À propos de la distribution statistique des cumuls pluviométriques annuels. Faut-il en finir avec la normalité? Statistical distribution of total annual rainfall. Is the Gaussian distribution inadequate?

Résumé

Il est communément admis que la distribution statistique des précipitations cumulées annuelles suit une loi de Laplace-Gauss. Les écarts entre cette loi et les distributions empiriques sont cependant un fait d'expérience : au-delà d'une probabilité au non dépassement correspondant à une période de retour d'une vingtaine d'années et pour les valeurs les plus fortes de pluie, l'ajustement n'est plus acceptable. Ce décrochage par rapport à la loi normale est mieux mis en évidence par l'étude des longues séries pluviométriques, plus riches en événements extrêmes. Pour étudier le comportement statistique de ces derniers, il est fait appel à un formalisme multifractal qui permet de mettre en évidence que, contrairement à ce qui est généralement admis, la décroissance de la probabilité au dépassement est de nature hyperbolique plutôt qu'exponentielle. Les probabilités des événements catastrophiques sont donc plus importantes que l'on ne le croyait jusqu'ici, ce qui peut avoir des conséquences particulièrement importantes. Cette approche appliquée à un ensemble de séries pluviométriques de longue durée permet de cerner le paramètre caractérisant la décroissance de la probabilité au dépassement. Les résultats obtenus jusqu'ici laissent à penser que ce paramètre pourrait être universel.

Mots-clés:

• Précipitations,
• événements extrêmes,
• loi normale,
• loi hyperbolique,
• multifractales,
• invariance d'échelle

Abstract

Up to now, annual rainfall accumulation have generally been modelled according to the Laplace-Gauss probability distribution. After a brief survey of the arguments for using this distribution to describe annual rainfall, which are mainly to consider annual rainfall accumulation as the sum of many independent individual rainfall events of similar magnitude identically distributed, we question its capacity to take into account the various characteristics of the rainfall events, in particular their magnitude, their number and their possible correlation or persistence. We have studied an alternative model based on the multifractal theory, well suited to model phenomena where matter and/or energy concentrate on a more and more sparse domain as the observation scale is decreasing. Some elements of the multifractal theory are briefly described. The main feature of the probabilistic model based on the assumption of a multifractal behavior is that for large enough accumulations, the probability distribution tail would have at any time scale an algebraic rather than an exponential behavior as it is the case for the Laplace-Gauss distribution. It is of value to note that such an algebraic behavior corresponds to a probability of occurrence decreasing much more slowly than in the case of an exponential one. It is also important to note that, unlike exponential distribution laws, all statistical moments of algebraic distribution laws are not defined, those of order greater than the exponent of the algebraic law diverging. This fact is of main importance in relation with sampling, as we cannot a priori be sure of the convergence of all statistical estimators and that the convergence of these estimators is likely to much more slow than that of exponential law estimators.

The application part of our study concerned 87 annual rainfall series spanning from 44 to 266 years with a mean of 116 years, gathered within the UNESCO FRIEND-AMHY project (Flow Regimes from International Experimental and Network Data - Alpine and Mediterranean Hydrology). For each series we have drawn on a log-log diagram the curve of the empirical probability of exceeding a given rainfall value (derived by the Weibull formula) versus the rainfall value. On such a diagram an algebraic behavior should be represented by a straight line the slope of which is the exponent characterizing the algebraic law tail.. Qualitatively the results tend to argue in favor of an algebraic rather than an exponential behavior of the probability distribution of high annual rainfall ( empirical return periods of more than 20 years ) but the exponent values are quite scattered, ranging from about 2 for the lowest values to more than 10. We then have concentrated our study on the basis of the 71 stations with series more than 90 years long. A diagram of the exponent value versus the corresponding series length suggest that there may be a slow convergence of exponent values towards a common value close to 3.8 as the length of the series increases ( This observation should be related to a possible divergence of high order moments which is the cause of poor estimation from small size samples ). This exponent would then be universal, related to the physical processes from which rainfall originates rather than to a geographical location. You have to recall here that the multifractal framework we used has been primarily designed to take into account the scaling symmetries of the Navier and Stokes equations. It is likely that the unknown partial differential equations governing rainfall processes share some properties with those governing atmospheric turbulence and it would not be too surprising that we can catch this way some physical feature of rainfall process.

These preliminary results, if confirmed by further researches, would have considerable theoretical and practical consequences. The algebraic behavior of heavy rainfall distribution, which is supposed, according to the multifractal theory, to arise at all time scales with the same exponent, should lead to reject the numerous empirical and ad-hoc distribution functions which are at present used mainly for practical purposes. These probability distribution functions have poor theoretical and/or physical basis and quite all of them exhibit more or less an exponential behavior. It is likely than a huge amount of efforts has been devoted for decades to the sophistication of fitting methods, using low and mean variate realizations, while large and extreme values were often understated and underused as extreme values are generally said to belong to another population or, equivalently, to be generated by an other random process than normal ones (outliers...). The new generation of distribution functions to design would be less empirical : it should take explicitly into account time scale invariance and should be able to reconcile into an unique model all the realizations of the rainfall accumulation variate, whatever their magnitudes are.

In order to size up the difference between algebraic and exponential statistical models one can note, for example, that the return period of the annual rainfall of four cities (Padova, Marseilles, Rome and Gibraltar) which were estimated at 1,000 years with the fitting of a Gaussian distribution, could be as low as 60 to 100 years with this new algebraic model. The return period is so divided by a factor of 10 ! All the same annual rainfall accumulations are said to have a rather " soft " behavior, the summation of numerous individual events being supposed to smooth their more " wild " behavior. This is probably not true and one can find in annual and even pluriannual rainfall accumulations the trace of extreme individual events. Whatever the time scale under consideration, it is easy to imagine the consequences of such a dramatic modification of return periods on engineering design. Such a revisiting of estimated return periods should be extended to hydrological events. As an example, the recent flooding of the river Oder in Germany, Poland, and Czech Republic (July 1997) for which preliminary information suggests a return period greater than 10,000 years (Gazowsky, personal communication), might not be so extreme if our new concepts are shown to be valid.

Keywords:

• Rainfall,
• extremes events,
• Gaussian law,
• hyperbolic law,
• multifractals,
• scale invariance