Logarithmic transformation and peak-discharge power-law analysis

The peak-discharge and drainage area power-law relation Q=alpha A(theta) has been widely used in regional flood frequency analysis for more than a century. The coefficients alpha and theta can be obtained by nonlinear or log-log linear regression. To illustrate the deficiencies of applying log-transformation in peak-discharge power-law analyses, we studied 52 peak-discharge events observed in the Iowa River Basin in the United States from 2002 to 2013. The results show that: (1) the estimated scaling exponents by the two methods are remarkably different; (2) for more than 80% of the cases, the power-law relationships obtained by log-log linear regression produce larger prediction errors of peak discharge in the arithmetic scale than that predicted by nonlinear regression; and (3) logarithmic transformation often fails to stabilize residuals in the arithmetic domain, it assigns higher weight to data points representing smaller peak discharges and drainage areas, and it alters the visual appearance of the scatter in the data. The notable discrepancies in the scaling parameters estimated by the two methods and the undesirable consequences of logarithmic transformation raise caution. When conducting peak-discharge scaling analysis, especially for prediction purposes, applying nonlinear regression on the arithmetic scale to estimate the scaling parameters is a better alternative.