The PESHMELBA model represents a catchment as a set of interconnected components that stand for landscape elements such as plots, vegetative filter strips (VFSs), ditches, hedges or rivers . In order to respect the spatial organization and the heterogeneity of the landscape, it deals with mesh elements that can be surfaces or lines. Surface mesh elements are called homogeneous units (HUs). A HU is a portion of landscape that is homogeneous in terms of hydrodynamic processes and agricultural practices. Linear mesh elements are called reaches. A reach is characterized by its nature (so far ditch, river or hedge) and by its neighbouring components: it is at most in contact with one elementary mesh element on each bank. In addition to its geometric or hydrodynamic properties, each mesh element is characterized by its one-way connections with the neighbouring components that stand at a lower altitude. One or several processes are represented on each element depending on its nature. Lateral transfers at surface and in the subsurface between elements are also integrated. Independent codes called units are used to simulate the different processes, depending on the knowledge the user has on the targeted catchment functioning. Then, the OpenPALM coupler is used to couple the units and to build the complete application. OpenPALM has adapted features to easily deal with spatial and temporal aspects of the model. For example, synchronization tools make it possible to couple processes with different time steps. The final structure is highly modular, and process representations can easily be added, upgraded or removed depending on the landscape description. These features make PESHMELBA particularly suitable for scenario exploration. PESHMELBA focuses on surface and subsurface transfers of water and pesticides. An extensive description of elements and processes already included can be found in . The PESHMELBA version used in this study integrates a representation of water and pesticide transfers on plots, VFSs and rivers. Each plot or VFS is represented by a unique column of soil divided into vertical cells. In such a column, vertical infiltration is simulated using a solution of the 1D Richards equation proposed by . An adapted set of parameters makes it possible to represent high infiltration rate, surface runoff reduction, and enhanced adsorption and degradation on VFSs. Root-water uptake is integrated based on . Surface runoff routing is represented based on the kinematic wave , and the Darcy law is used for lateral subsurface transfers. In addition to shallow groundwater tables, PESHMELBA also represents shallowly perched water tables and associated lateral transfers. Finally, the reactive transfer of solutes is represented: advection and degradation, based on a first-order law, and adsorption, based on linear or Freundlich isotherms, are integrated. Each river or ditch reach is represented by a unique tank. The River1D module solves the kinematic wave equation for water routing and pesticide advection in the network. Groundwater–river exchanges are represented by the Miles formulation adapted by .

A virtual scenario of limited size is set from a portion of La Morcille catchment (France) in order to explore the different GSA methods and to ease interpretation of spatialized results. The chosen portion is selected so as to remain representative of the global composition of La Morcille catchment in terms of soil, slope, type and size of elements, as well as interface length between them. The chosen scenario is composed of 10 vineyard plots, four vegetative filter strips and five river reaches that delimit a left and a right slope (see Fig. ).

Soils types on the catchment are mainly sandy . We use the classification from that groups soil types into three main soil units (SUs). Each SU is defined by the vertical succession of three or four soil layers, also called soil horizons: one surface horizon, one or two intermediary horizons, and one deep horizon, as depicted in Fig. . Note that interface depths can vary from one SU to another. The reader may refer to for further details on how soil types and soil horizons are represented in PESHMELBA. At the catchment scale, the classification results in the following SU (see Fig. ): sandy soil (SU1), sandy soil on clay on the right bank plateau (SU2), and heterogeneous sandy soils on lower slopes and thalwegs (SU3). Spatial arrangement is set in order to be as realistic as possible in terms of possible interfaces between SUs. Each SU is set at least on one vineyard plot and one VFS on the virtual scenario.

Considering the different soil horizons whose hydrodynamic behaviour and texture must be parameterized, the two types of vegetation (grassland and vineyard), and the different landscape element types (plots, VFSs and river reaches) that are simulated, the scenario results in 145 input parameters to be considered for sensitivity analysis. They are described in Table  as well as the spatial level on which they are set and their values for the nominal simulation.

For the nominal scenario, values for bulk density bd and organic carbon content moc are available from and . They are, as well as hydrodynamic parameters for each soil type, described in Table . Retention values measured by are used to fit retention curve using the SWRCfit tool . A Schaap–van Genuchten conductivity curve is used , whose matching points at saturation Ko and empirical pore connectivity L are derived from conductivity data and retention parameters from retention curve fitting. Surface organic carbon content is set equal to that of the first soil horizon on plots and VFSs. For each SU, only the first soil horizon on VFSs differs from vineyard plots so as to highlight enhanced infiltration capacities. The surface horizon on VFSs is characterized by a 2.8 % organic carbon content and a saturated hydraulic conductivity of 150 mm h-1 (or 4.31×10-5 m s-1) following . In the absence of data to characterize potential anisotropy of vertical and horizontal saturated conductivities Ksv and Ksh, isotropy is considered; thus the ratio Ksh/Ksv is set to 1 in the catchment.

The pesticide chosen in this study is tebuconazole as it is a fungicide widely used in the La Morcille catchment. It is a slightly mobile molecule, and we use a Freundlich isotherm to describe its adsorption equilibrium. Adsorption parameters are obtained from (Koc =769 mL g-1, Freundlich isotherm exponent = 0.84). The surface degradation coefficient is also taken from (DT50 =47.1 d), and a decreasing degradation rate as a function of depth is set as in . A 500 g application is considered at the beginning of the simulation on plots 2 and 7 (see Fig. ). Most of the transformation and adsorption of tebuconazole are supposed to happen on plots and VFSs at this modelling scale. Therefore, no adsorption or degradation is simulated in the river.

A no-flux boundary condition is applied on all sides except on the surface where rain and potential evapotranspiration are considered. Rain data are extracted from the BDOH database . A 3-month simulation is performed for a time period characterized by long and intense rain events (670 mm cumulated), allowing for significant water and pesticide transfers both by surface runoff and subsurface saturated transfers. Potential evapotranspiration (PET) data are obtained from MeteoFrance for the neighbouring site of Liergues . Data are averaged over 10 d periods and corrected by a factor of -11% to match the La Morcille site as recommended in and . Rain and PET data for the simulation are summarized in Fig. .

Although virtual, we aim at setting initial conditions as plausible as possible for this scenario. As running the model on a warm-up period is not possible due to data availability and computational cost limitation, initial water table levels are deduced from piezometric data on a neighbouring hillslope, and all soil columns are supposed to be in hydrostatic equilibrium at the beginning of the simulation. Data from several piezometers are available on a transect, perpendicular to the river. Data are extrapolated over the virtual hillslope width on both sides of the river. An upstream 0.177 m3 s-1 flow is considered in the river based on local measurements .

Two types of vegetation are represented in this scenario. Vineyard cover is considered on plots, while permanent grassland is simulated on VFSs. Considering the period of simulation (3 months), a fixed root depth (Zr =2.62 m) and a fixed root density in the first 10 % of the root depth (F10 =37%) are considered for vineyards following values reported in and confirmed by expert knowledge in the area. The root depth (Zr) is set to 0.9 m, and the root density in the first 10 % of the root depth (F10) is set to 33.5 % for grassland . For vineyards, the leaf area index (LAI) is assumed to increase from a minimum value LAImin from leaves formation until a maximum value LAImax before declining until harvest LAIharv. Dates and associated values for this development cycle are taken from , and they are detailed in Appendix . On grassland, the LAI is assumed to remain constant, and a nominal value of 5 is chosen based on . In Table , the reader should note that the LAI parameter relates to the constant LAI value for grassland, while LAImin, LAImax and LAIharv relate to vineyard plots. The remaining parameters for the root extraction model are also fixed to nominal values proposed by and . Manning coefficients are set from data reported in . A mature row crop value (0.033 s m-1/3) is chosen for vineyards, while a high grass pasture value (0.2 s m-1/3) is set for VFS cover.

Finally, ponding height is set to 0.01 m for vineyard plots, while an increased value is set for VFSs (0.05 m). According to and , the surface mixing layer thickness is set to 0.01 m in both plot and VFS domains. In the river, the distance between the river bed and the limit of impervious saturated zone (di) is set to 1.5 m (electrical resistivity tomography field measure, Rémi Dubois, personal communication, 2020), and the saturated hydraulic conductivity (Ks) is set to 2.38×10-5 m s-1 accordingly to local saturated conductivities in the neighbouring soil. The ponding height is set to 0.01 m, while the Manning coefficient is chosen equal to 0.079 s m-1/3, as suggested in for channels with limited obstruction.

The input factor distributions are set to be as representative as possible of the available data on the catchment and the associated uncertainties. Mean values are taken from the nominal scenario described in Sect. . Distributions and standard deviations are assigned based on experimental measurements from the catchment of application, available scientific literature or expert knowledge. All assigned distributions and corresponding statistics are summarized in Appendix .

As commonly found in the literature , a lognormal distribution is assigned to the saturated hydraulic conductivity Ks. A 20 % coefficient of variation (CV) is used so as to remain representative of each soil horizon hydrodynamic behaviour. Distributions for Schaap–van Genuchten parameters could not be found in the literature; thus the empirical pore-connectivity parameter L is assigned a uniform distribution ±20% around the mean value . As the matching point at saturation in the modified Mualem–van Genuchten conductivity curve Ko has the same physical meaning as Ks, a lognormal distribution is also assigned to this parameter and a 20 % CV is set. Saturated water content thetas, residual water content thetar, van Genuchten parameter mn and air-entry pressure hg are assigned normal distributions . A 10 % CV is set for thetas , and thetar is assigned a 25 % CV . A 10 % CV is set for mn and hg . A uniform distribution is assigned to organic carbon content moc . A triangular distribution is assigned to the Freundlich sorption coefficient Koc , and a normal distribution is assigned to the half-life DT50. A 60 % CV is assigned to Koc and DT50 distributions as such parameters are considered relatively uncertain . Triangular distributions are assigned to Manning's roughness (manning) on plots and for the river bed . A uniform distribution with a ±20% range around the mean value is assigned to remaining input factors as little information could be found in the literature .

Using a fully distributed model such as PESHMELBA raises the issue of sampling strategy. Indeed, in this case study, even if the site is only composed of 14 surface units, the large number of soil horizons on the catchment, considering the hydrodynamic distinction between plots and VFSs, also dramatically increases the number of parameters. Sampling all parameters on each spatial unit leads to a huge number of simulations that could not be computationally afforded. Moreover, such independent sampling on a very large number of parameters may lead to misinterpretation of the sensitivity analysis results as the influence of physical processes could not be distinguished from spatial arrangement. For each sample, one set of soil parameters is therefore sampled for each soil horizon, and those parameters are applied to all spatial units that contain this horizon which sets the number of parameters to be considered in the GSA at 145.

Although the PESHMELBA model is dynamic, model outputs considered in this paper are scalar quantities rather than temporal series to keep the problem simple. In order to investigate PESHMELBA abilities to properly represent transfers in a heterogeneous landscape, sensitivity analysis is performed on four hydrological and quality variables: (1) cumulated water volume transferred in the subsurface (saturated lateral transfers), (2) pesticide mass transferred in the subsurface (saturated lateral transfers) (3) cumulated water volume transferred on surface (surface runoff) and (4) cumulated pesticide mass transferred on surface (surface runoff). However, these quantities are spatialized, leading to multidimensional outputs. To deal with the spatialized aspect, GSA is first performed on scalar quantities, on each landscape element (see Sect.  to ), then sensitivity indices are aggregated in a second time to provide catchment-scale sensitivity indices (see Sect. ).

The full workflow used to perform GSA in the PESHMELBA model is summarized in Fig. . Considering the high number of input parameters, a screening step is first performed to decrease the dimension of the problem. Screening is performed with a statistical independence test based on the HSIC measure (see Sect. ) on an initial 4000-point Latin hypercube sample (LHS; ). Second, a new 2000-point LHS obtained from the reduced set of input parameters is computed to perform ranking. Sensitivity indices are computed from the 2000-point sample based on (1) variance decomposition (Sect. ), (2) the HSIC dependence measure (Sect. ) and (3) the feature importance measure obtained from random forests (Sect. ). These methods are all global and model-free, and all suit non-linear, non-monotonic models. In addition, they all belong to the category of “given-data” methods which means that the input sample they consider does not require a specific structure. This is of particular interest for models that are computationally costly as it allows us to take advantage of pre-existing simulations. That is why such methods have known a growing interest in recent years . However, such methods may not be equally costly to set, and they define the notion of sensitivity in contrasted ways. We compare them regarding the information they provide, their accuracy and their robustness.

In what follows, we denote Y∈R, a given scalar output from PESHMELBA. Y is function of a multivariate input random vector Y=M(X), where X=(X1,…,XM)∈RM contains the 145 input parameters considered in this case study and where M is the PESHMELBA model.

In order to assess the robustness of the calculated sensitivity indices, an additional 200-point test set is first used to assess PCE and RF metamodel performances. In a second time, 95 % confidence intervals on sensitivity indices are computed for all methods but by different means: the bootstrap resampling procedure provided in UQLab is used to calculate confidence intervals on PCE-based Sobol' indices (see , for justification). In the case of random forests, building extra bootstraps could affect the consistency of the OOB sample as the RF structure already includes a bootstrap step to build each tree. Then, we use a subsampling approach without replacement with a subsample size set to 80 % of the initial sample size to get error bounds on feature importance measures. The same procedure is applied to estimate error bounds on HSIC indices. As typically found in similar studies , 1000 bootstrap resamples are used for all methods. In addition, the reliability of each ranking method is assessed by monitoring its convergence properties. To do so, sensitivity indices and associated confidence intervals are calculated on samples of growing sizes. A new sample is generated for each size in [50,100,250,750,1000,2000], where the maximum size (2000) points corresponds to the maximum computational budget available for ranking we guess for a catchment-scale application. It is noted that we create a new sample for each sample size to keep all samples as independent as possible, but such an approach may be too costly in a catchment scale application. In such a case, another strategy could be to generate the sample for ranking from the initial LHS used for screening, dropping the dimensions corresponding to the non-influential inputs.

After computing local sensitivity indices for each landscape unit on a scalar quantity, aggregated indices are computed at the catchment scale (step 2b, Fig. ). Catchment-scale sensitivity indices are computed considering a multidimensional output Y∈Rd that gathers scalar outputs for each landscape unit. Sobol' indices are aggregated at the catchment scale following the formulation by for generalized Sobol' indices: 14Su=∑j=1dVar[Yj]Su,j∑j=1dVar[Yj], where u is a subset of {1,…,M}, Var[Yj] is the variance of the scalar jth component of Y and Su,j is the Sobol' indices of subset u on Yj. Equation () then formulates catchment-scale indices as an average of Sobol' indices on each landscape unit weighted by local output variances. First- and total-order Sobol' indices can be notably computed this way. Their computation can be made in a second step after performing local Sobol' analysis on each landscape unit, but a direct estimator is also proposed in avoiding numerous local analysis in the case of high-dimensional model output. For HSIC and RF sensitivity indices, the definitions for scalar output remain valid for vectorial output. Catchment-scale indices can thus be directly computed on Y when moving to a multidimensional case.

Screening is performed on the 4000-point LHS. Simulations are run on the HIICS cluster (26 nodes, 692 cores, 64 to 256 GO of RAM per server) available at INRAE, France, for a simulation time of 12000 CPU hours. Screening is performed at a site scale, on each HU individually, to remain as conservative as possible. Influential parameters at the catchment scale are then deduced from the union of influential parameters for each site. After screening and union, 42 influential parameters are selected for water subsurface flow, 54 parameters are selected for pesticide subsurface flow, 43 parameters are selected for water surface runoff, and 45 parameters are retained for pesticide surface runoff. The remaining parameters are given in Appendix .

For the ranking task, the three methods (Sobol' indices from PCE, HSIC and RF) are applied on each HU based on a 2000-point LHS generated for each scrutinized variable from the set of screened input parameters. Again, simulations are performed on the HIICS cluster, for a simulation time of 6000 CPU hours per variable. Columns 1 and 2 in Fig.  show Sobol' total- and first-order indices, while columns 3 and 4 show HSIC and RF sensitivity indices for the most influential parameters according to Sobol' total-order indices for the four variables on HU14.

In this section, we focus on Sobol' indices (first- and total-order) despite larger error bounds as it is the only method used in this study that allows us to get separate information on interactive effects. Site rankings such as those presented in Fig.  are gathered for all HUs in the form of sensitivity maps in Fig.  for water surface runoff and in Fig.  for pesticide surface runoff. Broadly speaking, both maps show strong spatial heterogeneities regarding influential parameters, and a contrasted behaviour between right and left banks can be identified. For both output variables, hydrodynamic parameters (thetas, thetar and mn) of deep horizon from soil 1 (i.e. 2 and 3) are mainly influential only on HUs characterized by soil 1 (i.e. 2 and 3) (see Fig.  for a reminder of soil types). The local hydrodynamic parameters are found to be dominating, explaining the output variable variance. A particular case is HU4 (indicated by an array on both Figs.  and ), which is characterized by soil 3, while parameters from soil 1 explained most of the variance of both water and pesticide surface runoff variables. The location of HU4, near the outlet, downstream several soil-1 HUs, may explain such spatial interactions. In addition to specific soil parameters, other parameters such as Manning's roughness on vineyard plots (manning_plot) or the coefficient of adsorption (Koc_pest) have a greater influence on HUs from the right bank (bottom part of the catchment in the figure). In addition, comparison of first-order and total-order maps shows quite similar results for water surface runoff on the one hand. It indicates that direct effects are significant for all influential parameters. On the other hand, direct effects are far from dominant on pesticide surface runoff. Once again, most parameters are influential nearly only in interaction with other parameters since the first-order indices are very low compared to the total-order indices.

Finally, Fig.  shows aggregated sensitivity indices for water and pesticide surface runoff variables following . Since the two banks have contrasted behaviours, aggregated indices are first calculated at the intermediary scale of the bank, then at the catchment scale. For both output variables, rankings strongly differ on each slope. As proposed at a local scale in Sect. , aggregated indices at this scale may constitute summarized information about the physical processes dominating in PESHMELBA to explain the output variable. For water surface runoff, hydrodynamic soil parameters related to vertical infiltration (thetas, thetar and mn) dominate. The influence of Ks_river is only significant in the right bank. The difference of altitude between the right and left bank may explain this contrast in the activation of saturated exchanges between water tables and the river. For pesticide surface runoff, deep horizon parameters from soil 1 and the pesticide adsorption coefficient (Koc) explain a major portion of the output variance on the left bank, while pesticide half-life time (DT50) and surface runoff parameter (hpond_plot) have lower or no impact. Conversely, surface parameters (manning_plot and hpond_plot) have a higher impact on the right slope. In that bank, soil horizons are characterized by lower permeabilities that may result in stronger surface runoff generation than on left bank. In addition, pesticide parameters (Koc and DT50) are also more influential. More broadly, these results show that pesticide surface runoff may result from the activation and interactions of more physical processes on the right bank than on the left bank.

For all variables considered, the number of input parameters retained after screening remained quite high, proving that performing screening on PESHMELBA variables is a challenging task. We can first incriminate the many physical processes interacting in PESHMELBA in a spatially distributed way, each of them with its own set of characteristic parameters. However, it can also be explained by the methodology that may not be discriminating enough. Many previous studies developed efficient screening techniques for complex environmental models e.g.. Exploring other screening approaches is beyond the scope of this study, but future work may focus on applying and critically comparing these techniques, especially with very limited sample size (inferior to 2000 points for example).

By exploring several methods for ranking, the aim was to analyse their specificities and the interest of each one for the sensitivity analysis of a complex environmental model, characterized by many parameters and a high computational cost such as PESHMELBA. Considering the results of this study, we believe that the choice of the method depends on the properties of the model, the objective of the sensitivity analysis and the sample size available:

The Sobol' method remains attractive when sensitivity analysis is used to gain knowledge about the model by finely analysing its behaviour. Indeed, Sobol' indices provide a clear interpretation of the calculated indices (percentages of variance explained) and explicit information about the interactions between parameters. These elements are particularly valuable when one wishes to use sensitivity analysis to understand the functioning of the model, and this is why this approach is still widely used in the hydrological community. In the case of variables that are reasonably complex and that are not characterized by too much interactions of physical processes (such as water variables in our case), using chaos polynomials to estimate Sobol' indices is particularly interesting and efficient since it allows the use of a pre-existing sample, of very limited size. Conclusions are much contrasted for complex variables such as pesticide variables as convergence results showed that 2000 points may not be enough to get fully robust Sobol' indices. This is particularly the case for parameters that are mainly characterized by interactive effects.

Beyond the comparison of the different methods, we also tried to evaluate if it was possible and useful to combine several of these methods. However, considering the results obtained, we believe that combining the tested methods is still of little interest for hydrologists to better understand the model functioning. Indeed, the differences we found in rankings remain difficult to interpret. This is particularly the case when combining Sobol' and HSIC indices, due to the fact that the results from the HSIC dependence measure remain fuzzy to interpret.

Results about landscape analysis showed that, on the one hand, sensitivity maps provide local, detailed information about influential parameters on each location of the catchment. However they are computationally costly as one GSA per HU must be performed. This approach may be hard or even impossible to transpose to a real catchment scale composed of several hundred elements. On the other hand, catchment-scale aggregated indices provide synthetic information at a lower computational cost, but the spatialized aspect of the GSA is lost. As pointed out in , both approaches are complementary and provide precious knowledge about the model functioning, but they cannot always be performed together. As a compromise, we propose to use the scale of the bank, or more generally of the hillslope, as it may constitute an adapted intermediary scale to meet both requirements of detailed results for physical interpretation and computational efficiency. Indeed, in this study Sobol' aggregated indices were directly computed from site sensitivity indices as they were available, but a pick–freeze estimator can be used for a direct computation of Sobol' generalized indices. In our case, such overview of sensitivity analysis allows us to focus calibration efforts on deep soil hydrodynamic parameters and pesticide adsorption coefficient to improve the quality of the simulation of both water and pesticide surface flows.