Suitability of Gamma, Chi-square, Weibull, and Beta distributions as synthetic unit hydrographs

Most available methods for synthetic unit hydrograph (SUH) derivation involve manual, subjective fitting of a hydrograph through a few data points. Because of this tedious procedure, the generated unit hydrograph is often left unadjusted for unit runoff volume. During recent decades, use of probability distribution functions (pdfs) in developing SUH has received much attention because of its similarity with unit hydrograph properties. In this study, the potential of four popular pdfs, i.e., two-parameter Gamma, three-parameter Beta, two-parameter Weibull, and one-parameter Chi-square distribution to derive SUH have been explored. Simple formulae are derived using analytical and numerical schemes to compute the distribution parameters, and their validity is checked with simulation of field data. The Gamma and Chi-square distributions behave analogously, and the Beta distribution approximates a Gamma distribution in a limiting case. Application to field data shows that the Beta and Weibull distributions are more flexible in hydrograph prediction than the Gamma, Chi-square, Gray [Gray, D.M., 1961. Synthetic hydrographs for small drainage areas. In: Proceedings of the ASCE, 87, HY4, pp. 33-54], SCS [SCS, 1957. Use of Storm and Watershed Characteristics in Synthetic Hydrograph Analysis and Application: V. Mockus. US Dept. of Agriculture, Soil Conservation Service, Washington, DC], and Snyder [Synder, F.F., 1938. Synthetic unit hydrographs. Trans. Am. Geophys. Union 19, 447-454] methods. A sensitivity analysis of pdf parameters on peak flow estimates of an UH indicated that Gamma and Chi-square distributions overestimate the peak flow value, for any overestimation in its parameter estimates. However, for the Beta and Weibull distributions a reverse trend was observed. Both were found to behave similarly at higher α (ratio of time to base and time to peak of UH) values. Further, an analogous triangular hydrograph approach was used to express the mean and variance of the UH in terms of time base and time to peak of the UH. This enabled a simple parameter estimation equation involving only time base and time to peak of the UH. Although the validity of this equation could not be evaluated with a proper amount of data, the results give an indication of the relationship between pdf and statistical properties of the UH to be further elaborated in future research.