Parameter identifiability of hydrological models with implicit structure: a numerical approach

This paper focuses on the identifiability problem of hydrological system modelling, taking implicit rainfall-runoff models as a target for study i.e. those models whose structures are nonlinear functions of their parameters. Because most implicit models and objective functions are very difficult as regards analytical solution or for finding high-order derivatives, a numerical analysis method is developed which not only can assess the identifiability problem but also can provide useful information for dealing with parameter optimization. The major studies involved are two-fold: (1) proposing a mathematical description and assessment method for the structural identifiability of hydrological models; and (2) providing a workable and robust identification technique using a rotating transformation of the parameter space. Using both generated error-free and watershed data in China, it has been verified that the approach is of significance and usefulness. Analyse de la capacité d'identification paramétrique de modèles hydrologiques avec une structure implicite: une méthode numérique Résumé Cet article étudie le problème de la capacité d'identification des modèles hydrologiques paramétriques, en considérant le modèle pluie-débit comme but. Puisqu'il est difficile de trouver des solutions analytiques pour la plupart des modèles implicites et des fonction empiriques, une méthode numérique est proposée. Elle peut non seulement évaluer la capacité d'identification du modèle, mais aussi peut fournir des informations utiles pour une optimisation paramétrique. Il s'agit de: (1) poser une méthode mathématique de la capacité d'identification de la structure du modèle hydrologique; et (2) présenter une technique d'identification du paramètre réalisable et robuste selon une transformation tournante dans l'espace paramétrique. Cette méthode a été vérifiée par les données d'un certain nombre de bassins. Elle est satisfaisante et valable. Open for discussion until 1 August 1989 1 Xia Jun 2 INTRODUCTION In rainfall-runoff modelling and analysis, two of the important problems are to determine model structure and to estimate model parameters. With respect to conceptual hydrological modelling, the structure may be suitably selected using physical knowledge or conceptualization, but values of most of the unknown parameters need to be estimated from measured data and other information. Since the 1970s, many investigators have reported difficulties associated with the identification of conceptual hydrological models (Ibbitt, 1970; Ibbitt & O'Donnell, 1974: Johnston & Pilgrim, 1976; Pickup, 1977; O'Donnell & Canedo, 1980; Brazil & Hudlow, 1981; Sorooshian & Gupta, 1983). For example, the conclusion of the work of Johnston & Pilgrim (1976) was their inability to find a "true optimum" and representative parameter set for the Boughton model applied to the Lidsdale 2 catchment in Australia. The major reasons for this failure were the existence of interdependence between model parameters and the presence of a long interacting valley (nearly a line optimum) in the response surface of the estimation criterion. Such features easily lead to arbitrary solutions. Therefore, in recent years, some researchers began to approach these problems (e.g. Sorooshian & Gupta, 1983). They found that besides the effects of the data, the structural identifiability of the model is a very important factor. At present, a number of analytical approaches have been developed by Sorooshian & Gupta (1985) and Xia Jun (1985a), which form a basis for, and play a notable role in, guiding theory. However, as far as actual situations are concerned, most conceptual hydrological models are nonlinear and have implicit structures, which means that the relationship between the output (runoff) and the unknown parameters of the models is nonlinear as well. The model function and estimation criteria (i.e. the objective function) are usually nondifferentiable or have very difficult high-order derivatives. Thus it is necessary to develop more workable and practical methods. In this paper, a numerical analysis approach is advanced in which two important concepts are (a) a numerical test of model parameters relative to the objective function; and (b) an interaction matrix, RQ € R, (also called a numerical information matrix) whose eigenvalues and elements measure the structural identifiability and correlation between model parameters. This approach does not have the particular requirement that the objective function should be differentiable or smooth. The application of the numerical analysis to rainfall-runoff modelling allows an assessment of the structural identifiability of models and their parameter calibration. NUMERICAL INFORMATION OF ESTIMATION CRITERIA According to the concept of system identification (Âstrôm & Eykhoff, 1971), parameter calibration of any conceptual hydrological model involves three basic elements. They are: (a) the model set, $ = {•}; (b) relevant data, DA = {•}; and (c) identification criteria, E = {•}. In general, the conceptual 3 Parameter identifiability of hydrological models model set has a known structure, $[•}, and unknown parameters, 9 e R. Let the variable ut be the input vector (rainfall), and yt be the output vector (runoff), i.e. DA = {ut, yj . The general form of a hydrological model can be set down as: yt = * {u9} + et (1) where € ; is the error vector representing the lack-of-fit with reality and t is the discrete time variable. As far as understanding the mechanism of mathematical modelling of hydrological systems is concerned, the distinction between linear and nonlinear models is of importance. A linear model is defined mathematically as a model whose operation is described by a linear differential equation i.e. there is said to be a linear relationship between the model input, uf and the model output, y(. The operation of a nonlinear model is represented by a nonlinear equation such as equation (1) in which yt is nonlinear in u . On the other hand, the distinction between an explicit structure model and an implicit structure model is also of paramount importance in understanding the identification of unknown parameters. If i{ •} is a linear function of the unknown parameter vector, i.e.: yt =*[",}• e + e, the model is said to have an explicit structure, e.g. the following equation refers to a nonlinear model with an explicit structure: yt = 9 o + 6 i M + 2 t 3 A t Further, if *[ •} is a nonlinear function of the parameter vector, 9, the model described by equation (1) will be said to have an implicit structure. Most conceptual rainfall-runoff models belong to this latter category, which means that, besides the relationship between output and input being nonlinear, the relationship between output and model parameters is nonlinear as well. The formulation of parameter identification will finally be done following nonlinear programming methodology, i.e. objective function £{9} = Il y £{8} $ E , where 8 ^ is the z'-th nodal vector of the parameter space, {8} e DQ, and Ec is a closure, 6 ^ € R , E{B}(') € i?, / = 1, 2,..., m, and m is an index set (Fig. 2). The second step requires normalization of the parameter set by selecting an appropriate reference point, &e.; The third step is the formation of an interaction matrix, RQ € R, from the test and normalized results, {8^, E(Q(')}, which can provide valuable information about the structural identifiability of a hydrological model.