A Unified Framework for Deriving Extremal Hypothesis Theories of Hydraulic Geometry

In a river cross section or segment that has attained an equilibrium flow regime, the relationship between geometric (river depth, width, hydraulic radius, bed slope, etc.) and/or hydraulic (velocity, roughness, shear stress, etc.) factors in relation to the river’s equilibrium discharge (i.e., flows of fixed nominal return period or at bankfull depth) is termed hydraulic geometry (HG). HG relations have been proposed from a multitude of approaches: empirical, statistical, physically based, and extremal. Furthermore, multiple extremal hypotheses exist, depending on the variable to be optimized (maximized or minimized). In this study, we (1) present a unified framework by invoking the principle of maximum entropy (or, equivalently, Laplace’s law of insufficient reason), that is, assuming that individual contributions by geometric and hydraulic factors to HG through the optimization of extremal hypotheses (Froude number, frictional resistance, mobility index, stream power, sediment flux, etc.) presumably stem from equiprobable contributions of these factors—in other words, entropy is maximized; (2) enumerate 26 combinations (groups of sizes two to five) of the five basic HG variables, namely depth, width, slope, velocity, and friction, and analyze their interrelations; and (3) relax our initial equiprobability assumption by plugging in a scaling (weighting) parameter and show that 11 special cases of the 26 unique combinations lead to reasonable approximations of benchmark empirical results in the literature in at least one instance for each of the five extremal hypotheses considered here. The benefits of the unified framework include providing a theoretical basis to bring together various hypotheses that result in HG relations and providing an analytical way to quantify (or explain) epistemic uncertainty therein. Ultimately, this unified framework improves our understanding of HG and provides an analytical framework for improving predictions of river flows.