Bivariate extreme value distributions: An application of the Gibbs sampler to the analysis of floods

The quantitative definition of the magnitude of a flood episode exclusively in terms of its peak rate of discharge is often insufficient. In common with most episodic hydrological phenomena, floods are intrinsically multivariate random events, characterized not only by their peak flow, but also by their volume and the duration of discharge above critical thresholds. A complete definition in terms of these three component random variables has not found its way into standard analytical practice. This may be partly because the theory of multivariate extremes is not straightforward, and partly because no intuitively attractive way of presenting the component variables in a probabilistic framework has been put forward. We propose that the Gibbs sampler, a member of the new generation of computer intensive methods in computational statistics, bypasses the theoretical difficulties. We demonstrate the Gibbs sampler, and provide graphical presentations of the results, for the annual flood hydrograph on the Mekong River at Vientiane, Laos. We extend the results to evaluate the potential distribution of flood damage to the city.