S. Nerantzaki, Propability distributions of extreme rainfall Application worldwide, Diploma thesis, 118 pages, Department of Water Resources and Environmental Engineering – National Technical University of Athens, Athens, October 2012.
The tail of a probability distribution, which is the upper part of it, governs both the magnitude and the frequency of extreme events. The probability distributions can be generally categorized into two families, based on their tail behavior: heavy-tailed and light-tailed distributions, with the latter having milder and less frequent extremes. In the case of rainfall, the study of the tail draws important conclusions concerning the return period of extreme events and its contribution in hydrologic design is obvious. In order to evaluate the behavior of rainfall extremes, 3 477 stations from all over the world with sample size over 100 years are examined. Two common graphical methods are applied in order to classify the empirical tails of rainfall to the above families. The first method, the log-log plot, demonstrates whether the tail is a power-type (heavy-tailed); however, a Monte Carlo study revealed that the method is unreliable for extreme values. The second method, the Mean Excess Function, is based on the mean value of a variable over a threshold and results in a zero slope line when applied and graphically depicted for the Exponential distribution (light-tailed). To apply the method we constructed confidence intervals for the slope of the Exponential distribution as functions of sample size. The validation of the method using Monte Carlo techniques revealed that it performs well especially for large samples. Additionally, four mean square error norms are compared in order to find the most reliable for fitting the distribution tail. The best performing norm is used to fit four different theoretical distribution tails to the empirical tails. The theoretical distributions tails are: the Pareto type II which is a power type, the Lognormal which is also heavy-tailed, the Weibull which has an intermediate tail, and the Gamma which is light-tailed and commonly used to describe daily rainfall. The results from this approach are well-matched with the ones obtained by the application of the Mean Excess Function method. The analysis shows that heavy-tailed distributions are in general in better agreement with the rainfall extremes compared to the commonly used light-tailed ones.