Trivariate copula to design coastal structures

Abstract

Some coastal structures must be redesigned in the future due to rising sea levels caused by climate change. The design of structures subjected to the actions of waves requires an accurate estimate of the long return period of such parameters as wave height, wave period, storm surge and more specifically their joint exceedance probabilities. The simplified Defra method that is currently used in particular for European coastal structures makes it possible to directly connect the joint exceedance probabilities to the product of the univariate probabilities by means of a single factor. These schematic correlations do not, however, represent all the complexity of the reality because of the use of this single factor. That may lead to damaging errors in coastal structure design. The aim of this paper is therefore to remedy the lack of robustness of these current approaches. To this end, we use copula theory with a copula function that aggregates joint distribution functions to their univariate margins. We select a bivariate copula that is adapted to our application by the likelihood method. In order to integrate extreme events, we also resort to the notion of tail dependence. The optimal copula parameter is estimated through the analysis of the tail dependence coefficient, the likelihood method and the mean error. The most robust copulas for our practical case with applications in Saint-Malo and Le Havre (in northern France) are the Clayton copula and the survival Gumbel copula. The originality of this paper is the creation of a new and robust trivariate copula with an analysis of the sensitivity to the method of construction and to the choice of the copula. Firstly, we select the best fitting of the bivariate copula with its parameter for the two most correlated univariate margins. Secondly, we build a trivariate function. For this purpose, we aggregate the bivariate function with the remaining univariate margin with its parameter. We show that this trivariate function satisfies the mathematical properties of the copula. We finally represent joint trivariate exceedance probabilities for a return period of 10, 100 and 1000 years. We finally conclude that the choice of the bivariate copula is more important for the accuracy of the trivariate copula than its own construction.