On the Mathematical Relationship Between RMSE and NSE Across Evaluation Scenarios

Model evaluation metrics play a crucial role in hydrology, where accurate prediction of continuous variables such as streamflow and rainfall--runoff is essential for sustainable water management. Among these metrics, the Nash-Sutcliffe efficiency (NSE) and the Root Mean Squared Error (RMSE) are widely used but can yield divergent rankings under certain conditions. This study analytically investigates three scenarios: (i) both metrics evaluated on the same dataset, (ii) both metrics re-evaluated on an expanded version of the same dataset, and (iii) metrics evaluated on different datasets. For each case, we derive mathematical conditions explaining when the RMSE and the NSE remain consistent and when contradictions arise. The results demonstrate that the RMSE and the NSE always align when metrics are evaluated on the same dataset (including the same expanded dataset), but discrepancies can emerge when metrics are evaluated on unequal datasets—for instance, when one metric is tested on the original dataset and the other on the expanded one. Two numerical demonstrations using real hydrological data from the Yufeng No. 2 torrential stream in Taiwan confirm these analytical results, illustrating how the NSE can be artificially inflated by dataset modification without improving actual prediction accuracy. These findings clarify the interpretation of the NSE and the RMSE in hydrological model assessment and provide practical guidance for reliable evaluation under SDG 6 (Clean Water and Sanitation) and SDG 13 (Climate Action).