Beyond deterministic hydrological modelling: a copula-based uncertainty framework

Conceptual hydrological models are widely used in both theoretical investigations and operational applications. Their flexibility and relative ease of implementation have contributed to their success in the last decades. Despite their widespread use, conceptual hydrological models are still predominantly applied in a deterministic form (D-model), without explicitly accounting for the inherent uncertainties affecting their predictions. To address this limitation, a broad range of uncertainty estimation methods has been proposed in the literature, spanning from simple resampling techniques to more complex Bayesian frameworks, from approaches that explicitly separate different sources of uncertainty, to methods that aggregate all sources into a single error term. The common objective of these approaches is the transition from a deterministic D-model to a probabilistic or stochastic representation (S-model), typically expressed through an ensemble of model output predictions.Recently, Koutsoyiannis and Montanari (Koutsoyiannis, D., & Montanari, A., 2022) introduced an innovative methodology, applied to river discharge model outputs, to tackle this problem, departing from the traditional residual-based paradigm adopted by most existing approaches. Their method, known as BLUECAT, instead exploits the dependence structure between D-model predictions and observed discharge, providing a local, data-driven characterization of predictive uncertainty. While the non-parametric nature of BLUECAT offers important advantages, it also entails intrinsic limitations, particularly in the representation of uncertainty near the extremes of the discharge distribution, especially when limited discharge records are available.This contribution builds upon the original BLUECAT framework by proposing a conceptually equivalent, yet operationally novel, parametric post-processing approach for conceptual rainfall–runoff uncertainty estimation. The method relies on the use of parametric copula models to describe the joint dependence between D-model predictions and discharge observations, enabling the analytical derivation of conditional predictive distributions. This formulation provides an elegant solution to several limitations of the non-parametric approach, including the definition of confidence bands in proximity to extreme flows. In addition, a second parametric variant specifically tailored to high-flow regimes is introduced, allowing for a focused characterization of uncertainty within a restricted range of D-model discharge predictions.The proposed methods are evaluated through a comparative study over 24 mountainous catchments in the Piedmont region (north-western Italy), considering both calibration and validation periods. The analysis includes reliability metrics for confidence bounds as well as performance indicators for key ensemble properties, such as the ensemble median. The results indicate that the parametric approaches, when short observation records are available, generally yield more robust and reliable uncertainty estimates during validation compared to the original non-parametric BLUECAT. Furthermore, the high-flow-tailored approach outperforms both the non-parametric method and the parametric approach applied over the full discharge range when focusing on extreme flows, thereby improving uncertainty quantification in high-risk hydrological scenarios.  Koutsoyiannis, D., & Montanari, A. (2022). Bluecat: A local uncertainty estimator for deterministic simulations and predictions. Water Resources Research, 58, e2021WR031215. https://doi.org/10.1029/2021WR031215