Functional Relationships and Asymptotic Properties of Distributions of Interest in Hydrologic Frequency Analysis

Abstract

The procedure of hydrologic frequency analysis involves fitting a theoretical probability distribution to a series of flows, water levels or rainfall. The data series must meet the criteria of being independent and identically distributed (iid). The theoretical distribution must be adequately chosen to reflect the nature of the phenomenon and the characteristics of the data being modeled (positive or negative skewness, range, etc.). A number of statistical distributions, with various numbers of parameters, have been proposed and used in a number of countries for the fitting of samples of hydrologic flood data (maximum annual discharge, for example). These distributions are reviewed in this paper, and the main characteristics of each distribution are briefly discussed. The functional relationships between these most commonly used distributions are highlighted. Proofs are derived for all the relationships that are established, and all necessary transformations of variables are identified. Special cases for each distribution are also discussed. This paper includes also a study and classification of the distributions according to their asymptotic properties (characteristics of the right tail of the distribution). One-, two-, three-, and four-parameter distributions are considered in this study. The final results are summarized in a diagram outlining the functional relationship between the variates of the different distributions, and a table detailing the probability density function (pdf) and the different forms and characteristics of each distribution. Another table classifying the asymptotic properties of these distributions is also presented.