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Abstract. 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.\ud Several international guidelines have recently recommended to select appropriate/relevant Hazard Scenarios\ud in order to tame the consequences of (extreme) natural phenomena. In particular, the scenarios should\ud be multivariate, i.e., they should take into account the fact that several variables, generally not independent,\ud may be of interest. In this work, it is shown how a Hazard Scenario can be identified in terms of (i) a specific\ud geometry and (ii) a suitable probability level. Several scenarios, as well as a Structural approach, are presented,\ud and due comparisons are carried out. In addition, it is shown how the Hazard Scenario approach\ud illustrated here is well suited to cope with the notion of Failure Probability, a tool traditionally used for\ud design and risk assessment in engineering practice. All the results outlined throughout the work are based\ud on the Copula Theory, which turns out to be a fundamental theoretical apparatus for doing multivariate risk\ud assessment: formulas for the calculation of the probability of Hazard Scenarios in the general multidimensional\ud case (d 2) are derived, and worthy analytical relationships among the probabilities of occurrence of\ud Hazard Scenarios are presented. In addition, the Extreme Value and Archimedean special cases are dealt\ud with, relationships between dependence ordering and scenario levels are studied, and a counter-example\ud concerning Tail Dependence is shown. Suitable indications for the practical application of the techniques\ud outlined in the work are given, and two case studies illustrate the procedures discussed in the pape

[1] The concept of return period is fundamental for the design and the assessment of many engineering works. In a multivariate framework, several approaches are available to its definition, each one yielding different solutions. In this paper, we outline a theoretical framework for the calculation of return periods in a multidimensional environment, based on survival copulas and the corresponding survival Kendall's measures. The present approach solves the problems raised in previous publications concerning the coherent foundation of the notion of return period in a multivariate setting. As an illustration, a practical hydrological application is presented.

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas.