these effects to the values
assigned to the uncertain model parameters is then
assessed. Finally, the results of the selected strategies should be presented to the decision-makers.
The selection process is depicted in Fig. 13.10.

13.4.3.1

Strategy Design and Impact
Assessment
Strategy design involves the development of
coherent combinations of promising measures to
satisfy the management objectives and meet the
management targets if possible. As there are
generally many criteria related to these objectives, and probably many expressed in different
units, strategy design is not a simple process.
Relations among combinations of measures and
their scores on the evaluation criteria are complex. The optimum combination may depend on
who is asked. Trade-offs among the values of
different criteria, and disagreements among various stakeholders, are inevitable.
The design of strategies is an iterative process.
One can start by developing strategies on the
basis of a single objective such as, for example,
reliability of food and energy production or
maximum net economic benefits. These strategies define the boundaries of the solution space.
Comparison of the impacts of these strategies can
lead to the construction of compromise strategies
by changing elements in the strategy. A resulting
loss with respect to one criterion is then compared with gains to another.
Evaluation of Alternative Strategies
Strategies can be compared based on their criteria
values or scores. To facilitate the comparison, the
number of evaluation criteria should be limited.
Criteria have to be comprehensive (sufficiently

13.4

Framework for Analysis and Implementation

593

Fig. 13.10 Activities in
the strategy building phase
promising measures

scenarios on future
developments

design of

impact assessment
analysis
comparison of strategies

ranking of strategies
based on achieving
and flexibillity

promising strategies
preferred strategy

indicative of the degree to which the objective is
met) and measurable, i.e., it should be possible to
assign a value on a relevant measurement scale.
Where possible, criteria should be aggregated;
for example, some financial criteria might be
processed into a single value when distribution
issues are not going to be important.
It is usually impossible to express all criteria
in a single measurement scale such as a monetary
value. (We say this recognizing the many
attempts to do so by highly respected economists.) Criteria related to environmental quality
or ecosystem vitality or the beauty of a scenic
view can often be expressed quantitatively but in
nonmonetary terms. This should, however, be
done in such a way that a ranking is possible on
the basis of the chosen criteria.

Generally, there will not be a single strategy
that is superior to all other ones with respect to
all criteria used in the assessment. That means
that an evaluation method is required for the
ranking of alternative strategies.
Scenario and Sensitivity Analysis
Before drawing conclusions from planning projects involving uncertain information, and indeed
predictions of possible futures, one should analyze the effects of changes in the uncertain
assumptions made throughout the analyses. If the
selection of a different scenario would significantly change the attractiveness of a selected
strategy, then additional study may be required to
reduce the uncertainties in that scenario. The
sensitivity of the results to changes in model

594

parameter values and assumptions should be
determined and addressed in a similar way.

13.4.3.2

Adaptive Management
Analysis
The analysis approach described in the previous
section is based on the assumption that it is
known what will happen in future. Predictions
are made on how population growth, economic
growth, spatial developments (e.g., urbanization)
and climate change will take place. Some of
these developments are quite certain, e.g., population growth for which one can make reasonable good projections. Other developments are
much more uncertain such as economic growth
and climate change. While we want to be prepared for these future conditions we do not want
to run the risk that huge infrastructural investments are being made which later appear to have
been overdesigned or even unnecessary.
The way to deal with future uncertainty is to
follow an adaptive management approach. An
adaptive management approach has to replace
the traditional approach of master plans for the

13

Project Planning: Putting It All Together

basin. The development of implementing
stand-alone projects to adaptive management is
illustrated in Fig. 13.11.
The message on how to follow an adaptive
management approach is given in the right two
columns of Fig. 13.11 and is the logical
follow-up from the project oriented developments in the two first columns. The figure
explains that:
• The project-based approach is straightforward
and easy to implement. This approach does
not consider the (positive and negative)
interaction of the project with other projects.
• The interaction is taken into account when
related projects are considered in a package of
projects. However, the overall system is not
integrated yet and not optimized.
• The traditional master planning tries to optimize the overall system. The projects are
implemented as components of an integrated
strategy. The implementation of the strategy
includes an optimization of the various projects over the planning period which is usually between 15 and 30 years, for which a
cost–benefit analysis usually applies. Such a

Fig. 13.11 Planning approaches in water resources management

13.4

Framework for Analysis and Implementation

master planning approach does not consider
the long-term uncertainties that are involved
in socioeconomic developments and climate
change. If the predicted changes in socioeconomic conditions and climate do not
materialize this might lead to “future regret.”
• To reduce future regret a planning period of
up to 50 or even 100 years needs to be considered. As the lifetime of most structural
measures (dikes, floodways, reservoirs, etc.)
are designed for a period of 50–100 years, it
is wise to incorporate future uncertainties in
boundary conditions in their designs and
make them part of a dynamic strategy. The
adaptive approach not only tells us what to do
now but also gives directions on what to do
when the conditions develop differently.
Adaptive pathways
Various methods have been developed that enable
us to deal with future uncertainties. Recent
methods include Decision Trees (Ray and Brown
2015) and Dynamic Adaptation Policy Pathways
(DAPP; Haasnoot et al. 2013). The Decision
Trees is a repeatable method for evaluation of

Fig. 13.12 Adaptive pathways approach

595

climate change risks to new development projects. DAPP identifies tipping points that determine in time when a certain policy or action is no
longer acceptable and (another) action is needed.
By exploring all possible actions you can develop
adaptation pathways that will minimize the regret.
The Adaptive Pathway Approach is illustrated in
Fig. 13.12. The approach requires that many
conditions are explored (pathways, scenarios,
long time series). For that reason the models used
in an adaptive pathway analysis are sometimes
limited versions (meta-models) of the ones
described in this book. See Haasnoot et al. (2014).
Following an adaptive pathways approach
basically means that two additional criteria
should be considered in decision-making:
• Robustness: how robust is the existing
strategy when the future develops differently
than expected? Will the strategy then still
achieve the objectives?
• Flexibility: how changeable is the strategy
when it appears that the future develops differently than expected and we need to change
the strategy?

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13

Robustness and flexibility often have a strong
relationship with costs. A robust strategy can be
more costly (big reservoirs, high dikes, etc.).
A flexible strategy (many small reservoirs, build
in time) can also appear to be more expensive in
the end. These costs need to be taken into
account when deciding on a strategy.

13.4.3.3

Presentation of Results—
Preferred Strategy
Presentation of the selected promising strategies
to decision-makers may be by means of briefings, presentations, and summary reports among
other means. The level of detail and the way
project results are presented should give an
overview of the results at an appropriate level of
detail for the audience involved. Visual aids such
as score cards and interactive computer presentations of study results are often very helpful for
promoting a discussion of the results of the
analysis.
The results of selected strategies can be presented in matrix form on “scorecards.” The columns of the scorecard represent the alternative
cases used in the analysis. The rows represent the
impact of different alternatives with respect to a
given criterion. An example is depicted in
Table 13.2. Scorecards can contain numbers
only, or the relative value of the criteria can be
expressed by plusses and minuses, or a color or
shading. The purpose of scorecard presentations
is to present a visual picture of the relative
attractiveness of the alternatives based on various
criteria. Scorecards can also help viewers detect
clusters of criteria for which alternatives have a
consistently better score. The presentation of the
results in scorecards allows a decision-maker to
give each impact the weight he considers most
appropriate.

13.4.4

Steps IV and V—Action
Planning
and Implementation

Once the preferred strategy has been selected this
strategy should be translated into concrete
actions. Careful planning and coordination is

Project Planning: Putting It All Together

required as many authorities will be involved in
the implementation. The action plan will have an
“open” and “rolling” character, meaning that it is
not static or prescriptive, and leaves room for
individual decision-makers to further elaborate
upon in relation to their own responsibilities. On
the other hand, the action plan should be concrete, by assigning clear responsibilities for carrying out the activities involved. It also should
include the budgetary requirements for the
implementation, including investments and
recurrent costs.

13.4.4.1 Investment and Action Plan
The action plan translates the selected strategy in
concrete actions. For each of these actions it
should be clear:
• what: concrete actions that have to be carried
out for each of the measures included in the
strategy to get it implemented?
• who: the prime decision-maker/stakeholder
responsible for carrying out the action and
who will take the lead in the implementation;
• how: the steps to be taken and the consultative process involved;
• when: the time planning; and
• financing: where will the money to implement
the action come from?
What
An integrated planning analysis is usually carried
out at pre-feasibility level. A rough description of
the measures will been included in the strategy
and the assessment was based on first estimates of
costs and benefits. Depending on the type of
measure, feasibility studies should be completed
before the measures can actually be implemented.
Often these feasibility studies are combined with
detailed (technical) design of the measures.
Who and How
The Action Plan aims to stimulate the coordinated development and management of the water
resources. This is illustrated in Fig. 13.13, which
presents the Implementation Plan for water
resource development in Central Cebu in the

13.4

Framework for Analysis and Implementation

597

Fig. 13.13 Implementation plan (taken from Cebu study)

Philippines. The measures included in the plan
will involve or affect many stakeholders. All
these stakeholders (based on the outcomes of the
stakeholder analysis and designed participatory
planning process) should therefore be included in
some way in the implementation process in order
to guarantee a successful implementation and a

sustainable benefit of the particular measure. In
general the following roles can be distinguished:
• Responsible: the stakeholder has the first
responsibility for the implementation of the
measure but will co-operate with and/or
consult other stakeholders in this process. In

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13

Project Planning: Putting It All Together

Fig. 13.13 this is indicated by the symbol:
“●”.
• Co-operate: the stakeholder has an important
say in the implementation of the measure but
is not the first responsible and is expected to
work with other stakeholders in this matter. In
the figure this is indicated by the symbol: “○”.
• Consult: the stakeholder has an interest in the
implementation of the measure and will be
consulted by the first responsible. In certain
cases permission will be needed before the
implementation can take place. In the figure
this is indicated by the symbol: “x”.

from local (province, municipality) budgets. In
some cases private funding can be considered in
PPP (Public Private Participation) constructions.
This seems in particular attractive when there is a
good possibility for payment by the stakeholders
of the services that will be provided. Examples
where PPPs can be considered are urban public
water supply and hydropower production.
The investment plan should also address how
the recurrent costs (operation and maintenance) of
the implemented projects will be recovered.
Preferably this should be done based on fees to be
paid by the people that benefit from the project.

When
The action plan should also specify the timing of
the implementation. When will (the preparation
of) the implementation start, and when should the
implementation be finalized. This information is
needed for the overall investment plan but also
because some measures will depend on the
completion of other measures.

13.4.4.3

13.4.4.2 Financing—Investment Plan
An important, if not the most important, part of the
Action Plan is to determine how the action will be
financed. The sources of the financing will largely
depend on the type and size of the measure. As
water resources management is mainly a governmental task, most of the finances will come from
public sources. These can be from the national
budget (possibly supported by donor funds) or
Fig. 13.14 Applying SEA
(OECD 2006)

Feasibility Studies
and Environmental
Impact Assessment
A feasibility study should include a more
detailed study of the projects (measures) proposed in the plan. Commonly a feasibility study
includes some 5 areas of feasibility:
•
•
•
•
•

technical
social/environmental
political/legal
financial/ economic
operational and scheduling

A feasibility study for a good implementation
planning will often include a more detailed
assessment of the possible socioeconomic and
environmental impacts of some of the measures

13.4

Framework for Analysis and Implementation

599

that comprise the preferred strategy. There are
several types of assessment depending on the
focus of the study. As depicted in Fig. 13.14 the
most well-known are: Environmental Impact
Assessment (EIA, for infrastructure projects),
Strategic Environmental Assessment (SEA,
mainly used in policy development) and Sustainability Appraisal (SA).

13.4.4.4 Promotion
After the action plan has been established one
needs to find ways to increase the influence of
stakeholder groups that favor the implementation
of the action but lack influence; to change the
attitude of influential groups that are opposing
this action; and to use the positive attitude of
influential groups that are in favor of this action.
The results of the stakeholder analysis are used
for the identification of the stakeholder groups.
As illustrated, the matrix highlights the strategy
toward project acceptability or appreciation and
therefore smooth implementation.
To create maximum awareness, enthusiasm
and support for selected projects within the
Action Plan the selected stakeholder groups need
to be provided with the right information on the
project. Additionally, involving a selection of
stakeholders in project preparation and

implementation will assist in making them
enthusiastic about the project. To do this effectively, a mix of marketing options can be used.
Appropriate marketing options might be:
• mass one-way communication for the general
public (such as newspapers, radio, television
plus more traditional media in the more rural
areas);
• selective one-way communication for selected
stakeholders groups (direct mail, brochures
with more specific information dedicated for
the selected group); and
• personal two-way communication between
the project promoter and selected stakeholders groups (education method, outreach
method or more risky word-of-mouth
method).

13.4.4.5 Monitoring and Evaluation
An overview of the implementation framework is
given in Fig. 13.15. This implementation
framework applies for both Steps IV (Action
Planning) and V (Implementation). The actual
implementation of most of the measures will take
place by decentralized agencies of national
ministries or at local governmental level and their
related utilities, districts, and associations. Where

Action Plan

Monitoring-evaluation
- progress
- effectivity

Promotion

Technical Secretariat
Feasibility studies /
project prep.

Implementation
Implementing
partners

Fig. 13.15 Implementation framework

600

13

needed feasibility and engineering studies will be
carried out before the actual implementation
and/or construction can take place.
Above the implementation level there should
be a guidance and coordination level, e.g., a
Technical Secretariat (TS) at basin level. Periodically a monitoring report compiled by the TS
can track the progress made in implementing the
measures of the Action Plan and the effectiveness
of these measures in meeting their objectives.
Insufficient progress may lead to an adjustment
of the Action Plan. The TS may also provide
assistance to the implementing partners, e.g. the
local government agencies, as they carry out
feasibility studies. The TS should be able to
support them by providing data and possibly
other relevant information from their Management Information System (MIS).

13.5

Making It Work

The framework of analysis presented in Fig. 13.3
includes next to the five steps of analysis two
crucial blocks that play a role in several of these
steps and deserve special attention. The first one
is the stakeholder engagement in the analysis.
Involving stakeholders and making sure that their
ideas and suggestions are taken into account is an
absolute requirement to develop a consensus and
support for the ultimate plan that is to be
implemented. There is no guarantee that a consensus will be reached, however. Involving
stakeholders in each stage of the planning
framework takes extra time and money, but if
any ultimate plan is to be accepted and prove
sustainable, there is no other choice. At a minimum, any plan that is derived from this process
should be an informed one, based on inputs from
all affected stakeholders and decision makers.

13.5.1

Stakeholder Engagement

The stakeholders that should be involved in a
planning process will depend on the specific basin

Project Planning: Putting It All Together

that is being addressed. In general the stakeholders will be all people and/or organizations that:
• will be effected by the plan; and
• are needed to implement the plan.
An integrated plan and its implementation
depend to a large extend on the acceptance and
ownership of the plan by the decision makers and
stakeholders at national and basin levels. A participatory planning process is therefore indispensable for sustainable WRM. A participatory
planning process is the results of a set of steps, as
depicted in Fig. 13.16. However, the order of the
steps can vary according to the local situation
and conditions. The prerequisite for the design of
a participatory planning process is a good
stakeholder analysis. The stakeholder analysis is
a supporting planning tool that supports the
identification of stakeholders and its engagement.
Particularly, this analysis technique supports the
task of identifying and in some occasions classifying the stakeholders according to their functions, capacities, interests, concerns and needs, as
well as their dependencies (including power
relations among them).
Based on the results of the stakeholder analysis
the participatory planning process is defined. First,
it is crucial to define the levels of participation of
the various stakeholders. The level of participation
of each group of stakeholders varies depending on
the stakeholder analysis and on the maximum
level of participation that the client of the study
wants to achieve. The second step is the design of
the participatory process. This will be adapted to
the agreed levels of participation and stakeholders
involved. The design of the participatory process
needs to take into account the modeling approach
(informed decision making) so it is carried out in a
participatory manner (step 3). Finally, as illustrated in Fig. 13.16, the design of the participatory
planning process needs to consider the information and communication tools used for disseminating and communicating the information to the
various groups of stakeholders as illustrated in the
power-interest matrix of Fig. 13.17.

Making It Work

Fig. 13.16 Steps in a
stakeholder analysis and
participatory planning
process

601

Stakeholder mapping

Stakeholder analysis

13.5

Interests and needs

Dependency analysis

Stakeholder analysis
A stakeholder analysis provides a better understanding of the perceptions, concerns, roles,
interests, and needs of the stakeholders and
contributes to a better approach to the solution. It
also helps reduce the possibility of forgetting
important risks. Finally, this technique increases
the chance that the various groups of stakeholders are willing to cooperate in solving the identified problems and issues.
A good stakeholder analysis should contain at
least the following steps:

(1) Situation analysis as point of departure.
(2) Inventory of the stakeholders involved (e.g.,
primary,
secondary
and
tertiary
stakeholders).
(3) Mapping of formal relations according to
their functions and responsibilities.
(4) Inventory of interests, perceptions, and
needs.
(5) Mapping of interdependencies.
Levels of Participation
The various stakeholders are grouped into the
different levels of participation according to the

602

13

Project Planning: Putting It All Together

(a)

(b)

(c)

(d)

Fig. 13.17 Power-interest matrix

outcomes of the stakeholder analysis, as illustrated in Fig. 13.18:
• Ignorance: where a stakeholder is not aware
of what is happening;
• Awareness: where a stakeholder is aware that
something is happening;
• Informed: where a stakeholder has been
specifically provided with information and is
left to decide what to do with it. The emphasis
is on the one-way provision of information,
with no formal option for the stakeholder to
provide feedback, negotiate, or participate in
the decision-making process;

• Consultation: where a stakeholder is asked to
provide information inputs to the planning
process. Information flows are likewise
one-way, but in the opposite direction. That
is, information is extracted from stakeholders
although no commitment is given to use it;
• Discussion: at this level are fully participating and are asked to give advice and recommendations. Here information flows in both
directions between stakeholders operating
with different interests and levels of influence,
and also between these stakeholders and the
organizing team (technical team). Since
two-way interactions occur, there is room for

13.5

Making It Work

603

Fig. 13.18 Levels of participation

alternative ideas, solutions and/or strategies to
emerge;
• Co-Design: at this level stakeholders are
actively involved in problem analysis and
problem design, which fosters ownership, but
where final decision-making powers reside
with the governing agencies;
• Co-Decision-Making: here decision making
powers are shared with those participating
stakeholders, leading to their empowerment
with respect to the policy/planning decision
taken. Typically decisions in these contexts
would emerge from a process of stakeholder
negotiation.
The first levels (from Ignorance to Consultation) could be thought of as top-down
management/planning approaches toward participation, where stakeholders have little control
over the decision-making process. The final three
levels are more appropriately considered as
bottom-up approaches toward participation
where stakeholders are much more active and
have
much
more
control
over
the
decision-making process.
Design of the participatory planning
process
The design of the participatory planning process
needs to take into consideration the River Basin
planning framework and the data and modeling

tools used. Participatory planning tools and
techniques enable participants (stakeholders) to
influence development initiatives and decisions
affecting them. The tools promote sharing of
knowledge, building up commitment to the process and empower the group to develop sustainable strategies.
The participatory and informed planning
process makes use of the “Circles of Influence”
model (Fig. 13.19) that enables to structure participation to limit numbers but not the influence
of specific groups of stakeholders (Cardwell et al.
2008; Bourget 2011). Under this model trust is
developed in concentric circles; planners and
managers work to develop trust with leaders and
organizations that other stakeholders already
trust. That is, those most directly involved in
policy analysis activities (i.e., planners, managers, and modelers who do most of the actual
work; Circle A) who communicate with trusted
leaders and major stakeholder representatives at
the next level (Circle B). These stakeholders then
in turn provide a trusted link to all other interested parties, who have much less direct
involvement (Circle C). Ideally, Circle B participants would be active in professional or
issues-oriented organizations and provide links to
others whose interests they represent. Hence,
Circle C stakeholders should see their interests
represented in Circle B, and have formal

604

13

Project Planning: Putting It All Together

Fig. 13.19 Participatory
planning structure based on
circles of influence (source
Cardwell et al. 2008)

opportunities to shape the work of Circles A and
B via these representatives. The levels of
involvement of those stakeholders in Circle C
can vary from Consultation to Awareness.
A fourth circle (Circle D) includes decision
makers such as agency heads and elected officials, who have been given the authority to
accept or reject the recommendations of the
policy analysis. For a good participatory and
informed planning process it should be clearly
identified and engaged throughout the planning
process with direction and information flows
possible to and from all circles.
Other aspects to be considered for the design
of the participatory planning process are:
• Timing of stakeholder involvement. This will
be dependent on the Circles of Influence and
levels of participation.
• Stakeholder participation in the modeling
process (Participatory Modeling). Mainly
those stakeholders in the Circles A and B will
be regularly involved in some of the phases of
the modeling process. The involvement can
be concentrated in (i) early and later stages of

the modeling process, (ii) construction of the
model, (iii) some of the activities prior to
model construction, or (iv) only after the final
model has been built.
• Type of stakeholder involvement. This can be
either individually, with homogeneous
(stakeholders with similar interests and
problem perceptions) or heterogeneous
groups.
• Information and communication tools. Information dissemination (e.g. face-to-face workshops or online platforms) and communication
tools need to be adapted to the background
conditions of the various groups of stakeholders. This is particularly important for
participatory model construction and use, as
well as, for the promotion of the plan. The
selected marketing options for creating
awareness, enthusiasm and support for selected projects within the action plan by stakeholders will vary depending on the results of
the stakeholder analysis (Fig. 13.17) and
levels
of
stakeholder
involvement
(Fig. 13.18). For more information about plan
promotion see Sect. 13.4.4.3.

13.5

Making It Work

13.5.2

Using Models in a Planning
Process

13.5.2.1 Managing Modeling Projects
There are some steps that, if followed in modeling projects, can help reduce potential problems
and lead to more effective outcomes. These steps
are illustrated in Fig. 13.20. Some of the steps
illustrated in Fig. 13.20 may not be relevant in
particular modeling projects and if so, these parts
of the process can be skipped. Each of these
modeling project steps is discussed in the next
several sections.
Creating a Model Journal
One common problem of modeling projects once
they are underway occurs when one wishes to go
back over a series of simulation results to see

Fig. 13.20 The modeling project process is typically an
iterative procedure involving specific steps or tasks

605

what was changed, why a particular simulation
was made or what was learned. It is also commonly difficult if not impossible for third parties
to continue from the point at which any previous
project terminated. These problems are caused by
a lack of information on how the study was
carried out. What was the pattern of thought that
took place? Which actions and activities were
carried out? Who carried out what work and
why? What choices were made? How reliable are
the end results? These questions should be
answerable if a model journal is kept. Just like
computer-programming documentation, project
documentation is often neglected under the
pressure of time and perhaps because it is not as
interesting as running the models themselves.
Initiating the Modeling Project
Project initiation involves defining the problem
to be modeled and the objectives that are to be
accomplished. There can be major differences in
perceptions between those who need information
and those who are going to provide it. The
problem “as stated” is often not the problem “as
understood” by either the client or the modeler.
In addition, problem perceptions and modeling
objectives can change over the duration of a
modeling project.
The appropriate spatial and time scales also
need to be identified. The essential natural system
processes must be identified and described. One
should ask and answer the question of whether or
not a particular modeling approach, or even
modeling in general, is the best way to obtain the
needed information. What are the alternatives to
modeling or a particular modeling approach?
The objective of any modeling project should
be clearly understood with respect to the domain
and the problem area, the reason for using a
particular model, the questions to be answered by
the model, and the scenarios to be modeled.
Throughout the project these objective components should be checked to see if any have
changed and if they are being met.
The use of a model nearly always takes place
within a broader context. The model itself can
also be part of a larger whole, such as a network

606

of models in which many are using the outputs of
other models. These conditions may impose
constraints on the modeling project.
Proposed modeling activities may have to be
justified and agreements made where applicable.
Any client at any time may wish for some justification of the modeling project activities.
Agreement should be reached on how this justification will take place. Are intermediate reports
required, have conditions been defined that will
indicate an official completion of the modeling
project, is verification by third parties required,
and so on? It is particularly important to record
beforehand the events or times when the client
must approve the simulation results. Finally, it is
also sensible to reach agreements with respect to
quality requirements and how they are determined or defined, as well as the format, scope
and contents of modeling project outputs (data
files) and reports.
Selecting the Model
The selection of an existing model to be used in
any project, as opposed to developing a new one,
depends in part on the processes that will be
modeled (perhaps as defined by the conceptual
model), the data available and the data required
by the model. The available data should include
system observations for comparison of the model
results. They should also include estimates of the
degree of uncertainty associated with each of the
model parameters. At a minimum this might only
be estimates of the ranges of all uncertain
parameter values. At best it could include statistical distributions of them. In this step of the
process it is sufficient to know what data are
available, their quality and completeness, and
what to do about missing or outlier data.
Determining the boundaries of the model is an
essential consideration in model selection and
use. These boundaries define what is to be
included in a model and what is not. Any model
selected will contain a number of assumptions.
These assumptions should be identified and justified, and later tested.
Project-based matters such as the computers
to be used, the available time and expertise, the
modeler’s personal preferences, and the client’s

13

Project Planning: Putting It All Together

wishes or requirements may also influence model
choice. An important practical criterion is whether there is an accessible manual for operating
the model program and if help is available to
address any possible problems.
The decision to use a model, and which model
to use, is an important part of water resources
plan formulation. Even though there are no clear
rules on how to select the right model to use, a
few simple guidelines can be stated:
• Use the simplest method that will yield adequate accuracy and provide the answer to
your questions.
• Select a model that fits the problem rather
than trying to fit the problem to a model.
• Question whether increased accuracy is worth
the increased effort and increased cost of data
collection.
• Consider model and computational cost.
Today computing costs are rarely an issue
except perhaps for some groundwater management problems.
• Do not forget the assumptions underlying the
model used and do not read more significance
into the simulation results than is actually
there.
Analyzing the Model
Once a modeling approach or a particular model
has been selected, its strengths and limitations
should be assessed. The first step is to set up a plan
for testing and evaluating the model. These tests
can include mass (and energy) balance checks and
parameter sensitivity analyses (see Chap. 8). The
model can be run under extreme input data conditions to see if the results are as expected.
Once a model is tested satisfactorily, it can be
calibrated. Calibration focuses on the comparison
between model results and field observations. An
important principle is: the smaller the deviation
between the calculated model results and the
field observations, the better the model. This is
indeed the case to a certain extent, as the deviations in a perfect model are only due to measurement errors. In practice, however, a good fit
is by no means a guarantee of a good model.

13.5

Making It Work

The deviations between the model results and
the field observations can be due to a number of
factors. These include possible software errors,
inappropriate modeling assumptions such as the
(conscious) simplification of complex structures,
neglect of certain processes, errors in the mathematical description or in the numerical method
applied, inappropriate parameter values, errors in
input data and boundary conditions, and measurement errors in the field observations. To
determine whether or not a calibrated model is
“good,” it should be validated or verified. Calibrated models should be able to reproduce field
observations not used in calibration. Validation
can be carried out for calibrated models as long
as an independent data set has been kept aside for
this purpose. If all available data are used in the
calibration process in order to arrive at the best
possible results, validation will not be possible.
The decision to leave out validation is often a
justifiable one especially when data are limited.
Philosophically, it is impossible to know if a
model of a complex system is sufficiently “correct”. There is no way to prove it. [“All models
are wrong but some are useful” Box (1976).]
Experimenting with a model, by carrying out
multiple validation tests, can increase one’s
confidence in that model. After a sufficient
number of successful tests, one might be willing
to state that the model is “good enough”, based
on the modeling project requirements. The model
can then be regarded as having been validated, at
least for the ranges of input data and field
observations used in the validation.
If model predictions are to be made for situations or conditions for which the model has
been validated, one may have a degree of confidence in the reliability of those predictions. Yet
one cannot be certain. Much less confidence can
be placed on model predictions for conditions
outside the range for which the model was validated. While a model should not be used for
extrapolations as commonly applied in predictions and in scenario analyses, this is often
exactly the reason for the modeling project. What
is likely to happen given events we have not yet

607

experienced? A model’s answer to this question
should also include the uncertainties attached to
these predictions.
Using the Model
Once the model has been judged ‘good enough’,
it may be used to obtain the information desired.
One should develop a plan on how the model is
to be used, identifying the input to be used, the
time period(s) to be simulated, and the quality of
the results to be expected. Again, close communication between the client and the modeler is
essential, both in setting up this plan and
throughout its implementation, to avoid any
unnecessary misunderstandings about what
information is wanted and the assumptions on
which that information is to be based.
Before the end of this model use step, one
should determine whether all the necessary
model runs have been performed and whether
they have been performed well. Questions to ask
include:
•
•
•
•
•
•
•
•
•
•

Did the model fulfill its purpose?
Are the results valid?
Are the quality requirements met?
Was the discretization of space and time
chosen well?
Was the choice of the model restrictions
correct?
Were the correct model and/or model program chosen?
Was the numerical approach appropriate?
Was the implementation performed correctly?
Are the sensitive parameters (and other factors) clearly identified?
Was an uncertainty analysis performed?

Some of these questions may not apply, but if
any of the answers to these questions is no, then
the situation should be corrected. If it cannot be
corrected, then there should be a good reason for
this.
Interpreting Model Results
Interpreting the information resulting from simulation models is a crucial step in a modeling

608

project, especially in situations in which the client may only be interested in those results and
not the way they were obtained. The model
results can be compared to those of other similar
studies. Any unanticipated results should be
discussed and explained. The results should be
judged with respect to the modeling project
objectives.
The results of any water resources modeling
project typically include large files of time series
data. Only the most dedicated of clients will want
to read those files, so the data must be presented in
a more concise form. Statistical summaries should
explicitly include any restrictions and uncertainties in the results. They should identify any gaps in
the domain knowledge, thus generating new
research questions or identifying the need for
more field observations and measurements.
Reporting Model Results
Although the results of a model should not be the
sole basis for policy decisions, modelers have a
responsibility to translate their model results into
policy recommendations. Policymakers, managers, and indeed the participating stakeholders
often want simple, clear and unambiguous
answers to complex questions. The executive
summary of a report will typically omit much of
the scientifically justified discussion in its main
body regarding, say, the uncertainties associated
with some of the data. This executive summary is
often the only part read by those responsible for
making decisions. Therefore, the conclusions of
the model study must not only be scientifically
correct and complete, but also concisely formulated, free of jargon, and fully understandable by
managers and policymakers. The report should
provide a clear indication of the validity,
usability and any restrictions of the model
results. The use of visual aids, such as graphs and
GIS, can be very helpful.
The final report should also include sufficient
detail to allow others to reproduce the model
study (including its results) and/or to proceed
from the point where this study ended.

13

Project Planning: Putting It All Together

13.5.2.2 Evaluating Modeling Success
There are a number of ways one can judge the
extent of success (or failure) in applying models
and performing analyses in practice. Goeller
(1988) suggested three measures as a basis for
judging success:
1. How the analysis was performed and presented (analysis success).
2. How it was used or implemented in the
planning and management processes (application success).
3. How the information derived from models
and their application affected the system
design or operation and the lives of those who
use the system (outcome success).
It is often hard to judge the extent to which
particular models, methods and styles of presentation are appropriate for the problem being
addressed, the resources and time available for
the study, and the institutional environment of
the client. Review panels and publishing in
peer-review journals are two ways of judging. No
model or method is without its limitations. Two
other obvious indications are the feelings that
analysts have about their own work and, very
importantly, the opinions the clients have about
the analysts’ work. Client satisfaction may not be
an appropriate indicator if, for example, the clients are unhappy only because they are learning
something they do not want to accept. Producing
results primarily to reinforce a client’s prior
position or opinions might result in client satisfaction, but, most would agree, this is not an
appropriate goal of modeling.
Application or implementation success
implies that the methods and/or results developed
in a study were seriously considered by those
involved in the planning and management process. One should not, it seems to us, judge success or failure on the basis of whether or not any
of the model results (the computer “printouts”)
were directly implemented. What one hopes for
is that the information and understanding

13.5

Making It Work

resulting from model application helped define
the important issues and identify possible solutions and their impacts. Did the modelling help
influence the debate among stakeholders and
decision-makers about what decisions to make or
actions to take? The extent to which this occurs is
the extent to which a modeling study will have
achieved application or implementation success.
Outcome success is based on what happens to
the problem situation once a decision largely
influenced by the results of modeling has been
made and implemented. The extent to which the
information and understanding resulting from
modeling helped solve the problems or resolve
the issues, if it can be determined, is a measure of
the extent of outcome success. It is clear that
success in terms of the second or third criteria
will depend heavily on the success of the preceding one(s). Modeling applications may be
judged successful in terms of the first two measures but, perhaps because of unpredicted events,
the problems being addressed may have become
worse rather than improved, or while those particular problems were eliminated, their elimination may have caused other severe problems. All
of us can think of examples where this has
happened.
For example, any river restoration project
involving the removal of engineering infrastructure is a clear indication of changing objectives
or new knowledge. Who knows whether or not a
broader systems study might have helped earlier
planners, managers, and decision-makers foresee
the adverse ecological consequences of converting rivers to canals, and whether or not anyone
will care. Hindsight is always clearer than foresight. Some of what takes place in the world is
completely unpredictable. We can be surprised
now and then. Given this, it is not clear whether
we should hold modelers or analysts, or even
planners or managers, completely responsible for
any lack of “outcome success” if unforeseen
events that changed goals, or priorities or
understanding did indeed take place.

609

Problem situations and criteria for judging the
extent of success will change over time, of
course. By the time one can evaluate the results,
the system itself may have changed enough for
the outcome to be quite different than what was
predicted in the analysis. Monitoring the performance of any decision, whether or not based on a
successfully analyzed and implemented modeling effort, is often neglected. But monitoring is
very important if changes in system design,
management and operation are to be made to
adapt to changing and unforeseen conditions.
If the models, data, computer programs, documentation and know-how are successfully
maintained, updated, and transferred to and used
by the client institutions, there is a good chance
that this methodology will be able to provide
useful information relevant to the changes that
are needed in system design, management, or
operation. Until relatively recently, the successful transfer of models and their supporting technology has involved a considerable commitment
of time and money for both the analysts and the
potential users of the tools and techniques. It has
been a slow process. Developments in interactive
computer-based data-driven decision support
systems that provide a more easily understood
human–model–data–computer interface have
substantially facilitated this technology transfer
process, particularly among model users. These
technology developments have had, and we think
will continue to have, a major impact on the state
of the practice in using models in support of
water resources planning and management
activities.

13.6

Conclusions

The effectiveness of strategies for dealing with
issues of water quantity and quality, and their
variability, has a major impact on the well-being of
living species, and even the survival of some. How
well water is managed also impacts the functioning and resilience of ecosystems, the vitality of

610

societies, and the strength and growth of economies. Fortunately we humans can determine
which water resources development and management strategy will work best in a given situation, not only for the immediate future but in the
long-run as well. And if conditions change, our
strategies can adapt. To accomplish this we need
to identify and evaluate the effectiveness of the
water resources development and management
alternatives available to us in an economic,
hydrologic and sociopolitical environment that
seems to be a constantly changing. We can do this
through the use of various models, developing
preferred strategies based in part on their results,
and informed by the concerns and objectives of
stakeholders and the decision making institutions.
This book has focused on ways of developing
and using various optimization and simulation
modeling methods for analyzing and evaluating
water resource development and management
alternatives. This final chapter has presented
some guidelines for carrying out water resources
planning projects, including its modeling components. Such projects are typically very complex and challenging.
Water management planning projects must
address a complex and interconnected web of
science, engineered infrastructure, legal regulations governing water use, societal expectations,
and institutional structures and authorities that
have evolved over time. Much of the current
complexity that exists in various regions of the
world has developed over time in response to
changing interests and objectives of water users
and environmental considerations. Although the
impacts of changes in the climate on water supplies and demands are generally recognized,

13

Project Planning: Putting It All Together

these ongoing changes as well as the linkages
between environmental and societal factors in
specific basins and regions all lead to major
uncertainties in the future.
The guidelines discussed in this chapter have
been developed and used by Dutch experts in
Deltares to assess water resources systems and to
develop plans and strategies for managing them.
Deltares has been actively involved in numerous
water resources planning and management projects throughout the world. The approach
described in this chapter illustrates how these
projects are conducted, and the major factors that
are considered while conducting them. The
effects and impacts of some of their projects have
been relatively local and required consideration
of only a few sectors of the economy. Other,
more comprehensive projects have had national
or international impacts, and led to transboundary (international) compacts.
Clearly each water resources system is unique
with respect to its management issues and problems and its institutional environment. Project
planning and analysis approaches must adapt to
these situations. Hence, each project will differ,
and will no doubt need to deviate from the suggested guidelines presented in this chapter. Other
approaches are possible and may be equally effective. What remains important in all cases is
the establishment of a comprehensive, systematic
process of planning and analysis together with
constant communication among planners,
decision-makers and the interested and affected
public. The end result should be an improved,
more sustainable, and equitable water resources
development plan and management policy,
appropriate for the region and its people.

13.6

Conclusions

611

Box 13.4. Example 1: Objectives and criteria adopted in West Java WRM study
Objectives

Evaluation criteria

Socio-economic objectives and criteria
1. Improve employment (–)

Increase of employment by WRM strategies
– Number of permanent jobs (#)
– Number of temp. jobs (mn-year)

2. Increase income of people
– Improve income position of farmers
– Improve equity in income distribution

∙ Farmer net income (Rp/year)
∙ Difference in benefits of WRM strategies per capita
between:
– Kabupatens (%)
– Urban/rural areas (%)
– Income groups (%)

3. Increase the non-oil export production
(shrimps, tea, and rubber)

– Export value (Rp/year)

4. Support economic development in an economically
efficient way

–
–
–
–
–
–
–
–
–
–
–

Total annual. benefits (Rp/year)
Total annualized costs (Rp/year)
B/C ratio (–)
IRR (%)
NPV (Rp/year)
Total capital required (Rp)
Foreign currency required (%)
Total construction costs (Rp)
Total O&M costs (Rp)
Sectoral value added (Rp/year)
GRP (Rp/year)

User-related (sectoral) objectives and criteria
1. Increase agricultural production
(3% per year)

– Padi (ton/year)
– Palawija (ton/year)
– Export value of crops (or import substitution)
(Rp/year)
– Unit costs water supply (Rp/m3)
– % failure meeting demand (%)

2. Increase power production (–)

–
–
–
–
–

3. Increase fish production (–)

– Fish produced (ton/year)
– Fish pond area (ha)
– Export value (Rp/year)

4. Support industrial development
∙ Water supply for industry (full supply)
∙ Provision of opportunity for discharge of waste water

–
–
–
–
–

Installed capacity (MW)
Power production (GWh/year)
Failure meeting firm power (%)
Price of power prod. (Rp/Kwh)
Energy production value (Rp)

Amount of supply (m3/s)
Cost of water supply (Rp/year)
Unit costs water supply (Rp/m3)
% failure meeting demand (%)
Cost to maintain water quality standards (Rp/year)

5. Enhance water-related recreation
Environmental and public health related objectives and criteria
1. Improve public health
∙ Improve drinking water supply
urban: BNA, IKK and major city programs:
60 l/cap/day, serving 70 %
rural: 55 %
∙ improve flushing
(1 L/s/ha in urban area)

–
–
–
–
–
–
–

Supply (1/day/ capital)
% of people connected (Rp/m3)
Price of drinking water (%)
% failure meeting demand (%)
Volume of flushing water (m3/s)
Unit costs (Rp/m3)
% failure meeting demand (%)
(continued)

612

13

Project Planning: Putting It All Together

Objectives

Evaluation criteria

2. Improve/conserve natural resources and environment
∙ Erosion and sedimentation control
(erosion <1 mm/year)
∙ Conservation of nature
∙ Water quality

–
–
–
–
–
–
–
–

3. Provide flood protection
(return period: depending on value of endangered area)

- return period [years]
- flood alleviation benefits (reduced damage)
[Rp/year]
- flood control cost [Rp/year]
- number of people in endangered areas [#]
- flooded area [ha]

Area severely eroding (ha)
Erosion (mm/year)
Sediment yield (tons/year)
Reafforestated area (ha)
Replanted area (ha)
Terraced area (ha)
% external wood supply to total wood demand (%)
Concentration water quality parameters (ppm)

Planning and implementation related objectives and criteria
1. Take care of maximum agreement with existing policies
in other fields of planning (e.g. economic regional planning)

– Deviations from/conflicts with existing policies

2. Maximize flexibility of proposed strategy

– Degree to which strategy can be adjusted to changes
in demands, standards, technological innovations

3. Maximize reliability of proposed strategy

– Degree of certainty with which proposed strategy
will meet the realization of objectives

4. Provide sufficient acceptance of proposed strategy by
public, interest groups and executing authorities

– Degree of acceptance by parties involved

5. Takes care of maximum agreement of proposed strategy
with existing competence and responsibilities of agencies
concerned

– Deviations from/conflicts with existing competence
and responsibilities

a

Kabupaten = Indonesian administrative unit
Padi = Rice crop
c
Paliwija = Non-rice crop
d
Rp = Rupiah
b

Box 13.5 Example 2: Score-card Egyptian National Water Resources Plan study
Unit

1997
base

2017 reference
case

Strategy facing the
challenge

Population

Million

59.3

83.1

83.1

Urbanization

Ratio

0.44

0.48

0.48

GDP at economic growth of 6%

Billion LE

246

789

789

General (middle scenario)

Economic development objectives
Agriculture: irrigation area

Mfeddan

7.985

11.026

10.876

Gross production value

Billion LE

34.46

35.76

38.50

Crop intensity

Ratio

2.1

1.5

1.7

Net value production per feddan

LE/feddan

2812

2075

2153

0.64

0.66

0.60

0.09

0.12

0.20

Net value production per unit of water

LE/m

Export/import value

Ratio

3

(continued)

13.6

Conclusions

613
Unit

1997
base

2017 reference
case

Strategy facing the
challenge

Industry: costs polluted intake water

LE/m3

0.65–
1.10

0.65–1.10

2.00

Wastewater treatment costs

LE/m3

0.22–
0.50

0.22–0.50

1.00

Fishery: production (index 100 in
1997)

Index

100

86

95

Tourism: navigation bottlenecks

Index

100

114

0

Create living space in desert areas

% of tot.
pop

1.5%

23%

22%

Employment and income
Employment in agriculture

M pers.
year

5.01

6.24

7.30

Employment in industry

M pers.
year

2.18

4.99

4.99

Social objectives

LE/year

5362

4629

4309

Drinking water supply
Coverage

Average income farmers

Percentage

97.3%

100%

100%

Sanitation
Coverage

Percentage

28%

60%

60%

Equity
Equity water distribution in
agriculture

−, 0, +

0

+

+

Self-sufficiency in food: cereals

Percentage

73%

53%

46%

BCM

55.8

55.5

55.5

Meeting water needs
Water resources development
Available Nile water

BCM

0.71

3.96

3.96

Water use efficiency Nile system
Outflow to sinks from Nile system

Abstraction deep groundwater

BCM

16.3

17.6

12.5

Overall water use efficiency Nile
system

Percentage

70%

67%

77%

Water in agriculture
Supply/demand ratio (1997 assumed
1.0)

Ratio

1.00

0.80

0.92

Water availability per feddan Nile
system

m3/
feddan/yr

4495

3285

3866

Public water supply
UFW losses

Percentage

34%

34%

25%

Ratio

0.67

0.76

1.00

Index

100

121

110

Supply/demand ratio
Health and environment
Pollution and health
E. coli standard violation
(1997 = 100)

(continued)

614

Water quality shallow groundwater
Ecology and sustainability
Sustainability: use of deep groundw.

13

Project Planning: Putting It All Together

Unit

1997
base

2017 reference
case

Strategy facing the
challenge

−, 0, +

0

−

−

Abstr/pot

0.15

1.00

1.00

Condition Bardawil (Ramsar site)

−, 0, +

+

−

+

Condition coastal lakes

−, 0, +

0

−

0

a

UFW = Unaccounted for water (the water that is lost in the system)
feddan = 0.42 ha
c
LE = Egyptian pound
b

References
Bourget, L. (Ed.). (2011). Converging waters: Integrating
collaborative modeling with participatory processes
to make water resources decisions. Institute for Water
Resources, U.S. Army Corps of Engineers.
Box, G. (1976). Science and statistics. Journal of the
American Statistical Association, 71, 791–799. doi:10.
1080/01621459.1976.10480949.
Cardwell, H., Langsdale, S., & Stephenson, K. (2008).
The shared vision planning primer: How to incorporate computer aided dispute resolution in water
resources planning. Alexandria, VA: Institute for
Water Resources. IWR Report 08-R-02 (2008).
Goeller, B. F. (1988). A framework for evaluating success
in systems analysis. Report P-7454. Santa Monica:
Rand Corporation.
GWP. (2000). Integrated water management, TEC Background Paper no. 4. Stockholm: GWP.
Haasnoot, M., Van Deursen, W. P. A., Guillaume,
J. H. A., Kwakkel, J. H., Van Beek, E., & Middelkoop, H. (2014). Fit for purpose? Building and
evaluating a fast, integrated model for exploring water
policy pathways. Environmental Modelling & Software, 60, 99–120. ISSN 1364-8152.
Haasnoot, M., Kwakkel, J. H., Walker, W. E., & Maat, J. T.
(2013). Dynamic adaptive policy pathways: A method
for crafting robust decisions for a deeply uncertain
world. Global Environmental Change, 23(2), 485–498.
OECD. (2006). Applying strategic environmental assessment
—Good practice guidance for development co-operation.
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Ray, P. A., & Brown, C. M. (2015). Confronting climate
uncertainty in water resources planning and project

design, the decision tree framework. Washington,
United States: World Bank.

Additional References
(Further Reading)
ADB, GIWP, UNESCO, WWF. (2013). River basin
planning, principles, procedures and approaches for
strategic basin planning.
Asian Water Development Outlook. (2013). Measuring
water security in Asia and the Pacific.
Bousset, J.-P., Macombe, C., & Taverne, M. (2005).
Participatory methods, guidelines and good practice
guidance to be applied throughout the project to
enhance problem definition, co-learning, synthesis
and dissemination. SEAMLESS.
Common Implementation Strategy for the Water Framework Directive. (2000/60/EC). Guidance document 8 on
public participation in relation to the water framework
directive. http://ec.europa.eu/environment/water/waterframework/facts_figures/guidance_docs_en.htm
GWP. (2005). Integrated water resources management
plans—Training manual and operational guide.
Stockholm: GWP.
GWP Toolbox www.gwp.org/ToolBox/
GWP, INBO. (2009). A handbook for IWRM in basins.
Stockholm/Paris: GWP/INBO.
NSMC. (2010). What is social marketing? www.thensmc.
com/content/what-social-marketing-1
UNESCO-IHP, NARBO, WWAP. (2009). IWRM guidelines at River Basin level. http://www.unesco.org/water/
Voinov, A., & Bousquet, F. (2010). Modelling with
stakeholders. Environmental Modelling and Software,
25(11), 1268–1281.

References

Open Access This chapter is distributed under the terms
of the Creative Commons Attribution-NonCommercial
4.0 International License (http://creativecommons.org/
licenses/by-nc/4.0/), which permits any noncommercial
use, duplication, adaptation, distribution and reproduction
in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a
link to the Creative Commons license and indicate if
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chapter are included in the work's Creative Commons
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such material is not included in the work’s Creative
Commons license and the respective action is not permitted by statutory regulation, users will need to obtain
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reproduce the material.

Index

A
Adaptive management, 12, 40, 43, 49, 88, 334, 410, 458,
459, 568, 578, 591, 594
Advective transport, 425, 426, 429
Aeration, 537
Agriculture, 12, 37, 41, 138, 469, 511, 567, 568, 570,
571, 574, 575, 585, 613
Algae eutrophication modelling, 438
Allocations, 30, 75, 83, 85, 97–99, 101, 102, 105–111,
113–115, 155, 166, 167, 169, 172, 175, 188–190,
197, 207, 209, 210, 305–307, 311–313, 315, 319,
329, 386, 387, 389, 458, 469, 505, 518, 583
Alternatives, 6, 34, 35, 38, 40, 45, 60, 69, 73, 76, 79, 81,
82, 88, 94, 96, 151, 163, 233, 270, 327, 346, 349,
366, 376–379, 394, 397, 398, 402, 414, 465, 490,
530, 596, 610
Anaerobic, 438, 441, 555, 557
Annual value, 96, 137, 313
Aquatic systems, 428
Aquifer, 2, 5, 6, 31, 38, 67, 71, 151, 258, 379, 568, 574
Artificial neural networks, 55, 63, 179–181, 183, 207
Autocorrelations, 251, 254, 256, 257, 260, 270, 299, 303
Autoregressive-moving average (ARMA) models, 269,
270
Average values, 215, 367, 425, 439
B
Benefit-cost analyses ratio, 165, 583
Benefits, 1, 16, 20, 24, 30, 33, 35, 37, 38, 41, 62, 75, 83,
85, 90, 94–96, 98, 99, 102, 107–109, 111, 114,
137, 155, 156, 158, 169, 171, 196, 210, 259, 288,
293–295, 307, 328, 373, 380, 383, 385–387, 393,
474, 493, 504, 505, 508, 512–514, 519, 520, 522,
558, 592
Bias, 183, 186, 211, 220, 222, 229, 256, 290, 297
Biochemical oxygen demand (BOD), 433, 436, 462, 537,
555
Boundary conditions, 104, 336, 338, 572, 573, 576, 585,
595, 607
Box-Jenkins models, 269
Budget, 58, 385, 512, 579, 580, 583, 587, 598

C
Calibration, 55, 80, 144, 145, 147, 179–181, 336–338,
340–342, 354, 360, 370, 425, 473, 536, 543, 559,
606
Capacity expansion, 75, 115, 116, 155, 174, 512, 514,
515
Capital, 24, 74, 95, 172, 216, 292, 387, 388, 559
Capital recovery factor, 96
Case studies, 2
Censored data, 245
Central limit theorem, 357
Chance constraints, 306, 327, 329
Climate change, 2, 527, 589, 590, 594, 595
Coasts beaches
management, 2, 5, 40, 41, 43, 207, 425
zones, 2, 40, 43, 580, 589
Coefficient of skewness, 220, 222, 223, 225, 235,
237–240, 244, 263, 290
Coefficient of variation, 220, 224, 225, 236, 240, 241,
248, 256, 290
Coliform bacteria, 537
Collective goods, 382
Compensation criterion, 382
Competitive conditions, 381
Concave functions, 98, 157, 160, 161
Concentration gradient, 426, 427
Confidence intervals, 231, 233, 277, 337
Constituents conservative
first-order, 433
Constraint method, 395–397, 412, 415
Constraints, 71, 75, 79, 85, 102–105, 121, 129, 138, 142,
149, 151, 158, 163, 176, 211, 243, 307, 317, 327,
339, 378, 389, 402, 464, 477, 482–484, 486, 506,
508, 511, 515, 525, 559, 576, 586, 588, 606
Continuous variables, 339, 353
Convex functions, 101, 159–161
Correlation, 233, 234, 248, 251, 255, 260–264, 266–270,
289, 292, 294, 298, 299, 302, 303, 308, 342,
357–359, 555
Cost, 2, 10, 31, 34, 37, 41, 73, 76–79, 94–96, 116, 117,
119, 120, 146, 148–150, 153, 155, 156, 165, 166,
168, 169, 172–174, 177, 196, 211, 292, 293, 338,

© The Editor(s) (if applicable) and The Author(s) 2017
D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,
DOI 10.1007/978-3-319-44234-1

617

618
340, 365, 373, 377, 379, 382, 384, 385, 411, 419,
458, 469, 494, 500, 510, 512, 514, 517, 522, 554,
559, 569, 575, 589, 596, 606
Cost-effectiveness, 380, 490
Count, 61, 308, 397, 420
Covariance, 193, 261, 263, 265, 268, 281, 298, 299
Critical value, 214, 231, 233, 234
Crop, 1, 5, 172–176, 297, 388, 421, 524–526, 571
Cross-correlation, 248, 267, 269
D
Data
collection, 39, 78, 151, 331, 338, 343, 458, 576, 587,
591, 606
management, 173
mining, 63
Dead storage, 474, 494, 510, 517, 518, 523
Decision-making, 35, 51, 63, 64, 69, 74, 79, 94, 108, 117,
120, 123, 151, 367, 399, 410, 422, 583, 602, 603
Decision support systems
uncertainty, 181
variables, 181
Deficit value, 175, 278, 292, 296, 462
Degrees of freedom, 269
Demand
agricultural, 12, 504
function, 101, 167, 170, 383–385
industrial, 1, 20, 31, 37, 267, 418, 504, 574, 585
municipal, 1, 2, 34, 37, 116, 267, 293, 469, 504, 582
Depression storage, 541, 543, 544
Depth, 10, 12, 33, 37, 121, 215, 218, 220, 226, 245, 257,
261, 282, 336, 339, 347, 349, 350, 389, 390, 402,
404, 405, 432, 447, 450, 452, 478, 499, 505, 523,
532, 540, 541, 543, 547, 570, 575
Design, 2, 25, 28, 39, 42, 51, 53, 57, 59, 63, 73, 75, 77,
78, 81, 93, 94, 138, 144, 166, 172, 183, 219, 258,
270, 280, 287, 301, 331, 334, 353, 366, 374, 375,
377, 390, 429, 431, 469, 470, 479, 494, 500, 504,
511, 515, 519, 522, 528, 536, 539, 541, 559, 565,
574, 591, 592, 595, 596, 600, 603, 604
Deterministic equivalent, 307, 327, 329
Deterministic modeling, 137, 213, 301, 311, 315, 317,
327, 341
Differential equations, 105, 166, 452, 456, 462, 463
Differential persistence, 264
Dimensionality, 115
Disaggregation, 265, 267–269, 298
Discount factors, 95, 168, 519
Dispersion
bulk, 1, 445, 536
Dispersive transport, 426
Dissolved oxygen, 54, 379, 389, 414, 418, 420, 422, 433,
435, 436, 438, 440, 449, 462, 463, 465, 537,
553–555
Distribution systems, 187, 527–529, 531, 536, 559, 561,
565
Diversions, 5, 97, 312, 469–471, 504, 506
Dominance, 394, 428, 555

Index
Droughts
El Niño and La Niña
indices, 259
management, 490
triggers, 31, 579
virtual exercises
Dynamic expansion, 512
Dynamic programming
backward-moving, 108, 109, 111, 112, 118, 120, 169,
312
deterministic, 312, 314
dimensionality, 115
forward-moving, 112, 113, 117, 119, 168
network, 105, 108, 112, 115, 169
stochastic, 315
E
Ecology
ecological criteria, 389
Economics
criteria, 165, 380
economies of scale, 117, 174
long-run benefits, 386, 388
loss functions, 387, 388, 519
short-run benefits, 386, 387
Ecosystems, 1, 2, 5, 12, 18, 30, 32, 36–38, 43, 56, 59,
281, 332, 378, 389, 392, 417–419, 423, 449, 457,
459, 469, 502, 527, 558, 567, 568, 570, 571, 581,
585, 609
Effective, 32, 35–37, 39, 41, 45, 54, 60, 93, 97, 121, 165,
169, 193, 248, 282, 327, 367, 368, 376, 379, 418,
422, 430, 458, 489, 503, 515, 530, 554, 558, 572,
576, 579, 605, 610
Environmental criteria
impacts, 389
Epilimnion, 450–452
Equity, 38, 204–206, 392, 393, 430, 611, 613
Equivalent, 22, 95, 96, 102, 159, 166, 176, 197, 218, 235,
240, 248, 264, 266, 293, 318, 368, 369, 381, 400,
427, 475, 476, 500, 544
Error, 79, 81, 105, 122, 144, 152, 156, 184, 188, 222,
255, 257, 286, 290, 293, 294, 299, 332–334,
336–338, 344, 345, 347, 349, 350, 352, 354, 356,
357, 360, 377, 421, 453, 455, 459, 485, 559, 606
Estimation
estimators, 219, 220, 222–224, 227, 229, 233, 235,
239, 243, 245–248, 250
Eutrophication, 1, 14, 57, 424, 437, 439, 440, 447, 555
Evaporation, 22, 123, 213, 226, 258, 282, 302, 373, 427,
432, 472–474, 478–480, 485, 486, 517–519, 523,
541, 570
Excess value, 154, 155
Expansion, 116, 118–120, 137, 154, 155, 168, 174, 289,
292, 293, 511, 512, 536
Expectations, 2, 45, 56, 218, 266, 268, 610
Expected value, 213, 214, 218, 221, 222, 256, 257, 259,
281, 287, 291, 303, 311, 315, 325, 350, 367, 374,
409, 481

Index
F
Failure, 1, 38, 39, 53, 59, 61, 215, 259, 275, 277, 278,
280, 409, 481, 485, 495–497, 499–501, 524, 608,
611
Feasible, 20, 41, 59, 79, 81, 83, 106, 107, 116, 118, 123,
127, 140, 142, 149, 150, 163, 188–190, 243, 274,
313, 315, 318, 377, 378, 381, 393, 394, 396, 398,
401, 410, 420, 515, 521
Finite difference, 452, 463
Firm yield, 481
First-order, 251, 261, 289, 309, 314, 345, 350–352, 370,
428, 433, 434, 436, 441, 445, 452, 457, 489, 523
Fixed costs, 117, 152, 153, 156, 159, 174
Flood
control reservoir, 491, 493
damage, 31, 37, 393, 470, 474, 490–495, 499–501,
517, 519, 522, 589
flood dikes and levees, 496, 497
flood storage capacity, 126, 210, 474, 488, 490,
492–494, 518–520, 522
management, 19
protection, 12, 16, 31, 74, 123, 287, 375, 383, 405,
490, 494, 495, 500, 501, 519, 612
return period, 492, 495, 500, 519
risk, 41, 250, 489, 490, 495, 500, 501, 589
storage, 75, 126, 209, 210, 387, 474, 487, 488, 490,
492–494, 500, 510, 518–520, 522
Flow augmentation, 377, 412, 465, 504, 505
Fractional factorial design, 353
Fuzzy models
optimization, 196, 204
sets, 196
G
Generalized likelihood estimates, 359
Genetic algorithms, 180, 181, 187, 189–193, 207, 210,
473
Genetic programming, 181, 187, 193, 195, 207
Global optimum, 104, 190
Goal attainment, 400, 401, 412, 414
Goal programming, 401, 412, 414
Goodness-of-fit, 224, 233, 234, 360
Groundwater
flow, 37, 260
interactions, 19, 207
management, 34, 606
modeling, 207
Growth, 33, 54, 248, 292, 390, 418, 420, 422, 427, 433,
439–441, 446, 447, 455, 463, 535, 553, 560, 573,
574, 585, 590, 594, 610
H
Hazen plotting position, 291
Heat, 432, 463, 543
Heavy metals, 32, 418, 441, 444, 449, 555, 557
Hill-climbing methods, 99, 170

619
Hydroelectric power, 3, 24, 37, 470, 474, 502, 504
Hydrological models
groundwater, 37, 207, 489
parameters, 75, 81, 207, 213, 336, 341, 470, 547
processes, 213, 267
surface water, 267, 489
Hypolimnion, 450–452
Hypothesis, 233, 296, 366, 550
I
Impervious, 421, 543, 544, 564
Incremental flow, 91, 99, 455, 472, 510, 521
Index-flood method, 247, 248
Indifference analysis, 369, 399, 415
Inequality constraints, 104
Infiltration, 373, 472, 527, 539, 543, 545–547, 549
Inflation, 95, 251
Interest, 2, 7, 16, 23, 30, 37, 38, 40, 44, 55, 63, 66, 67, 76,
84, 95, 96, 117, 165, 173, 226, 230, 243, 247, 265,
289, 333, 344, 351, 370, 376, 386, 397, 402, 405,
422, 456, 471, 507, 508, 514, 552, 553, 560, 575,
576, 579, 583, 591, 598, 600, 601, 603, 604, 610,
612
Internal rate of return, 165
Interquartile range, 219, 275, 287
Investments, 19, 38, 73, 96, 143, 166, 343, 378, 388, 580,
583, 587, 594, 596
Irreversibility, 368
Irrigation, 1–3, 5, 19, 37, 88, 175, 209, 272, 287, 380,
381, 383, 393, 412, 418, 464, 489, 504, 515, 524,
571, 575, 582, 612
J
Joules, 503
K
Kilowatts, 503, 504
Knowledge uncertainty, 334, 335, 337, 457
Kolmogorov–Smirnov, 231, 233, 234, 291
L
Lagrange multipliers, 97, 99, 103–105, 108, 110, 166
Lake
outflow, 33, 88–90, 211, 450, 441, 472, 473, 485, 510,
544
quality, 352, 425, 449
recreation, 1, 18, 24, 30, 34, 120, 123, 381, 382, 405,
493, 504, 505, 506
Land, 1, 3, 5, 8, 14, 16, 22, 25, 33, 37, 38, 41, 59, 74, 172,
176, 250, 341, 385, 405, 421, 469, 490, 500, 525,
527, 547, 550, 564
Latin hypercube sampling, 356, 360, 361, 374
Level, 1, 10, 16, 19, 20, 31, 39, 42, 45, 67, 69, 93, 139,
149, 184, 209, 214, 215, 251, 287, 294, 338, 343,

620
345, 353, 354, 356, 363, 365, 374, 381, 388, 405,
417, 421, 459, 492, 503, 522, 570, 576, 583, 585,
590, 596, 600, 602, 604
Lexicography, 398
Light, 41, 153, 334, 446, 449, 530, 539, 553, 557
Linear programming
linearizations, 151, 152
piecewise linearization, 154, 158
Loading, 1, 12, 54, 57, 342, 347, 418, 419, 422, 424, 428,
432, 450, 539, 549, 550, 552, 559
Local optima, 104
Log-likelihood, 227, 243
Long-run, 42, 386–388, 519, 610
Loss, 5, 24, 28, 32, 292, 353, 387, 401, 445, 472, 474,
478, 479, 486, 517, 519, 523, 533, 534, 536, 541,
543, 553, 569
M
Management
adaptive, 12, 40, 43, 49, 88, 334, 410, 458, 459, 568,
578, 591, 594
approaches, 34, 41, 43
models, 39, 53, 60, 145–147, 149, 150, 155, 206, 395,
421, 429, 457, 459, 471, 504, 545
Marginal, 95, 96, 98, 99, 103, 104, 126, 149, 150, 166,
211, 218, 261, 269, 317, 382, 385, 395, 397
Market prices, 381–384, 388
Markov
chains, 215, 229, 250, 251, 254, 265, 295, 309, 316,
317, 523
processes, 250, 251, 261, 263, 266, 308, 314
Mass diagram, 474–476, 478, 517
Mass transport, 427
Maximum likelihood estimates, 228, 236, 237, 239, 290
Mean, 1, 31, 37, 42, 75, 86, 123, 138, 140, 175, 203, 214,
219, 220, 222, 224, 228, 236, 241, 246, 249, 251,
255, 261, 262, 264, 269, 277, 278, 289, 294, 303,
308, 329, 347, 352, 357, 366, 404, 407, 408, 430,
472, 484, 505, 579, 593, 595, 606
Median, 213, 214, 219, 235, 236, 247, 275, 278, 287,
292, 296, 363, 366
Method of moments, 228, 236, 238, 243
Michaelis–Menten kinetics, 463
Mineralization, 434
Model
adequacy, 230, 466
calibration, 55, 80, 144, 179, 180, 337, 338, 340, 341,
343, 354, 360, 370, 536, 559
data, 338, 423
development, 40, 59, 65, 66, 70, 74, 75, 80, 93, 94,
151, 194, 344, 422, 457, 508
dynamic, 68, 452, 470, 512, 515, 560
errors, 269, 338, 352
fuzzy, 204
journal, 605
linear, 138, 143, 153, 156, 173, 226, 268, 328, 357

Index
optimization, 51, 77, 79, 81, 84, 85, 93, 94, 97, 111,
116, 117, 137, 138, 144, 146, 163, 175, 196, 199,
201, 204, 207, 250, 270, 307, 311, 314, 327, 394,
430, 450, 470, 476, 479, 481, 491, 504, 511, 517,
519, 522, 523, 528, 559, 565, 583, 591
output, 51, 59, 62, 65, 67, 179, 185, 187, 301,
331–333, 334, 336, 337, 339, 340, 342, 343, 345,
347, 352, 354, 367, 369, 373, 455
selection criteria, 421
shared-vision, 68
simulation, 40, 63, 73–75, 81, 85, 91, 93, 105, 138,
179, 187, 215, 258, 259, 270, 273, 282, 301, 304,
311, 325, 327, 341–343, 345, 352–354, 366, 392,
402, 452, 453, 491, 517, 519, 523, 528, 529, 541,
545–547, 552, 558, 560, 561, 588, 607
steady-state, 428, 560
structure, 51, 75, 194, 331, 334–338, 340–343, 349,
373, 587
success, 53, 61, 607–608
verification, 55, 81, 425, 606
Moments
L-moments, 224, 225, 228, 242, 243
Monitoring network
plan, 12
sampling
Monte Carlo
sampling, 354, 355, 370, 374
simulation, 304, 306, 327, 329, 353, 357, 359, 360
Multi-criteria analyses
constraint method, 395, 396, 397, 415
dominance, 394
goal-attainment, 400, 401, 414
goal-programming, 401, 412, 414
indifference analysis, 369, 399, 415
interactive methods, 397, 402
lexicography, 398
satisficing, 398, 399
weighting method, 395–397, 402, 415
Multiple criteria or objectives, 410
Multiple purposes, 30, 301, 411, 489, 504
Multiple yields, 480, 484, 485
Multiplier, 97, 99, 103–105, 164, 166, 171
Multivariate models, 265
N
National economic development, 393
Navigation, 7, 8, 16, 24, 37, 44, 380, 469, 504, 570
Network, 35, 67, 68, 105–108, 114, 117, 123, 127, 169,
172, 181, 183, 184, 210, 211, 328, 423, 512, 529,
531, 534, 535, 540, 559, 565, 574, 583, 605
Nitrogen
cycle, 433, 440, 441
models, 54, 422, 433, 438, 440, 463, 464, 466
Non-inferior, 394, 395, 397–399, 401, 402, 414, 415
Non-point
models, 418, 421, 424, 425, 454, 465

Index
quality, 418, 421, 424, 425, 454, 465
runoff, 418, 420, 425
Normal distributions, 234, 244, 264, 356
Nutrients
cycling, 441
models, 440
O
Objective, 2, 7, 34, 35, 38, 43, 51, 55, 57–59, 64, 69, 71,
73, 75, 76, 78, 81, 85, 94, 104, 112, 126, 137, 150,
169, 187, 189, 197, 201, 210, 312, 332, 370,
376–380, 393–396, 398–402, 410, 411, 414, 415,
494, 511, 524, 579, 581, 582, 592, 595, 605, 608,
609, 611
OMR costs, 96
Operating policy, 73, 75, 81, 82, 84, 94, 137, 167, 171,
270, 273, 274, 278, 281, 326, 328, 377, 491, 504,
521–523, 574
Operating rule, 39, 487, 524
Operations research, 71
Opportunity cost, 38, 385, 569, 570
Optimal, 42, 64, 69, 79, 85, 94, 105, 114, 115, 127, 151,
167, 197, 201, 293, 315, 319, 326, 328, 345, 414,
513, 514, 521, 526, 560
Optimization, 40, 51, 63, 73, 79, 81, 84, 85, 90, 93, 94,
97, 99, 105, 116, 137, 138, 163, 181, 196, 201,
213, 250, 307, 311, 314, 327, 345, 392, 412, 450,
476, 479, 491, 511, 515, 517, 522, 523, 529,
558–560, 583, 594, 610
Organics micropollutants, 441, 444, 445
Over-year storage, 258, 481, 482, 484, 485, 488, 508
Oxygen, 33, 54, 418, 422, 427, 433, 434, 436, 438, 439,
441, 447, 449, 462, 463, 553–555, 557
P
Parameter
calibration, 425
performance measures, 67, 73, 85, 122, 156, 195, 340,
341, 343, 344, 349, 363, 364, 366, 375, 389, 486,
512
uncertainty, 259, 333, 335, 338, 457, 478
value, 63, 74, 75, 78, 81, 145, 147, 148, 170, 181,
191, 203, 210, 211, 247, 259, 261, 265, 299, 302,
327, 334–338, 340–342, 345, 346, 354, 370, 607
Peak flow, 490, 491, 494, 496, 497, 500, 518
Performance criteria, 30, 35, 76, 135, 179, 195, 274, 340,
375, 377, 379, 380, 389, 392, 402, 410, 553
Phosphorus
cycle, 438, 441
model, 169, 339, 347, 352, 356, 357, 359, 389, 390,
438–440, 446
Photosynthesis, 436, 439, 557
Plan formulation
implementation, 607
selection, 75

621
Planning
approaches, 594
economic, 96
financial, 32, 38, 45, 56, 58, 79, 151, 380, 410, 511,
512, 568, 587, 593, 598
integrated, 34, 40, 45, 579, 581, 596, 600
Plant capacity, 168, 292, 503, 504, 511, 518
Plant factor, 503, 504
Plotting positions, 231, 233, 234, 246, 291
Population, 1, 5, 16, 36, 187, 189, 190, 193, 228, 290,
380, 384, 385, 490, 531, 574, 585, 590, 594, 612
Precipitation, 249, 260, 265, 267, 282, 332, 336, 373,
444, 446, 489, 539, 564
Present value, 95, 116, 118, 165, 168, 174, 293, 512, 514,
519
Primary treatment, 537
Principal of optimality, 115
Probability
beta, 230, 234, 243, 289
conditional distribution, 217, 218, 261, 310, 356, 360
cumulative, 219, 230, 241, 291, 303, 307, 329, 356,
374, 431, 496
density, 216–218, 227, 229, 230, 236, 237, 239, 289,
302, 307, 329, 339, 364, 390
exceedance, 246, 249, 481, 485, 491, 492, 497, 499,
500
exponential, 231, 233, 237, 240, 245, 249, 250, 290
functions, 497, 500
gamma, 222, 223, 226, 237–245, 263, 264, 289, 290,
292
Gumbel and GEV, 231, 241
joint, 216, 227, 289, 316, 328, 340
lognormal, 226, 227, 230, 232–237, 240, 244, 247,
264, 292
log-pearson Type 3, 239, 240
marginal, 95, 96, 98, 99, 103, 104, 126, 149, 150, 166,
211, 217, 218, 260, 261, 263, 264, 269, 289, 299,
317, 379, 382, 385, 387, 395, 397
normal, 236
transition, 252, 310, 315, 316
unconditional, 252, 307, 309, 314, 316
Project
management, 5, 43, 173, 567, 605
scheduling, 470, 511–513, 515
Public, 3, 12, 34, 44, 56, 96, 344, 367, 376, 381, 389,
407, 490, 561, 567, 575, 598, 611, 613
Q
Quality Impacts, 417, 418, 553
Quantity
flows, 12, 37, 472, 529
storage, 278, 381
velocities, 425, 469
Quantiles, 219, 223, 229, 232, 233, 239, 241, 246, 248,
360

622
R
Radioactive substances, 441, 446
Radionuclides, 446
Rainfall
design rainfall, 541
distribution in space and time, 281
models, 194, 229, 260, 336, 425, 471
synthetic rainfall, 539, 540
Random
independent, 217, 241, 255, 289
variables, 215–220, 226, 230, 235, 239, 250, 252,
256, 262–264, 267, 271, 277, 280, 288, 297, 302,
303, 305, 308, 327, 339, 374
Rate, 20, 94, 95, 117, 165, 166, 186, 214, 249, 275, 277,
291, 342, 352, 395, 428, 433, 434, 439, 445, 446,
452, 457, 463, 472, 503, 545, 550, 564, 591
Reaction, 7, 213, 334, 427, 428, 433, 436, 441, 444, 535
Re-aeration, 426, 433, 435, 438, 462, 463
Recreation, 7, 10, 12, 18, 24, 34, 120, 126, 172, 209, 214,
287, 375, 380, 381, 383, 384, 387, 413, 420, 469,
470, 474, 486, 490, 502, 505, 506, 510, 520, 522,
569, 574, 611
Recurrence, 481
Recursive, 105, 111, 114, 115, 118, 142, 167, 169, 263,
315, 318, 514
Redundant constraints, 105
Regional, 9, 19, 20, 37, 42, 63, 224, 229, 233, 241, 247,
248, 332, 353, 386, 396, 573, 575, 580, 581, 612
Regionalization, 224, 228, 241, 247, 248, 282
Regret, 40, 294, 576, 595
Reliability, 38, 59, 76, 135, 139, 260, 272, 274, 275, 277,
280, 282, 307, 331, 344, 355, 364, 366, 368, 374,
376, 390, 405, 409, 420, 469, 478, 480, 484–487,
523, 535, 579, 607, 612
Reservoir
active storage volume, 123, 139, 318, 412, 479, 485,
519
capacity, 75, 83, 84, 140, 169, 171, 174, 273, 312,
314, 317, 412, 482, 484, 485, 491, 494, 517–520,
524
dead storage volume, 519, 523
flood storage volume, 75, 126, 209, 210, 387,
487–495, 510, 517–519, 521
operation, 38, 67, 75, 108, 115, 120, 126, 127,
130–133, 137, 141, 167, 202, 207, 312, 480, 487
optimization, 199
over-year storage, 258, 481, 482, 484, 485, 487
release rule, 122, 135, 318, 326, 479
simulation, 294
within-year storage, 470, 479, 482, 483, 487, 488
yield, 138–141, 173, 476, 479, 481, 516, 517, 519,
523
Resilience, 135, 405, 409, 609
Respiration, 433, 436, 438
Response, 2, 36, 40, 52, 196, 203, 213, 347, 349, 352,
360, 377, 384, 388, 420, 421, 511, 547, 555, 561,
580, 581, 610
Return period, 249, 491, 492, 494, 495, 500, 510, 519,
539–541, 612

Index
Revenue, 38, 102, 104, 167, 170, 287, 570
River basins
management, 19, 33, 34, 64, 504
simulation, 91, 251, 258, 272
Rivers, 1, 3, 6, 20, 28, 49, 379, 390, 419, 425, 428, 430,
447, 452, 489, 570, 574, 581, 582, 609
Robust, 42, 81, 180, 246, 258, 334, 578, 595
Routing, 211, 470, 472, 473, 478, 491, 494, 543, 544
Root zone
Rule curves, 121, 480, 487
Runoff, 5, 12, 18, 31, 37, 59, 179, 207, 229, 247, 260,
377, 417, 418, 420, 425, 449, 462, 471, 472, 505,
527, 528, 539–545, 547, 549, 552, 557, 561, 564,
565
S
Safe yield, 480
Sampling
Monte Carlo, 354–356, 370, 374
uncertainty, 356
Satisficing, 398, 399
Scale
processes, 581, 593, 605
spatial, 33, 267, 335
temporal, 34, 67, 336
Scarcity, 22, 23, 30
Scheduling, 470, 511–513, 515, 559, 583, 598
Screening, 39, 63, 79, 86, 138, 151, 163, 327, 376, 486,
515, 576, 591, 592
Search, 6, 28, 63, 79, 115, 151, 181, 187, 191, 194, 243,
405, 559
Second-order conditions, 228
Sediment
bed load, 23
burial, 443
resuspension and sedimentation, 33, 209, 442, 444,
446, 449
settling, 35, 441, 442, 444, 446, 451
suspended load, 420, 427, 441–443, 446
Seepage, 123, 472–474, 517, 519
Sensitivity Analyses
coefficients, 345, 350–352
first-order analysis, 351
Sequent peak analyses, 476
Series, 94
Settling
detritus, 439, 441, 446
inorganic, 441, 442, 444, 445
sediment, 444, 451
Sewer systems, 187, 536, 539, 549, 551
Silica cycle
Simulation
accuracy, 277, 453, 455
methods, 79, 163, 452, 486
uncertainty, 277, 301, 457, 458
Social criteria, 42, 380, 392, 393
Stakeholder, 2, 11, 28, 34, 35, 39, 44, 45, 56, 61, 64–66,
68, 75, 173, 187, 282, 368, 376–380, 393, 397,

Index
407, 410, 421, 458, 490, 506, 527, 567, 575, 576,
579, 580, 583–585, 587, 589, 591, 592, 597–604,
610
State variable, 106, 115, 123, 295, 328, 446
Static analyses, 470
Stationarity, 250, 251, 259, 332
Statistical, 55, 115, 179–181, 184, 213, 224, 226, 230,
231, 251, 254, 258–261, 273, 280, 282, 297, 302,
304, 332, 336, 343–346, 360, 366, 367, 407, 421,
470, 560
Steady-state, 127, 129, 136, 137, 142, 252, 254, 316, 318,
432, 463, 465, 487, 511, 519, 559, 560
Step method, 402
Stochastic
model, 252, 259, 261, 282, 301, 314, 315, 317, 327,
332
optimization, 215, 311, 314, 327
processes, 215, 250–252, 254, 260, 282, 457
simulation, 215, 226, 270–272, 282
variables, 187, 193, 255, 261, 263, 271, 295, 301,
457, 560
Storage-yield functions, 139, 140, 476
Storage zones
active, 82, 123, 139, 140, 171, 312, 318, 412, 414,
474–476, 479–483, 485, 487, 488, 490, 494, 510,
517, 519, 523
dead, 474, 494, 510, 517, 518, 523
flood, 75, 126, 209, 387, 474, 487, 488, 490–495,
518, 519
over-year, 258, 470, 480–482, 484–488, 508, 523,
524, 580
within-year, 482, 486, 487, 523
Storms, 22, 539, 540, 550, 557, 560
Stormwater, 81, 418, 527, 539, 552, 553, 558, 565
Stratification, 33, 54, 422, 425, 447, 449–451
Streamflow
estimation, 471
gauge, 539, 540
generation, 257, 259, 260, 267, 297
routing, 472
synthetic values, 257, 258, 261, 264, 296, 298, 299
Streeter and Phelps, 433
Student’s t-distribution, 281
Substitution, 218, 255, 266, 395, 397, 611
Sustainability, 30, 35, 42, 43, 49, 88, 389, 410, 565, 581,
591, 599, 614
Synthetic streamflow, 257, 258, 261, 265, 273, 274, 296,
298, 299
Systems analysis, 12, 39, 71, 88, 571
Systems approach, 528, 571
T
Target, 37, 43, 73, 75, 82, 121, 123, 126, 127, 163, 166,
169, 171, 172, 174, 175, 198, 199, 209, 274, 275,
278, 295, 306, 307, 311, 318, 328, 329, 333, 343,
362–365, 368, 379, 386, 387, 389, 401, 412, 414,
505, 508, 511, 520, 522, 560, 569, 570, 581, 592

623
Temperature, 143, 144, 196, 197, 218, 257, 282, 336,
342, 420, 432–434, 436, 439, 444, 446, 449, 456,
457, 463, 543, 565
Thermocline, 450, 451
Thomas-Fiering model, 269, 273, 298
Time periods, 34, 81, 85, 95, 96, 106, 108, 117–119, 121,
137, 209, 304, 308, 310, 311, 313, 342, 388, 425,
454, 470, 473, 475, 479, 491, 504, 506, 517, 518,
539, 559, 560, 580
Time series, 67, 81, 89, 95, 114, 127, 215, 250, 251, 254,
255, 263, 267, 269, 282, 301, 303, 308, 336, 337,
402, 404, 405, 407–409, 473, 539, 550, 595
Time value, 96
Toxic chemicals, 441
Tradeoffs, 28, 35, 38, 41, 42, 69, 73, 340, 365, 367, 375,
376, 379, 380, 393, 396, 397, 402, 407, 410, 412,
430, 504, 508
Transition probability, 252, 310, 315, 316
Transpiration, 543
Transport, 9, 16, 18, 25, 194, 375, 417, 425–427, 432,
444, 446, 452, 502, 528, 529, 549, 552, 568, 574,
575
Trial and error, 78, 81, 122, 188, 270, 459, 559
U
Uncertainty
analyses, 81, 181, 213, 301, 332, 339, 343, 344, 352,
366, 367, 373, 374, 422, 458, 515, 607
model, 338, 457
parameter, 259, 333, 335, 338, 457, 478
Uniform, 96, 189, 271, 297, 302, 304, 309, 360, 362, 462,
472, 525, 536, 541
Urban Drainage
model, 187, 539, 540, 544
pollutant loading, 81, 418–420, 424, 425, 429, 449,
549, 550, 552
runoff, 41, 528
V
Validation, 81, 184, 211, 607
Value, 95
Variability, 2, 20, 33, 34, 44, 220, 234, 251, 258, 259,
262, 287, 332, 334–337, 345, 357, 367, 457, 511,
561, 580, 609
Variables
deterministic, 213, 327
random, 215–219, 221, 226, 227, 230, 237, 250, 252,
262, 264, 269, 271, 277, 281, 299, 302, 303, 306,
309, 327, 339, 340
Variance, 219–223, 227, 228, 230, 233, 236, 238, 240,
242, 247, 251, 254–257, 260, 262–264, 280, 287,
290, 297, 299, 302, 304, 329, 351, 352, 356, 366,
407
Verification, 16, 55, 81, 425, 606
Vulnerability, 135, 280, 405, 409, 489, 582

624

Index

W
Wastewater
collection, 378, 527, 528, 536, 569
discharge, 31, 143, 145, 177, 302, 377, 429, 430, 586
treatment, 37, 75, 76, 142, 143, 146, 149, 168–170,
172, 179, 187, 210, 288, 292, 378, 418, 420, 436,
465, 469, 529, 537, 538, 565, 568, 570, 613
Water
groundwater, 20, 31, 32, 34, 151, 152, 174, 191, 258,
270, 377, 380, 458, 469, 489, 528, 545, 561, 568,
574, 581, 613
polluted, 5
quality, 12, 19, 20, 30–32, 36, 56, 67, 74, 79, 142,
143, 145, 146, 148, 150, 185, 202, 206, 219, 246,
346, 377, 378, 389, 393, 404, 412, 414, 417–423,
425, 429, 432, 433, 439, 446–449, 452, 454,
457–459, 469, 502, 505, 535, 536, 553, 555, 560,
575, 582, 592, 612
quantity, 2, 12, 34, 373, 389, 417, 425, 426, 458, 459,
469, 585, 587, 609
surface water, 2, 5, 59, 376, 377, 389, 445, 449, 458,
473, 474, 478, 489, 528, 552, 574, 585

Watershed, 20, 32, 33, 37, 88, 121, 181, 247, 339, 344,
370, 380, 389, 419, 421, 469, 471, 472, 474, 517,
529, 564, 567, 582, 589
Weibull plotting position, 291
Weighting method, 395–397, 402, 415
Wetlands, 5, 10, 12, 23, 25, 32, 37, 38, 338, 390, 469
Wilcoxon test, 296
Willingness to pay, 95, 381–383, 385, 411
Wilson–Hilferty transformation, 239
Withdrawals, 1, 2, 6, 31, 305, 470, 504, 585
Within-year
periods, 34, 122, 123, 142, 311, 313, 317, 318, 412, 450,
470–472, 480–485, 486, 487, 508, 511, 518, 524
storage, 470, 479, 482–484, 487, 488

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Y
Yield, 25, 28, 41, 90, 95, 99, 103, 104, 138–142, 171,
175, 176, 219, 227, 237, 246, 250, 263, 269, 287,
296, 297, 338, 347, 358, 402, 476, 478–488, 499,
511, 517, 522–525, 551, 612
Zooplankton