Amélioration des performances d'un modèle stochastique de génération de hyétogrammes horaires: application au pourtour méditerranéen français Improvement of the results of a stochastic model for generating hourly hyetographs, applied to the french seaboard

Résumé

Depuis quelques années, un modèle stochastique de génération de hyétogrammes horaires est développé au groupement d'Aix-en-Provence du Cemagref, pour être couplé à une modélisation de la pluie en débit, fournissant ainsi une multitude de scénarios de crues analysés statistiquement et utilisés en prédétermination des débits de crues. L'extension de la zone d'application du modèle de pluies horaires au-delà de sa zone de conception, a fait apparaître une hétérogénéité dans les résultats. Ce constat a entraîné certaines modifications du modèle comme : la recherche d'une loi de probabilité théorique peu sensible aux problèmes d'échantillonnage pour une variable du modèle (intensité d'une averse), la prise en compte originale de la dépendance observée entre deux variables du modèle (durée et intensité d'une averse), et la modélisation de la persistance des averses au sein d'une même période pluvieuse. Ces différentes modifications apportées au modèle initial ont entraîné une très nette amélioration de ses performances sur la cinquantaine de postes pluviographiques du pourtour méditerranéen français. On obtient ainsi un outil beaucoup plus robuste et validé sur une zone étendue, capable de fournir de multiples formes de hyétogrammes, couvrant toute la gamme des fréquences, permettant ainsi de s'affranchir des pluies de projet uniques. On aborde aussi une nouvelle approche du comportement à l'infini des distributions de fréquences des pluies qui semble parfois supérieur à une tendance strictement exponentielle. De plus, l'étude de plusieurs événements par an dont chacun présente plusieurs réalisations des différentes variables du modèle augmente la taille des échantillons analysés, semblant rendre la méthode plus rapidement fiable qu'une approche statistique classique basée par exemple sur l'ajustement de valeurs maximales annuelles.

Mots-clés:

• Modèle stochastique,
• pluie horaire,
• pourtour méditerranéen français,
• variables dépendantes

Abstract

A stochastic model for generating hourly hyetographs has been recently developed, in the Cemagref of Aix-en-Provence, to be coupled with a rainfall runoff conversion modelling. Thus, by simulation of very long periods (1000 years for example), we obtain a large number of hourly hyetographs and flood scenarios that are statistically studied and used in flood predetermination problems. The rainfall model studied is based on the theory that rainfall can be linked to a random and intermittent process whose evolution is described by stochastic laws. It is also based on the hypothesis of independence between variables describing hyetographs and on the hypothesis of the stationary nature of the phenomenon studied. Generating a rainfall time series involves two steps : descriptive study of the phenomenon (nine independent variables are chosen to describe the phenomenon and these variables are defined by a theoretical law of probability fitted to the observations) and creation of a rainfall time series using descriptive variables generated randomly from their law of probability. Initially developed on the Réal Collobrier watershed data, the model has been applied to fifty raingauges located on the Mediterranean French seaboard. The extension of the model applying area has shown heterogeneousness in the results. Therefore, modifications have been made to the model to improve its performances. Among these modifications, three of them have presented notable improvements.

A study of the sensitivity of the parameters has been made. Parameters of shape variables and of some other variables had only a slight influence on depth of generated rainfalls. But, the law of mean rainfall intensities clearly differentiates the stations. Then, a theoretical probability distribution for the storm intensity variable, less sensitive to the sampling problems, has been searched. An exponential distribution is fitted to the value smaller than four times the mean of the variable. A slope breakage was then introduced to generate all the values beyond this limit. The breakage at the value four times the mean of the variable and modelling this breakage were based on a study of so-called "regional" distributions of the storm intensity variable. These distributions were designed by clustering the variable's homogenized values for all 50 studied stations.

A second modification has been made to develop new model for the observed dependence between two variables (duration and intensity of the storm). The study of this dependence has been considered directly based on the cumulative frequency of the two variables. Then, an additional parameter was defined to model the dependence between the probabilities of the two variables. This parameter characterises the cumulative frequency curve of the sum of the probabilities of the two variables. This point, neglected during a long time, has been very important in the improvement of the model.

Finally, the modelling of storm persistence in a same rainfall episode has been studied to generate some high 24 hours maximum rainfalls. Persistence modelling is entirely justified by the fact that "ordinary storms" cluster together around the "main storm" (the "main storm" is the greatest storm of an episode and the "ordinary storms" are the other storms of the episode). When the study of this phenomenon is extended, it can be observed that there is a certain positive dependency between occurrence probability of the "main storm" and occurrence probability of storms which come before or after it. Two combined effects occur : within one rainy episode, the strongest "ordinary storms" are preferentially clustered together around the "main storm", and considering the number of "ordinary storms" throughout all the episodes, the strongest storms close to the "main storm" are preferentially associated with the strongest "main storms" and vice versa. This modification improves the performances of the altitude raingauges, which are characterised by high daily rainfall accumulations.

The different modifications added to the initial model, give very important improvements on the calibration of the fifty raingauges studied on the French Mediterranean seaboard. Its aptitude to generate rains observed in Mediterranean climate, strongly variables, consolidates us in the idea of its application on a zone much larger. The generation of hyetographs makes it possible to use the maximum the temporal information of the rain. Thus, we obtain a reliable tool, validated on a large area, for simulating hyetographs and hourly flood scenarios at all frequencies, and used instead of a unique design storm and design flood. The approach allows a new cumulative probability curve extrapolation, which seems sometimes greater than an exponential behaviour. Moreover, the study of many events per year, with many occurrences of the different variables of the model, increase the analysed sample size and seems to make the method more reliable than a statistical approach simply based, for example, on the fitting of annual maximum values.

Keywords:

• Stochastic model,
• hourly rainfall,
• French Mediterranean seaboard,
• dependence of variables