Empirical Relations in Hydrology Derived Using Entropy Theory

Abstract

There is a multitude of relations employed in hydrologic modeling. These relations are either empirical or hypothesized. When tested against observed data, several of these relations do not seem valid. Nevertheless, they are often used for their simplicity, and hydrologic models using these relations perform reasonably well. The relations can be grouped into (1) proportional equality, (2) linear algebraic, (3) nonlinear algebraic or power, (4) gradient, and (5) mixed relations. A typical example of the proportional equality relation is the soil conservation service-curve number relation used for the computation of runoff. Its analogs have been employed in determining sediment yield, metal partitioning, Muskingum flow routing, and evapotranspiration. Examples of the linear algebraic relation are the storage-discharge relation used in rainfall-runoff modeling, the Muskingum relation used in flow routing in open channels, and the linear isotherm used in modeling solute transport in the vadose zone. Nonlinear algebraic relations include nonlinear storage-discharge relation, kinematic depth-discharge relation, rating curve, nonlinear isotherm, velocity-depth relation, permeability-grain size relation, sediment bed load-shear stress relation, and hydraulic geometry relations. Gradient relations can be represented by Darcy’s law and Fick’s laws, Newton’s viscosity law, and mixed type relations that combine linear and gradient laws, such as the Swartzendruber relation. The objective of this paper is to revisit these relations and derive them using the theory of entropy wherein their parameters can be expressed in terms of physically measurable quantities. Further, the entropy-based derivation leads to the derivation of the probability distribution underlying the empirical relation that can be used to determine its uncertainty. Using data from four watersheds, the entropy-based parameters of linear reservoir, nonlinear reservoir, Muskingum method, and depth-discharge relation were then compared with those computed using the least square method. When measured by the coefficient of correlation, percent bias, root mean square error, Nash–Sutcliffe efficiency, and Kling–Gupta efficiency, both methods were, in general, comparable. Entropy is a measure of information or uncertainty, and entropy theory allows the information contained in the design variable to be quantified. The implication here is that the design variable should be cast as a random variable or else it will have no uncertainty. Having defined the random variable, the theory can be applied to derive the probability distribution of the random variable. The derivation requires the maximization of entropy, subject to the information on the random variable that the probability distribution must satisfy. The information is often expressed in the form of moments, such as mean, variance, geometric mean, coefficient of skewness, and so on. If no information is available or known, then no constraints are specified, as is the case in some of the relations derived in this study. Thus, probability distributions are derived for given constraints. To derive physical relations, a flux-concentration relation connecting the cumulative distribution of the design variable to the concentration of the variable with the use of the Laplacian principle of insufficient reason is employed. These steps have been employed to derive the empirical relations reported in this study. These relations are frequently used in hydrologic applications. The advantage of applying the entropy theory in hydrologic practice is that it permits the computation of uncertainty of the design variable that is needed for decision making.