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: Population dynamics Reaction kinetics Water quality systems Longitudinal dispersion Time series analysis and forecasting Artificial neural networks Fractals and chaos Dynamical systems Support vector machines Fuzzy logic systems Genetic algorithms and genetic programming This book will be of great value to advanced students, professionals, academics and researchers working in the water environment.

Table of Contents:
Preface, Author, 1 Introduction, 1.1 Some definitions, 1.1.1 System, 1.1.2 State of a system, 1.2 General systems theory (GST), 1.3 Ecological systems (Ecosystems), 1.4 Equi-finality, 1.5 Scope and layout, References, 2 Historical development of hydrological modelling, 2.1 Basic concepts and governing equation of linear systems, 2.1.1 Time domain analysis, 2.1.1.1 Types of input functions, 2.1.1.2 System response function – convolution integral, 2.1.2 Frequency domain analysis, 2.1.2.1 Fourier transform – frequency response function (FRF), 2.1.2.2 Laplace transform, 2.1.2.3 z-Transform, 2.2 Linear systems in hydrological modelling, 2.2.1 Hydrological systems, 2.2.2 Unit hydrograph, 2.2.2.1 Unit hydrograph for a complex storm, 2.2.2.2 Instantaneous unit hydrograph (IUH), 2.2.2.3 Empirical unit hydrograph, 2.2.2.4 Unit pulse response function, 2.2.3 Linear reservoir, 2.2.4 Linear cascade, 2.2.5 Linear channel, 2.2.6 Time–area diagram, 2.3 Random processes and linear systems, 2.4 Non-linear systems, 2.4.1 Determination of the kernel functions, 2.5 Multilinear or parallel systems, 2.6 Flood routing, 2.6.1 Inventory method, 2.6.2 Muskingum method, 2.6.2.1 Estimation of the routing parameters K and c, 2.6.2.2 Limitations of the Muskingum method, 2.6.3 Modified Puls method, 2.6.4 Muskingum–Cunge method, 2.6.5 Hydraulic approach, 2.6.5.1 Solution of the St. Venant equations, 2.6.5.2 Diffusion wave approximation, 2.6.5.3 Kinematic wave approximation, 2.7 Reservoir routing, 2.8 Rainfall–runoff modelling, 2.8.1 Conceptual-type hydrologic models, 2.8.1.1 Stanford watershed model (SWM), 2.8.1.2 Tank model, 2.8.1.3 HEC series, 2.8.1.4 Xinanjiang model, 2.8.1.5 Variable infiltration capacity (VIC) model, 2.8.2 Physics-based hydrologic models, 2.8.2.1 Système Hydrologique Europèen (SHE) model, 2.8.3 Data-driven models, 2.8.3.1 Why data-driven models?, 2.8.3.2 Types of data-driven models, 2.9 Guiding principles and criteria for choosing a model, 2.10 Challenges in hydrological modelling, 2.11 Concluding remarks, References, 3 Population dynamics, 3.1 Introduction, 3.2 Malthusian growth model, 3.3 Verhulst growth model, 3.4 Predator–prey (Lotka–Volterra) model, 3.5 Gompertz curve, 3.6 Logistic map, 3.6.1 Specific points in the logistic map, 3.7 Cell growth, 3.7.1 Cell division, 3.7.2 Exponential growth, 3.7.3 Cell growth models in a batch (closed system) bioreactor, 3.8 Bacterial growth, 3.8.1 Binary fission, 3.8.2 Monod kinetics, 3.9 Radioactive decay and carbon dating, 3.10 Concluding remarks, References, 4 Reaction kinetics, 4.1 Introduction, 4.2 Michaelis–Menten equation, 4.3 Monod equation, 4.4 Concluding remarks, References, 5 Water quality systems, 5.1 Dissolved oxygen systems, 5.1.1 Biochemical oxygen demand (BOD), 5.1.2 Nitrification, 5.1.3 Denitrification, 5.1.4 Oxygen depletion equation in a river due to a single point source of BOD, 5.1.5 Reoxygenation coefficient, 5.1.6 Deoxygenation coefficient, 5.2 Water quality in a completely mixed water body, 5.2.1 Governing equations for a completely mixed system, 5.2.2 Step function input, 5.2.3 Periodic input function, 5.2.4 Fourier series input, 5.2.5 General harmonic response, 5.2.6 Impulse input, 5.2.7 Arbitrary input, 5.3 Water quality in rivers and streams, 5.3.1 Point sources, 5.3.2 Distributed sources, 5.3.3 Effect of spatial flow variation, 5.3.3.1 Exponential spatial flow variation, 5.3.4 Unsteady state, 5.3.4.1 Non-dispersive systems, 5.3.4.2 Dispersive systems, 5.3.5 Tidal reaches, 5.3.5.1 Special case of no decay, 5.3.5.2 Special case of no dispersion, 5.4 Concluding remarks, References, 6 Longitudinal dispersion, 6.1 Introduction, 6.2 Governing equations, 6.2.1 Some characteristics of turbulent diffusion, 6.2.2 Shear flow dispersion, 6.2.3 Taylor’s approximation, 6.2.4 Turbulent mixing coefficients, 6.3 Dispersion coefficient, 6.3.1 Routing method, 6.3.2 Time scale – dimensionless time, 6.4 Numerical solution, 6.4.1 Finite difference method, 6.4.2 Finite element methods, 6.4.3 Moving finite elements, 6.5 Dispersion through porous media, 6.6 General-purpose water quality models, 6.6.1 Enhanced Stream Water Quality Model (QUAL2E), 6.6.2 Water Quality Analysis Simulation Programme (WASP), 6.6.3 One Dimensional Riverine Hydrodynamic and Water Quality Model (EPD-RIV1), 6.7 Concluding remarks, References, 7 Time series analysis and forecasting, 7.1 Introduction, 7.2 Basic properties of a time series, 7.2.1 Stationarity, 7.2.2 Ergodicity, 7.2.3 Homogeneity, 7.3 Statistical parameters of a time series, 7.3.1 Sample moments, 7.3.2 Moving averages – low-pass filtering, 7.3.3 Differencing – high-pass filtering, 7.3.4 Recursive means and variances, 7.4 Tests for stationarity, 7.5 Tests for homogeneity, 7.5.1 von Neumann ratio, 7.5.2 Cumulative deviations, 7.5.3 Bayesian statistics, 7.5.4 Ratio test, 7.5.5 Pettit test, 7.6 Components of a time series, 7.7 Trend analysis, 7.7.1 Tests for randomness and trend, 7.7.1.1 Turning point test for randomness, 7.7.1.2 Kendall’s rank correlation test (t test), 7.7.1.3 Regression test for linear trend, 7.7.1.4 Mann–Kendall test, 7.7.2 Trend removal, 7.7.2.1 Splines, 7.8 Periodicity, 7.8.1 Harmonic analysis – cumulative periodogram, 7.8.2 Autocorrelation analysis, 7.8.3 Spectral analysis, 7.8.3.1 Hanning method (after J. von Hann), 7.8.3.2 Hamming method (after R.W. Hamming, 1983), 7.8.3.3 Lag window method (after Tukey, 1965), 7.8.4 Cross correlation, 7.8.5 Cross-spectral density function, 7.9 Stochastic component, 7.9.1 Autoregressive (AR) models, 7.9.1.1 Properties of autoregressive models, 7.9.1.2 Estimation of parameters, 7.9.1.3 First-order model (lag-one Markov model), 7.9.1.4 Second-order model (lag-two model), 7.9.1.5 Partial autocorrelation function (PAF), 7.9.2 Moving average (MA) models, 7.9.2.1 Properties of MA models, 7.9.2.2 Parameters of MA models, 7.9.2.3 MA(1) model, 7.9.2.4 MA(2) model, 7.9.3 Autoregressive moving average (ARMA) models, 7.9.3.1 Properties of ARMA(p,q) models, 7.9.3.2 ARMA(1,1) model, 7.9.4 Backshift operator, 7.9.5 Difference operator, 7.9.6 Autoregressive integrated moving average (ARIMA) models, 7.10 Residual series, 7.10.1 Test of independence, 7.10.2 Test of normality, 7.10.3 Other distributions, 7.10.4 Test for parsimony, 7.10.4.1 Akaike information criterion (AIC) and Bayesian information criterion (BIC), 7.10.4.2 Schwartz Bayesian criterion (SBC), 7.11 Forecasting, 7.11.1 Minimum mean square error type difference equation, 7.11.2 Confidence limits, 7.11.3 Forecast errors, 7.11.4 Numerical examples of forecasting, 7.12 Synthetic data generation, 7.13 ARMAX modelling, 7.14 Kalman filtering, 7.15 Parameter estimation, 7.16 Applications, 7.17 Concluding remarks, Appendix 7.1: Fourier series representation of a periodic function, References, 8 Artificial neural networks, 8.1 Introduction, 8.2 Origin of artificial neural networks, 8.2.1 Biological neuron, 8.2.2 Artificial neuron, 8.2.2.1 Bias/threshold, 8.3 Unconstrained optimization techniques, 8.3.1 Method of steepest descent, 8.3.2 Newton’s method (quadratic approximation), 8.3.3 Gauss–Newton method, 8.3.4 LMS algorithm, 8.4 Perceptron, 8.4.1 Linear separability, 8.4.2 ‘AND’, ‘OR’, and ‘XOR’ operations, 8.4.3 Multilayer perceptron (MLP), 8.4.4 Optimal structure of an MLP, 8.5 Types of activation function, 8.5.1 Linear activation function (unbounded), 8.5.2 Saturating activation function (bounded), 8.5.3 Symmetric saturating activation function (bounded), 8.5.4 Positive linear activation function, 8.5.5 Hardlimiter (Heaviside function; McCulloch– Pitts model) activation function, 8.5.6 Symmetric hardlimiter activation function, 8.5.7 Signum function, 8.5.8 Triangular activation function, 8.5.9 Sigmoid logistic activation function, 8.5.10 Sigmoid hyperbolic tangent function, 8.5.11 Radial basis functions, 8.5.11.1 Multiquadratic, 8.5.11.2 Inverse multiquadratic, 8.5.11.3 Gaussian, 8.5.11.4 Polyharmonic spline function, 8.5.11.5 Thin plate spline function, 8.5.12 Softmax activation function, 8.6 Types of artificial neural networks, 8.6.1 Feed-forward neural networks, 8.6.2 Recurrent neural networks, 8.6.2.1 Back-propagation through time (BPTT), 8.6.3 Self-organizing maps (Kohonen networks), 8.6.4 Product unit–based neural networks (PUNN), 8.6.4.1 Generation of the initial population, 8.6.4.2 Fitness function, 8.6.4.3 Parametric mutation, 8.6.4.4 Structural mutation, 8.6.5 Wavelet neural networks, 8.7 Learning modes and learning, 8.7.1 Learning modes, 8.7.2 Types of learning, 8.7.2.1 Error correction learning (optimum filtering), 8.7.2.2 Memory-based learning, 8.7.2.3 Hebbian learning (Hebb, 1949) (unsupervised), 8.7.2.4 Competitive learning (unsupervised), 8.7.2.5 Boltzmann learning, 8.7.2.6 Reinforced learning (unsupervised), 8.7.2.7 Hybrid learning, 8.7.3 Learning rate (.) and momentum term (a), 8.8 BP algorithm, 8.8.1 Generalized delta rule, 8.9 ANN implementation details, 8.9.1 Data preprocessing: Principal Component Analysis (PCA), 8.9.1.1 Eigenvalue decomposition, 8.9.1.2 Deriving the new data set, 8.9.2 Data normalization, 8.9.3 Choice of input variables, 8.9.4 Heuristics for implementation of BP, 8.9.5 Stopping criteria, 8.9.6 Performance criteria, 8.10 Feedback Systems, 8.11 Problems and limitations, 8.12 Application areas, 8.12.1 Hydrological applications, 8.12.1.1 River discharge prediction, 8.12.2 Environmental applications, 8.12.2.1 Algal bloom prediction, Hong Kong, 8.13 Concluding remarks, References, 9 Radial basis function (RBF) neural networks, 9.1 Introduction, 9.2 Interpolation, 9.3 Regularization, 9.4 Generalized RBFs, 9.5 Normalized radial basis functions (NRBFs) and kernel regression, 9.6 Learning of RBFs, 9.6.1 Fixed centres selection (random), 9.6.2 Forward selection, 9.6.3 Orthogonal least squares (OLS) algorithm, 9.6.3.1 Regularized orthogonal least squares (ROLS) algorithm, 9.6.4 Self-organized selection of centres, 9.6.5 Supervised selection of centres, 9.6.6 Selection of centres using the concept of generalized degrees of freedom, 9.6.6.1 Training of RBF networks, 9.6.6.2 Computational procedure, 9.6.7 Other methods of learning, 9.7 Curse of dimensionality, 9.8 Performance criteria, 9.9 Comparison of MLP versus RBF networks, 9.10 Applications, 9.11 Concluding remarks, References, 10 Fractals and chaos, 10.1 Introduction, 10.2 Fractal dimensions, 10.2.1 Topological dimension, 10.2.2 Fractal dimension, 10.2.3 Hausdorff dimension, 10.2.4 Box-counting dimension, 10.2.5 Similarity dimension, 10.2.6 Packing dimension, 10.2.7 Information dimension, 10.2.8 Capacity dimension, 10.2.9 Rényi dimension, 10.2.10 Correlation dimension, 10.3 Examples of some well-known fractals, 10.3.1 Cantor set, 10.3.2 Sierpinski (gasket) triangle, 10.3.3 Koch curve, 10.3.4 Koch snowflake (or Koch star), 10.3.5 Mandelbrot set, 10.3.6 Julia set, 10.4 Perimeter–area relationship of fractals, 10.5 Chaos, 10.5.1 Butterfly effect, 10.5.2 The n-body problem, 10.6 Some definitions, 10.6.1 Metric space, 10.6.2 Manifold, 10.6.3 Map, 10.6.4 Attractor, 10.6.4.1 Strange attractor, 10.6.5 Dynamical system, 10.6.6 Phase (or state) space, 10.7 Invariants of chaotic systems, 10.7.1 Lyapunov exponent, 10.7.2 Entropy of a dynamical system, 10.7.2.1 Kolmogorov–Sinai (K–S) entropy, 10.7.2.2 Modified correlation entropy, 10.7.2.3 K–S entropy and the Lyapunov spectrum, 10.8 Examples of known chaotic attractors, 10.8.1 Logistic map, 10.8.1.1 Bifurcation, 10.8.2 Hénon map, 10.8.3 Lorenz map, 10.8.4 Duffing equation, 10.8.5 Rössler equations, 10.8.6 Chua’s equation, 10.9 Applications areas of chaos, 10.10 Concluding remarks, References, 11 Dynamical systems approach of modelling, 11.1 Introduction, 11.2 Random versus chaotic deterministic systems, 11.3 Time series as a dynamical system, 11.3.1 Dynamical system, 11.3.2 Sensitivity to initial conditions, 11.4 Embedding, 11.4.1 Embedding theorem, 11.4.2 Embedding dimension, 11.4.2.1 False nearest neighbour (FNN) method, 11.4.2.2 Singular value decomposition (SVD), 11.4.3 Delay time, 11.4.3.1 Average mutual information, 11.4.4 Irregular embeddings, 11.5 Phase (or state) space reconstruction, 11.6 Phase space prediction, 11.7 Inverse problem, 11.7.1 Prediction error, 11.8 Non-linearity and determinism, 11.8.1 Test for non-linearity, 11.8.1.1 Significance, 11.8.1.2 Test statistics, 11.8.1.3 Method of surrogate data, 11.8.1.4 Null hypotheses, 11.8.2 Test for determinism, 11.9 Noise and noise reduction, 11.9.1 Noise in data, 11.9.2 Noise reduction, 11.9.3 Noise level, 11.10 Application areas, 11.11 Concluding remarks, Appendices, Appendix 11.1: Derivation of Equation 11.81, Appendix 11.2: Proof of Equation 11.82b, Appendix 11.3: Proof of Equation A1-4, References, 12 Support vector machines, 12.1 Introduction, 12.2 Linearly separable binary classificatio, 12.3 Soft-margin binary classification, 12.3.1 Linear soft margin, 12.3.2 Non-linear classification, 12.4 Support vector regression, 12.4.1 Linear support vector regression, 12.4.2 Non-linear support vector regression, 12.5 Parameter selection, 12.6 Kernel tricks, 12.7 Quadratic programming, 12.8 Limitations and problems, 12.9 Application areas, 12.10 Concluding remarks, Appendix 12.1: Statistical learning, Empirical risk minimization (ERM), Structural risk minimization (SRM), Appendix 12.2: Karush–Kuhn–Tucker (KKT) conditions, References, 13 Fuzzy logic systems, 13.1 Introduction, 13.2 Fuzzy sets and fuzzy operations, 13.2.1 Fuzzy sets, 13.2.2 Logical operators AND, OR, and NOT, 13.2.2.1 Intersection, 13.2.2.2 Union, 13.2.2.3 Other useful definitions, 13.2.3 Linguistic variables, 13.3 Membership functions, 13.3.1 Triangular, 13.3.2 Trapezoidal, 13.3.3 Gaussian, 13.3.4 Asymmetric Gaussian, 13.3.5 Generalized bell-shaped Gaussian, 13.3.6 Sigmoidal, 13.3.7 Singleton, 13.4 Fuzzy rules, 13.5 Fuzzy inference, 13.5.1 Fuzzy or approximate reasoning, 13.5.2 Mamdani fuzzy inference system, 13.5.2.1 Fuzzification of input, 13.5.2.2 Application of fuzzy operators ‘AND’ or ‘OR’, 13.5.2.3 Implication from antecedent to consequent, 13.5.2.4 Aggregation of consequents across the rules, 13.5.2.5 Defuzzification, 13.5.3 Takagi–Sugeno–Kang (TSK) fuzzy inference system, 13.5.3.1 Clustering, 13.5.4 Tsukamoto inference system, 13.5.5 Larsen inference system, 13.6 Neuro-fuzzy system, 13.6.1 Types of neuro-fuzzy systems, 13.6.1.1 Umano and Ezawa (1991) fuzzy-neural model, 13.7 Adaptive neuro-fuzzy inference systems (ANFIS), 13.7.1 Hybrid learning, 13.8 Application areas, 13.9 Concluding remarks, References, 14 Genetic algorithms (GAs) and genetic programming (GP), 14.1 Introduction, 14.2 Coding, 14.3 Genetic operators, 14.4 Parameters of GA, 14.5 Genetic programming (GP), 14.6 Application areas, 14.7 Concluding remarks, References, Index

About the 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 provider of expert opinion in several legal cases in Hong Kong.