From prediction improvement to uncertainty propagation structure: revisiting hybrid hydrological models

Abstract. Hybrid hydrological models integrating embedded neural networks (ENNs) have demonstrated strong potential for improving streamflow prediction. However, how uncertainties induced by ENNs propagate through hydrological processes and internal system states remains poorly understood. This limits a mechanistic understanding of the stability and reliability of hybrid hydrological systems. This study therefore moves beyond performance evaluation and focuses on the internal uncertainty propagation structure of fully coupled hybrid hydrological systems. We investigate the internal uncertainty propagation mechanisms of hybrid hydrological models with different levels of ENN embedding across 31 cold-region basins. An ensemble-based framework driven by stochastic optimization was used to quantify uncertainty across hydrological fluxes and internal state variables. Multiple complementary metrics were employed to characterize uncertainty magnitude, ensemble consistency, coverage and spread. Results show that ENNs improve streamflow simulation performance but alter the internal uncertainty structure of hydrological systems. Uncertainty amplification is primarily localized within ENN-replaced hydrological processes, while propagation to physically based unreplaced processes remains limited. Instead, internal state variables with memory effects act as key “uncertainty reservoirs”, where uncertainty accumulates over time. This reveals a structured rather than uniform pattern of uncertainty propagation in hybrid hydrological systems. Furthermore, increased ensemble coverage is accompanied by wider uncertainty bands, indicating a trade-off between predictive reliability and sharpness induced by stochastic optimization. Overall, this study provides new insights into how ENNs reshape uncertainty propagation pathways in hydrological models. The results highlight the importance of jointly considering predictive performance, uncertainty structure and internal system stability in hydrological modeling.