[1] In this paper we provide a general theoretical framework exploiting copulas for studying the return periods of hydrological events; in particular, we consider events depending upon the joint behavior of two nonindependent random variables, an approach which can easily be generalized to the multivariate case. We show that using copulas may greatly simplify the calculations and may even yield analytical expressions for the isolines of the return periods, both in the unconditional and in the conditional case. In addition, we show how a new probability distribution may be associated with the return period of specific events and introduce the definitions of sub-, super-, and critical events as well as those of primary and secondary return periods. An illustration of the techniques proposed is provided by analyzing some case studies already examined in literature.

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.

[1] Stochastic models of rainfall, usually based on Poisson arrivals of rectangular pulses, generally assume exponential marginal distributions for both storm duration and average rainfall intensity, and the statistical independence between these variables. However, the advent of stochastic multifractals made it clear that rainfall statistical properties are better characterized by heavy tailed Pareto-like distributions, and also the independence between duration and intensity turned out to be a nonrealistic assumption. In this paper an improved intensity-duration model is considered, which describes the dependence between these variables by means of a suitable 2-Copula, and introduces Generalized Pareto marginals for both the storm duration and the average storm intensity. Several theoretical results are derived: in particular, we show how the use of 2-Copulas allows reproducing not only the marginal variability of both storm average intensity and storm duration, but also their joint variability by describing their statistical dependence; in addition, we point out how the use of heavy tailed Generalized Pareto laws gives the possibility of modeling both the presence of extreme values and the scaling features of the rainfall process, and has interesting connections with the statistical structure of the process of rainfall maxima, which is naturally endowed with a Generalized Extreme Value law. Finally, a case study considering rainfall data is shown, which illustrates how the theoretical results derived in the paper are supported by the practical analysis.

The problem of selecting the appropriate design flood is a constant concern to dam engineering and, in general, in the hydrological practice. Overtopping represents more than 40% of dam failures in the world. The determination of the design flood is based in some cases on the T-year quantile of flood peak, and in other cases considering also the T-year quantile of flood volume. However, flood peak and flood volume have a positive (strong or weak) dependence. To model properly this aspect a bivariate probability distribution is considered using the concept of 2-Copulas, and a bivariate extreme value distribution with generalized extreme value marginals is proposed. The peak-volume pair can then be transformed into the correspondent flood hydrograph, representing the river basin response, through a simple linear model. The hydrological safety of dams is considered checking adequacy of dam spillway. The reservoir behavior is tested using a long synthetic series of flood hydrographs. An application to an existing dam is given.

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] Multivariate extreme value models are a fundamental tool in order to assess potentially dangerous events. The target of this paper is two-fold. On the one hand we outline how, exploiting recent theoretical developments in the theory of copulas, new multivariate extreme value distributions can be easily constructed; in particular, we show how a suitable number of parameters can be introduced, a feature not shared by traditional extreme value models. On the other hand, we introduce a proper new definition of multivariate return period and show the differences with (and the advantages over) the definition presently used in literature. An illustration involving flood data is presented and discussed, and a generalization of the well-known multivariate logistic Gumbel model is also given.Citation: Salvadori, G., and C. De Michele (2010), Multivariate multiparameter extreme value models and return periods:

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