Statistics of extremes and estimation of extreme rainfall: I. Theoretical investigation / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: I. Recherche théorique

Abstract

The Gumbel distribution has been the prevailing model for quantifying risk associated with extreme rainfall. Several arguments including theoretical reasoning and empirical evidence are supposed to support the appropriateness of the Gumbel distribution. These arguments are examined thoroughly in this work and are put into question. Specifically, theoretical analyses show that the Gumbel distribution is quite unlikely to apply to hydrological extremes and its application may misjudge the risk, as it underestimates seriously the largest extreme rainfall amounts. Besides, it is shown that hydrological records of typical length (some decades) may display a distorted picture of the actual distribution, suggesting that the Gumbel distribution is an appropriate model for rainfall extremes while it is not. In addition, it is shown that the extreme value distribution of type II (EV2) is a more consistent alternative. Based on the theoretical analysis, in the second part of this study an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100-154 years of data. This verifies the inappropriateness of the Gumbel distribution and the appropriateness of EV2 distribution for rainfall extremes.Key words design rainfall; extreme rainfall; generalized extreme value distribution; Gumbel distribution; hydrological design; hydrological extremes; probable maximum precipitation; risk Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: I. Recherche théoriqueRésumé La distribution de Gumbel a longtemps été le modèle régnant pour la quantification du risque associé aux précipitations extrêmes. Plusieurs arguments comprenant à la fois un raisonnement théorique et des faits empiriques sont censés soutenir la pertinence de la distribution de Gumbel. Ces arguments sont examinés exhaustivement dans ce travail et sont mis en question. Spécifiquement, des analyses théoriques montrent que la distribution de Gumbel est peu susceptible de s'appliquer aux valeurs extrêmes de variables hydrologiques et que son application peut conduire à une mauvaise estimation du risque par une sérieuse sous-estimation des plus grandes valeurs extrêmes de précipitations. En outre, il est montré que les séries hydrologiques de longueur typique (quelques décennies) peuvent montrer une image déformée de la distribution réelle, ce qui suggère que la distribution de Gumbel est le modèle approprié pour les précipitations extrêmes alors qu'elle ne l'est pas. En outre, on montre que la distribution de valeurs extrêmes du type II (EV2) est une alternative plus cohérente. Dans la deuxième partie de cette étude, une recherche empirique étendue est effectuée, en se basant sur l'analyse théorique, avec une collection de 169 séries de précipitations, parmi les plus longues de celles qui sont disponibles dans le monde entier, comportant chacune 100-154 ans de données. Ceci permet de vérifier la non-pertinence de la distribution de Gumbel et la pertinence de la dist...

Intensity–duration–frequency curves from remote sensing rainfall estimates: comparing satellite and weather radar over the eastern Mediterranean

“…The use of AMS ensures independency of the elements of the series and, rather than peak over threshold series, is suitable for this study because it does not require the definition of thresholds, problematic operation on highly variable climatic conditions, and potentially undermining the interpretation of the comparison between different datasets. The GEV distribution is a three-parameter extreme values distribution used worldwide to model rainfall extremes (Fowler and Kilsby, 2003;Gellens, 2002;Koutsoyiannis, 2004;Overeem et al, 2008). It is described by the location, scale, and shape parameters, representing mean, dispersion and skewness of the distribution, respectively.…”

Generalized extreme value shape parameter and its nature for extreme precipitation using long time series and the Bayesian approach

“…Koutsoyiannis (2004aKoutsoyiannis ( , 2004b has analysed the statistics of daily rainfall extremes and argued for the use of the EV2 distribution (with positive shape parameter) instead of the Gumbel distribution (EV1) when analysing rainfall data to avoid an underestimation of risk associated with extreme rainfall. L-moment estimation of the distribution's shape parameter, ξ, led Koutsoyiannis (2004aKoutsoyiannis ( , 2004b to conclude that ffi 0:15 and that it is "constant for all examined geographical zones (Europe and North America)". Recent work on the generalized Pareto (GP) distribution's shape parameter, GEV ffi GP (Serinaldi and Kilsby 2014), supports > 0 Gershunov 2015, Cavanaugh et al 2015).…”

Simulation of multisite precipitation using an extended chain‐dependent process

“…In this case, the general extreme value distribution is recommended for modeling the extremal behavior of precipitation [e.g., Koutsoyiannis, 2004;Furrer and Katz, 2008]. In this study, we showed that excluding the effect of the precipitation occurrence at the regional scale may be a major reason for extremes being underestimated.…”