an application to the model SED - Science Direct

Ecological Modelling, 68 (1993) 1-19 Elsevier Science Publishers B.V., Amsterdam

Environmental model calibration under different specifications: an application to the model SED Diederik T. Van der Molen and J~mos Pint6r 1 National Institute for Inland Water Management and Waste Water Treatment, Lelystad, Netherlands (Received 7 February 1992; accepted 18 September 1992)

ABSTRACT Van der Molen, D.T. and Pint6r, J., 1993. Environmental model calibration under different specifications: an application to the model SED. Ecol. Modelling, 68: 1-19. The subject of this paper is to draw attention to various criteria that can be used to calibrate a model (i.e. to assess its performance) and to study the differences in the results obtained when performing calibration using these criteria in a practical example. A recently proposed global optimization procedure is applied for parametrizing a simple dynamical model that describes the release of phosphorus from shallow, eutrophic lakes. In general, a number of tentative discrepancy measures can be analysed when a formal optimized calibration procedure is applied. The type of discrepancy measure depends, inter alia, on the number and type of available observations and the objective of the modelling exercise. Therefore, the results obtained should always be carefully verified. Special attention must be paid to the calibration of environmental models, as their "soft" character frequently demands the application of non-standard discrepancy measures.

LIST O F SYMBOLS

t =

T Pt

1,... ,T time-moments of observation sample size model output

Correspondence to: D.T. Van der Molen, National Institute for Inland Water Management and Waste Water Treatment (RIZA), P.O. Box 17, 8200 AA, Lelystad, Netherlands. 1 Present address: School for Resource and Environmental Studies, Dalhousie University, Halifax, N.S., Canada. 0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

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D.T. VAN DER MOLEN AND J. PINTER

Ot

observation (element of a homogeneous set of measured data) discrepancy measure (expresses the deviation of the sequences D {Pt} and {O/}) set of feasible model parametrizations, x ~ X is a real n-vector X j = 1 , . . . , J index of model output variables expected (mean) value operator E variance operator V weight vector (T-dimensional) W tolerance level random variable expressing relevant uncertainties (scalar or 4, vector, depending on the actual context) INTRODUCTION Environmental model development is an essential conceptual tool of the related theoretical and applied research (see, e.g., Loucks et al., 1981; Haith, 1982; Novotny and Chesters, 1982; Beck and Van Straten, 1983; JOrgensen, 1983; Orlob, 1983; Beck, 1985; Somly6dy and Van Straten, 1986). Generally speaking, the following main phases of quantitative analytical environmental modelling can be distinguished: - formulation of model objectives - setup of model structure calibration validation application (analysis, forecasting, control, management) Consequently, calibration - - finding the "best" or just "suitable" parametrizations - - is an important stage in model development. Confronting model results with expertise, background information and available observations, the following issues have to be properly addressed: - lack of "perfect" scientific knowledge (and, hence, of "perfect" models) - complicated model structure - necessary decomposition of reality, followed by (re)aggregation in the frame of a model inadequacy/errors in monitoring and data processing subjectivity in the interpretation of results According to the iterative model calibration procedure indicated by Fig. 1, one can attempt to approximate the "very best" theoretically admissible combination of the parameters which results in a model output that is "as close as possible" to the set of available observations or, alternatively, to find "acceptable" model parametrizations. The latter objective is especially relevant when "soft" systems are modelled. These systems are characterized by complex and ill-defined processes, and therefore parameters have -

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-

-

-

E N V I R O N M E N T A L M O D E L C A L I B R A T I O N A P P L I E D T O SED

OESCRIPTIVEMOOELOF ENVIRONMENTALSYSTEM

A

v Input: model parameters

Output: es~mated system

(selected iterativaly)

behavior/evolution

A T COMPARISON OF MODELOUTPUTAND MEASUREMENTS:ADJUSTEDCAUBRATION

(Evaluation of actual model parametedzation, in terms of a given discrepancy measure)

A ENWRONMENTALSYSTEM MEASUREMENTS,OBSERVATIONS

Fi'g. 1. Model calibration scheme (from Pint~r and Van der Molen, 1991).

wide defensible ranges and observations cannot unequivocal be compared to the system behaviour. In this paper particular attention will be devoted to the selection of diverse (scalar or vector, deterministic or stochastic) discrepancy measures, in order to express the "difference" between model output and corresponding observation data. These various model forms should reflect the intended model use, as there is no "universally best" calibration paradigm. In other words, the "goodness" of the model is to be judged in terms of those features that are important for a particular application. M E T H O D O L O G Y AND SCOPE OF APPLICATION

In mathematical terms, the objective of finding the optimal model parametrization can be expressed by the general problem statement minimize D ( x )

D ( x ) : = D { P t ( x ) , O , } , = ~..... ~r, x ~ X .

(1)

Due to the frequent nonlinearity of environmental models, in many cases the discrepancy measure D will be multiextremal, with respect to the parameter vector x. In other words, initiating a "standard" search procedure from different points of the feasible parameter domain X may often lead to markedly differing results, both in terms of the parametrization found and its "performance" (as being expressed by the function D). Therefore - - contrary to most "classical" model calibration approaches (see e.g. Box and Jenkins (1970)) - - , such numerical methodology is to be applied that is capable to find (approximate) the "very best" x in X. This objective necessitates the application of some properly chosen global optimization strategy. The specific theory and numerical methodology applied here is described elsewhere (see e.g., Pint6r, 1990, 1991b, 1992).

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D.T. V A N D E R M O L E N A N D J. PINTI~R

For illustrative purposes, several calibration model variants will be formulated with respect to the model SED (Van der Molen, 1991). The relatively simple model SED involves only a few parameters to be calibrated, hence facilitating the investigation of diverse discrepancy measures. PERFORMANCE CRITERIA

Single criterion analysis For a single state variable, the discrepancy measure is often derived in the form of D(x):=

J/T'[~t=I,T[Pt(x)-Ot[~] 1/~', 1 6 }, 6 > 0 . All previously presented model performance criteria are based on a coherent timing of model output and observations (represented by the indices t - - 1 , . . . , T ) . At the same time, in environmental models certain data (loads, meteorological forcing functions etc.) are often given on a daily, weekly or a decade time-scale. The larger time-steps often have a certain "equalizing" effect (possibly "hiding away" some extremes that, on the other hand, may be followed by the model output). Similar examples raise the issue of defining some other measures of "closeness" than those introduced earlier. For instance consider (19), based on (2), in which the objective function is defined by the distance between the measurement data and the linearly interpolated function passing through the (possibly discretized) model output points. P/(x) for t - 1,... ,T are those points of the linear interpolation curve that are "closest" to the corresponding observation in a geometric sense. For illustration, see Fig. 2 (note that the interpolation procedure suggested is scale dependent).

D{Pt(x),Ot}t= , ..... T:=I/T'[~.,t=,,TIP/(x)--Ot['] '/,,

l~'r~