Generalization of the power-law rating curve using hydrodynamic theory and Bayesian hierarchical modeling

The power-law rating curve has been used extensively in hydraulic practice and hydrology. It is given by $Q(h)=a(h-c)^b$, where $Q$ is discharge, $h$ is water elevation, $a$, $b$ and $c$ are unknown parameters. We propose a novel extension of the power-law rating curve, referred to as the generalized power-law rating curve. It is constructed by linking the physics of open channel flow to a model of the form $Q(h)=a(h-c)^{f(h)}$. The function $f(h)$ is referred to as the power-law exponent and it depends on the water elevation. The proposed model and the power-law model are fitted within the framework of Bayesian hierarchical models. By exploring the properties of the proposed rating curve and its power-law exponent, we find that cross sectional shapes that are likely to be found in nature are such that the power-law exponent $f(h)$ will usually be in the interval $[1.0,2.67]$. This fact is utilized for the construction of prior densities for the model parameters. An efficient Markov chain Monte Carlo sampling scheme, that utilizes the lognormal distributional assumption at the data level and Gaussian assumption at the latent level, is proposed for the two models. The two statistical models were applied to four datasets. In the case of three datasets the generalized power-law rating curve gave a better fit than the power-law rating curve while in the fourth case the two models fitted equally well and the generalized power-law rating curve mimicked the power-law rating curve.