Statistics of extremes and estimation of extreme rainfall: I. Theoretical investigation / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: I. Recherche théorique

Abstract Abstract The Gumbel distribution has been the prevailing model for quantifying risk associated with extreme rainfall. Several arguments including theoretical reasoning and empirical evidence are supposed to support the appropriateness of the Gumbel distribution. These arguments are examined thoroughly in this work and are put into question. Specifically, theoretical analyses show that the Gumbel distribution is quite unlikely to apply to hydrological extremes and its application may misjudge the risk, as it underestimates seriously the largest extreme rainfall amounts. Besides, it is shown that hydrological records of typical length (some decades) may display a distorted picture of the actual distribution, suggesting that the Gumbel distribution is an appropriate model for rainfall extremes while it is not. In addition, it is shown that the extreme value distribution of type II (EV2) is a more consistent alternative. Based on the theoretical analysis, in the second part of this study an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100–154 years of data. This verifies the inappropriateness of the Gumbel distribution and the appropriateness of EV2 distribution for rainfall extremes.

[1]  N. T. Kottegoda,et al.  Probability, Statistics, and Reliability for Civil and Environmental Engineers , 1997 .

[2]  M. Kendall,et al.  The Advanced Theory of Statistics, Vol. 1: Distribution Theory , 1959 .

[3]  V. Klemeš,et al.  Tall Tales about Tails of Hydrological Distributions. I , 2000 .

[4]  D. Wilks.  Comparison of three‐parameter probability distributions for representing annual extreme and partial duration precipitation series , 1993 .

[5]  James C. I. Dooge,et al.  Looking for hydrologic laws , 1986 .

[6]  Gene E. Willeke,et al.  Myths and Uses of Hydrometeorology in Forecasting , 1980 .

[7]  M. Taqqu,et al.  Estimating long-range dependence in the presence of periodicity: An empirical study , 1999 .

[8]  E. Zelenhasić,et al.  A Stochastic Model for Flood Analysis , 1970 .

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

[10]  J. Stedinger.  Frequency analysis of extreme events , 1993 .

[11]  D. L. Turcotte,et al.  Fractal theory and the estimation of extreme floods , 1994 .

[12]  J. V. Sutcliffe,et al.  Regional flood frequency analysis in arid and semi-arid areas , 1992 .

[13]  Malcolm R Leadbetter,et al.  On extreme values in stationary sequences , 1974 .

[14]  J. R. Wallis,et al.  Estimation of the generalized extreme-value distribution by the method of probability-weighted moments , 1985 .

[15]  V. Yevjevich.  Probability and statistics in hydrology , 1972 .

[16]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

[17]  Demetris Koutsoyiannis,et al.  Statistics of extremes and estimation of extreme rainfall: II. Empirical investigation of long rainfall records / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: II. Recherche empirique sur de longues séries de précipitations , 2004 .

[18]  J. Hosking.  L‐Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics , 1990 .

[19]  Mauro Fiorentino,et al.  Two‐Component Extreme Value Distribution for Flood Frequency Analysis , 1984 .

[20]  Scott A. Sisson,et al.  A fully probabilistic approach to extreme rainfall modeling , 2003 .

[21]  Renzo Rosso,et al.  Statistics, Probability and Reliability for Civil and Environmental Engineers , 1997 .

[22]  Demetris Koutsoyiannis.  The Hurst phenomenon and fractional Gaussian noise made easy , 2002 .

[23]  V. Klemeš.  The Hurst Phenomenon: A puzzle? , 1974 .

[24]  A. Jenkinson.  The frequency distribution of the annual maximum (or minimum) values of meteorological elements , 1955 .

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

[26]  Luis R. Pericchi,et al.  Anticipating catastrophes through extreme value modelling , 2003 .

[27]  Allen Hazen,et al.  Closure of "Storage to be Provided in Impounding Municipal Water Supply" , 1914 .

[28]  M. Fréchet.  Sur la loi de probabilité de l'écart maximum , 1928 .

[29]  Maurice G. Kendall,et al.  The Advanced Theory of Statistics, Vol. I. , 1945 .

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

[31]  Allen Hazen,et al.  Storage to be Provided Impounding Reservoirs for Municipal Water Supply , 1913 .

[32]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[33]  David M. Hershfield,et al.  Estimating the Probable Maximum Precipitation , 1961 .

[34]  D. Walshaw.  Modelling extreme wind speeds in regions prone to hurricanes , 2000 .

[35]  Demetris Koutsoyiannis,et al.  Analysis of a Long Record of Annual Maximum Rainfall in Athens, Greece, and Design Rainfall Inferences , 2000 .