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 λ and GED kurtosis coefficient β. 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 (β) value can cause low probability of large residuals and small β 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 β = 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.