3.9.1 Distribution Selection: Practical Considerations
Many different probability distributions have been proposed for application to hydrologic data. Some of them were proposed because the underlying concept of the distribution matched the goal of hydrologic frequency analysis. For example, the extremal distributions discussed in Sec. 2.6.4 have very favorable properties for hydrologic frequency analysis. Ang and Tang (1984, p. 206) noted that the asymptotic distributions of extremes in several cases tend to converge on certain limiting forms for large sample sizes n, specifically to the doubleexponential form or to two single-exponential forms. The extreme value from an initial distribution with an exponentially decaying tail (in the direction of the extreme) will converge asymptotically to the extreme-value type I (Gumbel) distribution form. Distributions with such exponentially decaying tails include the normal distribution and many others listed in Sec. 2.6. This is why Gumbel (1941) first proposed this distribution for floods, and it has gained considerable popularity since then. Also, the properties of the central limit theorem discussed in Sec. 2.6.2 have made the lognormal distribution a popular choice for hydrologic frequency analysis.
In the 1960s, as the number of different approaches to flood frequency analysis were growing, a working group of U. S. government agency hydrologic experts was formed by the U. S. Water Resources Council to evaluate the best/preferred approach to flood frequency analysis. Benson (1968) reviewed the results of this working group and listed the following key results of their study:
1. There is no physical rule that requires the use of any specific distribution in the analysis of hydrologic data.
2. Intuitively, there is no reason to expect that a single distribution will apply to all streams worldwide.
3. No single method of testing the computed results against the original data was acceptable to all those on the working group, and the statistical consultants could not offer a mathematically rigorous method.
Subsequent to this study, the U. S. Water Resources Council (1967) recommended use of the log-Pearson type 3 distribution for all flood frequency analyses in the United States, and this has become the official distribution for all flood frequency studies in the United States. There are no physical arguments for the application of this distribution to hydrologic data. It has added flexibility over two-parameter distributions (e. g., Gumbel, lognormal) because the skewness coefficient is a third independent parameter, and the use of three parameters generally results in a better fit of the data. However, a number of researchers have suggested that the use of data for a single site may be insufficient to estimate the skewness coefficient properly.
Beard (1962) recommended that only average regional skewness coefficients should be applied in flood frequency analysis for a single station unless that record exceeds 100 years. This led the U. S. Water Resources Council (1967) to develop maps of regional skewness coefficient values that are averaged with the at-a-site skewness coefficient as a function of the number of years of record. For details on the procedure, see Interagency Advisory Committee on Water Data (1982). Linsley et al. (1982) noted that although regional skewness coefficients may not make for more reliable analysis, their use does lead to more consistency between values for various streams in the region.
