A scale-invariant evaluation framework for geoscientific scalar time-series models: demonstration with wave model validation

Abstract. In the validation of geoscientific scalar time-series models, the selection of evaluation metrics is often based on experience and conventions, lacking a systematic design framework. Moreover, commonly used metrics are predominantly dimensional error measures whose values are inherently tied to the magnitude of the variable being assessed, making fair comparisons across different stations or datasets difficult. To address these issues, this study proposes a scale-invariant evaluation framework termed the "Golden Triangle", consisting of the range-normalized root mean square error (NRMSE), the range-normalized mean absolute error (NMAE), and the Pearson correlation coefficient (PCC). It is mathematically proven that all three metrics possess scale invariance, meaning their evaluation results are unaffected by the absolute magnitude of the variable, thereby enabling direct cross-station and cross-dataset comparisons. The three metrics capture extreme error sensitivity, average error level, and trend consistency, respectively, complementing one another to form a multi-dimensional diagnostic capability and providing a systematic design framework for metric selection. An empirical case study using data from four global deep-sea buoys and two wave reanalysis models demonstrates that the "Golden Triangle" framework effectively eliminates the influence of wave height magnitude differences among stations, overcomes the cross-station ranking distortion caused by dimensional dependence, and yields reasonable performance rankings. Furthermore, in the comparison between the full dataset and the high-energy-state dataset, the framework exhibits excellent consistency, confirming its reliability and sensitivity to extreme events. The "Golden Triangle" framework offers a logically coherent and generalizable evaluation tool for the validation of geoscientific scalar time-series models.