Fitting Power-Law Relations in Watershed Science and Analysis, with an Example Using the R Language

Abstract

Many quantitative relationships in the environmental sciences, and specifically in watershed science, can effectively be modelled using a power-law function. Such relationships are often estimated using ordinary least squares regression after linearizing the relationship by log-transforming both the x and y variables. Alternative approaches include nonlinear least squares regression and generalized nonlinear least squares regression. However, there are some differences in the underlying characteristics of  these models that can result in the generation of different relationships and associated prediction limits. This article provides an overview of the statistical models underlying these approaches, then illustrates their application using the R language for an example based on fitting a regional relationship to predict flood quantiles from catchment area. Keywords: law relationships, log-transformation, nonlinear least squares regression, generalized nonlinear least squares regression