Joint and Conditional Probability Distributions of Runoff Depth and Peak Discharge Using Entropy Theory

Abstract

A nonlinear relationship between peak discharge and runoff volume (or depth = volume per unit area or rainfall amount), reported in the literature, was derived based on the standardized peak discharge distribution (SPDD) with regression analysis. However, the SPDD regression-based runoff model may only predict the mean behavior of peak discharge for a given runoff depth (i.e., the conditional expectation of peak discharge for a given runoff depth). This study proposes the application of entropy theory to derive the joint frequency distribution of peak discharge and runoff depth and the distribution of peak discharge conditioned on runoff depth. The conditional expectation of peak discharge (i.e., predicted peak discharge) for a given runoff depth is then compared with that obtained from the second-order SPDD regression-based runoff model. The entropy-based method is validated using data from 27 watersheds of different areas located in different climate regions in the United States. The results show that (1) with properly defined constraints, the entropy-based method may properly model the joint distribution of runoff depth and peak discharge and conditional distribution of peak discharge given runoff depth, and (2) the proposed method performs better than the regression-based method in terms of representation of extreme values that otherwise may be considered outliers.