Monte Carlo Models DETERM I N ISTIC MODELS ]
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iComponent models Integrated models (Conceptual 1 Linear Non-linear
Linear
orLumped
orDiscrete or non-linear distributed continuous
I STATISTICAL MODELS AND METHODS I
Fig. 4.4
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Mathematical models hydrology (modified after Fleming, 1975).Students, who as beginners are becoming acquainted with hydrological models, often ask why there are so many models and what criteria are best for their classification, comparison and evaluation. It is quite clear that for the wide range of hydrological processes, frequently influenced by specific conditions which often make the dynamics of the particular process unique, a multiplicity of models is required.
'The feature that all hydrological models have in common is that the observed output variable y(t) deviates from its fitted value f (.) by a residual amount E(t) (see eq. 4.1);
the respects in which they differ are the assumptions made about the structure of a model f(.) and the assumptions made about a residual E(t)' (Clarke, 1973). The use of these residual errors in fitting mathematical models to prototype data is illustrated in Figure 4.5.
intercomparison and evaluation of the best known conceptual models proposed for operational use in hydrology was made by the WMQ in the period 1968 to 1974 (WMO, 197533). This project involved the testing of ten operational conceptual hydrological models on six standard river catchment data sets and their usefulness in providing a short term forecast of streamflow in
An
data yc(t)
Parameter adjustment Criterion or changes in of accuracy model structure
Fig. 4.5
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The mathematical model concept.various forecasting situations. The details of the tested models including specifications of their purpose, data requirements, fitting criteria and references together with a description of the model structure are contained in the quoted WMO monograph, which in this sense may be used as a teaching aid. There are a number of other publications discussing mathematical models in hydrology, and serving well the needs of teaching in hydrology (Fleming,, 1975;
Clarke, 1973; Kutchment, 1972 etc.).
However, not all models developed far use in operational hydrology or intended for scientific research are well suited as teaching aids. The use of the models in the teaching/
learning process will require the teacher to follow a set methodological approach relating to the model's computer implementation. This approach largely consists of:
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the mathematical analysis of the problem (numerical or analytical solution)-
the aonstruction of a flow-chart-
the writing of the program for a digital or analogue computerThe procedure for the construction of a mathematical (simulation) model is illustrated by a flow chart such as that shown in Figure 4.6.
The mathematical simulation of hydrological phenomena shares the same phases of model development and implementation:
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Identification phase is the primary phase of model work. Its aim is to identify the model-
Simulation phase is the secondary phase in model implementation, and may consist of short structure and estimate model parametersterm prediction in real-time for operational purposes, or long term prediction and data generation for planning and design.
Conceptual hydrological models can be used...
'to considerable advantage as teaching aids in integrating the interaction between hydrological processes, e.g. to demonstrate the effect of various hydrological phenomena on catchment
response. Use of hydrological models in teaching, however, requires a sound understanding of
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Formulate problem-examine system's behaviour, analyse real-world data
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Define problem reauirina simulationI
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Formulate mathematical model L
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Develop algorithms
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RejectAccept Program computer model
Reject
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Check program AcceptReject
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Accept
Conduct simulation experiments Check results
?eject model
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Accept model
Reject simulation
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Accept si m u la t i on
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Fig. 4.6
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Phases in the construction of a mathematical model (modified after Hille, 1977).the individual components of the whole system. Therefore, the use of models should be introduc- ed into a course only after basic hydrological principles have been already taught with a para- llel teaching of related subjects, such as hydraulics, soil science, geology and meteorology.'
(Fleming, 1975).
4.2.1 Basic Deterministic Concepts
The degree of equivalence of the model and prototype response is dictated by the deterministic principles encompassed in a given approach.
mination of physical parameters which describe the system, the deterministic approach is frequently called the parametric concept.
It is always difficult to classify deterministic models according to the principles governing their intrinsic structure or the simulation methods used.
'The formulation of a model is a continual process of modification, testing and remodification.
The model may begin by being largely empirical; as more is learnt about the physical behaviour of the system under study, empirical components for the model will be replaced by others, more firmly based in theory, and modification must be tested, where possible, by comparison of forecast with observation'. (Clarke, 1973).
For this reason students must be told repeatedly that hydrological models must be considered as something subject to constant supplementation and improvement and that it is possible to combine both deterministic and stochastic approaches.
Since the model solution is rooted in the deter-
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According to the implementation technique, deterministic models in hydrology are based on Mathematical models in which the system behaviour is represented by a set of equations and statements expressing relations between the variables and parameters.
Analogue models in which simulation of hydrological processes is based on the analogy between a response of the hydrological prototype (or its physical element), and its electric, hydraulic or thermal counterpart.
Physical models which respect the scale similarity with the prototype.
or more principles:
4.2.1.1 Mathematical Models
Two basic approaches should be emphasised in the teaching process:
a. Component Modelling representing a mathematical simulation of a small component in the hydrological cycle, while mutual interactions between other hydrological zones are substantially suppressed.
the component. Such a major component might be infiltration, evapotranspiration, streamflow routing, etc. Modelling of streamflow routing was among the first means of introducing
systems theory into hydrology (Dooge, 1959). In particular, linear systems theory substantiated the scigntific basis of the unit hydrograph method and its broad application in hydrological engineering, and also led to the rapid development of storage models. The superposition princ- iple, mathematically expressed by the convolution integral, forms the basis of these models.
Different combinations of equal/unequal linear reservoirs in series or parallel can be introduc- ed, showing how more complex models may lead to better prediction of prototype behaviour but at the cost of an increasing number of estimated parameters. The students should be told emphatic- ally that the so-called black-box approach in simulation of a physical system results in the development of a relationship between the input and the output, without introducing any physical relevance to the equations and parameters of the model. A clear example is needed in any
explanation of the structure and function of the black-box models (see examples 4.3 and 4.4).
Various structural schemes of linear and non-linear runoff and flood routing models are described in the literature (e.g. TNO, 1966; DFG,1975; Kutchment, 1972). Examples from the group of linear runoff models suitable for teaching purposes should include the Muskingum method, Nash model, Kalinin-Miljukov model, etc. Suitable non-linear rainfall-runoff models are the Kutchment-Borshevskij scheme, O'Donnell-Mandeville model, etc. (Kutchment, 1972; O'Donnell and Mandeville, 1975; Fleming, 1975).
The teaching of these component models should be concluded with the procedures required to determine the parameters using optimisation or estimation techniques followed by implementation procedures with actual hydrological data. The application of component models of surface runoff runs into the difficulties of mutual interaction between hydrological processes. This drawback is reflected in a problem common to this group of models, namely the determination of time and space development of effective (net) rainfall for individual isolated flood events. These draw- backs limit the application scope of the models. For these reasons there is a tendency to favour integrated (conceptual) modelling.
The development has been towards an understanding of the physical laws governing
b. Integrated System Modelling involves the simulation of the whole hydrological system. Here- in, the component theories are conceptually combined to represent a time-variant interaction of processes constituting the hydrological cycle. This approach is often referred to in various publications as conceptual modelling. An example of a conceptual system representation is given in Figure 4.7;
continuous time basis.' (Fleming, 1975, see example 4.5).
set of analytical relationships between hydrological processes, and may never do so.
approach, is the representation of the hydrological cycle as a determinate system quantitatively expressed by mathematical functions governing the process. Model parameters are obtained either from direct measurement, analyses of measured records, by trial and error, or by an automatic optimisation procedure (see example 4.6). It is important to specify the criteria for model performance and accuracy. A common criterion of accuracy is to minimise the sum of squared errors between the recorded output from the catchment, and the simulated output from the model.
'The conceptual approach is an integration of the component theories on a
Empirical relationships are still necessary since the subject has not yet produced a complete A necessary prerequisite of the deterministic approach, compared with the stochastic
Lack
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of fit might be caused by:
errors resulting from an over-simplified description of the physical processes in the model, in contrast to the prototype,
time and spatial variability of the input and the output data, errors in measurement of the input and output data.