A Bayesian Hierarchical Approach to Regional Frequency Analysis of Extremes
Résumé
Rainfall and runoff frequency analysis is a major issue for the hydrological community. The distribution of hydrological extremes varies in space and possibly in time. Describing and understanding this spatiotemporal variability are primary challenges to improve hazard quantification and risk assessment. This presentation proposes a general approach based on a Bayesian hierarchical model, following previous work by Cooley et al. [2007], Micevski [2007], Aryal et al. [2009] or Lima and Lall [2009; 2010]. Such a hierarchical model is made up of two levels: (1) a data level modeling the distribution of observations, and (2) a process level describing the fluctuation of the distribution parameters in space and possibly in time. At the first level of the model, at-site data (e.g., annual maxima series) are modeled with a chosen distribution (e.g., a GEV distribution). Since data from several sites are considered, the joint distribution of a vector of (spatial) observations needs to be derived. This is challenging because data are in general not spatially independent, especially for nearby sites. An elliptical copula is therefore used to formally account for spatial dependence between at-site data. This choice might be questionable in the context of extreme value distributions. However, it is motivated by its applicability in spatial highly dimensional problems, where the joint pdf of a vector of n observations is required to derive the likelihood function (with n possibly amounting to hundreds of sites). At the second level of the model, parameters of the chosen at-site distribution are then modeled by a Gaussian spatial process, whose mean may depend on covariates (e.g. elevation, distance to sea, weather pattern, time). In particular, this spatial process allows estimating parameters at ungauged sites, and deriving the predictive distribution of rainfall/runoff at every pixel/catchment of the studied domain. An application to extreme rainfall series from the French Mediterranean area is presented. This case study highlights the applicability of the Bayesian hierarchical framework, and yields encouraging results. In particular, the derived predictive distributions appear reliable, with an honest quantification of the uncertainties affecting the analysis. Several avenues for improvement are also already apparent, for instance the use of alternative models to describe spatial dependence between data, or the inclusion of temporal covariates to describe the temporal variability of the distribution of extremes.