The Gumbel Mixed Model for Flood Frequency Analysis

Hydro-Québec, Varennes, Quebec, Canada
Journal of Hydrology (Impact Factor: 3.05). 12/1999; 228(3-4):88-100. DOI: 10.1016/S0022-1694(99)00168-7


Many hydrological engineering planning, design, and management problems require a detailed knowledge of flood event characteristics, such as flood peak, volume and duration. Flood frequency analysis often focuses on flood peak values, and hence, provides a limited assessment of flood events. This paper proposes the use of the Gumbel mixed model, the bivariate extreme value distribution model with Gumbel marginals, to analyze the joint probability distribution of correlated flood peaks and volumes, and the joint probability distribution of correlated flood volumes and durations. Based on the marginal distributions of these random variables, the joint distributions, the conditional probability functions, and the associated return periods are derived. The model is tested and validated using observed flood data from the Ashuapmushuan river basin in the province of Quebec, Canada. Results indicate that the model is suitable for representing the joint distributions of flood peaks and volumes, as well as flood volumes and durations.

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    • "The data used in this study were derived from the work ofYue et al. (1999). The data refer to the annual peak flows and maximum flood volumes of the Ashuapmushuan river basin of the(1999). "
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    ABSTRACT: Floods are multidimensional phenomena which are conventionally studied considering the pairwise dependence between peak flow and flood volume or peak flow and duration. In this paper, the flood phenomenon is analysed based on the peak flow-flood volume dependence. The paper presents a comparison between two methodologies for the double frequency analysis using a bivariate probability distribution and the copulas approach. The comparison is performed with data from a case study example of the Ashuapmushuan river basin in Quebec, Canada. In the presented example the bivariate Extreme Value distribution type I (Gumbel) is used for comparison with the bivariate Gumbel-Hougaard Archimedean copula. For the parameters estimation of the the latter, the Kendall method and the maximum likelihood method are employed. Based on the derived results of the analysed example, it can be concluded that for engineering purposes, the copulas approach, regardless of the method of its parameter estimation, provides a simple and accurate approach for the frequency analysis of floods and the estimation of the design variables thereof. The Kendall estimation parameter method, as simpler method, is easier to apply.
    Full-text · Article · Sep 2015
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    • "Prominent examples are the analysis of floods (Hosking and Wallis, 2005), heavy rainfalls (Cooley et al., 2007) and extreme temperatures (Katz and Brown, 1992). Many of these problems are intrinsically multivariate; for instance, the severity of a flood depends not only on its peak flow, which is considered in many univariate flood studies, but also on its volume and its duration (Yue et al., 1999). Catastrophic flood events typically occur when more than one of these variables is taking a high value and therefore, the analysis of the joint behavior is of key importance. "
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    ABSTRACT: In environmental sciences, it is often of interest to assess whether the dependence between extreme measurements has changed during the observation period. The aim of this work is to propose a statistical test that is particularly sensitive to such changes. The resulting procedure is also extended to allow the detection of changes in the extreme-value dependence under the presence of known breaks in the marginal distributions. Simulations are carried out to study the finite-sample behavior of both versions of the proposed test. Illustrations on hydrological data sets conclude the work.
    Preview · Article · May 2015
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    • "Moreover, it is not invariant to monotonic transformations to Kendall's and Spearman's measures (Favre et al., 2004). Appropriateness of the choice of the distribution function was based on the "Gringonten plotting-position", which represents the empirical probability (Cunnane, 1978; Gringorten, 1963; Yue et al., 1999; Zhang and Singh 2007 "
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    ABSTRACT: The study is focused on the analysis and statistical evaluation of the joint probability of the occurrence of hydrological variables such as peak discharge (Q), volume (V) and duration (t). In our case study, we focus on the bivariate statistical analysis of these hydrological variables of the Danube River in Bratislava gauging station, during the period of 1876-2013. The study presents the methodology of the bivariate statistical analysis, choice of appropriate marginal distributions and appropriate copula functions in representing the joint distribution. Finally, the joint return periods and conditional return periods for some hydrological pairs (Q-V, V-t, Q-t) were calculated. The approach using copulas can reproduce a wide range of correlation (nonlinear) frequently observed in hydrology. Results of this study provide comprehensive information about flood where a devastating effect may be increased in the case where its three basic components (or at least two of them) Q, V and t have the same significance.
    Full-text · Article · Aug 2014 · Journal of Hydrology and Hydromechanics
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