Environmental and Hydrological Systems Modelling

Environmental

and Hydrological

Systems Modelling

A. W. Jayawardena

Jayawardena

ISBN-13: 978-0-415-46532-8

9 780415 465328

9 0 0 0 0

RU54409

EnvironmEntal EnginEEring

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:

• Linear Systems

• Conceptual Models

• Data Driven Models

• Process-Based Models

• Risk-Management Models

• 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

RU54409_Cover_mech.indd All Pages 12/3/13 8:53 AM

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

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the

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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© 2010 Taylor & Francis Group, LLC v

Contents

Preface xvii

Author xix

1 Introduction 1

1.1 Some definitions 1

1.1.1 System 1

1.1.2 State of a system 2

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

2.1.1.1 Types of input functions 10

2.1.1.2 System response function – convolution integral 12

2.1.2 Frequency domain analysis 12

2.1.2.1 Fourier transform – frequency response function (FRF) 12

2.1.2.2 Laplace transform 14

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

2.2.2.1 Unit hydrograph for a complex storm 18

2.2.2.2 Instantaneous unit hydrograph (IUH) 20

2.2.2.3 Empirical unit hydrograph 20

2.2.2.4 Unit pulse response function 21

2.2.3 Linear reservoir 21

2.2.4 Linear cascade 23

2.2.5 Linear channel 25

2.2.6 Time–area diagram 26

2.3 Random processes and linear systems 27

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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

2.6.2.1 Estimation of the routing parameters K and c 33

2.6.2.2 Limitations of the Muskingum method 35

2.6.3 Modified Puls method 35

2.6.4 Muskingum–Cunge method 35

2.6.5 Hydraulic approach 37

2.6.5.1 Solution of the St. Venant equations 37

2.6.5.2 Diffusion wave approximation 38

2.6.5.3 Kinematic wave approximation 38

2.7 Reservoir routing 41

2.8 Rainfall–runoff modelling 43

2.8.1 Conceptual-type hydrologic models 44

2.8.1.1 Stanford watershed model (SWM) 44

2.8.1.2 Tank model 44

2.8.1.3 HEC series 45

2.8.1.4 Xinanjiang model 47

2.8.1.5 Variable infiltration capacity (VIC) model 49

2.8.2 Physics-based hydrologic models 51

2.8.2.1 Système Hydrologique Europèen (SHE) model 51

2.8.3 Data-driven models 52

2.8.3.1 Why data-driven models? 53

2.8.3.2 Types of data-driven models 53

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

3.5 Gompertz curve 65

3.6 Logistic map 66

3.6.1 Specific points in the logistic map 67

3.7 Cell growth 68

3.7.1 Cell division 69

3.7.2 Exponential growth 70

3.7.3 Cell growth models in a batch (closed system) bioreactor 70

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3.8 Bacterial growth 72

3.8.1 Binary fission 73

3.8.2 Monod kinetics 73

3.9 Radioactive decay and carbon dating 74

3.10 Concluding remarks 75

References 76

4 Reaction kinetics 77

4.1 Introduction 77

4.2 Michaelis–Menten equation 78

4.3 Monod equation 81

4.4 Concluding remarks 84

References 84

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

5.2.6 Impulse input 101

5.2.7 Arbitrary input 101

5.3 Water quality in rivers and streams 106

5.3.1 Point sources 106

5.3.2 Distributed sources 108

5.3.3 Effect of spatial flow variation 109

5.3.3.1 Exponential spatial flow variation 110

5.3.4 Unsteady state 111

5.3.4.1 Non-dispersive systems 111

5.3.4.2 Dispersive systems 111

5.3.5 Tidal reaches 113

5.3.5.1 Special case of no decay 113

5.3.5.2 Special case of no dispersion 114

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

6.2.3 Taylor’s approximation 120

6.2.4 Turbulent mixing coefficients 120

6.3 Dispersion coefficient 121

6.3.1 Routing method 123

6.3.2 Time scale – dimensionless time 124

6.4 Numerical solution 126

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

References 136

7 Time series analysis and forecasting 139

7.1 Introduction 139

7.2 Basic properties of a time series 139

7.2.1 Stationarity 139

7.2.2 Ergodicity 140

7.2.3 Homogeneity 140

7.3 Statistical parameters of a time series 140

7.3.1 Sample moments 140

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

7.5.2 Cumulative deviations 145

7.5.3 Bayesian statistics 148

7.5.4 Ratio test 148

7.5.5 Pettit test 150

7.6 Components of a time series 151

7.7 Trend analysis 151

7.7.1 Tests for randomness and trend 151

7.7.1.1 Turning point test for randomness 152

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

7.7.1.3 Regression test for linear trend 154

7.7.1.4 Mann–Kendall test 155

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

7.8.3.1 Hanning method (after J. von Hann) 171

7.8.3.2 Hamming method (after R.W. Hamming, 1983) 171

7.8.3.3 Lag window method (after Tukey, 1965) 172

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

7.9.1.1 Properties of autoregressive models 175

7.9.1.2 Estimation of parameters 176

7.9.1.3 First-order model (lag-one Markov model) 177

7.9.1.4 Second-order model (lag-two model) 179

7.9.1.5 Partial autocorrelation function (PAF) 180

7.9.2 Moving average (MA) models 181

7.9.2.1 Properties of MA models 182

7.9.2.2 Parameters of MA models 182

7.9.2.3 MA(1) model 183

7.9.2.4 MA(2) model 184

7.9.3 Autoregressive moving average (ARMA) models 185

7.9.3.1 Properties of ARMA(p,q) models 185

7.9.3.2 ARMA(1,1) model 185

7.9.4 Backshift operator 186

7.9.5 Difference operator 187

7.9.6 Autoregressive integrated moving average (ARIMA) models 187

7.10 Residual series 188

7.10.1 Test of independence 188

7.10.2 Test of normality 188

7.10.3 Other distributions 189

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

7.13 ARMAX modelling 197

7.14 Kalman filtering 198

7.15 Parameter estimation 202

7.16 Applications 204

7.17 Concluding remarks 204

Appendix 7.1: Fourier series representation of a periodic function 205

References 207

8 Artificial neural networks 211

8.1 Introduction 211

8.2 Origin of artificial neural networks 212

8.2.1 Biological neuron 212

8.2.2 Artificial neuron 212

8.2.2.1 Bias/threshold 213

8.3 Unconstrained optimization techniques 215

8.3.1 Method of steepest descent 215

8.3.2 Newton’s method (quadratic approximation) 216

8.3.3 Gauss–Newton method 216

8.3.4 LMS algorithm 217

8.4 Perceptron 218

8.4.1 Linear separability 219

8.4.2 ‘AND’, ‘OR’, and ‘XOR’ operations 220

8.4.3 Multilayer perceptron (MLP) 221

8.4.4 Optimal structure of an MLP 222

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

8.6.2.1 Back-propagation through time (BPTT) 235

8.6.3 Self-organizing maps (Kohonen networks) 237

8.6.4 Product unit–based neural networks (PUNN) 239

8.6.4.1 Generation of the initial population 242

8.6.4.2 Fitness function 242

8.6.4.3 Parametric mutation 242

8.6.4.4 Structural mutation 244

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

8.7.2.1 Error correction learning (optimum filtering) 249

8.7.2.2 Memory-based learning 249

8.7.2.3 Hebbian learning (Hebb, 1949) (unsupervised) 250

8.7.2.4 Competitive learning (unsupervised) 250

8.7.2.5 Boltzmann learning 251

8.7.2.6 Reinforced learning (unsupervised) 251

8.7.2.7 Hybrid learning 251

8.7.3 Learning rate (η) and momentum term (α) 252

8.8 BP algorithm 252

8.8.1 Generalized delta rule 256

8.9 ANN implementation details 256

8.9.1 Data preprocessing: Principal Component Analysis (PCA) 256

8.9.1.1 Eigenvalue decomposition 259

8.9.1.2 Deriving the new data set 260

8.9.2 Data normalization 260

8.9.3 Choice of input variables 262

8.9.4 Heuristics for implementation of BP 262

8.9.5 Stopping criteria 262

8.9.6 Performance criteria 263

8.10 Feedback Systems 264

8.11 Problems and limitations 265

8.12 Application areas 265

8.12.1 Hydrological applications 265

8.12.1.1 River discharge prediction 266

8.12.2 Environmental applications 276

8.12.2.1 Algal bloom prediction, Hong Kong 276

8.13 Concluding remarks 279

References 279

9 Radial basis function (RBF) neural networks 287

9.1 Introduction 287

9.2 Interpolation 287

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9.3 Regularization 288

9.4 Generalized RBFs 291

9.5 Normalized radial basis functions (NRBFs) and kernel regression 294

9.6 Learning of RBFs 296

9.6.1 Fixed centres selection (random) 297

9.6.2 Forward selection 298

9.6.3 Orthogonal least squares (OLS) algorithm 298

9.6.3.1 Regularized orthogonal least squares (ROLS) algorithm 302

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

9.6.6.1 Training of RBF networks 308

9.6.6.2 Computational procedure 312

9.6.7 Other methods of learning 313

9.7 Curse of dimensionality 314

9.8 Performance criteria 315

9.9 Comparison of MLP versus RBF networks 315

9.10 Applications 316

9.11 Concluding remarks 318

References 318

10 Fractals and chaos 321

10.1 Introduction 321

10.2 Fractal dimensions 322

10.2.1 Topological dimension 322

10.2.2 Fractal dimension 322

10.2.3 Hausdorff dimension 324

10.2.4 Box-counting dimension 324

10.2.5 Similarity dimension 325

10.2.6 Packing dimension 325

10.2.7 Information dimension 325

10.2.8 Capacity dimension 326

10.2.9 Rényi dimension 326

10.2.10 Correlation dimension 327

10.3 Examples of some well-known fractals 328

10.3.1 Cantor set 328

10.3.2 Sierpinski (gasket) triangle 330

10.3.3 Koch curve 332

10.3.4 Koch snowflake (or Koch star) 333

10.3.5 Mandelbrot set 333

10.3.6 Julia set 335

10.4 Perimeter–area relationship of fractals 335

10.5 Chaos 337

10.5.1 Butterfly effect 337

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10.5.2 The n-body problem 338

10.6 Some definitions 339

10.6.1 Metric space 339

10.6.2 Manifold 339

10.6.3 Map 339

10.6.4 Attractor 340

10.6.4.1 Strange attractor 340

10.6.5 Dynamical system 340

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

10.8.3 Lorenz map 352

10.8.4 Duffing equation 356

10.8.5 Rössler equations 359

10.8.6 Chua’s equation 360

10.9 Applications areas of chaos 362

10.10 Concluding remarks 362

References 362

11 Dynamical systems approach of modelling 365

11.1 Introduction 365

11.2 Random versus chaotic deterministic systems 366

11.3 Time series as a dynamical system 367

11.3.1 Dynamical system 368

11.3.2 Sensitivity to initial conditions 369

11.4 Embedding 369

11.4.1 Embedding theorem 370

11.4.2 Embedding dimension 372

11.4.2.1 False nearest neighbour (FNN) method 372

11.4.2.2 Singular value decomposition (SVD) 375

11.4.3 Delay time 378

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

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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

11.9.1 Noise in data 390

11.9.2 Noise reduction 392

11.9.3 Noise level 396

11.10 Application areas 401

11.11 Concluding remarks 402

Appendices

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

12.5 Parameter selection 427

12.6 Kernel tricks 427

12.7 Quadratic programming 428

12.8 Limitations and problems 428

12.9 Application areas 428

12.10 Concluding remarks 429

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

References 433

13 Fuzzy logic systems 437

13.1 Introduction 437

13.2 Fuzzy sets and fuzzy operations 438

13.2.1 Fuzzy sets 438

13.2.2 Logical operators AND, OR, and NOT 441

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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

13.3 Membership functions 444

13.3.1 Triangular 444

13.3.2 Trapezoidal 445

13.3.3 Gaussian 446

13.3.4 Asymmetric Gaussian 446

13.3.5 Generalized bell-shaped Gaussian 447

13.3.6 Sigmoidal 447

13.3.7 Singleton 447

13.4 Fuzzy rules 448

13.5 Fuzzy inference 450

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

13.7.1 Hybrid learning 472

13.8 Application areas 472

13.9 Concluding remarks 485

References 485

14 Genetic algorithms (GAs) and genetic programming (GP) 489

14.1 Introduction 489

14.2 Coding 490

14.3 Genetic operators 491

14.4 Parameters of GA 492

14.5 Genetic programming (GP) 492

14.6 Application areas 494

14.7 Concluding remarks 494

References 494