The study, led by Patricio Yeste of the University of Potsdam together with colleagues from Wageningen University, the Federal University of Rio Grande do Sul, the University of Granada, Pompeu Fabra University, the University of Bergen, and the University of Naples Federico II, examines the practical relationship between Sobol’ total-effect indices and feature importance measures derived from machine learning metamodels. The work was published as a technical note and highlighted by the journal’s editorial team as bringing important insights to the hydrological community on the opportunities that explainable artificial intelligence offers for sensitivity analysis, while directly addressing the computational efficiency barrier that has long limited its application to complex models.

The theoretical foundation of the approach rests on a mathematical link established in earlier work. When the input factors of a model have independent probability distributions, the total-effect Sobol’ index of a given parameter is directly related to permutation variable importance, a model-agnostic measure widely used in explainable artificial intelligence. Permutation variable importance quantifies how much a predictive model’s error increases when the association between one input and the output is deliberately destroyed, for example by randomly shuffling that input’s values. The paper builds on this connection and provides systematic numerical evidence that the link holds in practice for real hydrologic models, provided the surrogate faithfully reproduces the behaviour of the original model.

To test the idea, the researchers ran a comprehensive numerical experiment involving three hydrologic models of varying structural complexity: the conceptual rainfall-runoff models HBV and HyMod, and the physically based land-surface model VIC version 5. These models were applied to three headwater catchments in northern Germany, the Sude, Treene, and Trave rivers, all part of the CAMELS-DE dataset. The catchments experience semi-humid climatic conditions, are dominated by groundwater-driven runoff generation, and range in area from 136 to 675 square kilometres. Mean annual precipitation varies from 661 millimetres per year in the Sude catchment to 909 millimetres per year in the Treene, and the catchments are free of major flow regulation, allowing analysis under near-natural conditions.

For each model and catchment combination, the team conducted a Monte Carlo experiment using Sobol’ quasi-random sampling with 10,000 samples, generating a total of n(p+2) model evaluations per combination, where n is the sample size and p the number of free parameters. The models have 13, 5, and 9 free parameters respectively, and the target output was the Kling-Gupta Efficiency of daily streamflow over the period from October 2001 to September 2020. Parameters were treated as independent and uniformly distributed within prescribed bounds, an assumption that is required for the theoretical relationship between total-effect indices and permutation variable importance to hold. In the VIC model, where two parameters share a dependence relationship, this was explicitly handled so that the resulting input space satisfied the independence assumption.

Three machine learning algorithms were then trained as surrogates on the Monte Carlo samples: a gradient boosting decision tree implemented in XGBoost, a feed-forward neural network with two hidden layers implemented in TensorFlow, and a simple linear model implemented in scikit-learn. The choice of gradient boosting over a conventional random forest was motivated by computational efficiency, since both share a tree-based architecture but the boosting implementation is advantageous. The neural network used ReLU activation, and the authors note that recurrent architectures such as long short-term memory networks were not appropriate because the surrogate task involves static parameter vectors rather than sequential data. Each metamodel was then used to estimate sensitivities through permutation variable importance and through SHAP values, a model-agnostic explanation method rooted in cooperative game theory.

The results were striking. Across all three hydrologic models, the gradient boosting and neural network metamodels reliably reproduced both the ranking and the relative magnitude of the variance-based total-effect indices using permutation variable importance, with only minor discrepancies. This provides clear empirical support for the theoretical connection between the two measures and extends earlier findings that had focused primarily on random forests. The linear model told a different story: its performance as a surrogate was markedly weaker, particularly for the 13-parameter HBV model, and its sensitivity estimates agreed well with the variance-based benchmark only for HyMod and VIC, where the model response exhibits a stronger degree of linearity. The lesson is that the validity of the machine learning route depends not on the specific algorithm but on the fidelity of the regression surrogate to the original model.

Beyond reproducing classical results, the study explored what machine learning surrogates can offer that traditional sensitivity analysis cannot. Mean absolute SHAP values showed a consistent monotonic relationship with the total-effect indices and preserved nearly the same ranking of influential parameters, although with greater dispersion. More importantly, sample-specific SHAP values enable a distributed evaluation of sensitivities across both the parameter space and the target variable space. This matters because global averages can emphasise parameter sensitivity in regions of poor model performance while obscuring the controls that dominate in well-performing regions of the parameter space. The authors show, for example, that for the HyMod model the region of highest performance, indicated by larger Kling-Gupta Efficiency values, is associated with positive and comparatively larger SHAP contributions for the most influential parameter. This capability is conceptually related to the Distributed Evaluation of Local Sensitivity Analysis, though the authors caution that neither SHAP values nor that approach should be regarded as equivalent to classical global sensitivity measures.

The computational argument is compelling. Classical variance-based estimation of total-effect indices requires n(p+2) model evaluations, which grows rapidly with the number of uncertain inputs. In contrast, once a surrogate model is trained on the n evaluations of the sampling matrix, permutation variable importance can be computed at negligible additional cost, replacing thousands of additional model runs with surrogate predictions that exploit highly optimised machine learning libraries. The authors also verified that their sensitivity estimates had converged, with stabilisation occurring at subsample sizes well below the full set of 10,000 samples, and that deliberately overfitted metamodel configurations did not distort the resulting feature importance estimates, since the goal was faithful emulation within the sampled parameter space rather than generalisation to unseen data.

The authors are careful to state the limitations of their work. All three models were run in lumped configurations with daily time steps, which limits the structural complexity relative to fully distributed applications; they note that a previous study using a random forest surrogate for a distributed pesticide transfer model with 145 input parameters found the relationship between permutation importance and total-effect indices was not well respected, likely because of poor surrogate performance. The analysis was also confined to catchments with semi-humid conditions and groundwater-dominated runoff, which may influence the generality of the findings. Future research, they suggest, could extend the approach to catchments with contrasting hydroclimatic characteristics, to settings with dependent inputs, and to applications where metamodels support calibration or uncertainty exploration. Even so, the study establishes a clear, empirically validated bridge between two research communities, showing that the tools of explainable artificial intelligence can serve as a computationally efficient and conceptually sound framework for global sensitivity analysis in hydrology and beyond.

Subject of Research: Machine learning metamodelling as an efficient alternative to variance-based global sensitivity analysis in hydrologic modelling

Article Title: Technical note: Machine learning metamodelling for global sensitivity analysis

Article References: Yeste, P., Melsen, L. A., Brêda, J. P. L. F., Tacoronte, N., Saltelli, A., Vannucci, G., Siciliano, R., & Bronstert, A. (2026). Technical note: Machine learning metamodelling for global sensitivity analysis. Hydrology and Earth System Sciences, 30(20), 6579-6597. https://doi.org/10.5194/hess-30-6579-2026

Image Credits: AI Generated

DOI: 10.5194/hess-30-6579-2026

Keywords: global sensitivity analysis, machine learning, metamodelling, hydrologic modelling, Sobol indices, permutation variable importance, SHAP values, surrogate models, HBV, HyMod, VIC, explainable AI