# Bivariate return periods and their importance for flood peak and volume estimation

Brunner, Manuela Irene; Favre, Anne-Catherine; Seibert, Jan (2016). Bivariate return periods and their importance for flood peak and volume estimation. Wiley Interdisciplinary Reviews: Water, 3(6):819-833.

## Abstract

Estimates of flood event magnitudes with a certain return period are required for the design of hydraulic structures. While the return period is clearly defined in a univariate context, its definition is more challenging when the problem at hand requires considering the dependence between two or more variables in a multivariate framework. Several ways of defining a multivariate return period have been proposed in the literature, which all rely on different probability concepts. Definitions use the conditional probability, the joint probability, or can be based on the Kendall’s distribution or survival function. In this study, we give a comprehensive overview on the tools that are available to define a return period in a multivariate context. We especially address engineers, practitioners, and people who are new to the topic and provide them with an accessible introduction to the topic. We outline the theoretical background that is needed when one is in a multivariate setting and present the reader with different definitions for a bivariate return period. Here, we focus on flood events and the different probability concepts are explained with a pedagogical, illustrative example of a flood event characterized by the two variables peak discharge and flood volume. The choice of the return period has an important effect on the magnitude of the design variable quantiles, which is illustrated with a case study in Switzerland. However, this choice is not arbitrary and depends on the problem at hand.

## Abstract

Estimates of flood event magnitudes with a certain return period are required for the design of hydraulic structures. While the return period is clearly defined in a univariate context, its definition is more challenging when the problem at hand requires considering the dependence between two or more variables in a multivariate framework. Several ways of defining a multivariate return period have been proposed in the literature, which all rely on different probability concepts. Definitions use the conditional probability, the joint probability, or can be based on the Kendall’s distribution or survival function. In this study, we give a comprehensive overview on the tools that are available to define a return period in a multivariate context. We especially address engineers, practitioners, and people who are new to the topic and provide them with an accessible introduction to the topic. We outline the theoretical background that is needed when one is in a multivariate setting and present the reader with different definitions for a bivariate return period. Here, we focus on flood events and the different probability concepts are explained with a pedagogical, illustrative example of a flood event characterized by the two variables peak discharge and flood volume. The choice of the return period has an important effect on the magnitude of the design variable quantiles, which is illustrated with a case study in Switzerland. However, this choice is not arbitrary and depends on the problem at hand.

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