A Simple approximation to multifractal rainfall maxima using a generalized extreme value distribution model

Define Id to be the average rainfall intensity inside an interval of duration d, and denote by Iyr,d the maximum of Id in a year. Based on a standard asymptotic result from extreme value (EV) theory, assuming independence and distributional identity between the variables Id, Iyr,d is typically assumed to follow a generalized extreme value (GEV) distribution with shape parameter k(d) that depends on the extreme upper tail of the distribution of Id. Estimation of k(d) from either at-site or regional rainfall data is generally difficult for two reasons. The first is the poor knowledge of the upper tail of the distribution of Id, even for long rainfall records. The other is more theoretical and it is related to the applicability of the asymptotic EV result, when the number n = 1yr/d of the d-intervals in a year (or, equivalently, the number n of the Id variables over which the maximum Iyr,d is taken) is finite. In a recent study, Veneziano et al. (Water Resour. Res., doi:10.1029/2009WR008257) showed that for multifractal rainfall and typical values of d, 1yr/d is too small for convergence of Iyr,d to a GEV distribution. Hence, k(d) cannot be derived from asymptotic arguments and it is influenced by a region of the distribution of Id that is close to the body thereof, rather than its extreme upper tail. Here, we propose a simple method to theoretically calculate the shape parameter k(d) of a GEV distribution model fitted to Iyr,d, as a function of the averaging duration d. We do so by assuming that rainfall is stationary multifractal below some maximum temporal scale D, and estimate k(d) by fitting a GEV distribution to the maximum of n = 1yr/d independent and identically distributed Id variables. To keep the method simple and suitable for practical applications, we analytically approximate the distribution of Id and estimate k(d) by recursively solving a non-linear equation. The suggested method, to theoretically constraint the shape parameter of a GEV distribution model and then fit the model to the recorded annual rainfall maxima, is compared to the classical annual maximum (AM) approach, where all GEV parameters are calculated from data. The relative performance of the methods is evaluated by comparing the bias, variance and root mean square error (RMSE) of each approach, using the differences between the empirical annual maxima and those calculated theoretically from the fitted distribution models and the empirical exceedance probabilities.