SPatial Efficiency And Kmoments (SPEAK): Evaluating Spatial Consistency in (Semi)Distributed Rainfall–Runoff Models

Abstract. We introduce the Spatial Efficiency and Kmoments (SPEAK) metric, a novel objective function for the spatial calibration of hydrological models. SPEAK is built on Kmoment-based statistics, including a Kmoment-based: i) correlation, ii) coefficient-of-variation ratio, and iii) probability density function. This novel formulation is explicitly designed to overcome key limitations of existing spatial performance metrics, such as sensitivity to binning strategies, grid resolution, and sample heterogeneity. By relying on distributional properties rather than grid-to-grid correspondence, SPEAK provides a statistically robust framework for evaluating spatial patterns in gridded hydrological variables. The proposed metric is implemented in both semi-distributed and fully distributed configurations of the TUW hydrological model and tested across 99 near-natural Chilean catchments that encompass strong climatic and physiographic gradients. Actual evapotranspiration (ETa) from GLEAM v4.2a is used as an independent spatial benchmark, allowing the assessment of model performance beyond streamflow reproduction. Calibration using SPEAK is compared with a conventional streamflow-only calibration based on the Kling-Gupta Efficiency (KGE) and an ETa-only calibration based on the Spatial Efficiency metric (SPAEF). Model performance is evaluated using the normalised root-mean-square error (NRMSE), the spatial Pearson correlation coefficient, the Fraction Skill Score (FSS), and sensitivity to catchment attributes. Results demonstrate that while streamflow-only calibration leads to satisfactory runoff simulations (KGE ≥ 0.25 for all catchments and cases analysed; whereas the mean and median KGE are 0.80 and 0.85, respectively), it fails to reproduce the spatial patterns of ETa. When ETa is used as a calibration target, SPEAK consistently outperforms SPAEF, exhibiting lower NRMSE (number of catchments with lower NRMSE: 85 and 92 in fully and semi-distributed configuration, respectively), reduced internal component dispersion, and improved representation of spatial patterns across seasons and hydroclimatic zones. Importantly, SPEAK shows limited dependence on catchment characteristics. These findings highlight SPEAK as a methodologically robust spatial performance metric, with clear potential for improving the calibration and diagnosis of distributed hydrological models and other gridded environmental variables.