A log-sinh transformation for data normalization and variance stabilization

When quantifying model prediction uncertainty, it is statistically convenient to represent model errors that are normally distributed with a constant variance. The Box-Cox transformation is the most widely used technique to normalize data and stabilize variance, but it is not without limitations. In this paper, a log-sinh transformation is derived based on a pattern of errors commonly seen in hydrological model predictions. It is suited to applications where prediction variables are positively skewed and the spread of errors is seen to first increase rapidly, then slowly, and eventually approach a constant as the prediction variable becomes greater. The log-sinh transformation is applied in two case studies, and the results are compared with one- and two-parameter Box-Cox transformations.