WATER RESOURCES BULLETIN VOL. 22, NO. 5 AMERICAN WA TER RESOURCES ASSOCIATION OCTOBER 1986 BIAS IN HYDROLOGIC PREDICTION USING LOG-TRANSFORMED REGRESSION MODELS1 Roy W. Koch and Gaiy M. Smillie2 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 streamfiow. Parameters are usually estimated using least squares analysis on a linear relationship between a logarithmic trans- formation 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 streamfiow forecasting show the mag- nitude 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. (KEY TERMS: regression modeling; transformation bias; bias cor- rection.) INTRODUCTION The understanding of hydrologic processes and prediction of hydrologic variables is becoming increasingly important for proper management as our water resources are developed for multiple, and often potentially conflicting uses. In many instances, these hydrologic variables such as streamflow, water quality or sediment concentration have a record of widely spaced samples. On the other hand, if reasonably continuous data (e.g., on a daily basis) are available, the period of record may be short from the point of view of statistical evaluation. Stream discharge is the most widely measured variable and relatively long records are often available in concert with shorter, more infrequent data on the related variables. If rela- tionships between streamflow and the other variables can be developed, these can be used to either fill in the record, ex- tend it in time or both. Certainly, many hydrologic variables such as water quality are physically related to streamflow; however, the processes are even more complex than those producing runoff. They are a function of the watershed and channel properties, including the supply of the particular constituent as well as transport and dispersion. Even if these processes were reasonably well understood at a point, heterogeneity and scale of river basins make them extremely difficult to define theoretically. As a result, hydrologists often re.sort to empirical models relating hydrologic variables to stream discharge. One common ex- pression is the regression equation with discharge as the in- dependent variable. Hydrologic variables are usually not linearly related to discharge, however, and nonlinear functions such as the power function: 0p = aQb or the exponential function: = a exp(bQ) are used for prediction. In these equations (1) (2) = the predicted variable, e.g., water quality, sedi- ment concentration, streamfiow at some other location or some future time; Q = the independent variable, usually streamflow; and a and b = statistically derived parameters. In order to use simple least squares estimation techniques for the model parameters, logarithmic transformations of the variables can be used to "linearize" these equations. For example, Lystrom, et al. (1978), used this type of relation- ship to describe various water quality constituents as a func- tion of discharge. Andrews (1978) and others have also used 1Paper No. 85071 of the Water Rerources Bulletin. Discussions are open until June 1, 1987. 2Respectively, Associate Professor, Department of Civil Engineering, Portland State University, P.O. Box 751, Portland, Oregon 97207; and Na- tional Park Service, Applied Research Branch, Colorado State University, Fort Collins, Colorado 80523. 717 WATER RESOURCES BULLETIN