Non-parametric multidimensional uncertainty estimation employing a hybrid approach

A good understanding of uncertainty is of paramount importance in the hydrological sciences, notably in streamflow prediction. Over recent decades, hydroinformatics has played a key role in advancing hydrological prediction through the exploration of physically inspired and conceptual models and data-driven approaches. In particular, machine learning (ML) models demonstrate strong predictive skills. Despite this, limited interpretability and potentially weak extrapolation under extreme conditions remain major disadvantages of ML applications [1,2].

To address these limitations, a hybrid framework that combines conceptual hydrological modelling with machine learning–based probabilistic forecasting is proposed. The  so-called Generalized Pareto Uncertainty (GPU) framework can be used to train an ensemble of models (potentially physically based) so that it reliably reproduces the predictive uncertainty of the output [3]. In this case, GPU is employed with the conceptual HYdrological Predictions for the Environment (HYPE) model. By embedding hydrological knowledge into a data-driven uncertainty framework, the proposed approach seeks to improve robustness, generalization, and physical consistency of streamflow forecasts.

GPU relies on finding a multi-objective optimal surface (something akin to a double Pareto surface) that selects model parameters that span the full range of exceedance of simulations—at the extremes, forcing some models to always underpredict and others to always overpredict—while simultaneously searching for optimal error metrics (e.g., Nash-Sutcliffe efficiency, King-Gupta efficiency, mean absolute error, etc.). One promising feature of the framework is that it is not constrained to one type of error metric or even two dimensions (exceedance and error metric), potentially even opening avenues for addressing equifinality challenges.

The methodology is applied to the Nabão and Douro river basin in Portugal, one basin in Sweden, and one in Ireland. The performance of three modelling strategies is compared: (i) a standalone conceptual model (HYPE), (ii) GPU combined with artificial neural networks (missing indirect foreknowledge about hydrological processes), and (iii) a hybrid approach that incorporates HYPE models as ensemble members. Results show that the inclusion of conceptual hydrological information leads to clear improvements in the quality of the predictive uncertainty estimates, including its resolution, reliability, and aggregate metrics (e.g., CRPS).

In this work, we clarify the concept behind GPU, demonstrate its results, address challenges, and discuss potential innovative applications.

[1] Baste, S., Klotz, D., Acuña Espinoza, E., Bardossy, A., & Loritz, R. (2025). Unveiling the limits of deep learning models in hydrological extrapolation tasks. Hydrology and Earth System Sciences, 29(21), 5871–5891. https://doi.org/10.5194/hess-29-5871-2025

[2] Nearing, G. S., Kratzert, F., Sampson, A. K., Pelissier, C. S., Klotz, D., Frame, J. M., Prieto, C., & Gupta, H. v. (2021). What Role Does Hydrological Science Play in the Age of Machine Learning? In Water Resources Research (Vol. 57, Issue 3). Blackwell Publishing Ltd. https://doi.org/10.1029/2020WR028091

[3] Matos, J. P., Hassan, M. A., Lu, X. X., & Franca, M. J. (2018). Probabilistic Prediction and Forecast of Daily Suspended Sediment Concentration on the Upper Yangtze River. Journal of Geophysical Research: Earth Surface, 123(8), 1982–2003. https://doi.org/10.1029/2017JF004240