A signal-processing-based interpretation of the Nash–Sutcliffe efficiency

Abstract. The Nash–Sutcliffe efficiency (NSE) is a widely used score in hydrology, but it is not common in the other environmental sciences. One of the reasons for its unpopularity is that its scientific meaning is somehow unclear in the literature. This study attempts to establish a solid foundation for the NSE from the viewpoint of signal progressing. Thus, a simulation is viewed as a received signal containing a wanted signal (observations) contaminated by an unwanted signal (noise). This view underlines an important role of the error model between simulations and observations. By assuming an additive error model, it is easy to point out that the NSE is equivalent to an important quantity in signal processing: the signal-to-noise ratio. Moreover, the NSE and the Kling–Gupta efficiency (KGE) are shown to be equivalent, at least when there are no biases, in the sense that they measure the relative magnitude of the power of noise to the power of the variation in observations. The scientific meaning of the NSE suggests a natural way to define NSE=0 as the threshold for good or bad model distinction, and this has no relation to the benchmark simulation that is equal to the observed mean. Corresponding to NSE=0, the threshold of the KGE is given by approximately 0.5. In the general cases, when the additive error model is replaced by a mixed additive–multiplicative error model, the traditional NSE is shown to be prone to contradiction in model evaluations. Therefore, an extension of the NSE is derived, which only requires one to divide the traditional noise-to-signal ratio by the multiplicative bias. This has a practical implication: if the multiplicative bias is not considered, the traditional NSE and KGE underestimate or overestimate the generalized NSE and KGE when the multiplicative bias is greater or smaller than one, respectively. In particular, the observed mean turns out to be the worst simulation from the viewpoint of the generalized NSE.