Bias in Hydrologic Prediction Using Log‐transformed Regression Models1

ABSTRACT: Hydrologic variables are related through a complex set of dynamic processes. Due to this complexity, empirical, usually statistical, models are used for the synthesis of records or extentions of short‐term data. Two statistical models applied are the power function and the exponential function of a hydrologic variable expressed in terms of streamflow. Parameters are usually estimated using least squares analysis on a linear relationship between a logarithmic transformation of the variables. This procedure produces biased results when used to predict an individual value or the long‐term mean. Assuming the errors of the linear model are normally distributed, the bias is derived and is shown to result from the inverse transformation process. For cases where the errors are not normal, a nonparametric approach is used to estimate the bias. Evaluation of the implications for water quality, sediment and streamflow forecasting show the magnitude of the bias to vary with the particular application and to be significant in a few cases. Use of this simple technique for sediment discharge did not provide accurate results and should be questioned in general. Since no general conclusions can be drawn from this study as to when the bias is significant, evaluation in each situation is recommended as standard practice in hydrologic regression when a transformation is applied.