STOCHASTIC WATER RESOURCES TECHNOLOGY Titles of related interest B. Henderson-Sellers: Reservoirs D. M. McDowell and B. A. O'Connor: Hydraulic Behaviour ofE stuaries E. M. Wilson: Engineering Hydrology, Second Edition Stochastic Water Resources Technology N. T. Kottegoda Department of Civil Engineering University of Birmingham © N. T. Kottegoda 1980 Softcover reprint of the hardcover 1st edition 1980 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission First published 1980 by THE MACMILLAN PRESS LTD London and Basingstoke Associated companies in Delhi Dublin Hong Kong Johannesburg Lagos Melbourne New York Singapore and Tokyo British Library Cataloguing in Publication Data Kottegoda, N T Stochastic water resources technology. I. Hydraulic engineering-Mathematical models 2. Stochastic processes I. Title 627'.01'84 TC153 ISBN 978-1-349-03469-7 ISBN 978-1-349-03467-3 (eBook) DOI 10.1007/978-1-349-03467-3 This book is sold subject to the standard conditions of the Net Book Agreement Contents Preface xi 1 INTRODUCTION AND CLIMATE 1 1.1 Hydrological Uncertainties 2 I.2 Climate and Climatic Change 7 1.2.1 Climatic fluctuations 8 1.2.2 Main causes of climatic change 9 1.2.3 Records of climate II 1.2.4 Climate and water resource planning I5 1.3 Scope of Book I5 References I7 2 ANALYSIS OF HYDROLOGIC TIME SERIES 20 2.I General Definitions of Time Series and Stochastic Processes 20 2.2 Components and Main Properties of Time Series 22 2.3 Trend 26 2.3.I Filtering methods 28 2.3.2 Tests for randomness and trend 3I 2.4 Periodicity 34 2.5 Autocorrelation 4I 2.5.I Periodic time series 43 2.5.2 Non-periodic dependent time series 45 2.5.3 Independent time series 45 2.6 Spectral Methods 46 2.6.I Estimation of smoothed spectrum through autocorre- lation function 48 2.6.2 Some theoretical spectra 53 2.6.3 Prewhitening 54 2.6.4 Confidence limits 54 2.6.5 Fast Fourier transform 55 2. 7 Further Remarks on Spectrum and Serial Correlogram 57 Vi CONTENTS 2.8 Cross Correlogram 58 2.9 Cross Spectral Analysis 59 2.10 Concluding Remarks 61 References 62 3 PROBABILITY FUNCTIONS AND THEIR USE 67 3.1 Probability Distributions of Hydrological Data 67 3.2 Pearson Density Functions 71 3.2.1 Type I function 73 3.2.2 Type III function 74 3.2.3 Type V function 74 3.2.4 Type VI function 75 3.3 Estimation by Method of Moments 75 3.3.1 Graphical procedure 76 3.3.2 Sampling errors in estimates of skewness and kurtosis 77 3.3.3 Reasons for frequent use of type III function 78 3.3.4 Method of moments applied to type III function 78 3.4 Maximum Likelihood Method of Estimation 80 3.4.1 Maximum likelihood method applied to type III function 81 3.4.2 Comparison of maximum likelihood method and method of moments 83 3.5 Goodness-of-fit Tests 86 3.5.1 Chi-squared test 86 3.5.2 Kolmogorov-Smirnov goodness-of-fit test 89 3.5.3 Kolmogorov-Smirnov two-sample test 93 3.5.4 Remarks on cur.ve fitting 95 3.6 Other Families of Probability Functions 95 3.6.1 General transformations 95 3.6.2 Johnson's system 95 3.6.3 Use of polynomial functions 96 3.6.4 Additional types and methods 97 3. 7 Random Numbers: Generation and Transformations 98 3.7.1 Generation of uniform random numbers 98 3.7.2 Transformation to normal distribution 102 3.7.3 Generation of gamma variates 102 3.7.4 Random variates with other densities 104 3.8 Other Comments and Further Reading 104 References 10 6 4 LINEAR STOCHASTIC MODELS 111 4.1 Introduction to Data Generation and Assumptions 111 4.2 Linear Autoregressive Models 113 4.2.1 pth-order model 114 4.2.2 Estimation of autoregressive parameters 114 CONTENTS vii 4.2.3 Variance of independent variables 116 4.2.4 First-order model 116 4.2.5 Second-order model 117 4.2.6 General recursive formulae 118 4.3 Partial Autocorrelations 120 4.4 Moving-average Models 123 4.5 Box-Jenkins Models: Formulation and Identification 125 4.5.1 Autoregressive moving-average ARMA(p, q) model 126 4.5.2 Autoregressive moving-average ARMA(l, 1) model 127 4.5.3 Autoregressive integrated moving-average ARIMA(p, d, q) model 128 4.5.4 Model identification and estimation 129 4.6 Application to Non-normal Series 134 4.6.1 Preservation of skewness in Markov model 136 4.6.2 Box-Cox transformations 136 4.6.3 Data generation using Markov model and lognormal distribution 136 4.6.4 Dealing with non-normal distributions in general 138 4.7 Seasonal Models 138 4.7.1 Autoregressive seasonal models 139 4.7.2 Thomas-Fiering seasonal model 140 4.7.3 Box-Jenkins seasonal model 143 4.7.4 Models for daily flows: problems and references 144 4.8 Multisite Data Generation 145 4.8.1 Two-site models J 46 4.8.2 First-order multisite autoregressive model 147 4.8.3 Higher-order multisite autoregressive models 152 4.8.4 Further comments on multisite models 154 4.9 Short-term Forecasting 154 4.9.1 Forecasting with ARMA(p, q) model 155 4.9.2 Filtering methods of real-time forecasting 162 4.10 Practical Applications and General Comments 165 4.1 0.1 Infilling missing flow data 165 4.10.2 Comments on testing generated data 165 4.10.3 Applications of data generation methods 166 4.10.4 Other properties and models 166 References 166 5 SPECIAL PROPERTIES AND MODELS 173 5.1 Runs and Crossing Properties 173 5.1.1 Theoretical aspects 175 5.1.2 Some properties of independent discrete series 178 5.1.3 Runs in hydrology: references to other works 180 5.2 Rippl Diagram and Reservoir Storage 180 5.3 Range Analysis 184 viii CONTENTS 5.3.1 Hurst phenomenon 184 5.3.2 Estimation of Hurst coefficient h 188 5.3.3 Range and rescaled range in small samples 188 5.3.4 Explanations of Hurst phenomenon and experiments 189 5.4 Fractional Gaussian Noise 192 5.4.1 Joseph and Noah effects 192 5.4.2 Fractional brownian motion 193 5.4.3 Approximations to fractional gaussian noise 194 5.4.4 Type II approximation to fractional gaussian noise 194 5.4.5 Fast fractional gaussian noise 195 5.5 Broken-line Models 199 5.6 Final Comments 202 References 203 6 STATISTICAL TREATMENT OF FLOODS 208 6.1 Annual Maximum Series and Return Periods 209 6.2 Distribution of Extreme Values 209 6.3 Gumbel Distribution 210 6.3.1 Moment-generating function 211 6.3.2 Statistical properties 211 6.3.3 Definition of return period 213 6.3.4 Relationship between Gumbel variate and return period 214 6.3.5 Probability paper 214 6.3.6 Plotting positions 215 6.3.7 Method-of-moments fitting procedure 216 6.3.8 Gumbel's fitting method 218 6.3.9 Frequency factors for Gumbel distribution 219 6.3.10 Confidence limits 220 6.3.ll Maximum likelihood method of estimation 222 6.3.12 Limitations in Gumbel method 223 6.4 General Extreme Value Distribution 224 6.4.1 Type II extreme value distribution 224 6.4.2 Type III extreme value distribution 225 6.4.3 General formula for extreme value distribution 225 6.5 Lognormal distribution 229 6.5.1 Theoretical considerations 229 6.5.2 Probability paper 231 6.5.3 Frequency factors 232 6.5.4 Confidence limits 233 6.5.5 Bias in skewness and Hazen's correction 235 6.5.6 Regional skewness 235 6.6 Pearson Type III Function Applied to Extreme Values 236 6.6.1 Frequency factors 236 6.6.2 Twp-parameter gamma function 238 6.6.3 Log Pearson type III function 239 CONTENTS ~ 6.7 Discussion on Frequency Methods of Flood Estimation 241 6.8 Binomial, Poisson and Multinomial Distributions 244 6.8.1 Binomial distribution 244 6.8.2 Poisson distribution 245 6.8.3 Multinomial distribution 246 6.8.4 Limitations 246 6.9 Peaks-over-threshoid Method 247 6.10 Regional Flood Frequency Analysis 250 6.11 Probable Maximum Precipitation 252 6.12 Other Methods and Comments 256 6.13 Final Remarks and Summary 257 References 258 7 PROBABILITY THEORY APPLIED TO RESERVOIR STORAGE 264 7.1 Queueing Theory and Water Storage 264 7.2 Definition of Markov Chain 265 7.3 Moran's Theory of Reservoirs 266 7.3.1 Application of Markov chains and assumptions made 266 7.3.2 Unconditional probabilities and n-step transition probabilities 269 7.3.3 Steady state probabilities 270 7.4 Gould Method for Failures within a Year 278 7.5 Serial Correlation and Seasonal Changes in Inflows 282 7.5 .I Serially correlated inflows 282 7.5.2 Effect of seasonality on probability of emptiness 283 7.5.3 Application of bivariate Markov chain 283 7.5.4 Other practical considerations 286 7.6 Associated Topics, Other Works and Comments 287 7. 6.1 Applications in meteorology 287 7.6.2 Summary of other works and scope for future 289 References 290 8 STOCHASTIC PROGRAMMING METHODS IN SYSTEMS ENGINEERING 294 8.1 Historical Background 294 8.2 Introduction to Systems Engineering 295 8.3 Linear Programming 299 8.3.1 General considerations for water resource applications 299 8.3.2 Programme formulation 300 8.3.3 Simplex algorithm 303 8.3.4 Chance-constrained linear programming 305 8.3.5 Limitations of linear decision rule method 311 8.3.6 Other applications of linear programming 312 8.3.7 Concluding remarks 313