Markov models for coherent estimation of rainfall extremes

A fundamental challenge in hydrological risk assessment is the accurate estimation of the probabilities of rainfall events exceeding given (typically high) intensities. Since the design of infrastructures must take into account rainfall extremes at different temporal scales, coherence between estimated exceedance probabilities across different durations is desirable. Intensity duration frequency (IDF) curves, describing the expected frequency of extreme rainfall intensities measured at different durations, are an important tool in this context. Most existing methods for IDF estimation introduce adequate shape constraints through duration-dependent Generalized Extreme Value (dGEV) distributions. Efficient estimation requires the unrealistic assumption that rainfall accumulations over different durations are independent of each other. Proposed models to introduce dependence between rainfall intensities at different durations come at a heavy computational burden and often resort to approximate inference. This thesis proposes an alternative model, based on a first-order Markov assumption, which incorporates dependence between consecutive (discretely defined) rainfall durations via bivariate extreme distributions, while the marginal distributions are dGEV, satisfying shape constraints. Two frameworks are applied for the estimation of the IDF relationship: i) Frequentist framework, where uncertainty is captured by bootstrapping; ii) Bayesian decision-making framework, where Bayes risk functions provide alternative point estimators. In both cases, we investigate the usefulness of the proposed model via simulation studies. Results show that ignoring dependence across durations can result in overdispersed estimates. Finally, the proposed methods are showcased with an application to four stations in the German state of Brandenburg. The proposed model demonstrates strong performance, even when the data is derived from independent processes, without significantly increasing the uncertainty of the estimates. This flexibility allows the model to effectively capture various scenarios while avoiding the restrictive independence assumption.