Improved Estimators of Model Performance Efficiency for Skewed Hydrologic Data

The Nash-Sutcliffe efficiency (NSE) and the Kling-Gupta efficiency (KGE) are now the most widely used indices in hydrology for evaluation of the goodness of fit between model simulationsSand observationsO. We introduce two theoretical (probabilistic) definitions of efficiency,EandE ', based on the estimatorsNSEandKGE, respectively, which enable controlled Monte Carlo experiments at 447 watersheds to evaluate their performance. AlthoughNSEis generally unbiased, it exhibits enormous variability from one sample to another, due to the remarkable skewness and periodicity of daily streamflow data. However, use ofNSEwith logarithms of daily streamflow leads to estimates ofEwith almost no variability from one sample to the next, though with high upward bias. We introduce improved estimators ofEandE ' based on a bivariate lognormal monthly mixture model that are shown to yield considerable improvements overNSEand slight improvements overKGEin controlled Monte Carlo experiments. Our new estimators ofEshould avoid most previous criticisms ofNSEimplied by the literature. Improved estimators ofEthat account for skewness and periodicity are needed for daily and subdaily streamflow series becauseNSEis not suited to such applications.