^{1}

^{2}

^{1}

<p>The Nash-Sutcliffe efficiency (NSE) is a widely used score in hydrology but 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 NSE from the viewpoint of signal progressing. Thus, a forecast 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 forecasts and observations.</p> <p>By assuming an additive error model, it is easy to point out that NSE is equivalent to an important quantity in signal processing: the signal-to-noise ratio. Moreover, 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 variation of observations. The scientific meaning of NSE explains why it is reasonable to choose NSE=0 as the boundary between skilful and unskilful forecasts in practice, and this has no relation with the benchmark forecast that is equal to the mean of observations. Corresponding to NSE=0, the critical values of KGE is given approximately by 0.5.</p> <p>In the general cases, when the additive error model is replaced by a mixed adaptive-multiplicative error model, the traditional NSE is shown not to be a well-defined notion. Therefore, an extension of NSE is derived, which only requires to divide the traditional noise-to-signal ratio by the multiplicative factor. This has a practical implication: if the multiplicative factor is not considered, the traditional NSE and KGE underestimate (overestimate) the generalized ones when the multiplicative factor is greater (smaller) than one. In particular, the benchmark forecast turns out to be the worst forecast under the view of the generalized NSE.</p>