On the Cumulative Distribution Function for Entropy-Based Hydrologic Modeling

Abstract

In spatial or temporal physically based entropy-based modeling in hydrology and water resources, the cumulative distribution function (CDF) of a design variable (e.g., flux, say discharge) is hypothesized in terms of concentration (e.g., stage of flow). Thus far, a linear hypothesis has been employed when applying entropy to derive relationships for design variables, but without empirical evidence or physical justification. Examples of such relationships include velocity distribution as a function of flow depth, wind velocity as a function of height, sediment concentration profile along the flow depth, rating curve, infiltration capacity rate as a function of time, soil moisture profile along the depth below the soil surface, runoff as a function of rainfall amount, unit hydrograph, and groundwater discharge along the horizontal direction of flow. This study proposes a general nonlinear form of the CDF that specializes into commonly used linear forms. The general form is tested using empirical data on velocity, sediment concentration, soil moisture, and stage-discharge and compared with those reported in the literature. It is found that a simpler form of the general nonlinear hypothesis seems satisfactory for the data tested, and it is quite likely that this simple form will suffice for other data as well. The linear hypothesis does not seem to hold for the data employed in the study.