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

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.

[1]  Jery R. Stedinger,et al.  Plotting positions for historical floods and their precision , 1987 .

[2]  George Kuczera,et al.  Combining site and regional flood information using a Bayesian Monte Carlo approach , 2009 .

[3]  Blanca Adriana Botero Hernández.  Estimación de crecidas de alto período de retorno mediante funciones de distribución con límite superior e información no sistemática , 2006 .

[4]  V. Yevjevich.  Misconceptions in hydrology and their consequences , 1968 .

[5]  David M. Hershfield,et al.  Method for Estimating Probable Maximum Rainfall , 1965 .

[6]  J. R. Wallis,et al.  Regional Frequency Analysis: An Approach Based on L-Moments , 1997 .

[7]  Robin Naulet,et al.  Utilisation de l'information des crues historiques pour une meilleure prédétermination du risque d'innondation. Application au bassin de l'Ardèche à Vallon Pont-d'Arc et St-Martin d'Ardèche. , 2002 .

[8]  Félix Francés,et al.  Incorporating Non-Systematic Information to Flood Frequency Analysis Using the Maximum Likelihood Estimation Method , 2001 .

[9]  Luis R. Pericchi,et al.  A case for a reassessment of the risks of extreme hydrological hazards in the Caribbean , 2006 .

[10]  V. Klemes,et al.  Probability of extreme hydrometeorological events - a different approach , 1993 .

[11]  J. R. Wallis,et al.  Estimation of parameters and quantiles of Wakeby Distributions: 1. Known lower bounds , 1979 .

[12]  Eric V. McDonald,et al.  Late Holocene upper bounds of flood magnitudes and twentieth century large floods in the ungauged, hyperarid alluvial Nahal Arava, Israel , 2008 .

[13]  V. Klemeš,et al.  Hydrological and Engineering Relevance of Flood Frequency Analysis , 1987 .

[14]  Neil S. Grigg,et al.  Dam Safety Policy for Spillway Design Floods , 1996 .

[15]  Jose D. Salas,et al.  Flood frequency analysis with systematic and historical or paleoflood data based on the two‐parameter general extreme value models , 1994 .

[16]  Demetris Koutsoyiannis,et al.  A probabilistic view of hershfield's method for estimating probable maximum precipitation , 1999 .

[17]  J. C. Houghton.  Birth of a parent: The Wakeby Distribution for modeling flood flows , 1978 .

[18]  Richard M. Vogel,et al.  An assessment of exceedance probabilities of envelope curves , 2007 .

[19]  Dani Gamerman,et al.  Statistical inference : an integrated approach , 1999 .

[20]  Melvin J. Dubnick.  Army Corps of Engineers , 1998 .

[21]  V. Baker.  Paleoflood hydrology and extraordinary flood events , 1987 .

[22]  Daniel R. H. O'Connell,et al.  Bayesian flood frequency analysis with paleohydrologic bound data , 2002 .

[23]  Steven G. Vick.  Degrees of Belief: Subjective Probability and Engineering Judgment , 2002 .

[24]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[25]  Allan McRobie,et al.  The Bayesian View of Extreme Events , 2004 .

[26]  Andrew Thomas,et al.  WinBUGS - A Bayesian modelling framework: Concepts, structure, and extensibility , 2000, Stat. Comput..

[27]  Jery R. Stedinger,et al.  Bayesian MCMC flood frequency analysis with historical information , 2005 .

[28]  Jonas Eliasson.  A statistical model for extreme precipitation , 1997 .

[29]  Dennis P. Lettenmaier,et al.  Initiative for Risk‐Based Flood Design , 1987 .

[30]  Stuart G. Coles,et al.  Bayesian methods in extreme value modelling: a review and new developments. , 1996 .

[31]  J. Stedinger,et al.  Generalized maximum‐likelihood generalized extreme‐value quantile estimators for hydrologic data , 2000 .

[32]  Michel Lang,et al.  The flood probability distribution tail: how heavy is it? , 2008 .

[33]  J. Stedinger,et al.  Flood Frequency Analysis With Historical and Paleoflood Information , 1986 .

[34]  M. Lang,et al.  Statistical analysis of extreme events in a non-stationary context via a Bayesian framework: case study with peak-over-threshold data , 2006 .

[35]  Victor R. Baker,et al.  Paleoflood evidence for a natural upper bound to flood magnitudes in the Colorado River Basin , 1993 .

[36]  Joesron Loebis.  Frequency Analysis Introducing Probable Maximum Hydrologic Events--Preliminary Studies in Japan and in Indonesia , 1997 .

[37]  J. J. Slade,et al.  An Asymmetric Probability Function , 1934 .