Calculating return periods and design quantiles in a multivariate environment is a difficult problem: this paper tries to make the issue clear. First, we outline a possible way to introduce a consistent theoretical framework for the calculation of the return period in a multi-dimensional environment, based on Copulas and the Kendall's measure. Secondly, we introduce several approaches for the identification of suitable design events: these latter quantities are of utmost importance in practical applications, but their calculation is yet limited, due to the lack of an adequate theoretical environment where to embed the problem. Throughout the paper, a case study involving the behavior of a dam is used to illustrate the new concepts outlined in this work.

This paper is of methodological nature, and deals with the foundations of Risk Assessment. Several international guidelines have recently recommended to select appropriate/relevant Hazard Scenarios in order to tame the consequences of (extreme) natural phenomena. In particular, the scenarios should be multivariate, i.e., they should take into account the fact that several variables, generally not independent, may be of interest. In this work, it is shown how a Hazard Scenario can be identified in terms of (i) a specific geometry and (ii) a suitable probability level. Several scenarios, as well as a Structural approach, are presented, and due comparisons are carried out. In addition, it is shown how the Hazard Scenario approach illustrated here is well suited to cope with the notion of Failure Probability, a tool traditionally used for design and risk assessment in engineering practice. All the results outlined throughout the work are based on the Copula Theory, which turns out to be a fundamental theoretical apparatus for doing multivariate risk assessment: formulas for the calculation of the probability of Hazard Scenarios in the general multidimensional case ( ) are derived, and worthy analytical relationships among the probabilities of occurrence of Hazard Scenarios are presented. In addition, the Extreme Value and Archimedean special cases are dealt with, relationships between dependence ordering and scenario levels are studied, and a counter‐example concerning Tail Dependence is shown. Suitable indications for the practical application of the techniques outlined in the work are given, and two case studies illustrate the procedures discussed in the paper.

Key Points: New consistent and well‐founded definition of Multivariate Return Period Easy‐to‐use formulas for performing Multivariate design and risk assessment The new approach is so general that all hydrological sciences may benefit of it

Salvadori, G., Tomasicchio, G.R.., D'Alessandro, F. 2013. Multivariate approach to design coastal and off-shore structures. A frequent statistical problem in many coastal and off-shore engineering situations is to estimate the probability of structural failure expressed in terms of Return Period and Design Quantile. Usually, an univariate analysis is carried out to quantify the risk of failure. However, coastal and off-shore structures typically fail because of the occurrence of a critical combination of all the variables at play in a sea storm: thus, it may be important to consider the joint occurrence of dangerous conditions. The present manuscript outlines an original approach, exploiting Copulas, in order to provide a consistent framework for the calculation of bivariate Return Periods and Design Quantiles, which can be generalized to the multivariate case.