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

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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. WIREs Water 2016, 3:819–833. doi: 10.1002/wat2.1173 This article is categorized under: Engineering Water > Methods Engineering Water > Planning Water Science of Water > Water Extremes
(a) Illustration of joint probabilities. Quadrant I shows the case when both variables X and Y exceed the values x and y. Quadrant II shows the case where Y but not X exceeds the reference value. Quadrant III shows the case where neither X nor Y exceed their reference values. Finally, Quadrant IV shows the case when X but not Y exceeds the reference value. (Modified from Ref ; Copyright 2002) (b): Hydrological example. The red line stands for the peak discharge threshold x, the red hydrograph for the threshold value of total flood volume y. For each quadrant in figure a, one example event is given. The flood event in Quadrant I has a higher peak discharge and a higher flood volume than given by the thresholds. The event in Quadrant II has a higher volume than the threshold but a lower peak discharge. The event in Quadrant IV has a lower volume than the threshold but a higher peak discharge.
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Illustration of different flood hydrograph characteristics.
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Kendall's critical level divided into a central (green) and two naïve parts (red). The black dots stand for the observed flood events for the Birse at Moutier‐la‐Charrue and the gray dots are 10,000 randomly generated pairs using the bivariate distribution of the peak discharges and flood volumes. The two possibilities of choosing one design realization are displayed. Namely, these are the most‐likely design realization and the component‐wise excess realization. It is also shown how a subset of realizations can be chosen with the ensemble approach.
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Design variable quantiles for different return period definitions. The black dots stand for the observed flood events for the Birse at Moutier‐la‐Charrue and the gray dots are 10,000 randomly generated pairs using the bivariate distribution of the peak discharges and flood volumes. The black square stands for the univariate quantile. The triangles represent the design variable pairs resulting from the Qmax‐ and V‐conditional approaches applied to the joint OR isoline. The isolines represent the return level curves for the two joint approaches AND and OR, and the approaches using the Kendall's and survival Kendall's distribution function. The squares on the isolines stand for the most‐likely design realizations on these isolines.
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