Probability Distribution of Rainfall-Runoff Using Entropy Theory

Abstract

Using the entropy theory, the Shannon entropy-based joint probability distribution of rainfall amount and runoff depth (runoff volume per unit area) is derived. In order to evaluate the accuracy of the entropy-based joint distribution, the study applies the following approaches: (1) comparison of the entropy determined from the derived joint distribution with that of the empirical joint entropy calculated based on the kernel density with the optimized kernel bandwidth, (2) the root mean square error (RMSE) of the joint probability density derived from the joint distribution and the empirical joint density, and (3) testing the derived joint distribution using the chi-square goodness-of-fit test for multivariate distributions. In addition, the marginal distributions of rainfall and runoff derived from the joint distribution are compared with the empirical marginal and fitted gamma distributions and are evaluated using the Kolmogorov-Smirnov (K-S) goodness-of-fit test and RMSE. With the joint distribution appropriately determined, the conditional distribution of runoff depth is derived for a given rainfall depth. The derived distributions are evaluated using rainfall-runoff data from small agricultural experimental watersheds near Riesel (Waco), Texas, maintained by the USDA-ARS. Results of the study indicate that: (1) the Shannon entropy theory can be satisfactorily applied to model the bivariate rainfall and runoff distribution; (2) it can be applied to model the bivariate random variables when convenient bivariate distributions are not able to capture the dependence between them; and (3) because the Shannon entropy uses constraints to formulate the joint distribution, it may be applied to the bivariate random variables incorporating different types of marginal distributions.