The Future of Extreme Event Risk Assessment: A Look at Multivariate Return Periods in More than Three Dimensions
Authors: Urrea Méndez Diego Armando , Gómez Rave Dina Vanesa , Del Jesus Peñil Manuel 
Published: February 2023

The multivariate return period is a measure of the frequency with which simultaneous sets of variables are expected to occur in a given area. So far, most approaches to calculate the multivariate return period of various hydrological variables have used copulas in two and three dimensions. (Salvadori et al., 2011) proposed a methodology for calculating the return period based on Archimedean copulas and the Kendall measure in 2 and 3 dimensions. (Gr&#228;ler et al., 2013) proposed the calculation of the trivariate return period based on Vine copulas and Kendall distribution functions to describe the characteristics of the design hydrogram, considering the annual maximum peak discharge, its volume and duration. (Tosunoglu et al., 2020) applied three-dimensional Archimedean, Elliptical and Vine copulas to study the characteristics of floods. These studies have shown that the use of copulas can improve the accuracy of the risk measure of extreme events compared to univariate approaches, that only consider one variable at a time.One of the limitations in describing the occurrence of multivariate extreme involving more than three simultaneous variables is the complex mathematical model to be solved (highest probability density point of a hypersurface) and the high computational cost that this imposes. However, in some areas of hydrology, developing more robust analyses that consider more than three variables can further improve risk assessments. For example, considering multiple rainfall stations in a watershed may help to properly capture the spatial structure of extremes -instead of relying on other spatial distribution procedures-. This improvement can provide a more accurate measure of the return period in the design of critical infrastructure, flood prediction, risk plans, etc.In this context, we present an application where an extreme characterization of 5 rain gauges is considered simultaneously, using vine copulas based on Kendall distribution functions. More specifically, we analyze which measures are suitable for explaining the spatial and temporal correlation of rain events in different locations within a network of stations; which families and structures of vine copulas can optimally capture the spatial dependence structure within a region; how to solve the complex mathematics that is imposed when expanding the dimensionality; what is a computationally reasonable alternative to improve the computational cost involved.; and how multivariate analysis can improve the precision of the extreme event risk measure compared to univariate approaches.These questions are answered by applying the proposed methods to a pilot case, which is developed in a basin located in northern Spain. Multivariate modeling is becoming increasingly relevant in the field of hydrology due to its ability to model extreme stochastic events, which are key to mitigating the risk and damages caused by floods.ReferencesGr&#228;ler, B., Berg, M. J. van den, Vandenberghe, S., Petroselli, A., Grimaldi, S., De Baets, B. & Verhoest, N. E. C., 2013. Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation. Hydrol. Earth Syst. Sci., 17(4), 1281&#8211;1296.Salvadori, G., De Michele, C. & Durante, F., 2011. On the return period and design in a multivariate framework. Hydrol. Earth Syst. Sci., 15(11), 3293&#8211;3305.