Improved Estimators of Model Performance Efficiency for Skewed Hydrologic Data

Abstract 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 simulations S and observations O . We introduce two theoretical (probabilistic) definitions of efficiency, E and E′ , based on the estimators NSE and KGE , respectively, which enable controlled Monte Carlo experiments at 447 watersheds to evaluate their performance. Although NSE is generally unbiased, it exhibits enormous variability from one sample to another, due to the remarkable skewness and periodicity of daily streamflow data. However, use of NSE with logarithms of daily streamflow leads to estimates of E with almost no variability from one sample to the next, though with high upward bias. We introduce improved estimators of E and E′ based on a bivariate lognormal monthly mixture model that are shown to yield considerable improvements over NSE and slight improvements over KGE in controlled Monte Carlo experiments. Our new estimators of E should avoid most previous criticisms of NSE implied by the literature. Improved estimators of E that account for skewness and periodicity are needed for daily and subdaily streamflow series because NSE is not suited to such applications.