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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    • "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.
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    • "SERINALDI: MULTIVARIATE FREQUENCY ANALYSIS WITH UNCERTAINTY whereas two different models were applied to model the copula, namely, a bivariate Gumbel mixed model [Yue et al., 1999] and a Gumbel logistic model [Chebana and Ouarda, 2011]. In this study, the latter is considered. "
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    ABSTRACT: [1] Moving from univariate to multivariate frequency analysis, this study extends the Klemeš' critique of the widespread belief that the increasingly refined mathematical structures of probability functions increase the accuracy and credibility of the extrapolated upper tails of the fitted distribution models. In particular, we discuss key aspects of multivariate frequency analysis applied to hydrological data such as the selection of multivariate design events (i.e., appropriate subsets or scenarios of multiplets that exhibit the same joint probability to be used in design applications) and the assessment of the corresponding uncertainty. Since these problems are often overlooked or treated separately, and sometimes confused, we attempt to clarify properties, advantages, shortcomings, and reliability of results of frequency analysis. We suggest a selection method of multivariate design events with prescribed joint probability based on simple Monte Carlo simulations that accounts for the uncertainty affecting the inference results and the multivariate extreme quantiles. It is also shown that the exploration of the p-level probability regions of a joint distribution returns a set of events that is a subset of the p-level scenarios resulting from an appropriate assessment of the sampling uncertainty, thus tending to overlook more extreme and potentially dangerous events with the same (uncertain) joint probability. Moreover, a quantitative assessment of the uncertainty of multivariate quantiles is provided by introducing the concept of joint confidence intervals. From an operational point of view, the simulated event sets describing the distribution of the multivariate p-level quantiles can be used to perform multivariate risk analysis under sampling uncertainty. As an example of the practical implications of this study, we analyze two case studies already presented in the literature.
    10/2013; 49(10). DOI:10.1002/wrcr.20531
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    • "At this stage, probabilities of occurrence of certain sea level are calculated, associated with a rainfall range using a Gumbel mixed-model as suggested by Yue et al. [9]. Table 4 illustrates this step. "
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    ABSTRACT: The North Coastal Region of the State of São Paulo, which comprises the Municipalities of Caraguatatuba, São Sebas-tião, Ilhabela and Ubatuba, is one of the most prone to flooding and debris flow deposition Brazilian areas, owing to hydrological extreme rainfall events usually coupled with extreme tidal levels. This risk is also high due to human lives and material assets, with increasing population rates and the establishment of large Companies such as the Oil industry, with reduced defense/prevention measures and works. The catastrophic scenario of the city of Caraguatatuba, in March 1967, resulting from one of the most serious natural disasters in Brazil, fosters discussions about probabilities of heavy rainfall-caused events and rise in the sea level in coastal areas. Hence, this research is a consequence of this reality. The research is founded on an innovative methodology based on the analysis of past data of rainfall and tidal stations, complemented with debris flow registers in the region of the North coastal zone of the State of São Paulo (Brazil). The analysis developed involved the meteorological, hydraulic, geotechnical and statistical knowledge areas. Practical results are intended to be used for urban planning, designs of macro-drainage, fluvial, maritime projects and debris flow retention structures. These practical applications will then associate the probability of occurrence of certain types of heavy rainfall-caused events such as flooding or debris flow coupled with a corresponding increase in tidal levels.
    International Journal of Geosciences 09/2013; 4(5B). DOI:10.4236/ijg.2013.45B006 · 0.26 Impact Factor
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