Maximum likelihood parameter estimation for fitting bedload rating curves

Abstract

Fluvial sediment loads are frequently calculated with rating curves fit to measured sediment transport rates. Rating curves are often treated as statistical representations in which the fitted parameters have little or no physical meaning. Such models, however, may produce large errors when extrapolation is needed, and they provide no insight into the sediment transport process. It is shown that log-linear least squares, the usual method for fitting rating curves, does not generally produce physically meaningful parameter values. In addition, it cannot accommodate data that include zero-transport samples. Alternative fitting methods based nonlinear least squares and on maximum likelihood parameter estimation are described and evaluated. The maximum likelihood approach is shown to fit synthetic data better than linear or nonlinear least squares, and to perform well with data that include zero-transport samples. In contrast, nonlinear least squares methods produce large errors in the parameter estimates when zero-transport samples are present or when the variance structure of the data is incorrectly specified. Analyses with fractional bedload data from a mountain stream suggest that bedload transport rates are gamma distributed, that the arrivals of bedload particles in a sampler conform to a Poisson distribution, and that the variance of nonzero samples can be expressed as a power function of the mean. Preliminary physical interpretations of variations in the rating curve parameters fit to fractional bedload data with the maximum likelihood method are proposed, and their relation to some previous interpretations of rating curve parameters are briefly discussed.