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This paper introduces a novel data-driven methodology named Evolutionary Polynomial Regression (EPR), which permits the multi-purpose modelling of physical phenomena, through the simultaneous solution of a number of models. Multipurpose modelling or "multi-modelling", enables the user to make a more robust choice of those models aimed at (a) the knowledge based on data modelling, (b) on-line and offline forecasting, and (c) data augmentation (i.e. infilling of missing data in time series). This methodology is particularly useful in modelling environmental phenomena, for which it is usually impossible to obtain physical data at a laboratory scale. In particular, the non-linearity of phenomena and non Gaussian nature of background noise make on-line forecasting complex, and where data are available, they often contain discontinuities (i.e. missing data). The use of EPR in modelling and analysis is illustrated by application to a case study containing all these limitations. The application of EPR to thermal behaviour of a stream gives not only a good physical insight of the phenomenon, but also allows infilling of missing data, resulting in good models that forecast the water temperature.

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Abstract:The data-driven technique, evolutionary polynomial regression, has been tested and used for the study of water temperature behaviour in the River Barle (south-west England). The study aimed to produce multiple models for forecasting water temperature, using air temperature as input. In addition, river discharge data were used to describe the hydrological regime of the study stream, even if they are not involved in the modelling phase. The availability of data sampled at hourly intervals allowed behaviour to be studied at several time scales, including short-term lags between air temperature and water temperature. The approach to model building differs from previous studies in that the relationship between air temperature and water temperature is not evaluated on the basis of a multi-parameter regression, nor does it identify particular structures; rather the evolutionary technique identifies the model by itself. In fact, the non-linear relationship between air temperature and water temperature is investigated by an evolutionary search in the space of particular pseudo-polynomials structures.

The problem of identifying and reproducing the hydrological behaviour of groundwater systems can often be set in terms of ordinary differential equations relating the inputs and outputs of their physical components under simplifying assumptions. Conceptual linear and nonlinear models described as ordinary differential equations are widely used in hydrology and can be found in several studies. Groundwater systems can be described conceptually as an interlinked reservoir model structured as a series of nonlinear tanks, so that the groundwater table can be schematized as the water level in one of the interconnected tanks. In this work, we propose a methodology for inferring the dynamics of a groundwater system response to rainfall, based on recorded time series data. The use of evolutionary techniques to infer differential equations from data in order to obtain their intrinsic phenomenological dynamics has been investigated recently by a few authors and is referred to as evolutionary modelling. A strategy named Evolutionary Polynomial Regression (EPR) has been applied to a real hydrogeological system, the shallow unconfined aquifer of Brindisi, southern Italy, for which 528 recorded monthly data over a 44-year period are available. The EPR returns a set of non-dominated models, as ordinary differential equations, reproducing the system dynamics. The choice of the representative model can be made both on the basis of its performance against a test data set and based on its incorporation of terms that actually entail physical meaning with respect to the conceptualization of the system. Key words groundwater; conceptual model; ordinary differential equations; evolutionary modelling; shallow aquifer Inférer la dynamique du système hydrogéologique à partir de séries des données hydrologiques Résumé Le problème de l'identification et de la reproduction du comportement hydrologique des systèmes hydrogéologiques peut souvent être posé en termes d'équations différentielles ordinaires relatives aux entrées et aux sorties de leurs composantes physiques, avec des hypothèses simplificatrices. Des modèles conceptuels linéaires et non-linéaires décrits sous forme d'équations différentielles ordinaires sont largement utilisés en hydrologie et peuvent être trouvés dans plusieurs études. Les systémes hydrogéologiques peuvent être décrits sur le plan conceptuel par un modèle de réservoirs interdépendants structuré comme une série de réservoirs nonlinéaires, de sorte que le niveau de la nappe peut être schématisé comme étant le niveau d'eau dans léun des réservoirs interconnectés. Dans ce travail, nous proposons une méthodologie permettant d'inférer la dynamique de la réponse d'un systéme hydrogéologique aux précipitations, sur la base de données temporelles enregistrées. L'utilisation de techniques évolutives pour déduire les équations différentielles à partir de données afin d'obtenir leur dynamique phénoménologique intrinséque a été étudiée récemment par quelques auteurs et est appelée modélisation évolutive. Une stratégie...

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