Deterministic Methods in Systems Hydrology - Chapter 1 pdf

- 1 - A.A. BALKEMA PUBLISHERS / LISSE / ABINGDON / EXTON (PA) / TOKYO IH E DEL F T LE CTU R E NOT E SE RIES Deterministic Methods in Systems Hydrology JAMES C.I. DOOGE J. PHILIP O’KANE - 2 - Cover Design: Typesetting: Charon Tec Pvtt. Ltd, Chennai. India. Printed in the Netherlands @ 2003 Swets 4. Zeitlinger B.V., Lisse All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, eithout written prior permission from the publishers. Although all care is taken to ensure the integrity and quality of this publication and the information herein, noresponsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained rerein. Published by: A.A. Balkema Publishers, amember of Swets & Zeitlinger Publishers www.balkema.nl and www.szp.swetz.nl ISBN 90 5809 391 3 hardbound edition ISBN 90 5809 392 2 paperback edition To the memory of Eamonn Nash - 3 - Table of Contents PREFACE 1 THE SYSTEMS VIEWPOINT XIII 1 11 Nature of systems approach 1 1.2 Systems terminology 3 1.3 Linear time - invariant systems 7 1.4 Discrete forms of convolution equation 13 1.5 Suggestions for further reading 15 2 NATURE OF HYDROLOGICAL SYSTEMS 17 2.1 The hydrological cycle as a system 17 2.2 Unit hydrograph methods 20 2.3 Identification of hydrological systems 26 2.4 Simulation of hydrological systems 28 3 SOME SYSTEMS MATHEMATICS 35 3.1 Matrix methods 35 3.2 Optimisation 37 3.3 Orthogonal functions 41 3.4 Application to systems analysis 45 3.5 Fourier and Laplace transforms 47 3.6 Differential equations 53 3.7 References on systems mathematics 55 4 BLACK-BOX ANALYSIS OF DIRECT STORM RUNOFF 59 4.1 The problem of system identification 59 4.2 Outline of numerical experimentation 61 4.3 Direct algebraic m ethods of identification 64 4.4 Optimisation methods of unit hydrograph derivation 67 4.5 Unit hydrograph derivation through z-transforms 71 4.6 Unit hydrograph derivation by harmonic analysis 74 4.7 Unit hydrograph derivation by Meixner analysis 76 4.8 Overall comparison of identification methods 78 5 LINEAR CONCEPTUAL MODELS OF DIRECT RUNOFF 81 5.1 Synthetic unit hydrographs 81 5.2 Comparison of conceptual models 85 5.3 Cascades of linear reservoirs 88 5.4 Limiting forms of cascade models 94 6 FITTING THE MODEL TO TUE DATA 101 6.1 Use of moment matching 101 6.2 Effect of data errors on conceptual models 103 6.3 Fitting one-parameter models 105 6.4 Fitting two- and three-parameter models 110 6.5 Regional analysis of data 118 - 4 - PROBLEM SET 201 Runoff prediction System identification 201 System identification 201 Unit hydrograph derivation 202 Conceptual models 204 Comparing models 205 ACKNOWLEDGEMENTS 207 ENCOMIUM 209 REFERENCES 211 Appendix A - PICOMO: A Program for the Identification of Conceptual Models 225 Appendix B - Inverse Problems are III-Posed 261 Appendix C - The Non-Linearity of the Unsaturated Zone 273 Appendix D – Unsteady Flow in Open Channels 287 INDEX 303 7 SIMPLE MODELS OF SUBSURFACE FLOW 127 7.1 Flow through porous media 127 7.2 Steady percolation and steady capillary rise 133 7.3 Formulae for ponded infiltration 137 7.4 Simple conceptual models of infiltration 148 7.5 Effect of the water table 152 7.6 Groundwater storage and outflow 155 8 NON-LINEAR DETERMINISTIC MODELS 163 8.1 Non-linearity in hydrology 163 8.2 The problem of overland flow 169 8.3 Linearisation of non-linear systems 178 8.4 Non-ling black-box analysis 188 8.5 Concept of uniform non-linearity 192 - 5 - List of Figures 1.1 The concept of system operation. 3 2.1 The hydrological cycle. 18 2.2 Block diagram of the hydrological cycle. 18 2.3 Simplified catchment model. 19 2.4 Models of hydrological processes. 20 2.5 Superposition of unit hydrographs. 21 2.6 Hydrograph response. 23 2.7 Effective precipitation. 25 2.8 Classical methods of unit hydrograph derivation. 26 2.9 Typical regression model. 30 2.1 Coaxial correlation diagram (Becker, 1996). 31 2.11 Stanford Model Mark IVA 33 2.12 Schematic diagram of the overall model of the hydrological cycle. 34 4.1 Shape of unit hydrograph in numerical experimentation. 62 4.2 Input shapes. 62 4.3 The relationship between optimisation methods. 69 4.4 The procedure in transform methods. 71 5.1 Development of synthetic unit hydrographs. 84 5.2 Comparison of conceptual models with 2 parameters. 87 5.3 Limiting forms of unimodal cascade. 93 5.4 Simulation of GUM (Geomorphic Unit Hydrograph). 93 5.5 Limiting form for U-shaped inflow. 98 6.1 One-parameter conceptual models. 107 6.2 Shape factor comparison of conceptual models. 11I 6.3 The model inclusion graph. 115 6.4 The model inclusion graph with the RMS errors for the Big 117 Rivet data (A). 6.5 The model inclusion graph with the RMS errors for the Ashbrook 117 Catchment data (B). 6.6 The general synthetic scheme. 119 6.7 Shape factor plotting of regional data. 120 6.8 Typical unit hydrograph parameters. 121 7.1 Variation of soil moisture suction (Yolo light clay). 130 7.2 Variation of hydraulic conductivity (Yolo light clay ). 131 7.3 Variation of hydraulic diffusivity (Yolo fight clay). 131 7.4 Comparison of profiles at ponding. 139 8.1 Hydrograph for a 27 acre catchment. 165 8.2 Laboratory experiment. 166 8.3 Typical results of laboratory experiment. 167 8.4 Time to peak versus rainfall intensity (Yura river). 167 - 6 - 8.5 Time of travel versus discharge. 168 8.6 Storage versus discharge. 170 8.7 Hydrograph of overland flow. 170 8.8 Overland flow 178 8.9 Linear flood routing. 184 8.1 Effect of reference discharge. 184 8.11 Attenuation and phase shift in LCR. 188 8.12 Dimensionless plot of laboratory data. 198 8.13 Dimensionless plot of field data. 198 A.1 The structure of PICOMO. 226 A.2a Echo-check and normalization of data set A. 232 A.2b Echo-check and normalization of data set B. 733 A.3a Moments and shape-factors for data set A. 235 A.3b Moments and shape-factors for data set B. 235 A.4a Parameters for several models of data set A. 235 A.4b Parameters for several models of data set B. 236 A.5.a Tableau output for model 20 (Convective-diffusion reach) of data 736 A.5b Tableau output for model 20 (Convective-diffusion reach) of data set B. 237 - 7 - List of tables 1.1 Classification of basic problems in systems analysis and synthesis. 6 1.2 Definition of linearity. 7 1.3 The integral equation for linear systems. 8 4.1 Effects of constraints on forward substitution solution. 65 4.2 Effects of rainfall pattern on error in unit hydrograph. 66 4.3 Comparison of direct algebraic solutions. 66 4.4 Effects of input pattern on least squares solution. 68 4.5 Effects of error type on least squares solution. 68 4.6 Comparison of optimisation methods. 70 4.7 Root-matching solution for 10% random error. 73 4.8 Effect of length of harmonic series. 75 4.9 Harmonic analysis for 10% random error. 75 4.10 Effect of length of series on Meixner analysis (forward substitution). 77 4.11 Effect of series length on Meixner analysis (least squares). 78 4.12 Meixner analysis for 10% random error. 78 4.13 Summary of transform methods. 79 4.14 Overall comparison of identification methods. 79 4.15 Comparison of relative CPU times for different methods. 80 4.16 Effect of level of error. 80 6.1 Effect on unit hydrograph of 10% error in the data. 104 6.2 Effect of level of random error on unit hydrograph. 105 6.3 One-parameter conceptual models. 106 6.4 One-parameter fitting of Sherman's data. 109 6.5 One-parameter fitting of Ashbrook data. 109 6.6 Two-parameter conceptual models. 110 6.7 Two-parameter fitting of Sherman's data. 112 6.8 Two-parameter fitting of Ashbrook data. 113 6.9 Best models for Sherman's Big Muddy River data. 114 6.10 Best models for Nash's Ashbrook Catchment data. 115 7.11 Richards equation. 133 7.12 Solutions for concentration boundary condition. 138 7.13 Solutions for flux boundary condition. 139 A.1 Name table. 231 A.2 The test data sets. 232 - 8 - Preface This work is intended to survey the basic theory that underlies the multitude of parameter-rich models that dominate the hydrological literature today. It is concerned with the application of the equation of continuity (which is the fundamental theorem of hydrology) in its complete form combined with a simplified representation of the principle of conservation of momentum. Since the equation of continuity can be expressed in linear form by a suitable choice of state variables and is also parameter- free, it can be readily formulated at all scales of interest. In the case of the momentum equation, the inherent non-linearity results in problems of parameter specification at each particular scale of interest. The approach is that of starting with a simplified but rigorous analysis in order to gain insight into the essential characteristics of the system operation and then using this insight to decide which restrictive simplification to relax in the next phase of the analysis. The benefits of this approach have been well expressed by Pedlosky (1987) 1 "One of the key features of geophysical fluid dynamics is the need to combine approximate forms of the basic fluid-dynamical equations of motion with careful and precise analysis. The approximations are required to make any progress possible, while precision is demanded to make the progress meaningful”. The replacement of empirical correlation analysis by complex parameter-rich models represents an improvement in the matching of predictive schemes to individual known data sets but does not advance our basic knowledge of hydrological processes firmly based on hydrologic theory. The original version of the text was prepared at the invitation of Professor Mostertman some twenty-five years ago for the benefit of international postgraduate students at UNESCO-IHE Delft and has been used as a basis for lectures in subsequent years. It deals with the basic principles of some important deterministic methods in the systems approach to problems in hydrology. As such, it reflects the classical period of development in the application of systems theory to hydrology. In these lectures attention was confined to deterministic inputs as the methods appropriate to stochastic inputs were dealt with elsewhere. The objectives of the course of lectures on "Deterministic Methods in Systems Hydrology" were (1) To introduce the elements of systems science as applied to hydrologic problems in such a way that students can appreciate the nature of the approach and can, if they wish, extend their knowledge of it by reading the relevant literature; (2) To approach flood prediction and the hydrologic methods of flood routing as problems in linear systems theory, so as to clarify the basic assumptions inherent in these methods, to extend the scope of these classical methods, and to evaluate their accuracy; 1 Paragraph 2 in Joseph Pedlosky's "Preface to the First Edition" in his book "Geophysical Fluid Dynamics", second edition, Springer-Verlag, pp. vii and viii. - 9 - (3) To review and evaluate some deterministic models of components of the hydrologic cycle, with a view to assembling themost appropriate model model of catchment response, for a particular problem in applied hydrology. The material is developed in two parts. The four chapters in the first part present the systems viewpoint, the nature of hydrologic systems, some sys- tems mathematics and their application to the black-box analysis of direct storm runoff. Four additional chapters form the second part and cover linear conceptual models of direct runoff, the fitting of conceptual models to data, simple models of subsurface flow, and non-linear deterministic models. A set of exercises completes the exposition of the material. It was not anticipated that the student would be able as a result of these lectures to master the complexities of the theory and all the details of individual models. Rather it was hoped that he or she would gain a general appreciation of the systems approach to hydrologic problems. Such an appreciation could serve as a foundation for a more complete understanding of the details in this text and in the cited references. The original version of the text has been extensively edited. New material has also been added: the equivalence theorem of linear cascades in series and parallel, and the limiting cases of cascades, with and without lateral inflow, as seen in shape factor diagrams. Four new appendices present additional material extending the treatment of various topics. Appendix B shows that de-convolution of linear systems, and by extension the inversion of non-linear systems, is in general an ill-posed problem. Imposing mass conservation is not sufficient to ensure that the problem is well-posed. Additional assumptions are required. To this end, we include in appendix A, a detailed description of the computer program PICOMO, which is referred to extensively in the text. It contains approx- imately twenty linear conceptual models built using various assumptions on lateral inflow, translation in space, and storage delay. These all lead to well-posed problems of system identification. The reader is encouraged to experiment with the program, which can be downloaded through the IHE http://www.ihe.nl/. The reader may wish to compare or combine PICOMO with other more recent hydrological toolboxes, which can be requested by e-mail from http://ewre.cv.icAac.uk/software/toolkit.htm, or http://www.nuigalway.ie/hydrology/. Linear methods of analysis require a clear understanding of the nature and occurrence of strong non-linearities in the relevant processes. Appendices C and D address these questions. Appendix C presents the non-linear theory of isothermal movement of liquid water and water vapour through the unsaturated zone. In the case of bare soil at the scale of one meter, two pairs of non-linearities present themselves as switches in the surface boundary conditions of the governing partial differential equation. The outer pair represents alternating wet and dry periods when the atmosphere switches the surface flux of water either into or out of the soil. The inner pair represents the intermittent switching to soil control of the surface flux. Appendix D discusses the linearisation of the non-linear equations of open channel flow, their solution as a problem in linear systems theory, and the errors of linearisation. The cited references have also been supplemented to cover subsequent developments in the topics dealt with in the original text and in the new appendices. These are not intended to provide a comprehensive review of current literature but [...]... hn 1 (t ) (1. 15f)  a2 x (t ) * hn (t ) * hn 1 (t ) * * h2 (t )  a1 x(t ) * hn (t ) * hn 1 (t ) * h2 (t ) * h1 (t ) Sub -systems in parallel or, in more compact notation,  n  yn (t )  x (t ) *  a j g j (t )   j 1  (1. 15g) where the index j runs over n subsets of the cascade, starting with all n sub -systems in Ss (j = 1) , and ending with the last sub-system in Ss On its own (j = n): g1 (t... (t), we must solve (n - 1) - 20 - problems of de-convolution to find all the hk(t) in expression (1. 15i) Consequently, the canonical form for a linear time-invariant system, which can be decomposed into subsystems, is the cascade form, Ss, consisting of sub -systems in series For a particular case see Diskin and Pegram (19 87) 1. 4 DISCRETE FORMS OF CONVOLUTION EQUATION In practice, input and output data... i in Ss by definition is (1. 15b) yi (t )  hi (t ) *[ ai x (t )  yi 1 (t )], y00 i =1, ,n Where the weights are non-vegative real numbers that sum to one: n ai  0, a i (1. 15c) 1 1 Iterating this recurrence relation gives y1 (t )  h1 (t ) *[a1 x(t )] y2 (t )  h2 (t ) *[ a2 x(t )  y1 (t )] yn 1 (t )  hn 1 (t ) *[ an 1 x (t )  yn  2 (t ) yn (t )  hn (t ) *[ an x(t )  yn 1 (t )] (1. 15d)... to j gn(t) = hn(t) g k (t) = hk(t) * g k +1( t), k = n - 1, , 1 De-convolution Canonical form (1. 15i) Given the impulse response, hk(t), of each sub-system in Ss we can find the impulse response, gk(t), of the corresponding sub-system in Sp, by successive convolution in (1. 15i) taken in reverse order using the index k However, the inverse problem: "given Sp, find Ss", is considerably more difficult Given... where both if the input is an isolated one, the infinite limits in equation (1. 18) can be replaced by finite limits, as in the case of equation (1. 13) for continuous time, thus becoming s y ( sD )   X ( ) hD ( s   )   0 ,1, 2, (1. 19a)  0 which can be written without ambiguity as s y ( s )   X ( ) hD ( s   ) s,  0 ,1, 2, (1. 19b)  0 in which the standard sampling interval D has been... Doebelin (19 66) deals with the more special subject of instrumentation systems Other special subjects on which books are available are chemical engineering systems (Franks, 19 67) and estuarine and ma systems (Nihoul, 19 75) Good introductions to the adaptive nature of economic, social and biological systems are contained in works by Tustin (19 53), Bellman (19 61) , and Forrester (19 68) A series of readings... significant extensions of the material - 10 - CHAPTER 1 The Systems Viewpoint 1. 1 NATURE OF SYSTEMS APPROACH System Mathematical physics approach Before commencing our discussion of deterministic methods in hydrologic systems it is necessary to be clear as to what we mean by a system The word is much used nowadays both in scientific and non-scientific writing Even if we confine ourselves to the scientific... equation (1. 19) is completely discrete in form, it represents a set of simultaneous linear algebraic equations This set of equations can be written in matrix form as - 21 - Y = Xh (1. 20) where y is the column vector (yo, y1 yp -1 , yp), formed from the known output ordinates sampled at interval D, h is the column vector of unknown ordinates of the pulse response (h0, h1, ,hn -1 , hn), for the sampling interval... basic systems approach to the mathematical modelling of hydrologic processes and systems has been discussed by a number of authors particularly between 19 60 and 19 90 The general role of deterministic methods has been dealt with in papers by Amorocho and Han (19 64) Dooge (19 68), Vemuri and Vemuri (19 70), and in monographs by Becker and Glos (19 69), Kuchment (19 72) - 22 - Dooge (19 73) and Singh (19 88)... x ( )h(t   )d (1. 12) If the input is an isolated one i.e if the time between successive inputs exceeds the memory of the system then x(t) and y(t) will be zero for negative arguments so that equation (1. 11) can be written in the form t y (t )   x ( )h(t   )d (1. 13) 0 Equations (1. 12) and (1. 13) can be combined by writing Cascades of sub -systems y (t )   t [t  M ] (1. 14) x( )h(t   )d . - 3 - Table of Contents PREFACE 1 THE SYSTEMS VIEWPOINT XIII 1 11 Nature of systems approach 1 1. 2 Systems terminology 3 1. 3 Linear time - invariant systems 7 1. 4 . Linearisation of non-linear systems 17 8 8.4 Non-ling black-box analysis 18 8 8.5 Concept of uniform non-linearity 19 2 - 5 - List of Figures 1. 1 The concept of system operation. 3 2 .1 The hydrological. y     (1. 15b) Where the weights are non-vegative real numbers that sum to one: 1 0, 1 n i i a a    (1. 15c) Iterating this recurrence relation gives 1 1 1 2 2 2 1 1 1 1 2 1 ( ) ( )