DOI: 10.5176/2251-1911_CMCGS18.10

Authors: Nicholas A. Coleman, Leong Lee, Gregory S. Ridenour

Abstract:Hydraulic geometry equations show that the width, mean depth, and mean velocity of a stream are power functions of discharge (varying in time or space) whose exponents sum to 1. The governing equations include an extremal hypothesis, which can be expressed as an objective function with constraints. In this study, variables in an objective function based on minimum variance theory were assigned random numbers from a normal distribution followed by optimization using an algorithm known as the method of steepest descent (gradient method), which iterated a golden section line search. Output from a model with a single random variable was confined to a straight line on a ternary diagram. Chi-square analysis of one of the hydraulic exponents indicated that the simulated values were normally distributed. Such one-dimensional output would be of value in virtual, controlled experiments and sensitivity analyses. Output from a model randomizing two variables produced variation in two dimensions on a ternary diagram. Compositional data analysis of ratios of logarithms of hydraulic exponents indicated that the simulated values were nearly multivariate normal and produced a distribution with a statistically identical logratio mean vector but a different logratio covariance matrix than experimental flume data. Stochastic modeling reveals the influence of other variables on hydraulic geometry; compositional data analysis measures how well a stochastic extremal hypothesis model simulates natural distributions.

Keywords: rivers/streams, fluvial processes, geomorphology, open channel flow, hydraulic geometry, stochastic models, statistics, minimum variance theory


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