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A Bayesian approach for estimating extreme flood probabilities with upper-bounded distribution functions

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Abstract

Some recent research on fluvial processes suggests the idea that some hydrological variables, such as flood flows, are upper-bounded. However, most probability distributions that are currently employed in flood frequency analysis are unbounded to the right. This paper describes an exploratory study on the joint use of an upper-bounded probability distribution and non-systematic flood information, within a Bayesian framework. Accordingly, the current PMF maximum discharge appears as a reference value and a reasonable estimate of the upper-bound for maximum flows, despite the fact that PMF determination is not unequivocal and depends strongly on the available data. In the Bayesian context, the uncertainty on the PMF can be included into the analysis by considering an appropriate prior distribution for the maximum flows. In the sequence, systematic flood records, historical floods, and paleofloods can be included into a compound likelihood function which is then used to update the prior information on the upper-bound. By combining a prior distribution describing the uncertainties of PMF estimates along with various sources of flood data into a unified Bayesian approach, the expectation is to obtain improved estimates of the upper-bound. The application example was conducted with flood data from the American river basin, near the Folsom reservoir, in California, USA. The results show that it is possible to put together concepts that appear to be incompatible: the deterministic estimate of PMF, taken as a theoretical limit for floods, and the frequency analysis of maximum flows, with the inclusion of non-systematic data. As compared to conventional analysis, the combination of these two concepts within the logical context of Bayesian theory, contributes an advance towards more reliable estimates of extreme floods.

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Abbreviations

LB:

Lower-bounded floods as referring to the floods that are larger than a given low threshold LB

UB:

Upper-bounded floods as referring to the floods that are smaller than a given high threshold UB

DB:

Double-bounded floods as referring to the floods that are comprised within the discharge interval (LR, UR)

EX:

Annual flood-peaks with exact values

α :

Upper bound of the LN4 distribution

ε :

Lower bound of the LN4 distribution

σ Y :

Scale parameter of the LN4 distribution

μ Y :

Location parameter of the LN4 distribution

ρ:

Scale parameter of the prior distribution of parameter

β:

Shape parameter of the prior distribution of parameter

μ μ :

Location parameter of the prior distribution of parameter μ

σ μ :

Scale parameter of the prior distribution of parameter μ

ρ σ :

Scale parameter of the prior distribution of parameter σ

β σ :

Shape parameter of the prior distribution of parameter σ

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Acknowledgements

The authors wish to acknowledge the support to this research from FAPEMIG (“Fundação de Amparo à Pesquisa do Estado de Minas Gerais”), through grant CRA APQ 4683-5.04/07 (PPM), and from CNPq (“Conselho Nacional de Desenvolvimento Científico e Tecnológico”), through grants 305870/2006-8, 304505/2006-4, 472877/2006-2, and a scholarship for W. Fernandes. The authors also wish to thank the anonymous reviewers for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Hydraulic and Water Resources Engineering, Federal University of Minas Gerais, Escola de Engenharia, Av. Antônio Carlos 6627 – Bloco 1 – Sala 4609, Belo Horizonte, MG, 31270-901, Brazil

    Wilson Fernandes

  2. Department of Hydraulic and Water Resources Engineering, Federal University of Minas Gerais, Escola de Engenharia, Av. Antônio Carlos 6627 – Bloco 1 – Sala 4506, Belo Horizonte, MG, 31270-901, Brazil

    Mauro Naghettini

  3. Department of Statistics, Federal University of Minas Gerais, Belo Horizonte, MG, 30110-060, Brazil

    Rosângela Loschi

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  1. Wilson Fernandes
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Correspondence to Wilson Fernandes.

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Fernandes, W., Naghettini, M. & Loschi, R. A Bayesian approach for estimating extreme flood probabilities with upper-bounded distribution functions. Stoch Environ Res Risk Assess 24, 1127–1143 (2010). https://doi.org/10.1007/s00477-010-0365-4

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