Approche statistique du coefficient d'écoulement et utilisation pour la prédétermination des crues

A statistical approach for determination of runoff coefficients and their use for forecasting flood flow

¹ SRAE Lorraine
² Subdivision Hydrologie CTGREF, Antony

Abstract

Rainfall-discharges methodes generally used for flood predetermination take into account a runoff coefficient which is a number comprised between 0 and 1. This approach denies the strongly random nature of the ability of a basin for accelerated runoff and therefore it results erroneous to use a constant runoff coefficient for flood estimation. It seems better to define this coefficient as being a random variable a priori. More precisely, a sample of this variable can be executed in the following way : for each reference period, the year for instance, the strongest rainfall P and the strongest discharge Q,, expressed as depth of runoff on rainfall duration are considered symetrically. An execution of the random variable K is thus obtained and defined as : K = In Q / P - Q Which inversely gives : Q = P / 1 + e - K From this formulation, we can deduce that it is necessary to know the law of the pair of random variables (P, K) so as to be able to determine the law of the Q variable and solve the flood pre-determination cases. If the two random variables P and K are independent, the determination of their law becomes a difficult problem. However, for certain small basins and for frequencies which can be observed, independence between the two variables was shown.It is then sufficient to know the law followed by each variable to use easily the previous relationship. As rainfall records are generally known for a longer period than discharge records, one can therefore take advantage of the information they contain. If often happens that P follows a Gumbel's law and K a Gauss's. In that case, charts permitting the calculation of the distribution of Q without the help of a computer have been drawn up. Two possible applications are outlined. The first one is an attempt to obtain a full sample of K variable when one has only a limited number of record years at his disposal. The second application tries to determine a posteriori the law of K, from an experimental distribution of discharges and to use it to extrapolate this distribution until a return period large enough to permit the application of the gradex method (Guillot, Duband, EDF-Grenoble) without any danger of overestimation of floods. These are only attempts to evaluate the possible developments of this new approach to the runoff coefficient. In the different parts of this study, a numerical application with real data available on a representative basin is described. It can also be noticed that the proposed technique gives rather different results from those obtained by the gradex method. Another relationship is then presented which could permit a better agreement with the hypotheses obtained by the gradex method. However, it would remain to check up that the independence of random variables is well respected in so far as one wants to keep its operational feature to the method. To conclude and without prejudice to the possibility of generalizing the methods born of this statistical approach to the runoff coefficient, it nevertheless remains certain that it is an error to consider this coefficient as constant. The example shown in this paper effectively proves that it can vary from 1 to 15 when one goes from quantile 0.01 to quantile 0.99.