Using maximum likelihood to derive various distance-based goodness-of-fit indicators for hydrologic modeling assessment.

Currently used goodness-of-fit (GOF) indicators (i.e. efficiency criteria) are largely empirical and different GOF indicators emphasize different aspects of model performance; a thorough assessment of model skill may require the use of robust skill matrices. In this study, based on the maximum likelihood method, a statistical measure termed BC-GED error model is proposed, which firstly uses the Box-Cox (BC) transformation method to remove the heteroscedasticity of model residuals, and then employs the generalized error distribution (GED) with zero-mean to fit the distribution of model residuals after BC transformation. Various distance-based GOF indicators can be explicitly expressed by the BC-GED error model for different values of the BC transformation parameter <italic>λ</italic> and GED kurtosis coefficient <italic>β.</italic> Our study proves that (1) the shape of error distribution implied in the GOF indicators affects the model performance on high or low flow discharges because large error-power (<italic>β</italic>) value can cause low probability of large residuals and small <italic>β</italic> value will lead to high probability of zero value; (2) the mean absolute error could balance consideration of low and high flow value as its assumed error distribution (i.e. Laplace distribution, where <italic>β</italic> = 1) is the turning point of GED derivative at zero value. The results of a study performed in the Baocun watershed via comparison of the SWAT model-calibration results using six distance-based GOF indicators show that even though the formal BC-GED is theoretically reasonable, the calibrated model parameters do not always correspond to high performance of model-simulation results because of imperfection of the hydrologic model. However, the derived distance-based GOF indicators using the maximum likelihood method offer an easy way of choosing GOF indicators for different study purposes and developing multi-objective calibration strategies. [ABSTRACT FROM AUTHOR]