Given a deterministic model of the form Y=f(X), where Y is a scalar model output and X=(X1,…,Xp) is a vector of p uncertain input factors, variance-based sensitivity analysis aims to quantify how the uncertainty in Y can be attributed to the uncertainties in the individual input factors and their interactions.

The first-order Sobol' index of the i-th factor, Xi, is defined as e.g, : 1Si=VarXi(EX∼i[Y|Xi])Var(Y) where Var(.) and E[.] are the variance and expectation operators, respectively, and X∼i denotes the vector of all input factors except Xi.

Si is normalized between 0 and 1, with larger values indicating a stronger additive effect of Xi on the model output. The inner expectation in VarXi(EX∼i[Y|Xi]) represents the conditional expectation of Y given Xi, and intuitively averages out the effect of all other factors, isolating the contribution of Xi. The outer variance measures how much this contribution varies across all possible values of Xi.

When ∑i=1pSi=1, the model is additive, which means that Var(Y) can be decomposed as the sum of all the first-order effects as there are no interactions among the input factors. However, in the presence of non-negligible interactions ∑i=1pSi≤1, and the first-order indices would not be informative enough to quantify sensitivities. This is generally the case for any real-world model.

The total-effect Sobol' index of Xi measures the total contribution of Xi to the output variance, including all interactions with other factors, and is given by e.g. : 2Ti=1-VarX∼i(EXi[Y|X∼i])Var(Y)

One intuitive interpretation is that the term VarX∼i(EXi[Y|X∼i]) represents the variance explained by all factors except Xi. Subtracting this from the total variance Var(Y) isolates the contribution of Xi and all its interactions. Ti satisfies the condition Si≤Ti≤1, and for additive models Si=Ti as there are no factor interactions.

Further details and a complete derivation of the Sobol' indices can be found, for example, in . It is important to note that the definitions presented above assume that the input factors have independent probability distributions. Under this assumption, Var(Y) can be uniquely decomposed into contributions from individual factors and their interactions. When the inputs are not independent, this decomposition is no longer unique. Extensions of Sobol' indices to dependent inputs have been proposed in the literature e.g.,, but these are beyond the scope of this work.

The estimation of Sobol' indices requires two essential components: a sampling strategy to generate points in the space of the input factors, and an estimator to compute sensitivity . Here, we follow the Monte Carlo–based numerical approach recommended by to estimate Si and Ti using a single set of model evaluations. For a comprehensive overview of available estimators, we refer the reader to .

Let A and B be two independent sampling matrices, each containing n realizations of the p input factors. For each input factor, we construct the hybrid matrix AB(i) by replacing the ith column of A with the corresponding column of B. Si and Ti can be estimated as: 3S^i=1n∑v=1nf(B)vfAB(i)v-f(A)vVar(Y)4T^i=1n∑v=1nf(A)v-f(AB(i))v22Var(Y)

In the previous expressions, Var(Y) can be estimated as : 5Var^(Y)=12n-1∑v=1n(f(A)v-f0)2+(f(B)v-f0)2 where f0=12n∑v=1nf(A)v+f(B)v. Here and throughout the remainder of the paper we employ the hat notation for estimators.

Equation () was originally proposed in as an improvement to previous approaches, although it has been outperformed by better practices, see e.g. . Equation () was introduced in and represents the best practice thus far to estimate Ti. As a result, the estimation of Si and Ti based on the triplet of matrices A, B, and AB(i) has a total cost of n(p+2) simulations: 2n simulations are required for matrices A and B, and pn simulations for the hybrid matrices AB(i).

Let Y be a real-valued random variable and X=(X1,…,Xp) a vector of real-valued random variables. We assume that X and Y are jointly distributed according to some joint distribution encoding a general relationship between X and Y, which may be deterministic or stochastic.

To assess the contribution of each variable Xi to the variability of Y, we define the modified vector X(i)=(X1,…,Xi′,…,Xp), where Xi′ is an independent replication of Xi that is also independent of Y and of all other variables. The permutation variable importance (PVI) of the ith variable is formally defined as : 6PVIi=E(Y-E[Y|X(i)])2-E(Y-E[Y|X])2 where E[Y|X] denotes the conditional expectation of Y given X.

E[Y|X] represents the best function of X for predicting Y in the least-square sense , and therefore the expression E(Y-E[Y|X])2 corresponds to the smallest possible mean squared error (MSE) achievable if one could use the true conditional expectation as a predictor. This quantity represents the irreducible uncertainty in Y, that is, the part of its variability that cannot be explained by X, no matter what model is used.

On the other hand, the term E(Y-E[Y|X(i)])2 measures the expected MSE when Xi is replaced by an independent copy Xi′ and the information carried by Xi has been destroyed. Hence, PVIi measures the increase in MSE when Xi is made independent of Y, with a large PVIi indicating that Xi contributes substantially to predicting Y.

If we restrict our attention to the case when the dependence between X and Y is purely deterministic, the relationship between X and Y can be written as Y=f(X). This is the case of the generic model defined in the previous section. Here, E[Y|X]=f(X), and Eq. () reduces to: 7PVIi=E(f(X)-f(X(i)))2

PVIi measures the average change in f(X) when the ith component is permuted. Then, if the components of X are independent, established the following relationship for Ti and PVIi: 8Ti=PVIi2Var(Y)=E(f(X)-fX(i))22Var(Y)

This relation constitutes an important connection between sensitivity analysis and ML: Ti is central to GSA, while PVIi provides a model-agnostic approach widely used in explainable artificial intelligence (XAI) to quantify feature importance for ML models. Although beyond the scope of this work, interested readers are referred to the work of , which proposed a modified version of PVIi to calculate Si and provided an in-depth analysis of the dependent case using a Rosenblatt transformation of the input variables.

Based on Eq. (), an estimator of Ti can be constructed with the matrices A, B, and AB(i) introduced in the previous section. The generation of AB(i) by replacing the i-th column of A with the i-th column of B mimics the independent replication of Xi in X(i). Therefore, Ti can be estimated with the following expression: 9T^i=1n∑v=1nf(A)v-f(AB(i))v22Var(Y) where Var(Y) can be estimated using Eq. ().

Equation () is identical to the total-order estimator proposed by (see Eq. ). Our goal now is to demonstrate how the estimation of PVIi is carried out in practice for ML models. This provides an alternative way for estimating Ti based on ML metamodelling, which is the focus of this work.

Let Dn={(X1,Y1),…,(Xn,Yn)} denote a learning set consisting of n independent observations from the joint distribution of X and Y, where X and Y are defined as in the previous section. The goal in a regression setting is to find an estimator f^ of E[Y|X] . Broadly speaking, this involves fitting a function to the learning set that minimizes a loss function of choice, thereby producing a reliable approximation of the underlying relationship between X and Y.

The most common approach to estimate PVIi in practice is the importance measure proposed in for random forests . Let f^k denote the k-th tree in an ensemble of ntree trees and let D‾k be its out-of-bag (OOB) sample, namely the subset of observations not used to train that tree. The prediction error R^ of f^k, expressed as MSE, is given by the following estimator over D‾k : 10R^f^k,D‾k=1|D‾k|∑i:Xi,Yi∈D‾k(Yi-f^k(Xi))2 where |D‾k| is the number of samples in D‾k.

For each input factor Xi, let us now define a perturbed version of the OOB sample, denoted D‾ki, by randomly permuting the values of Xi while keeping all other variables unchanged. The importance of Xi is then estimated through the degradation in prediction error induced by this perturbation averaged over all trees in the forest : 11PVI^i=1ntree∑k=1ntree[R^(f^k,D‾ki)-R^(f^k,D‾k)]

This expression is the empirical counterpart of the theoretical definition of PVIi presented in Eq. (). Although random forests provide a natural setting for estimating PVIi, the permutation principle is not restricted to this class of models. Given any regression algorithm, one may estimate the importance of Xi by evaluating the predictive performance of the fitted model f^ and comparing it with the performance obtained after permuting the values of Xi. This general procedure is a model-agnostic approach widely used across ML algorithms . In all cases, the resulting PVI^i quantifies how much the predictive accuracy of the particular model deteriorates when the information carried by Xi is destroyed.

Finally, when the learning set is generated via a Monte Carlo-based approach for GSA – for example, using the sampling matrix A and the corresponding model outputs f(A) defined in Sect.  – the regression setting becomes a metamodelling approach. In this case, the regression algorithm provides a surrogate model f^ for the true model f, and PVI^i can be used to approximate Ti. The quality of this approximation depends directly on the predictive accuracy of the surrogate: only when f^ adequately captures the behaviour of f on the learning set can PVI^i be expected to reproduce the corresponding theoretical quantity, and consequently yield reliable estimates of Ti. This approach yields a substantial computational advantage over classical estimation of Ti, which requires n(p+2) model evaluations (see Sect. ). The surrogate is trained on the n evaluations of matrix A alone and subsequently used to perform the additional evaluations required for PVI^i computation across all p parameters, replacing n(p+1) model runs with surrogate predictions that take advantage of highly optimized ML libraries at negligible cost.

Specific objectives in this technical note are:

To provide a representative hydrologic modelling context in which to generate numerical evidence for the relationship between Ti and PVIi expressed in Eq. ().

It is important to note that the specific choice of hydrologic models, catchments, and model output of interest is not central to the objectives of this study.

In this study, three hydrologic models that are frequently used in hydrologic modelling were employed: Hydrologiska Byråns Vattenbalansavdelning HBV,, HyMod and Variable Infiltration Capacity (VIC) version 5 . The number of free parameters p for these models is set to 13, 5, and 9, respectively. It should be noted that VIC is a physically based land-surface model and therefore includes a more complex parameterization than conceptual hydrologic models such as HBV and HyMod, with numerous soil and vegetation parameters describing land–atmosphere interactions. In this study, only a subset of VIC parameters was treated as free, while the remaining soil and vegetation parameters were obtained from the VICGlobal dataset and kept fixed. Definitions of each parameter and their feasible ranges, as informed by literature ( for HBV and HyMod; and for VIC), are provided in the Supplementary Material. For the purpose of this experiment, all models were implemented in a lumped configuration with a daily time step.

The study focuses on three headwater catchments located in northern Germany, namely the Sude River (gauge Garlitz), the Treene River (gauge Augaard), and the Trave River (gauge Schackendorf). These catchments have semi-humid climatic conditions. Runoff generation is predominantly groundwater-driven, while during wet and high-flow conditions saturation-excess processes also play a role. The landscape is characterized by glacial deposits with deep, sandy to loamy soils, and land use is predominantly agricultural. The catchments are free of major flow regulation, allowing for the analysis of hydrologic dynamics under near-natural conditions. Key physiographic and hydroclimatic characteristics of the catchments are summarized in Table . Catchment areas range from 136 to 675 km². Mean annual precipitation ranges from 661 mm yr⁻¹ (Sude) to 909 mm yr⁻¹ (Treene), while snow contributions are minor, with fractions below 0.05 for all catchments. Aridity indices range from 0.59 (Treene) to 0.96 (Sude), and runoff ratios are relatively similar (0.32–0.38), although the Sude catchment is somewhat drier, receiving about 20 % less precipitation and yielding lower specific annual runoff. Overall, these groundwater-dominated catchments display similar annual hydrologic variability, suggesting comparable long-term water balance characteristics.

The study catchments are part of the CAMELS-DE dataset : Garlitz (gauge ID DE812200), Schackendorf (gauge ID DEF13760), and Augaard (gauge ID DEF10190). They were selected based on the performance of the HBV model reported in the original CAMELS-DE data publication, in which HBV was applied across the full dataset and the Nash–Sutcliffe Efficiency (NSE) was provided as a performance metric. The three catchments with the highest NSE values were retained for the present analysis.

For each model–catchment combination, a Monte Carlo experiment was conducted using the sampling matrices A, B, and AB(i) described in Sect. . The sampling matrices were generated using the Sobol' quasi-random sequence scheme implemented in the SALib Python package after setting the number of samples n to 10000. This design resulted in a total of n(p+2) model evaluations per model and catchment combination. The model output of interest was the Kling–Gupta Efficiency (KGE) of daily streamflow over the period 1 October 2001 to 30 September 2020.

For sensitivity analysis and metamodelling, model parameters were treated as uncertain inputs and assumed to be independent and uniformly distributed within their prescribed bounds. This choice corresponds to the most common practice in hydrologic sensitivity analysis and model calibration, where parameter uncertainty is represented by specifying lower and upper bounds and assigning a uniform distribution to each parameter . Importantly, the assumption of independent inputs is required for the theoretical relationship between Ti and PVIi expressed in Eq. (). In the VIC model, there is a dependence relationship between two parameters. This dependency was explicitly accounted for and handled so that the resulting input space satisfies the independence assumption required for the analyses conducted here. Details of this procedure are provided in the Supplement.

To emulate the behaviour of the hydrologic models, a learning set was constructed using the parameter columns in matrix A as input variables and the corresponding KGE values from the Monte Carlo experiment as the target variable. This setup defined a regression problem, enabling the ML models to act as surrogate models for the hydrologic simulations. The learning set constructed from the sampling matrix A was used exclusively for training the metamodels, without applying cross-validation or a separate test set, as would be common in a predictive ML workflow. This choice reflects the purpose of the metamodels in this study, which is not to generalize to unseen data, but to accurately emulate the behaviour of the original hydrologic models within the sampled parameter space. From a sensitivity analysis perspective, faithful reproduction of the input–output relationship on the training domain is the primary requirement for obtaining reliable sensitivity estimates. Accordingly, model configurations that would typically be regarded as overfitting in a predictive setting are acceptable here, provided that they do not distort the resulting sensitivity measures. The implications of this choice and its consistency are further examined through a dedicated sanity check, as described in Sect. .

Three ML algorithms were tested as metamodelling approaches. A gradient boosting decision tree (GBDT) was implemented in XGBoost with three configurations of 100, 150, and 200 trees. Choosing GBDT over a conventional random forest was motivated by computational efficiency: while both share the same tree-based architecture, the boosting implementation in XGBoost is advantageous over the bagging approach of random forests. A neural network (NN) was implemented in TensorFlow with two hidden layers, three configurations of 100, 150, and 200 neurons per layer, and ReLU activation. The choice of a conventional feed-forward network over recurrent architectures such as long short-term memory (LSTM) networks is motivated by the nature of the surrogate task: each Monte Carlo simulation produces a single scalar KGE value from a fixed parameter set, meaning the surrogate's input space consists of static parameter vectors rather than sequential data. LSTM networks are designed to capture long-range dependencies in sequential data through their recurrent architecture . In the present study, however, the input consists of a fixed-length vector of p parameter values without any temporal or other intrinsic sequential ordering. Therefore, the recurrent structure of an LSTM is not directly motivated by the structure of the input data, and the problem is instead treated as a conventional tabular regression task. A feed-forward network with ReLU activations is therefore a natural and computationally efficient choice for this purpose. Finally, a linear model (LM) was implemented in scikit-learn . For all three ML algorithms, MSE was selected as the loss function, which is a crucial step in assessing the applicability of Eq. (). These metamodelling approaches allowed us to efficiently reproduce the hydrologic simulations, thus providing a computationally cheap framework for GSA.

While Sect.  presents the theoretical definitions and general estimation procedures for the sensitivity measures, this section describes their specific implementation in the context of the numerical experiment.

Variance-based sensitivity T^i was calculated using the Python library SALib , which implements Eq. () described previously, and ML-based PVI^i was computed from the three ML metamodelling approaches using the implementation available in scikit-learn .

Feature importance was also evaluated using SHapley Additive exPlanations SHAP,, a model-agnostic method rooted in cooperative game theory. SHAP decomposes each individual model prediction into additive contributions associated with each input feature, ensuring local accuracy and global consistency. The formal mathematical definition of SHAP values is provided in Appendix . In this work we used the SHAP Python library, which provides architecture-specific explainer implementations: GPUTreeExplainer (or TreeExplainer when CUDA is not available) was used for the GBDT metamodel, DeepExplainer for the NN metamodel, and LinearExplainer for the LM metamodel.

In the present application, where the inputs correspond to hydrologic model parameters, we denote by SHAPi the contribution of parameter Xi to the target variable. From a sensitivity analysis perspective, SHAPi values enable two complementary levels of analysis. First, a global measure of parameter importance can be obtained by averaging the absolute SHAPi values across all samples, yielding mean SHAPi, which provides a scalar summary of the overall importance of Xi on the model response. Second, the sample-specific SHAPi values allow for a distributed evaluation of parameter importance across both the parameter space and the model output space. This property is particularly relevant when assessing sensitivities with respect to model performance metrics. In such cases, global averages may emphasize parameter sensitivity in regions associated with poor model performance, potentially obscuring the controls that dominate in well-performing regions of the parameter space. By retaining sample-level contributions, SHAPi values allow parameter importance to be examined across behavioural and non-behavioural regions of the model response surface.

Finally, standardized regression coefficients were computed for the LM as an additional sensitivity measure. The standardized coefficient for parameter Xi is defined as 12β^i=b^iσXiσY where b^i is the estimated regression coefficient, and σXi and σY are the sample standard deviations of Xi and the model output Y, respectively. Standardized regression coefficients have been used in previous hydrologic sensitivity analyses as a computationally inexpensive alternative to variance-based methods e.g.,, at the cost of assuming linear model behaviour.

If the relationship between X and Y is given by a linear model of the form Y=b0+∑i=1pbiXi, the model is additive and Si and Ti coincide. In this case, the following equality holds : 13Si=Ti=βi2

Moreover, combining this result with the theoretical relationship between Ti and PVIi (Eq. ) yields 14βi2=PVIi2Var(Y)

Accordingly, the ability of the LM to approximate Ti through β^i2 or PVI^i depends directly on the degree to which the hydrologic model response can be represented by a linear relationship.

To assess the uncertainty of the sensitivity estimates, 95 % confidence intervals were computed for all sensitivity measures. For T^i, bootstrap resampling was performed using the implementation available in SALib (1000 resamples). For PVI^i and mean SHAPi, confidence intervals were obtained by evaluating the trained metamodels on resampled versions of the training data (1000 resamples). For the GBDT metamodel, subsampling without replacement with a subsample size of 80 % was used, following a similar approach to for random forests, to avoid interference with the internal subsampling mechanism of XGBoost. For the NN and LM metamodels, standard bootstrap with replacement was used.

To evaluate the consistency of the ML metamodelling approach with GSA, two a priori assessments were performed: a sanity check of the bias–variance tradeoff for the important measures, and a convergence test for all sensitivity measures. Simply put, the bias–variance tradeoff establishes that increasing model flexibility produces smaller training error, but may lead to larger test error due to overfitting e.g.,. In the present context, however, the objective was not to generalize to unseen data, but to accurately emulate the true model response. As discussed in Sect. , PVI^i can provide a reliable approximation of Ti only if the metamodel is able to adequately reproduce the behaviour of the original model. From this perspective, a low training error was a necessary condition, and configurations that would be otherwise regarded as overfitting could be acceptable provided that they do not affect the feature importance estimates. Based on this, PVI^i and mean SHAPi were calculated for the GBDT and the NN metamodels using five overfitting configurations, as reported in Table . For the subsequent analysis, the less complex configuration was selected in each case, as it provided feature importance estimates comparable to the more complex configurations while achieving similar predictive performance.

The convergence test was performed to verify that the sensitivity measures stabilized as the number of Monte Carlo samples increased. The test was carried out using a random subsampling strategy. Each subsample was drawn as an independent random sample without replacement from the full set of Monte Carlo simulations, with subsample sizes incremented in steps of 100 up to the full set of 10,000 samples, resulting in 100 subsampling matrices. For each subsampling matrix, T^i, PVI^i, mean SHAPi, and β^i2 were computed, and convergence was assessed by comparing their values between consecutive subsample sizes. Sensitivity estimates corresponding to each subsampling matrix were normalized by dividing by the sum of all sensitivities. The Euclidean distance between the resulting sensitivity vectors at consecutive subsample sizes was then used as the convergence metric. This procedure ensured that the sensitivity results were robust with respect to the size of the training dataset. The purpose of this test was to verify that the full set of 10 000 samples was sufficient for the sensitivity estimates to stabilize, rather than to identify an exact sample size at which convergence was achieved. Since the subsamples were drawn from the same finite pool, consecutive draws of increasing size inevitably share some samples, leading to smoother variations as the subsample size approaches the full set. Despite this overlap, the test provided a reliable assessment of the stability of the sensitivity estimates across the range of subsample sizes considered.

Figure  summarizes the predictive performance of the GBDT, NN, and LM metamodels by comparing fitted and original KGE values across the three hydrologic models. For GBDT and NN, only configuration 1 is shown (see Table ), as all five configurations exhibit similar predictive performance. Both GBDT and NN closely match the reference simulations for all three hydrologic models, indicating that they capture the dominant structure of the parameter–output relationship. The high R2 values are expected, as the metamodels are evaluated on the same points used for training. This confirms that the surrogates adequately capture the behaviour of the original models, which is a necessary condition for the validity of Eq. () (see Sect. ). In contrast, LM exhibits markedly weaker performance, particularly for HBV, with HyMod and VIC showing the highest linear response. The suitability and limitations of using an LM metamodel in contrast to GBDT and NN are discussed in the next section.

Figure  shows the sensitivity estimates obtained for the GBDT and NN metamodels across five flexible configurations, using both PVI^i and mean SHAPi. All five configurations correspond to deliberately overfitted metamodels (see Table ), with coefficients of determination exceeding R2>0.98 for all hydrologic models. The importance curves corresponding to the different configurations largely overlap, indicating that the magnitude of the importance estimates remains remarkably consistent across varying model complexity and random initialization. In addition, panel titles in Fig.  report the mean ± standard deviation of pairwise Kendall τ and Spearman ρ between the importance vectors of the five configurations, which consistently show high values across all model–metamodel–importance combinations. This behaviour is observed for both PVI^i and mean SHAPi, suggesting that potential overfitting does not affect the resulting importance estimates. For a given importance measure, GBDT and NN exhibit nearly identical magnitudes and rankings, highlighting the robustness of the results with respect to the choice of metamodel. When comparing PVI^i and mean SHAPi, the relative ranking of influential parameters is generally preserved, while small differences in magnitude are observed. These differences are expected given the distinct theoretical foundations of the two approaches, but their close agreement indicates a consistent identification of dominant parameters across methods. Based on these results, configuration 1 was selected for GBDT and NN for the subsequent analysis, as it is the least complex while providing feature importance estimates comparable to the more complex configurations (see Sect. ).

Finally, Fig.  presents the convergence behaviour of all sensitivity measures as a function of the number of Monte Carlo samples during the subsampling experiment. For all three hydrologic models, T^i exhibits stable and monotonic convergence as the subsample size increases. A similar convergence pattern is observed for PVI^i and mean SHAPi derived from the ML metamodels, indicating that the importance estimates stabilize with increasing training data. For all sensitivity measures, stabilization occurs at subsample sizes well below the full set of 10 000 samples, confirming that the number of Monte Carlo simulations used in this study was sufficient. Convergence is generally faster and smoother for HyMod and VIC, whereas HBV shows comparatively slower convergence and higher variability across sensitivity measures. This behaviour is consistent with the higher structural complexity of HBV, which involves 13 parameters, compared to the more parsimonious HyMod (5 parameters) and VIC (9 parameters), and reflects the increased sampling requirements associated with higher-dimensional parameter spaces. In addition, mean SHAPi values computed from the GBDT metamodel display slightly reduced stability for HBV relative to the other cases.

The theoretical relationship between Ti and PVIi expressed in Eq. () was examined empirically for the HBV, HyMod, and VIC models using the importance measures derived from GBDT, NN, and LM metamodels. Figure  shows the normalized T^i together with normalized PVI^i estimates and normalized mean SHAPi values obtained from the three metamodels. As in the convergence analysis (Fig. ), values were normalized by dividing by the sum of all sensitivities in order to facilitate a direct comparison and produce a consistent ranking of parameters. For the GBDT and NN metamodels, PVI^i reproduces both the ranking and the relative magnitude of T^i across all three hydrologic models, reflecting only minor discrepancies. In contrast, LM-based PVI^i provides a good approximation of T^i for HyMod and VIC, but exhibits larger deviations for HBV, particularly in the relative importance assigned to dominant parameters. This behaviour is consistent with the comparatively poor predictive performance of the LM for HBV (Fig. ). This expected limited fit of the LM underscores that the ability of PVI^i to approximate T^i depends critically on the accuracy with which the metamodel captures the underlying model response.

The correspondence between T^i and PVI^i is further illustrated in Fig.  (panels a–c), which directly compares the two quantities for each hydrologic model parameter and metamodel. GBDT- and NN-based estimates closely follow the theoretical relationship, providing clear numerical support for the connection established in Eq. (). For the LM, agreement with the theoretical relationship is observed only for HyMod, whereas a weaker correspondence is observed for HBV and VIC – in line with how well the LM is able to capture hydrologic model behaviour. The close agreement observed between T^i and PVI^i for GBDT is in line with previous theoretical and numerical results under independence assumptions . While earlier work primarily focused on random forest, the present results extend these findings by demonstrating that alternative ML metamodels can also provide reliable approximations of T^i, provided that they accurately reproduce the underlying input–output relationship. In particular, the strong performance of the NN metamodel, and the satisfactory behaviour of the LM for HyMod, indicate that the validity of PVI^i-based sensitivity estimates is not tied to a specific algorithm, but rather to the fidelity of the regression surrogate. This observation corroborates the insight of , who emphasized that the empirical estimation of PVIi strongly depends on the algorithm used to approximate the regression function. In contrast, reported that the relationship between PVI^i and T^i was not well respected when using a random forest surrogate for a distributed, physically based pesticide transfer model with 145 input parameters, and after applying a screening procedure to reduce the dimension of the problem. In their study, the poor predictive performance of the surrogate was identified as the likely cause, further underscoring that the practical applicability of Eq. () depends on surrogate fidelity. It should be noted, however, that in our work all three models were used in lumped configurations: HBV and HyMod are lumped conceptual rainfall-runoff models, while VIC, though a land-surface model with a more complex parameterization including surface and energy balance components, was likewise run in a lumped setup. This limits the degree of structural complexity relative to fully distributed applications such as that of , and may partly explain why high surrogate fidelity was more readily achieved in our study.

In recent hydrologic studies, tree-based ensemble approaches and feature importance measures have been widely used as exploratory tools to analyse large-sample datasets and to identify dominant drivers of hydrologic behaviour, for example, in the context of intercatchment groundwater flow , hydrologic signatures , and flood-generating processes . In these applications, different XAI approaches are used to help elucidate empirical relationships learned by a random forest. ML metamodels are also efficient alternatives to complex hydrologic models, such as VIC and ParFlow.CLM , and have already been applied to accelerate studies on climate impacts as well as for regionalization and calibration of conceptual hydrologic models . A closer connection to sensitivity analysis is found in the work of , who employed feature importance as a diagnostic approach for global hydrologic and land-surface models using a random forest surrogate. The present study builds directly on this emerging line of research by explicitly linking feature importance measures to variance-based total-effect sensitivity indices across multiple hydrologic models and metamodel classes.

As an alternative to PVI^i, Fig.  also examines the possibility of using mean SHAPi and β^i2 for GSA. The normalized mean SHAPi values included in Fig.  further illustrate this comparison at the parameter level. Panels (d)–(f) of Fig.  contrast mean SHAPi with T^i, revealing a clear monotonic relationship across all hydrologic models, with larger dispersion than PVI^i (panels a–c). Results obtained with GBDT and NN are very similar for all hydrologic models, whereas, as observed for PVI^i, LM-based results are satisfactory for HyMod and VIC, where the model response exhibits a stronger degree of linearity. Despite the increased dispersion, mean SHAPi preserves nearly the same ranking of influential parameters as identified by PVI^i and T^i. Panels (g)–(i) compare PVI^i with β^i2 for the LM metamodel, confirming the proportionality between the two measures established in Eq. ().

Beyond providing a global ranking, SHAPi values enable a more detailed and informative sensitivity analysis across both the parameter space and the target variable space, as illustrated in Fig.  for HyMod using the GBDT metamodel. Panel (a) reports a global importance ranking based on mean SHAPi, which coincides with the results shown in Fig.  (panel e). Panel (b) shows the distribution of SHAPi values across the parameter space, highlighting that parameters with similar global importance can exhibit markedly different local contributions depending on their values (e.g., Sm and beta). Panel (c) focuses on the most influential parameter, alfa, illustrating how its contribution to the model response varies across its range. Finally, panel (d) displays SHAPi values for alfa as a function of KGE, showing that the region of highest model performance (i.e., larger KGE values) is associated with positive and comparatively larger SHAPi values for alfa.

The ability of SHAPi values to characterize how parameter influence varies across both the parameter space and the target variable space is conceptually closely related to the Distributed Evaluation of Local Sensitivity Analysis (DELSA) introduced by . DELSA was developed to move beyond single summary measures of sensitivity by providing distributions of parameter sensitivities across the parameter space, thereby revealing regimes in which parameters may alternate between being influential or negligible. As emphasized by , the ability to understand how parameter sensitivities vary throughout parameter space is critical for informing decisions related to data collection and model development, particularly when sensitivities are evaluated with respect to performance metrics. This perspective aligns with the broader critique of GSA articulated by , who demonstrated that global aggregation can obscure important behaviour when sensitivity is assessed using performance metrics. The SHAPi-based analyses presented here address these limitations by providing sample-specific contributions that can be examined jointly in parameter space and target variable space.

In this sense, SHAPi values provide a complementary perspective to distributed sensitivity approaches such as DELSA, while naturally integrating with ML metamodelling frameworks. It is important to emphasize that neither SHAPi nor DELSA should be regarded as equivalent to classical GSA measures such as T^i. DELSA is a hybrid local-global approach that provides distributions of parameter sensitivities across the parameter space, rather than the single summary measures characteristic of GSA see alsofor a non-aggregated approach based on copula density-driven metrics. Similarly, SHAPi operates as a local prediction-explanation method that attributes the contribution of each input to individual predictions, and the global importance derived from mean absolute SHAPi values is an aggregation of these local explanations rather than a variance decomposition index. This complementary relationship between SHAP and SA-based approaches has been explored in recent studies. compared an SA-based explainer, the Explainer Variogram Analysis of Response Surface (XVARS), with SHAP, showing that both approaches mostly align in their feature rankings. However, they differ in that XVARS characterizes the entire response surface of ML models, whereas SHAP provides local explanations that are generalized by averaging over sampled data points and may miss behaviour in regions without training data. extended this comparison, finding that XVARS and SHAP are broadly consistent in identifying dominant features but differ in scope and robustness: SA-based methods are independent of training sample size and more robust to sampling variability, whereas SHAP scales linearly with sample size and its importance estimates depend on the underlying ML architecture.

Last, despite the absence of a formal link between SHAPi and T^i, both measures share a common theoretical root in the Shapley value framework from cooperative game theory, which provides a principled approach for allocating contributions among interacting variables. In the context of GSA, the same principle has been adopted through Shapley effects, which allocate contributions to the variance of model outputs among input variables. demonstrated that, for independent inputs, Shapley effects are bounded by the first-order and total-effect Sobol indices. Although SHAP and Shapley effects address different attribution problems, both rely on the same theoretical foundation. Hence, when SHAP values are aggregated over a representative sample of the input space, as in the mean SHAPi values used in this work, they provide a global measure of variable influence that, while not equivalent to Sobol indices or Shapley effects, can be considered a complementary global sensitivity measure.