Stochastic modeling of flood peaks using the generalized extreme value distribution

Abstract

[1] The generalized extreme value (GEV) distribution is a standard tool for modeling flood peaks, both in annual maximum series (AMS) and in partial duration series (PDS). In this paper, combined maximum likelihood estimation (MLE) and L moment (LMOM) procedures are developed for estimating location, shape, and scale parameters of the GEV distribution. Particular attention is given to estimation of the shape parameter, which determines the ''thickness'' of the upper tail of the flood frequency distribution. Mixed MLE-LMOM methods avoid problems with both MLE (estimator variance) and LMOM (estimator bias) estimators of the shape parameter. The mixed MLE-LMOM procedure is extended to use the two largest flood peaks in a year. This extension is developed in a PDS framework. Estimation procedures are applied to flood peak observations from 104 central Appalachian basins. The estimated values of the shape parameter for the central Appalachian basins are more negative than has been considered physically reasonable, independent of the estimation procedure that is used. Twenty-eight percent of mixed method estimators of the shape parameter have values less than À0.5, implying that the moments of order 2 and above are infinite. The estimated shape parameters for the central Appalachian basins do not depend on basin morphological parameters (such as drainage area) or land cover properties (such as percent urban, forest, or agricultural land use). Estimated values of the location and scale parameters for the central Appalachian watersheds correspond well with GEV-based simple scaling theory. Estimated values of the shape parameter for central Appalachian watersheds are shown to differ markedly from those of southern Appalachian watersheds and the difference is shown to be linked to contrasting properties of extreme floods. To conclude the paper, analyses of mixture distribution models are presented to address the question of whether flood peaks really have extreme ''heavy tail'' behavior or whether the GEV distribution is not the appropriate model for flood peaks.

Analyses of extreme flooding in Austria over the period 1951–2006

“…Previous studies (Morrison and Smith, 2002;Northrop, 2004;Lima and Lall 2010;Villarini et al, 2011aVillarini et al, , 2011d found that the location and scale parameters scale linearly in the log-log domain (power-law behaviour) when plotted against drainage area. The power-law coefficients from these studies ranged between 0.5 and 0.8.…”

“…The GEV parameters are estimated by means of maximum likelihood estimators (refer to Hosking, 1990;Martins and Stedinger, 2000;Coles, 2001, Morrison andSmith, 2002 among others for a discussion about other estimation techniques). Because of possible non-stationarities in the data which would require time-varying GEV parameters, the GEV distribution is fit only to those stations without statistically significant abrupt and slowly varying changes.…”

“…The location and scale parameters exhibit a linear behaviour in the log-log domain when plotted as a function of drainage area (Morrison and Smith, 2002;Northrop, 2004;Lima and Lall, 2010;Villarini et al, 2011aVillarini et al, , 2011d, while the shape parameter tends to depend log-linearly on basin size Villarini et al, 2011aVillarini et al, , 2011d. Moreover, comparison with respect to the results in Villarini et al (2011d) for central Europe (they did not include Austria in their study area) would allow examining the possible regional differences in the dependence of the GEV parameters on drainage area.…”

“…This is a continuous distribution with three parameters (location, scale, and shape), and represents the limiting distribution resulting from maxima of identically distributed and independent or weakly dependent random variables (Leadbetter, 1983). In previous studies (Morrison and Smith, 2002;Northrop, 2004;Lima and Lall, 2010;Villarini et al, 2011aVillarini et al, , 2011d, the location and scale parameters were found to scale linearly in the log-log domain (power law behaviour). Moreover, these flood records exhibited heavy tail behaviour (Bernardara et al, 2008;Villarini et al, 2011aVillarini et al, , 2011d.…”

Regional scale spatio‐temporal consistency of precipitation variables related to water resource management and planning

“…The use of the GEV distribution has gained widespread application in recent literature because of its flexibility and ability to capture the frequency of extremes (Martins and Stedinger, 2000;Morrison and Smith, 2002;Nadarajah and Shiau, 2005). The GEV is the generalized form of three commonly applied extreme value distributions: the Gumbel, the Frechet and the Weibull.…”