Environmental and hydrological systems modelling

The earliest application of systems theory may have been the introduction of the Dirac delta function and the unit step function as input functions for linear systems.. Time and frequenc

Environmental and Hydrological Systems Modelling

Mathematical modelling has become an indispensable tool for engineers,

scientists, planners, decision makers and many other professionals to make

predictions of future scenarios as well as real impending events As the

modelling approach and the model to be used are problem specific, no

single model or approach can be used to solve all problems, and there are

constraints in each situation Modellers therefore need to have a choice

when confronted with constraints such as lack of sufficient data, resources,

expertise and time

Environmental and Hydrological Systems Modelling provides the tools

needed by presenting different approaches to modelling the water

environment over a range of spatial and temporal scales Their applications

are shown with a series of case studies, taken mainly from the Asia-Pacific

Region Coverage includes:

• Model Parameter Estimation

• Model Calibration, Validation and Testing

This book will be of great value to advanced students, professionals,

academics and researchers working in the water environment

A W Jayawardena is an Adjunct Professor at The University of Hong Kong and

Technical Advisor to Nippon Koei Company Ltd (Consulting Engineers), Japan

Environmental and Hydrological Systems Modelling

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

Boca Raton London New York

Environmental and Hydrological Systems Modelling

A W Jayawardena

accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

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1.2 General systems theory (GST) 3

1.3 Ecological systems (Ecosystems) 4

1.4 Equi-finality 4

1.5 Scope and layout 5

References 7

2 Historical development of hydrological modelling 9

2.1 Basic concepts and governing equation of linear systems 9

2.1.1 Time domain analysis 9

2.1.2 Frequency domain analysis 12

2.1.2.3 z-Transform 15 2.2 Linear systems in hydrological modelling 16

2.2.1 Hydrological systems 16

2.2.2 Unit hydrograph 17

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2.4 Non-linear systems 29

2.4.1 Determination of the kernel functions 29

2.5 Multilinear or parallel systems 31

2.6 Flood routing 31

2.6.1 Inventory method 31

2.6.2 Muskingum method 32

2.6.3 Modified Puls method 35

2.6.4 Muskingum–Cunge method 35

2.6.5 Hydraulic approach 37

2.7 Reservoir routing 41

2.8 Rainfall–runoff modelling 43

2.8.1 Conceptual-type hydrologic models 44

2.8.2 Physics-based hydrologic models 51

2.8.3 Data-driven models 52

2.9 Guiding principles and criteria for choosing a model 53

2.10 Challenges in hydrological modelling 54

2.11 Concluding remarks 56

References 56

3 Population dynamics 61

3.1 Introduction 61

3.2 Malthusian growth model 61

3.3 Verhulst growth model 63

3.4 Predator–prey (Lotka–Volterra) model 64

© 2010 Taylor & Francis Group, LLC

5 Water quality systems 85

5.1 Dissolved oxygen systems 85

5.1.1 Biochemical oxygen demand (BOD) 85

5.1.2 Nitrification 88

5.1.3 Denitrification 88

5.1.4 Oxygen depletion equation in a river due

to a single point source of BOD 89 5.1.5 Reoxygenation coefficient 92

5.1.6 Deoxygenation coefficient 94

5.2 Water quality in a completely mixed water body 94

5.2.1 Governing equations for a completely mixed system 95

5.2.2 Step function input 96

5.2.3 Periodic input function 97

5.2.4 Fourier series input 98

5.2.5 General harmonic response 99

5.3.3 Effect of spatial flow variation 109

5.3.4 Unsteady state 111

5.3.5 Tidal reaches 113

5.4 Concluding remarks 114

References 114

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6 Longitudinal dispersion 117

6.1 Introduction 117

6.2 Governing equations 117

6.2.1 Some characteristics of turbulent diffusion 118

6.2.2 Shear flow dispersion 119

6.4.1 Finite difference method 127

6.4.2 Finite element methods 128

6.4.3 Moving finite elements 130

6.5 Dispersion through porous media 131

6.6 General-purpose water quality models 134

6.6.1 Enhanced Stream Water Quality Model (QUAL2E) 134

6.6.2 Water Quality Analysis Simulation Programme (WASP) 135

6.6.3 One Dimensional Riverine Hydrodynamic and

Water Quality Model (EPD-RIV1) 135 6.7 Concluding remarks 136

7.3.2 Moving averages – low-pass filtering 141

7.3.3 Differencing – high-pass filtering 142

7.3.4 Recursive means and variances 142

7.4 Tests for stationarity 143

7.5 Tests for homogeneity 144

7.5.1 von Neumann ratio 145

7.7.1 Tests for randomness and trend 151

© 2010 Taylor & Francis Group, LLC

7.7.1.2 Kendall’s rank correlation test ( τ test) 153

7.7.2 Trend removal 156

7.7.2.1 Splines 157 7.8 Periodicity 159

7.8.1 Harmonic analysis – cumulative periodogram 159

7.8.2 Autocorrelation analysis 164

7.8.3 Spectral analysis 167

7.8.4 Cross correlation 173

7.8.5 Cross-spectral density function 173

7.9 Stochastic component 174

7.9.1 Autoregressive (AR) models 175

7.9.2 Moving average (MA) models 181

7.9.3 Autoregressive moving average (ARMA) models 185

7.10.4 Test for parsimony 190

7.10.4.1 Akaike information criterion (AIC) and

Bayesian information criterion (BIC) 190 7.10.4.2 Schwartz Bayesian criterion (SBC) 190 7.11 Forecasting 191

7.11.1 Minimum mean square error type difference equation 191

7.11.2 Confidence limits 193

7.11.3 Forecast errors 193

7.11.4 Numerical examples of forecasting 193

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7.12 Synthetic data generation 196

8.3.1 Method of steepest descent 215

8.3.2 Newton’s method (quadratic approximation) 216

8.5 Types of activation functions 223

8.5.1 Linear activation function (unbounded) 223

8.5.2 Saturating activation function (bounded) 223

8.5.3 Symmetric saturating activation function (bounded) 228

8.5.4 Positive linear activation function 228

8.5.5 Hardlimiter (Heaviside function; McCulloch–

Pitts model) activation function 229 8.5.6 Symmetric hardlimiter activation function 229

8.5.7 Signum function 229

8.5.8 Triangular activation function 229

8.5.9 Sigmoid logistic activation function 229

8.5.10 Sigmoid hyperbolic tangent function 230

8.5.11 Radial basis functions 230

8.5.11.1 Multiquadratic 230 8.5.11.2 Inverse multiquadratic 231 8.5.11.3 Gaussian 231

8.5.11.4 Polyharmonic spline function 231 8.5.11.5 Thin plate spline function 231 8.5.12 Softmax activation function 231

8.6 Types of artificial neural networks 232

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8.6.1 Feed-forward neural networks 233

8.6.2 Recurrent neural networks 234

8.6.3 Self-organizing maps (Kohonen networks) 237

8.6.4 Product unit–based neural networks (PUNN) 239

8.6.5 Wavelet neural networks 245

8.7 Learning modes and learning 248

8.7.1 Learning modes 248

8.7.2 Types of learning 249

8.8.1 Generalized delta rule 256

8.9 ANN implementation details 256

8.9.1 Data preprocessing: Principal Component Analysis (PCA) 256

8.9.2 Data normalization 260

8.9.3 Choice of input variables 262

8.9.4 Heuristics for implementation of BP 262

© 2010 Taylor & Francis Group, LLC

9.6.3 Orthogonal least squares (OLS) algorithm 298

9.6.4 Self-organized selection of centres 304

9.6.5 Supervised selection of centres 306

9.6.6 Selection of centres using the concept of

generalized degrees of freedom 307

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10.5.2 The n-body problem 338

10.6.6 Phase (or state) space 341

10.7 Invariants of chaotic systems 341

10.7.1 Lyapunov exponent 341

10.7.2 Entropy of a dynamical system 342

10.7.2.1 Kolmogorov–Sinai (K–S) entropy 343 10.7.2.2 Modified correlation entropy 344 10.7.2.3 K–S entropy and the Lyapunov spectrum 347 10.8 Examples of known chaotic attractors 348

10.8.1 Logistic map 348

10.8.1.1 Bifurcation 351 10.8.2 Hénon map 352

11.2 Random versus chaotic deterministic systems 366

11.3 Time series as a dynamical system 367

11.4.3.1 Average mutual information 378 11.4.4 Irregular embeddings 379

11.5 Phase (or state) space reconstruction 380

11.6 Phase space prediction 382

11.7 Inverse problem 384

11.7.1 Prediction error 385

11.8 Non-linearity and determinism 386

© 2010 Taylor & Francis Group, LLC

11.8.1 Test for non-linearity 386

11.8.1.1 Significance 386 11.8.1.2 Test statistics 387 11.8.1.3 Method of surrogate data 387 11.8.1.4 Null hypotheses 388

11.8.2 Test for determinism 389

11.9 Noise and noise reduction 390

Appendix 11.1: Derivation of Equation 11.81 403

Appendix 11.2: Proof of Equation 11.82b 407

Appendix 11.3: Proof of Equation A1-4 407

References 408

12 Support vector machines 413

12.1 Introduction 413

12.2 Linearly separable binary classification 413

12.3 Soft-margin binary classification 418

12.3.1 Linear soft margin 418

12.3.2 Non-linear classification 421

12.4 Support vector regression 424

12.4.1 Linear support vector regression 424

12.4.2 Non-linear support vector regression 426

Appendix 12.1: Statistical learning 429

Empirical risk minimization (ERM) 430

Structural risk minimization (SRM) 431

Appendix 12.2: Karush–Kuhn–Tucker (KKT) conditions 432

© 2010 Taylor & Francis Group, LLC

13.2.2.1 Intersection 441 13.2.2.2 Union 441 13.2.2.3 Other useful definitions 442 13.2.3 Linguistic variables 444

13.5.1 Fuzzy or approximate reasoning 450

13.5.2 Mamdani fuzzy inference system 451

13.5.2.1 Fuzzification of inputs 451 13.5.2.2 Application of fuzzy operators ‘AND’ or ‘OR’ 453 13.5.2.3 Implication from antecedent to consequent 453 13.5.2.4 Aggregation of consequents across the rules 456 13.5.2.5 Defuzzification 456

13.5.3 Takagi–Sugeno–Kang (TSK) fuzzy inference system 459

13.5.3.1 Clustering 461 13.5.4 Tsukamoto inference system 463

13.5.5 Larsen inference system 463

13.6 Neuro-fuzzy systems 465

13.6.1 Types of neuro-fuzzy systems 467

13.6.1.1 Umano and Ezawa (1991) fuzzy-neural model 468 13.7 Adaptive neuro-fuzzy inference systems (ANFIS) 469

Preface

Systems science, or systems theory, perhaps has its origin in the general systems theory tulated by biologist Ludwig von Bertalanffy in the 1930s as a modelling tool that accom-modates the interrelationships and overlap between separate disciplines In the early 20th

pos-century, systems modelling started with two approaches, an a priori approach in which

some assumptions are made followed by proposing some mathematical equations that when

solved lead to some deductions, and an a posteriori approach in which mathematical

equa-tions that potentially explain the phenomena are fitted to observaequa-tions The advantage of

the systems theory approach in problem solving is that it does not require a priori

under-standing of the detailed mechanisms of the underlying processes that govern the system.Systems science is an interdisciplinary field of studies that covers a broad range of areas, including nature, society, science, engineering, and medicine It covers well-defined fields such as cybernetics, dynamical systems and chaos, control theory, operational research, ecology, and many others In view of the diversity of fields covered, systems theory has developed from different fields and its applications extend from microscopic scale to very large scale

Most environmental and hydrological problems are quite complex, involve the tion of many variables, and are often little understood, at least at the present time The databases available for model development and calibration are also of a coarse resolution There are also unresolved issues of scale effect Therefore, it is difficult, if not impossible,

interac-to model such systems starting from first principles Systems approach offers an alternative The earliest application of systems theory may have been the introduction of the Dirac delta function and the unit step function as input functions for linear systems They enable the determination of responses to arbitrary input functions via the principles of superposi-tion and proportionality Linear approaches have stood the test of time owing to their simplicity, generality, and extrapolation capability However, they are not close to reality More recently, researchers, in addition to attempts made to work from first principles, are moving into data-driven, non-linear approaches, which of course also have limitations and lack generality but can be made closer to reality Approaches such as artificial neural networks, support vector machines, fuzzy logic, fuzzy neural systems, genetic algorithms, and genetic programming are becoming popular particularly when other approaches are difficult to implement or infeasible Many such approaches that require repeated calcula-tions of simple recursive equations could not have been possible some years ago With the advent of high-speed computers, such calculations have now become feasible This book attempts to bridge the gap between the linear assumption and reality It is targeted towards

a readership of researchers, graduate students, and professional with general interest in the hydro-environment and specialist interest in data-driven techniques of environmental and hydrological systems modelling

The contents of the book are built on graduate courses given by the author in the Department

of Civil Engineering of the University of Hong Kong and the International Centre for Water Hazard and Risk Management (ICHARM) under the auspices of UNESCO, Public Works Research Institute, Tsukuba, Japan, and the related research work carried out by the author and his co-workers and graduate students

The author would like to record his appreciation to his co-workers at different times,

in particular to Pengcheng Xu of the Chinese Academy of Sciences, Beijing, China; Bellie Sivakumar, now of the University of New South Wales, Australia; I.W Johnson, formerly

of the Institute of Hydrology, Wallingford, UK; Dulakshi S.K Karunasingha of the Faculty

of Engineering, University of Peradeniya, Sri Lanka; and N Muttil of Victoria University, Australia The author would also like to pay tribute to his graduate students at different times, both at the Department of Civil Engineering of the University of Hong Kong and at the International Centre for Water Hazard and Risk Management (ICHARM) In particular, the contributions by P.H Lui, Lai Feizhou, Achela D.K Fernando, T.M.K.G Fernando, A.B Gurung, Zhou Maichun, S.P.P Mahanama, Tian Ying, Somchit Amnatsan, Robin Kumar Biswas, Manish Maharjan, A.K.M Saifudeen, Prem Raj Ghimire, J.D Amarasekara, and Zhu Bing are highly appreciated Sincere appreciation also goes to K Takeuchi, Director of the International Centre for Water Hazard and Risk Management, for providing an oppor-tunity to be academically active after leaving the University of Hong Kong

The author also benefitted from the published work of others Appropriate ments have been made in citing such work in relevant parts of the text Last but not least, the author expresses his gratitude to Tony Moore, senior editor of the publisher Taylor & Francis, who initially proposed the idea, and gave continuous encouragement His patience

acknowledge-is gratefully appreciated Thanks are also due to two Taylor & Francacknowledge-is staff members: Simon Bates, who looked after the project for some time, and Stephanie Morkert, who did the production work from the author’s manuscript, and Amor Nanas from Manila Typesetting Company, who skillfully did the copy editing

The book contains some 1072 equations, some simple and some not so simple Despite the care taken to ensure the correctness of these equations and other material presented, it is still possible that there may be typographical errors and/or omissions due to oversight The author would be grateful if the readers would kindly bring to his attention if they find any such errors and/or omissions After all, to err is human and to forgive is divine

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Author

A.W Jayawardena obtained his undergraduate degree BSc (Eng) Hons from the University

of Ceylon (now Sri Lanka) and postgraduate degrees MEng from the University of Tokyo,

MS from the University of California at Berkeley, and PhD from the University of London

He is a Chartered Engineer, a Fellow of the UK Institution of Civil Engineers, a Fellow of the Hong Kong Institution of Engineers, and a Life Member of the American Society of Civil Engineers His academic career includes many years of teaching in the Department

of Civil Engineering of the University of Hong Kong; he was also a Research and Training Advisor to the International Centre for Water Hazard and Risk Management (ICHARM) under the auspices of UNESCO and hosted by the Public Works Research Institute of Japan,

a Professor at the National Graduate Institute for Policy Studies, Japan, and an Honorary Professor in the Department of Statistics and Actuarial Sciences of the University of Hong Kong He is currently an Adjunct Professor in the Department of Civil Engineering of the University of Hong Kong, Technical Advisor to the Research and Development Centre, Nippon Koei Co Ltd (Consulting Engineers), Japan, and a Guest Professor of Beijing Normal University, China He has been a specialist consultant for UNESCO and for several engineering consulting companies in Hong Kong, including as an expert witness and pro-vider of expert opinion in several legal cases in Hong Kong

In reality, however, it is difficult, if not impossible, to compartmentalize problems in nature into any of the above tightly bounded disciplines They invariably involve interac-tions across disciplines Therefore, a holistic approach is necessary to understand and solve real-life problems Systems theory provides an approach that can be considered as a field of inquiry rather than looking from a specific discipline.

The advantage of the systems theory approach in problem solving is that it does not

require a priori understanding of the detailed mechanisms of the underlying processes in

the system What it needs is a set of inputs and a set of corresponding outputs from which the system parameters (or functions) are estimated The relationship between the input vari-ables and the corresponding output variables may be assumed to be linear, in which case the analysis and subsequent generalization is easy, piece-wise linear, or completely non-linear

1.1 SOME DEFINITIONS

1.1.1 System

Several definitions of a system have come at different times For example,

• A system is a group of interacting components that conserves some identifiable set of relations with the sum of the components plus their relations (i.e., the system itself) conserving some identifiable set of relations to other entities (including other systems)

• A system is a complex of interacting components together with the relationships among them that permit the identification of a boundary-maintaining entity or process

• A system is a combination of interacting elements that performs a function not ble with any of the individual elements The elements can include hardware, software, bioware, facilities, policies, and processes

possi-• A system is a set of social, biological, technological, or material partners cooperating

on a common purpose

© 2010 Taylor & Francis Group, LLC

A system accepts inputs, over which it has no direct control, and transforms them into outputs A system should have a well-defined boundary Fitting into this single definition are many types of systems, some with states and some without

Systems can be categorized as memoryless or dynamic In a memoryless system, the puts depend only on the present values of its inputs In a dynamic system, the outputs depend on the present and past values of its inputs For dynamic systems, the concept of a

out-state must be defined.

1.1.2 State of a system

The state of a system makes the system’s history irrelevant The state of the system contains all the information needed to calculate responses to present and future inputs without refer-ence to the history of inputs and outputs The state of the system, the present inputs, and the sequence of future inputs allow computation of all future states (and outputs)

Some dynamic systems are modelled best with state equations, while others are modelled best with state machines.1 State-equation systems are modelled with equations For example,

a projectile’s motion can be modelled with state equations for position and velocity, which are functions of time State-machine systems focus less on physical variables and more on logical attributes Therefore, such systems have memory and are modelled with finite state machines.2 Most computer systems are modelled with finite state machines

At each instant of time, a dynamic system is in a specific state State-equation systems can have one or many state variables At any time, the system’s state is defined as the unique values for each of the state variables State-machine systems can be modelled with one or many concurrent state machines At any time, each of the concurrent state machines must

be in one and only one state A state is a unique snapshot that is specified by values for a set of variables, characterizes the system for a period of time, and is different from other states Each state is different from other states in either the inputs it responds to, the outputs

it produces, or the transitions it take A transition is a response to an input that may cause

a change of state

A closed system is one where interactions occur only among the system components and not with the external environment An open system is one that receives input from the

external environment and/or releases output to the environment The basic characteristic of

an open system is the dynamic interaction of its components, while the basis of a cybernetic model is the feedback cycle Open systems can tend towards higher levels of organization (negative entropy), while closed systems can only maintain increasing entropy

Systems modelling in the early 20th century started using two basic approaches: First, used

mainly by mathematicians, is the a priori approach in which some assumptions are made, a

set of mathematical equations are proposed, some deductions are made from the solutions

to the mathematical equations, and finally such results are tested against observations This

1 A state machine is a device that stores the status of something at a given time that changes its status upon ing inputs For example, a computer is a state machine Each machine instruction is input that changes one or more states and may cause other actions to take place A simple example of a state machine is a turnstile It has three arms and it can be in a locked or an unlocked state When a coin or token is inserted, the turnstile is unlocked and the arm can be moved one-third of a rotation It will not rotate more than one-third to prevent more than one person passing through After a one-third rotation, the arms are locked again until a second coin or token is inserted It has two inputs, inserting a coin or token and rotating the arm In the locked state, the arms do not rotate In the unlocked state, inserting additional coins has no effect (does not allow the arm

receiv-to rotate) When a person passes through the turnstile, the machine again reverts back receiv-to the locked position Other examples of simple state machines include vending machines, elevators, traffic lights, etc.

2 Finite state machines are controllers of machines very common in daily life For example, a traffic light signal that has three states, red, amber, and green, for vehicles and two states for pedestrians, red and green.

© 2010 Taylor & Francis Group, LLC

approach seems to work in situations where there are well-defined patterns in the behaviour

of the system variables The second approach, mainly used by statisticians, is the a riori approach, which begins with observations, fitting mathematical equations to the obser-

poste-vations, and attempting to explain the underlying phenomena Some degree of randomness

is inherent in phenomena fitting into this category

1.2 GENERAL SYSTEMS THEORY (GST)

The general systems theory (GST) was originally proposed by the biologist Ludwig von Bertalanffy in the 1930s as a modelling tool that accommodates the interrelationships and overlap between separate disciplines His idea of the concept of systems is exemplified in the following passage:

The 19th century and the first half of the 20th century conceived of the world as chaos

Chaos was the oft-quoted blind play of atoms, which, in mechanistic and positivistic philosophy, appeared to represent ultimate reality, with life as an accidental product

of physical processes, and mind as an epi-phenomenon It was chaos when, in the rent theory of evolution, the living world appeared as a product of chance, the outcome

cur-of random mutations and survival in the mill cur-of natural selection In the same sense, human personality, in the theories of behaviourism as well as of psychoanalysis, was considered a chance product of nature and nurture, of a mixture of genes and an acci-dental sequence of events from early childhood to maturity

Now we are looking for another basic outlook on the world – the world as tion Such a conception – if it can be substantiated – would indeed change the basic cat-

organiza-egories upon which scientific thought rests, and profoundly influence practical attitudes This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory They agree, however, in being concerned, in one way or another, with “systems,” “wholes” or “organizations”; and in their totality, they herald a new approach (Lilienfeld, 1978, pp 7–8)

As highlighted in the second paragraph above, the reality is that new disciplines begin

to emerge as the knowledge base expands When scientists and philosophers first tried to explain how things worked in the universe, there were no separate disciplines There were questions to be answered and problems to be solved With the progress in understanding

of the universe, science branched out into disciplines such as chemistry, physics, biology, mathematics, etc., and the investigations of a component of a problem were confined to the boundaries of the relevant discipline With exponential increase in the size of the knowledge base, well-established disciplines were further subdivided into specialized fields This pro-cess is likely to continue with time

Real-world phenomena are usually too complex to be understood by modelling all their parts and interactions Traditionally, scientists have simplified natural complexity by view-ing individual items of observation in isolation Since the time of the French mathematician and philosopher Renè Descartes (1596–1650), scientific methods had progressed under two assumptions One was that a system could be broken down into individual parts and ana-lysed as independent entities, and the other was that the parts so broken down can be linearly connected together to describe the whole The logic at the time has been that it is better to

© 2010 Taylor & Francis Group, LLC

have a deeper understanding of the parts than a not-so-deep understanding of the whole The interactions of the parts that together constituted the whole were not the main concern Bertalanffy (1934, 1968), on the contrary, postulated that a system is characterized by the interactions of its parts and the non-linearity of their interactions The systems approach attempts to view the whole as well as the interrelationships among the constituent parts.Systems theory and cybernetics have often been used synonymously, although cybernetics refers to a subset of systems that involve feedback loops Bertalanffy identified cybernetics

as the theory of control mechanisms in technology It is founded on the concepts of mation and feedback, but as part of a GST Bertalanffy reiterated that although it has wide applications, it should not be identified with ‘systems theory’ in general Cybernetics arose from engineering, whereas GST arose from biology

infor-Systems theory is a philosophical doctrine of describing systems as abstract tions independent of substance, type, time, and space Systems theory has influenced many sciences, including engineering fields, management science, mathematics, political science, psychology, sociology, life sciences, and many more In the chapters that follow, the descrip-tion and analysis are confined to environmental and hydrological fields only

organiza-1.3 ECOLOGICAL SYSTEMS (ECOSYSTEMS)

An ecosystem is a community of living and non-living things that work together In nature, everything is connected It can be as small as a microorganism or as large as the entire universe The first attempt to make predictive generalization of the whole world as an eco-system with large cyclic entities interacting with each other and with the environment at the natural level of integration was perhaps by Odum (1950) in his PhD dissertation, although the idea of ecosystem was conceived much earlier (Tansley, 1935) He sought to give a gen-eral statement of natural selection applicable to large entities in the same way as it was

to small entities such as those traditionally studied in biology His first published papers (Odum, 1960a,b) on passive analogues set out the theoretical proposition that Ohm’s law from electrical engineering was analogous to the thermodynamic functioning of ecosys-tems He further explored this idea by constructing and simulating an electrical circuit of the Silver Springs ecosystem in which resistances at locations were considered as analogous

to the producing and consuming populations, batteries as analogous to the energy sources from the sun and organic matter from the external environment, and wires of electrical cir-cuits as analogous to the flow of food energy to consumers In the electric circuit, electron flow represented the material flow in the ecosystem, the charge in the capacitor represented the storage of a material, and the scale of the model was determined by adjusting the sizes

of electrical components This concept became the foundation of Odum’s approach to the development of ecological systems Passive analogues had been used since the 1930s for simulating water flows, neurons, and other systems where Ohm’s law could act as a basis for modelling; however, Odum was the only ecologist to use them for simulating ecosystems (Kangas, 2004)

1.4 EQUI-FINALITY

A system can be represented by a simple single-parameter model or by a more complex parameter model A model with all the characteristics of the system lumped into a single parameter is easy to calibrate As the number of parameters increases, the degree of diffi-culty in calibration also increases In multiparameter systems, the calibration is usually done

multi-© 2010 Taylor & Francis Group, LLC

using some kind of optimization technique A problem that arises in such situations is that the same results may be arrived at with different sets of parameters This is attributed to the principle of equi-finality, which states that in open systems, a given state can be reached in many potential paths or trajectories This principle was postulated by Ludwig von Bertanffy (1934, 1968) in his GST In a closed system, a direct cause–effect relationship exists between the initial and final states of the system

As a calibration strategy that has the ability to simultaneously incorporate several tive functions, multiobjective optimization has its roots in late 19th century welfare eco-nomics in the original work of Francis Ysidro Edgeworth (1881) and later generalized by Vilfredo Pareto (1906) The Pareto set of solutions represent trade-offs with the property that moving from one solution to another results in the improvement of one objective while causing deterioration in one or more others The Pareto set represents the minimum uncer-tainty that can be achieved for the parameters via calibration

objec-Equi-finality for multiparameter optimization means that the same final result may be arrived at from different initial conditions and in different ways Two models are said to be equi-final if they lead to equally acceptable results It is a key concept to assess the uncer-tainty of real-world predictions There is no unique set of parameter values, but rather a feasible parameter space from which a Pareto set of optimal solutions is sought A Pareto set

is a set of states of objective parameters satisfying the criterion of Pareto optimality for

multi-objective optimization problems A state A (a set of target parameters) is said to be Pareto optimal if there is no other state B dominating the state A with respect to a set of objective functions A state A dominates a state B, if A is better than B in at least one objective func-

tion and not worse with respect to all other objective functions

1.5 SCOPE AND LAYOUT

The contents of this book are arranged in 14 chapters Chapter 2 gives a review of the historical development of hydrological modelling, outlining the concepts originated under the assumption of linearity Time and frequency domain analysis and their applications to linear systems through standard input functions such as Dirac delta and unit step functions are described next, followed by a description of the concepts of unit hydrograph, linear reservoir, linear channel, linear cascade, and time–area diagram, all in the context of linear hydrological systems A brief introduction to random processes and linear systems, non-linear systems, and multilinear systems is given next The chapter ends with a description

of flood routing using the hydrologic approach as well as the hydraulic approach, reservoir routing, a review of some rainfall–runoff models, and highlighting the challenges that lie ahead in hydrological modelling

Chapter 3 gives a description of population dynamics starting from the Malthusian nential growth theory Although it has been modified, refuted, or superseded by the works

expo-of subsequent researchers, it still stands as the backbone expo-of many population dynamics models The studies of Verhulst, Pearl and Reed; the Lotka-Volterra predator–prey model; and the many oscillations of the logistic map, including bifurcation and chaos, are briefly reviewed The chapter ends with a description of exponential cell growth, cell division by binary fission, mitosis and meiosis, cell growth in a bioreactor, bacterial growth and binary fission, Monod kinetics, and radioactive decay and carbon dating In summary, the chapter gives a review of the historical development of the study of population dynamics, including those of some microbiological species

Chapter 4 describes reaction kinetics, which is the process of concentration variation

in a chemical or biological reaction The concept of half-life, the relationship between the

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reaction rate and the substrate concentration, the relationship between the growth and sumption rates of the biomass produced and the substrate consumed, as well as two of the widely used models for describing the kinetics of such reactions, the Michaelis Menten equa-tion and the Monod equation, are introduced

con-Chapter 5 describes water quality systems Starting with dissolved oxygen systems, the chapter describes biochemical oxygen demand, nitrification and de-nitrification, oxygen sag curve, re-oxygenation and de-oxygenation coefficients, as well as the dynamics of a completely mixed body of water, including the governing equations and their solutions to specific input functions The chapter ends with a description of the water quality variation

in rivers and streams due to specific waste loads

Chapter 6 describes longitudinal dispersion, an important topic in environmental neering Starting from Fick’s law of diffusion, the chapter first describes turbulent diffusion, shear flow dispersion, Taylor’s approximations, and turbulent mixing coefficients Next, a detailed description of the longitudinal dispersion coefficient, including how it is estimated

engi-by various methods; the analytical solution of the dispersion equation for certain specific inputs, boundary, and initial conditions; and the numerical solution of the dispersion equa-tion using the finite difference method, finite element method, and moving finite element method are given The chapter ends with a brief description of dispersion through porous media and an introduction to some general-purpose water quality models

Chapter 7 describes time series analysis and forecasting It gives the basic properties of time series, homogeneity tests, decomposition of a time series, tests for identification of trends and periodicities, time and frequency domain analyses including the convolution integral and Fourier transforms, correlation and spectral analysis, representation of the dependent stochastic component by various stochastic models, generation of synthetic data using the probability distribution of the independent residuals, forecasting, and basic con-cepts of Kalman filtering Some illustrative examples of time series analysis are also given.Chapter 8 describes artificial neural networks as one of the widely used data-driven tech-niques in environmental and hydrological systems modelling Starting from the biological neuron, the chapter describes the development of the artificial neuron, perceptron, multi-layer perceptron, types of activation functions, types of artificial neural networks such

as feed-forward, recurrent, Kohonen, product unit, different types of learning, back- propagation algorithm, back-propagation through time, data preprocessing methods, and application areas in hydrology and environmental systems

Chapter 9 describes radial basis functions as another form of artificial neural networks It

is an approach that can be used to solve interpolation problems in multidimensional space Different types of radial basis function networks are introduced together with hybrid learn-ing methods in which the selection of centres is done in an unsupervised mode, whereas the optimization of the weights is done in a supervised mode Selection of centres is a key com-ponent of radial basis function networks, and a review of both supervised and unsupervised method of centre selection is given Finally, some example applications in the hydrological context are given

Chapter 10 is about fractals and chaos Fractals are geometric objects that can be divided into parts, each of which is a copy of the whole, a property known as self-similarity They have fine structures at infinitely small scales and can be generated by simple and well-defined recursive equations Examples in nature include clouds, mountains, coastlines, trees, ferns, river networks, cauliflowers, system of blood vessels, snowflakes, etc In the chapter, basic concepts of fractals and chaos, definitions and estimation methods of fractal dimension, and invariant measures for chaotic systems, as well as some examples of well-known fractals such as Cantor set, Sierpinski (gasket) triangle, Koch curve, Koch snow-

sub-flake (or Koch star), Mandelbrot set, Julia set, and the perimeter–area (P–A) relationship of

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fractals, are given In the context of chaos, an introduction, some basic definitions, the terfly effect, the ‘n’-body problem, and the invariants of chaotic systems such as Lyapunov exponent and various measures of entropy are described Finally, some examples of chaotic maps, such as the logistic map, Hénon map, Lorenz map, Duffing, Rössler and Chua’s equa-tions, including some application areas, are given

but-Chapter 11 is about dynamical systems approach to modelling Topics covered include

a comparison between random and chaotic deterministic systems, dynamical systems and their sensitivity to initial conditions, embedding, embedding dimension and methods of estimating embedding dimension, phase–space reconstruction and phase–space predic-tions, some descriptions about non-linearity and determinism, including tests for deter-minism, noise and methods of noise reduction, noise level estimation, and some application areas

Chapter 12 is about support vector machines, which belong to the class of supervised learning and which can be used for optical character recognition; pattern recognition such

as handwriting, speech, images, etc.; classification and regression analysis; and time series prediction Issues such as binary classification, linear and non-linear soft margin classifica-tion, linear support vector regression, non-linear support vector regression, parameter selec-tion, kernel tricks, and some application areas are presented

Chapter 13 describes fuzzy logic systems Basic concepts of fuzzy logic; types of ship functions; rule bases; fuzzy inference systems such as Mamdani, Takagi–Sugeno–Kang, Tsukamoto, and Larsen; fuzzification and de-fuzzification methods; neuro-fuzzy systems; and adaptive neuro-fuzzy systems, including some application areas, are presented in the chapter

member-Chapter 14, which is the last chapter, gives a brief introduction to genetic algorithms and genetic programming As two of the relatively recent variations of evolutionary programs, genetic algorithms and genetic programming have become popular search techniques for parameter optimization in complex non-linear spaces The chapter gives the basic compo-nents of genetic algorithms, coding, genetic operators, and the basic differences between genetic algorithms and genetic programming Some application areas for both types are also given

REFERENCES

Bertalanffy, L von (1934): Untersuchungen über die Gesetzlichkeit des Wachstums I Allgemeine Grundlagen der Theorie; mathematische und physiologische Gesetzlichkeiten des Wachstums bei

Wassertieren Archives Entwicklungsmech, 131, 613–652.

Bertalanffy, L von (1968): General System Theory: Essays on its Foundation and Development, Revised

Edition George Braziller, New York.

Edgeworth, F.Y (1881): Mathematical Psychics P Keagan, London.

Kangas, P (2004): The role of passive electrical analogs in H.T Odum’s systems thinking Ecological Modelling, 178, 101–106.

Lilienfeld, R (1978): The Rise of Systems Theory: An Ideological Analysis Wiley, New York.

Odum, H.T (1950): The biogeochemistry of strontium with discussion on the ecological integration of elements PhD dissertation, Yale University, New Haven 373 pp.

Odum, H.T (1960a): Ecological potential and analogue circuits for the ecosystem American Science,

48, 1–8.

Odum, H.T (1960b): Ten classroom sessions in ecology American Biology Teacher, 22, 71–78 Pareto, V (1906): Manuale di Economia Politica Societa Editrice Libraria, Milan, Italy Translated into English by A.S Schwier as Manual of Political Economy Macmillan, New York, 1971.

Tansley, A.G (1935): The use and abuse of vegetational terms and concepts Ecology, 16(3), 284–307.

Deterministic systems can be considered as lumped parameter systems or as distributed parameter systems In general, from a mathematical point of view, lumped parameter sys-tems are represented by ordinary differential equations and distributed parameter systems are represented by partial differential equations In a linear system, the transformation from input to output takes place via a linear operator.

2.1 BASIC CONCEPTS AND GOVERNING

EQUATION OF LINEAR SYSTEMS

A linear system can be represented by an ordinary differential equation of the form

where x(t) represents an input function, y(t) represents an output function, a i ’s and b i’s are

system parameters, and the superscripts in x and y indicate the corresponding derivatives

with respect to time

All linear systems satisfy the principles of proportionality and superposition; that is, if

y1(t) and y2(t) are output functions corresponding to the input functions x1(t) and x2(t), then

y1(t) + y2(t) will be the output function corresponding to the input function x1(t) + x2(t) and αy1(t) will be the output corresponding to the input function αx1(t) (α is a constant) Linear systems are relatively easier to analyse as the methods of analysis are well estab-lished Analysis can be carried out in the time domain or in the frequency domain

2.1.1 Time domain analysis

To make use of the principles of proportionality and superposition, the input functions

to any linear system can be described as a linear summation of standard input functions Several types of standard input functions can be identified

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2.1.1.1 Types of input functions

a Unit impulse (or Dirac delta) function

The unit impulse (Figure 2.1) is of the Dirac delta function (Dirac, 1958) type, which has the following properties:

b Unit step function

The unit step function, u(t) (Figure 2.2), has the following properties:

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the S-curve Two S-curves separated by a known time interval can be used to convert a

unit hydrograph of one unit duration to that of another unit duration (see Section 2.2.2)

where the x( )τ represents the derivative

In discrete form for real systems, Equations 2.7 and 2.8 can be written as

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2.1.1.2 System response function – convolution integral

The input and output functions of a linear system are related to each other by the tion integral, which takes the form

where h(t) is the system response function For real systems, the variables in the convolution

integral are interchangeable; that is,

The system response function h(t) corresponds to the output function when the input

function is of the impulse (delta) type It is therefore called the impulse response function (IRF) A linear system is completely known if the IRF is known

If the system is linear time variant, then the corresponding equation is of the form

2.1.2 Frequency domain analysis

2.1.2.1 Fourier transform – frequency response function (FRF)

If a linear time invariant system is physically realizable and stable,1 then it can be described

by a frequency response function (FRF) in the frequency domain The FRF H(f) is the Fourier

transform (after Jean Baptiste Joseph Fourier, 1768–1830) of the IRF and is a function of

1 If the input function is bounded, then the output function is also bounded; that is, if |x(t)| < N, then |y(t)| < M;

N, M are constants.

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the frequency f in the same way as h(t) is a function of t Fourier transform enables

decom-position of any wavy signal into sine and cosine waves, thus forming an alternative way of representing the behaviour of the function This is possible because any time domain signal can be represented by a series of sines and cosines

For a complex function, the Fourier transforms2 are given by (Kreyszig, 1999, pp 565–570)

In these equations, H(f) represents the amplitude of the real and imaginary parts of the

sinusoids at each frequency

For an even function, the Fourier cosine transformation is given by

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For an odd function, the Fourier sine transformation is given by

engineering The Laplace transform H(s) of h(t) is given by (Kreyszig, 1999, pp 251)

where s is the variable of the Laplace transform, which has the dimension [T]−1 H(s) is

dimensionless Similar to Fourier transform, the convolution integral transforms to an braic equation by Laplace transformation as well

alge-Y(s) = X(s)H(s) (2.19) where Y(s) and X(s) are the Laplace transforms of y(t) and x(t).

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H s

s

H s t

( )= ( ) is negative

The problem with Laplace transforms is that it is not easy to find the inverse transform to

obtain h(t) Usually H(s) is obtained in a discrete form, and an empirical equation is fitted to

the discrete form before inverse transformation is carried out

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2.2 LINEAR SYSTEMS IN HYDROLOGICAL MODELLING

hydrolog-of such processes can be considered as a system in its own right or as components hydrolog-of a much larger system However, the main concern of hydrologists is the catchment scale in which a relationship between the input rainfall and the corresponding output runoff is sought Some

of the important processes and how they are modelled is first briefly described next

The principal processes in the hydrological cycle are precipitation, evaporation and transpiration, interception, infiltration, runoff as overland flow, interflow and/or baseflow, and subsurface flow under saturated conditions or as moisture flow under partially satu-rated conditions

evapo-At any instant, the atmosphere contains about 13 × 103 km3 (Table 10.4, World Water Development Report 3 [2009]) of water as vapour, liquid, or solid Precipitation is the process

by which this water is deposited on the earth’s surface It can take place in one of several forms, such as rain, snow, mist, hail, sleet, dew, and frost, depending on the prevailing environmental conditions The effect of liquid precipitation on the hydrological cycle is immediate, whereas that of solid precipitation is slow and attenuated Evaporation, which is the process of convert-ing liquid water to gaseous water, or water vapour, returns the water deposited on the earth’s surface back to the atmosphere These two are the processes of exchange of water between the earth’s surface and the atmosphere and forms a bridge between hydrology and meteorology Evaporation is an energy-absorbing process, whereas condensation, which is the process of changing phase from a gaseous phase back to liquid phase, is energy releasing Many of the extreme events taking place in the atmosphere are fuelled by the energy from the sun and the latent heat of condensation released when water vapour becomes liquid water

The next important process in the hydrological cycle is infiltration, which transfers the water on the surface of the earth to the subsurface The infiltrated water reappears as base flow, percolate into deeper layers of the subsoil and form groundwater, or increase the water content stored in the soil

Quantitative information about these three processes is usually obtained by ment, and as such, they become input data to any hydrological modelling system They can, however, be empirically estimated using measurements of the predominant factors that affect them Predictions based on the laws of physics are too complicated, and it is doubtful whether such laws that are defined for an infinitesimally small spatial domain can be applied

measure-to a larger domain typically of interest measure-to hydrologists The scale issue is yet unresolved.The most important catchment process of interest to hydrologists is the transformation of rainfall to runoff Rainfall, which is measurable to a high degree of accuracy, becomes the

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input and the corresponding runoff, which can also be measured but not so easily and not with the same degree of accuracy and reliability, becomes the corresponding output The problem, therefore, is to predict runoff as the output from a hydrological model that takes in rainfall as the input This is the focus of the hydrological systems modelling part of this book

2.2.2 Unit hydrograph

The unit hydrograph method is one of the basic tools in hydrological computations It was originally presented by L.K Sherman in 1932 It is founded upon the assumption that the rainfall excess–direct runoff process is linear, which implies that the principles of superposi-tion and proportionality hold It is defined as the discharge hydrograph resulting from a unit depth of rainfall excess lasting for a specified period called the unit duration and assumed to

be uniformly distributed over the catchment Rainfall excess is that portion of the rainfall that comes out as direct runoff Direct runoff is the difference between the total runoff and base flow It is that component directly resulting from rainfall

Under the principles of the unit hydrograph, the hydrograph corresponding to any tive rainfall excess can be obtained via the convolution integral The limitation for the use

effec-of the convolution integral is that the unit hydrograph duration should be the same as the unit interval of the rainfall excess hyetograph Otherwise, methods of converting a unit hydrograph of one unit duration to that of a different unit duration should be followed If the required unit duration is an integer multiple of the known unit duration, superposition

of a number of unit hydrographs of the given unit duration, each displaced by the given unit duration, and dividing the cumulative hydrograph by the number of unit hydrographs superposed will give the unit hydrograph of the required unit duration For example, a 4-h unit hydrograph can be obtained by the superposition of two unit hydrographs each of 2-h unit duration and dividing by two, or by the superposition of four unit hydrographs each

of 1-h unit duration and dividing by four, etc If the required unit duration is not an integer

multiple of the known unit duration, the S-curve method, which is obtained by summing

up an infinite number of unit hydrographs of the known unit duration, each displaced by the given unit duration, shifting it by the time interval of the required unit duration, tak-

ing the difference between the two S-curves so obtained and diving by the ratio of the two

unit durations, should be used The summation of a large number of unit hydrographs is equivalent to the hydrograph resulting from a continuous rainfall excess of intensity equal to

the reciprocal of the given unit duration For example, if the given unit duration is t0 hours, the summation is equivalent to a continuous rainfall excess of intensity 1

0

t The resulting

hydrograph obtained by subtracting one S-curve from another is not a unit hydrograph and

therefore should be divided by ′t

t00, where 0′t is the required unit duration A property of the

S-curve is that it will level off after a finite number of superpositions.

The principles of proportionality and superposition lead to very useful practical tions of the unit hydrograph concept For example, a hydrograph of discharge resulting from

applica-a series of rapplica-ainfapplica-all excesses mapplica-ay be constructed by summing up the hydrograpplica-aphs due to eapplica-ach single unit of rainfall excess They also imply that the time base of direct runoff hydrographs resulting from rainfall excesses of same unit duration is the same regardless of the intensity.The unit duration corresponds to the rate at which the catchment is filled up to a rainfall excess

of unit depth Application of a given amount of rainfall excess at different rates gives different responses For example, the consequence of a 100-mm rain falling in a duration of 1 h is quite different from the same amount of rain falling in 1 day The rate of application of the rainfall excess therefore plays a significant role in the runoff generation characteristic of the catchment

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The catchment can be thought of as a tank or reservoir that receives the rainfall excess and releases it through an outlet The catchment characteristics are reflected in the discharge coefficient of the tank or reservoir outlet, which affects the recession part of the hydrograph The rising part is affected by the rate of application of the unit rainfall excess

2.2.2.1 Unit hydrograph for a complex storm

Using the convolution method, which is a linear superposition involving multiplication, translation in time, and addition, the discharge corresponding to a multiple unit rainfall excess can be obtained The convolution integral is given as follows:

because u t = 0 for t ≤ 0 and t ≥ m Expanding Equations 2.26a and 2.26b, the following set

of equations can be obtained:

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In Equation 2.27, x1, x2,…, x r are the rainfall excess values, u1, u2,…,u m are the unit

hydrograph ordinates, y1, y2,…,y n are the direct runoff hydrograph ordinates, and n = r +

m − 1 With this system of equations, it is relatively easy to determine the direct runoff

hydrograph if the input rainfalls and the unit hydrograph ordinates are known by direct substitution In practice, however, the main issue is deriving the unit hydrograph ordinates from a given set of rainfall excess and corresponding direct runoff The following procedure can be adopted for this purpose

Equation 2.27, in matrix form, can be written as

x

u u u

u

y y y

y y

m

r r

which is of the form

[x][u] = [y] (2.28a)

where

[x] is a matrix of dimension (m + r – 1) × m

[u] is a vector of dimension m

[y] is a vector of dimension m + r – 1

The solution of Equation 2.28 is not straightforward because [x] is not a square matrix

There are more equations than the number of unknowns in the system To get the optimal

It would be a square matrix of dimension m × m because, [x] T is of dimension m × (m +

r − 1) and [x] is of dimension (m + r − 1) × m, and can therefore be inverted.

Then,

from which

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Equation 2.30 does not lead to an exact solution to the problem but gives a unique tion Solutions could also be obtained in a sequential manner as follows:

solu-First row of Equation 2.27 gives u y

accu-or from the end There are maccu-ore equations than unknowns A unit hydrograph can be made dimensionless by dividing the discharge by a reference discharge and the time by a reference time

A variety of unit hydrographs and dimensionless unit hydrographs can be found in the literature: the general dimensionless unit hydrograph based on the cascade of linear res-ervoirs and dependent only on the number of linear reservoirs’ and the Courant number, which in the context of unit hydrograph is equal to the ratio of the duration of the unit hydrograph and the storage coefficient of the reservoir (Ponce, 1989a,b); a geomorphologi-cal unit hydrograph in which Horton’s stream laws have been used to integrate the delay effects of streams and expresses the unit hydrograph as a probability density function of travel time to the catchment outlet (Rodríguez-Iturbe and Valdés, 1979); a unit hydrograph formulated by combining a time–area diagram with a linear reservoir to take into account both translation and storage effects (Clark, 1945); a spatially distributed unit hydrograph that is similar in concept to the geomorphological unit hydrograph but uses GIS to describe the connectivity of the links in the streamflow network, thereby eliminating the need to use

a probability density function (Maidment, 1993); a synthetic unit hydrograph defined by the time to peak, time base, peak discharge, and the widths of the hydrograph at 50% and 75%

of the peak discharge, which are all related to catchment characteristics (Snyder, 1938); and

a dimensionless unit hydrograph obtained from a large number of unit hydrographs from a number of catchments of different sizes and locations in which the dimensionless discharge

is expressed as the ratio of discharge to peak discharge and dimensionless time is expressed

as the ratio of time to the time to peak of the hydrograph (US Soil Conservation Service,

1972, 1985)

2.2.2.2 Instantaneous unit hydrograph (IUH)

The instantaneous unit hydrograph (IUH) u(0,t) for a catchment is the hydrograph of direct

runoff resulting from a finite volume (or depth) of rainfall excess falling in an infinitesimally

short time In terms of the delta function, such a rainfall excess may be expressed as V0δ(0), where V0 is the volume of rainfall excess A t0 hour unit hydrograph u(t0, t) is the direct runoff hydrograph resulting from a finite volume (or depth) of rainfall excess falling in t0

hours The conversion of the IUH to one of finite duration can be done using superposition

2.2.2.3 Empirical unit hydrograph

The actual unit hydrograph derived from a particular storm, also referred to as the cal unit hydrograph, can be obtained by selecting a storm that is isolated, intense, and uni-form over the catchment and time by scanning through rainfall and runoff records with the duration of the storm not greater than the period of rise (time of concentration), separating the base flow and determining the direct runoff hydrograph, finding the depth (or volume)