Deterministic Methods in Systems Hydrology - Chapter 1 pdf

It deals with the basic principles of some important deterministic methods in the systems approach to problems in hydrology.. The objectives of the course of lectures on "Deterministic M

A.A BALKEMA PUBLISHERS / LISSE / ABINGDON / EXTON (PA) / TOKYO

IHE DELFT LECTURE NOTE SERIES

Deterministic Methods in Systems Hydrology

JAMES C.I DOOGE

J PHILIP O’KANE

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

4.4 Optimisation methods of unit hydrograph derivation 67

Appendix C - The Non-Linearity of the Unsaturated Zone 273

List of Figures

2.12 Schematic diagram of the overall model of the hydrological cycle 34 4.1 Shape of unit hydrograph in numerical experimentation 62

5.2 Comparison of conceptual models with 2 parameters 87

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)

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

8.4 Time to peak versus rainfall intensity (Yura river) 167

8.5 Time of travel versus discharge 168

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

4.7 Root-matching solution for 10% random error 73

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

4.14 Overall comparison of identification methods 79 4.15 Comparison of relative CPU times for different methods 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

6.4 One-parameter fitting of Sherman's data 109 6.5 One-parameter fitting of Ashbrook data 109

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

7.12 Solutions for concentration boundary condition 138

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

(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

rather to highlight key publications that deal with significant extensions of the material

The Systems Viewpoint

1.1 NATURE OF SYSTEMS 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 literature, there is a bewildering range of definitions of what

is meant by a system and what is meant by the systems approach For the purposes of our present discussion, a system may be defined as (Dooge, 1968, p 58)

Any structure, device, scheme, or procedure, real or abstract, that inter-relates

in a given time reference

an input, cause or stimulus,

of matter, energy or information and

an output, effect or response

of information, energy or matter

The first line of the above definition emphasises that anything at all that consists of connected parts may be considered as a system The second line emphasises that a system may be real as in the case of an actual catchment area or abstract as in the case of a model of the operation of that catchment based on digital simulation The following lines emphasise the important characteristics of dynamic systems

that link input and output The latter terminology is that usually used by the engineer but the physicist would tend to refer to them as cause and effect and a biologist as

stimulus and response The terms are used as alternatives in the above definition in

order to stress that, from the systems viewpoint, there is no basic difference between the particular systems studied by the engineer, the physicist and the biologist In the definition, the input and the output may consist of matter, or energy, or information These alternative categories are listed in different order for the input and the output respectively, to emphasise that input and output need not belong to the same category

In the case of dynamic systems, input and output must be defined in terms of a given

time reference

The scale of the time reference in which a dynamic system is considered, may vary according to the particular aspect of the system under study Thus, in the case of

hydrologic catchments, the flood producing power of the catchment may be considered

in a time reference of hours or days On the other hand, consideration of the sustained baseflow may require a time scale of months and possibly years Consideration of the

development of the drainage network From a geomorphological viewpoint, may involve a time

reference expressed in centuries

Figure 1.1

The concept

of system

operation

In applied science our concern is predicting the output from the system of interest This

problem can be approached from a number of points of view Firstly, one can adopt what

might he called the mathematical physics approach, which seeks (a) to establish differential

equations governing the physical phenomena involved, (b) to formulate the set of equations and boundary conditions for the particular system under study and (c) to solve the resulting problem for a given input An example would be the use of the St Venant equation, to solve problems of overland flow or unsteady flow in a channel network Freeze ( 1972) provides

further discussion with particular reference to subsurface flow

A second and sharply contrasting approach is that known as black-bar analysis In this approach, we do not use (at least initially) our knowledge of either (a) the physics of the

processes involved, nor (b) the exact nature of the system Instead an attempt is made to extract From past records of input-output events on the system under examination, enough knowledge of the operation of that particular system, to serve as the basis for predicting its output due to other specified inputs Dooge (1973, PP 75-101) has reviewed the development of this approach, as it relates to the classical medthods of applied hydrology Between these two extremes, lies the approach based on what are termed conceptual models Even though both extremes represent "conceptual models" in the broad sense of that phrase, the term is usually limited in systems hydrology to a particular type of model

Conceptual models in hydrology consist of simple arrangements of elements whose structure and parameter values are chosen to simulate the behaviour of the hydro logic system under study Such conceptual models may take the form of closed systems or open systems depending on the nature of the prototype (Vertalanffy, 1968, pp 39-41) It is important to include (a) feedbacks between the components of the model (Bertalanfry, 1968,

pp 42 - 44), and (b) thresholds, i.e points of concentrated non-linearity, which may switch the control of the system operation from one component to another Rather than three isolated classes of model – black-box, conceptual models, equations of mathematical physics – there is in fact a complete spectrum of models ranging from pure black-box analysis, which makes almost no physical assumptions, to a highly complex approach based

on equations derived from continuum mechanics

The concept of system operation is illustrated in Figure 1.1 (Dooge 1968, p 59) In

the classical approach based on mathematical physics, our knowledge of the physical laws involved and the information available to us concerning the nature of the system, enables us to describe the system operation in terms of a set of mathematical