The new version of the VarKarst model should be appropriate to assess the impact of climate and land cover change on karst groundwater recharge. It should also consider the range of challenges related to modelling ET at large scales, namely a lack of ET observations to compare with model predictions and a lack of observations of vegetation properties (e.g. rooting depth, stomatal resistance and canopy interception storage capacity) to constrain the model parameters (see Sect. S1 in our Supplement). We define the three following criteria to develop an enhanced version of the VarKarst model with explicit representation of land cover properties:

The new model should assess the three main ET components (bare soil evaporation in presence of sparse canopy, transpiration and evaporation from canopy interception) separately and explicitly. In fact, these fluxes exhibit different dynamics and sensitivity to environmental conditions; therefore, they are likely to respond differently to climate and land cover changes (Gerrits, 2010; Maxwell and Condon, 2016; Savenije, 2004; Wang and Dickinson, 2012).

With respect to previous modelling studies of karst systems, to our knowledge, only four have used models that explicitly include land cover properties, all of which were applied at the local scale with detailed on-site information. Three of these studies (Canora et al., 2008; Doummar et al., 2012; and Zhang et al., 2011) used generic hydrological models that were not specifically developed for karst areas but included enough flexibility in their spatially distributed parameters to represent the variability in soil and bedrock properties. The large number of parameters in these models hampers their application at large scales and does not comply with criterion 3 (parsimony). The fourth study (Sauter, 1992) used a lumped model that is much more parsimonious but does not represent soil evaporation and uses empirical PET equations, which does not allow for the separation of the effect of climate and land cover (criteria 1 and 2).

As for large-scale models, we can identify two main types: hydrological models, which focus on the assessment of hydrological fluxes, and land surface models, which also evaluate many other fluxes such as sensible heat, latent heat, ground heat flux, radiation and carbon fluxes (for a review, see Bierkens, 2015). Land surface models do not usually comply with criterion 3 because they have many parameters, including a number of empirical parameters that are difficult to constrain (Mendoza et al., 2015). Moreover, it has been shown that land surface models could be simplified when the objective is to assess hydrological fluxes only. For instance, Cuntz et al. (2016) demonstrated that a large number of parameters of the Noah land surface model are non-influential or have very little influence on simulated runoff. In contrast, hydrological models focus on the representation of hydrological processes and include far fewer parameters. However, our literature review (summarised in Tables A1–A3) showed that we cannot directly adopt any of their ET representations into VarKarst. In fact, as shown in Tables A1–A3, the most parsimonious models (WBM, WaterGAP and mHM) neglect some ET components and/or use empirical PET equations, which contradicts criteria 1 and 2, while models that comply with criteria 1 and 2 (PCR-GLOBWB, VIC and the model of Kergoat, 1998) use heavily parameterised schemes, such as a Jarvis type parameterisation of surface resistance (Jarvis, 1976; Stewart, 1988); therefore, these models do not satisfy criterion 3 (parsimony). Moreover, we found that large-scale models include empirical schemes with no clear origin, such as the reference crop formulation used in the PCR-GLOBWB model for PET calculation or the interception model used in the LPJ dynamic global vegetation model and in the model of Kergoat (1998). Importantly, our review revealed the tremendous variability of approaches used in large-scale models to represent ET processes. A detailed list of all parameters involved in the representation of ET in the models in Tables A1–A3 can be found in Sect. S2 of our Supplement. Consequently, no clear indication has emerged regarding a “best way” to parameterise the different ET processes at large scales, which leaves us with a large range of different formulations to choose from to implement an explicit representation of land cover processes into VarKarst.

VarKarst (Hartmann et al., 2015) is currently the only karst recharge model developed for large-scale applications. It is a conceptual semi-distributed model that simulates karst potential recharge (Fig. 1a). The representation of karst processes in VarKarst is based on a previous karst model developed for applications at the local scale introduced in Hartmann et al. (2012b). VarKarst includes two horizontal subsurface layers, a top layer called “soil” and a deeper layer called “epikarst”. The soil layer corresponds to the layer from which ET can occur. The epikarst layer corresponds to the uppermost layer of weathered carbonate rocks where it is assumed that water cannot be lost through ET. VarKarst represents karst processes because for each model grid cell, the water balance is evaluated separately over a number of vertical compartments with varying soil and epikarst properties. Recharge is the flux leaving the bottom of the epikarst and is produced in saturated or unsaturated compartments where there is moisture stored in the epikarst. Recharge can occur at faster rates in deeper compared to shallower compartments. Additionally, lateral flow concentrates the infiltrating water from saturated to unsaturated compartments. Conceptually, the direct contribution of precipitation to infiltration and recharge can be associated with the diffuse fraction, while the contribution of lateral flow can be associated with the concentrated fraction. The model has been shown to be more appropriate for applications over karst areas than other large-scale hydrological models that do not represent karst processes (Hartmann et al., 2017). The ET component of the VarKarst model is very simple and does not include explicit representation of land cover properties. ET is lumped in the soil layer, is estimated from PET and is reduced by a water stress factor, which is estimated as a linear function of soil moisture. The PET rate is calculated with the empirical Priestley–Taylor equation (Priestley and Taylor, 1972) using a spatially and temporally uniform value of the empirical coefficient. This approach does not allow for the separation of the effects of climate and land cover, as the empirical coefficient reflects both climate and vegetation characteristics simultaneously. Therefore, the ET component of VarKarst needs to be modified if the model is to be used for large-scale land cover change impact assessment.

In this section, we propose a new version of the VarKarst model, called V2Karst (Sarrazin et al., 2018; Fig. 1b). In accordance with criteria 1 and 2 defined in Sect. 2.1, the new V2Karst model includes a physically based PET equation, separates the evapotranspiration flux into three components (transpiration, bare soil evaporation and evaporation from canopy interception) and comprises three soil layers. In agreement with criteria 3 of Sect. 2.1 (parsimony), we sought to parsimoniously represent the different ET processes in VarKarst. In fact, V2Karst uses 12 parameters to represent ET and vegetation seasonality, namely 11 newly introduced parameters that replace the Priestley–Taylor empirical coefficient α used in Varkarst and the soil water capacity parameter Vsoi already present in VarKarst (model parameters are described in Table 1 and Fig. 1). This is less than other existing large-scale models that use the Penman–Monteith equation and separate the three ET components, as these models have over 15 parameters in their ET component (PCR-GLOBWB, VIC and the model of Kergoat, 1998, shown in Tables A1–A3). We assumed homogeneous above ground vegetation properties across model compartments.

The new model is forced by time series of precipitation (P), air temperature (T) and net radiation (Rn) in the same manner as VarKarst. Additionally, time series of relative humidity (RH) and wind speed (WS) are now needed for PET calculation. The V2Karst model can be run at both daily and sub-daily time steps. In this study, we present simulation results obtained using a daily time step, while results from hourly simulations are reported in Sect. S8 of our Supplement. We did not find significant differences between daily and hourly simulations for assessing recharge at monthly and annual timescales, which is the focus of our study. Therefore, it is reasonable to apply the model at a daily time step, which significantly reduces the computational requirements.

We test the model with plot scale measurements from sites of the FLUXNET network (Baldocchi et al., 2001). We identified four FLUXNET sites across European and Mediterranean carbonate rock areas for which sufficient data were available to force V2Karst and to test the model (see Sect. 3.2). A short summary of each site's characteristics is provided in Fig. 2, and more detailed information can be found in Table B1.

The sites have different climate and land cover properties. The first site (Hainich site, referred to as “German site”) is located in the protected Hainich National Park, Thuringia, central Germany, and is characterised by a suboceanic–submountain climate and a tall and dense deciduous broadleaf forest. The second site (Llano de los Juanes site, referred to as “Spanish site”) is located on a plateau of the Sierra de Gádor mountains, south-eastern Spain, has a semi-arid mountain Mediterranean climate and is an open shrubland. The third site (Font-Blanche site, referred to as “French 1 site”) is located in south-eastern France, has a Mediterranean climate and its land cover is medium-height mixed evergreen forest. The fourth site (Puéchabon site, referred to as “French 2 site”) is located in southern France and is characterised by a Mediterranean climate with a short evergreen broadleaf forest. Overground vegetation properties are well characterised at all sites, but subsurface properties are more uncertain. In particular, the rooting depth water capacity was only well investigated at the French 2 site. The four sites are appropriate for testing V2Karst, as they satisfy the model assumptions: a karstified or fissured and fractured bedrock, overall high infiltration capacity with limited surface runoff and high clay content in the soil (Table B1).

Data available at the four FLUXNET sites include measurements of precipitation, temperature, net radiation, relative humidity and wind speed to force the model. Eddy-covariance measurements of latent heat flux (density) and at the German and Spanish sites measurements of soil moisture are also available to estimate the model parameters (Sect. 4.1). Specifically, at the German site, soil moisture was measured in one vertical soil profile at three different depths (5, 15 and 30 cm) with Theta probes (Knohl et al., 2003). We selected the measurement at 30 cm depth; we deemed this depth to be most representative of the entire soil column which has a depth between 50 and 60 cm. At the Spanish site, soil moisture was assessed at a depth of 15 cm using a water content reflectometer (Pérez-Priego et al., 2013).

Data to force the model were gap-filled and aggregated from a 30 min to a daily timescale. V2Karst output observations, namely latent heat flux and soil moisture measurements, were aggregated from a 30 min to a monthly timescale and we discarded the months when more than 20 % of the 30 min data were missing. We discarded the monthly observations of latent heat flux and soil moisture for months in which the forcing data contained many gaps; therefore, the impact of the gap-filling of the data on the simulation results is likely to be too significant to sensibly compare simulated and observed soil moisture and latent heat flux. Additionally, we removed monthly aggregated latent heat flux measurements when the mismatch in the energy balance closure was higher than 50 %, similar to Miralles et al. (2011). We corrected latent heat flux measurements to force the closure of the energy balance assuming that measured latent heat flux (LE; MJ m-2 month-1) and sensible heat flux (H; MJ m-2 month-1) have similar errors. The corrected evapotranspiration estimate (Eact,cor; mm month-1) is calculated as proposed in Foken et al. (2012) and Twine et al. (2000): Eact,cor=Rnλ1+HLE.

Further details on the data processing and on the analysis of the uncertainty in latent heat flux measurements are reported in Sect. S4 of our Supplement. In particular, we showed that, while the actual value of the latent heat flux can have large uncertainties, we have a much higher confidence in its temporal variations.

Table 2 reports the simulation period and the number of monthly latent heat flux and soil moisture observations that were used to estimate the model parameters at the four FLUXNET sites. We extracted a continuous time series of forcing data covering about 10 years at the German site, 7 years at the Spanish site, 3 years at the French 1 site and 8 years at the French 2 site, while latent heat flux and soil moisture measurements were not available over the entire simulation time series. All model simulations were performed using a 1-year warm-up period, which we found to be sufficient to remove the impact of the initial conditions on the simulation results (see Sect. S5 of our Supplement).

We investigate whether it is possible to estimate parameter values that produce plausible simulations based on information available at each FLUXNET site. To this end, and similarly to Hartmann et al. (2015), we use “soft rules” to accept or reject parameter combinations based on the consistency between monthly model simulations on one side, and monthly observations and a priori information on model fluxes on the other side. Using soft rules instead of “hard rules” (i.e. the minimisation of the mismatch between observations and simulations) allows for the identification of a set of plausible parameter sets and accounts for the fact that (1) the observed soil moisture is not strictly commensurate with simulated soil moisture, (2) observations are affected by uncertainties (see Sect. 3.2) and (3) it is not expected that V2Karst simulations closely match site-specific data, as the model structure is based on general understanding of karst systems for large-scale applications and may not account for some site specificities. We define five soft rules to identify acceptable (“behavioural”) parameter combinations:

The bias between observed and simulated actual ET is below 20 %:Bias=∑t∈MET(Eact,sim(t)-Eact,cor(t))∑t∈METEact,cor(t)<20%,where Eact,sim(t) (mm) is the simulated actual ET for month t (sum of transpiration, soil evaporation and evaporation from canopy interception), Eact,cor(t) (mm) is the corrected observed actual ET (Eq. 19) and MET is the set of months for which latent heat measurements are available. This rule allows one to constrain the simulated water balance.

For each site, we derived a sample size of 100 000 for the 15 parameters of V2Karst using Latin hypercube sampling and unconstrained (wide) ranges for the model parameters to explore a large range of soil and vegetation types. We applied the above rules in sequence to either reject or accept the sampled parameter combinations. We sampled the constrained parameter ranges used in rule 5 more densely so that a sufficiently large number of parameterisations remain after applying rule 5. Similarly to Hartmann et al. (2015), a priori information on parameter ranges (rule 5) is applied last so that we can first assess the constraint of the parameter space based on information on model output only (rules 1–4); we then assess the consistency of the constraint using a priori information (rule 5).

We also note that the thresholds used in rules 1 to 3 are stricter than in the study by Hartmann et al. (2015), in which the threshold for the bias rule 1 was set to 75 % and for the correlation rules 2 and 3 was set to 0. The reason for this is that in Hartmann et al. (2015) behavioural parameter sets had to be consistent with observations at all sites within each climate zone defined in the study, while here we perform the parameter estimation for each site separately and therefore we expect better model performances.

We use a global sensitivity analysis called the elementary effect test (Saltelli et al., 2008), also referred to as the method of Morris (1991), to analyse sensitivity across the entire space of parameter variability (Saltelli et al., 2008). We chose this method in particular because it is well suited for identifying uninfluential parameters (Campolongo et al., 2007; Saltelli et al., 2008), which is one of the objectives of our analysis, and because it can be applied to dependent parameters (in V2Karst it is assumed that Ve≤Vr≤VS,n as explained in Sect. 2.3.1).

The method requires the computation of the elementary effects (EEs) of each parameter in n different baseline points in the parameter space. The EE of the ith parameter xi at a given baseline point x1j,x2j,…,xi-1j,xij,…,xMj and for a predefined perturbation Δ is assessed as follows: EEij=yx1j,x2j,…,xi-1j,xij+Δ,…xMj-yx1j,x2j,…,xi-1j,xij,…xMjΔ, where M is the number of parameters and y is the model output (simulated annual recharge in our case). We analyse the mean of the absolute values of the EEs (denoted by μi*) introduced in Campolongo et al. (2007), which is a measure of the total effect of the ith parameter; we also analyse the standard deviation of the EEs (σi) proposed in Morris (1991), which is an aggregate measure of the intensity of the interactions of the ith parameter with the other parameters and of the degree of non-linearity in the model response to changes in the ith parameter.

The total number of model executions required to compute these two sensitivity indices is n(M+1), where n is the number of baseline points chosen by the user. The baseline points and the perturbation Δ of Eq. (21) were determined following the radial design proposed by Campolongo et al. (2011). The baseline points and the perturbation were randomly selected using Latin hypercube sampling for the 15 parameters of V2Karst, and dropping the parameter sets that did not meet the condition Ve≤Vr≤VS,n. In our application, we used n=500 points, which means that we needed 8000 model executions for each sensitivity analysis for each of the four FLUXNET sites. We derived confidence intervals on the sensitivity indices using 1000 bootstrap resamples, and checked the convergence of the results at the chosen sample size, as suggested in Sarrazin et al. (2016).

Our last analysis consists of a set virtual experiments to investigate the sensitivity of recharge simulated by V2Karst to changes in (1) the precipitation properties, specifically monthly total precipitation and daily temporal distribution (i.e. frequency and intensity), which are likely to change (IPCC, 2013) and (2) land cover, specifically from forest to shrub or vice versa.

Virtual experiments permit full control of experimental conditions, thus, changes in model outputs can be unequivocally attributed to changes in model inputs (see e.g. Pechlivanidis et al., 2016; Weiler and McDonnell, 2004). Different studies have used virtual experiments to analyse the impact of precipitation spatial and temporal variability on hydrologic model outputs. In fact, using historical precipitation time series or future projections only allows for the exploration of a limited range of possible realisations, which makes it difficult to disentangle the effects of different precipitation properties on model outputs. Instead, synthetic precipitation time series can be tailored to analyse the impact of specific precipitation characteristics, for instance precipitation spatial distribution (Pechlivanidis et al., 2016; Van Werkhoven et al., 2008) and precipitation temporal distribution, namely frequency and intensity (Jothityangkoon and Sivapalan, 2009; Porporato et al., 2004), storminess (Jothityangkoon and Sivapalan, 2009), and seasonality (Botter et al., 2009; Jothityangkoon and Sivapalan, 2009; Laio et al., 2002; Yin et al., 2014).

In this study, we create a synthetic precipitation time series where the same precipitation event is periodically repeated. The precipitation time series is characterized by the intensity of precipitation events Ip (mm d-1) and the interval between two wet days Hp (days). The duration of each precipitation event is set here to 1 day. Therefore, the average monthly precipitation Pm (mm month-1) for an average month with 30 days is equal to: Pm=30Ip1+Hp.

To determine the possible range of variation of the three variables, Pm, Ip and Hp, we analysed their distributions at the four FLUXNET sites and over all European and Mediterranean carbonate rock areas using Global Land Data Assimilation System (GLAS) data (Rodell et al., 2004; distributions are reported in Sect. S6 of our Supplement). We found that wide but plausible ranges are as follows: Pm varies between 0 and 500 mm month-1, Ip varies between 0 and 200 m d-1 and Hp varies between 0 and 89 days (note that Hp=0 means that it rains every day). We then derived a set of 2266 precipitation time series by deterministically sampling Pm and Hp within those ranges (and consequently deriving a sampled value of Ip from Eq. 22). We sampled more densely closer to the lower bound of the ranges as this is where lower values of Pm and Hp are more likely to occur. For each of the precipitation time series obtained in this manner, we ran the V2Karst model until a steady-state was reached (i.e. periodic oscillations of all state and flux variables) and we analysed the steady-state monthly average of recharge.

The experiments are conducted at two “virtual sites” that are designed based on the characteristics of the FLUXNET sites. Specifically, we use a virtual “forest site” that has the characteristics of the German site (i.e. its behavioural parameterisations for the soil, epikarst and vegetation parameters, and its climate characteristics) as the baseline and a virtual “shrub site” that has the characteristics of the Spanish site. We do not investigate the effects of varying temperature, net radiation, relative humidity or wind speed characteristics as we did for precipitation, because these variables are correlated (see e.g. Ivanov et al., 2007) and, therefore, they cannot be varied independently. We account for their overall combined effect and we vary them jointly to reproduce two conditions, i.e. winter (low energy for ET) and summer (high energy for ET). For winter (summer) conditions we forced the model using constant values of temperature, net radiation, relative humidity and wind speed that were taken as the average values measured at the FLUXNET sites during winter (summer). To investigate the impact of a change in land cover, we swapped the vegetation parameters between the two virtual sites. Table 4 reports the values of the parameters and the values of the weather variables used at the two virtual sites.

We first present the results of applying the soft rules defined in Sect. 4.1 at the four FLUXNET sites. Figure 3 shows that behavioural parameterisations consistent with all rules can be identified at all sites, although the number of parameterisations is very different from one site to another. Specifically, out of the initial 100 000 randomly generated parameter samples, we found 36 838 behavioural parameterisations at the German site, 147 at the Spanish site, 6354 at the French 1 site and 4077 at the French 2 site. From Fig. 3, we also see that the application of each rule reduces the number of behavioural parameterisations, except for rule 4 (value of total surface runoff <10 % of precipitation), as all model simulations produce less than 7 % surface runoff at all sites. This can be explained by the fact that V2Karst gives priority to recharge production over surface runoff. Therefore, the latter only occurs under extremely wet conditions when all model compartments are saturated.

Figure 4 reports a parallel coordinate plot of the behavioural parameter sets and associated values of the output metrics after sequential application of the soft rules. The application of rules 1 to 4 does not significantly reduce the parameter ranges, but it allows for low values of parameters Vr and Vsoi to be discarded at all sites (dark blue lines in Fig. 4). In contrast, the application of rule 5 (a priori parameter ranges, black vertical lines in Fig. 4) permits for a significant reduction in parameter ranges, not only for the parameters that are directly constrained by this rule (hveg, rst, LAImin, LAImax, Vr and Vsoi) but also for the spatial variability coefficient a. Specifically, behavioural values of the a parameter are found to be between 0 and 3.2 at the French 1 site and between 0 and 2.8 at the French 2 site. At the Spanish site, we also observe that the behavioural simulations (red lines) cover some portions of the ranges more densely, specifically higher values of parameters rs,soi and a, and lower values of z0 and Ve. This means that the value for these parameters is more likely to be within these sub-ranges.

We also analyse the repartition of the simulated water fluxes when using the behavioural parameterisations. The top panel (Fig. 5a) compares the total simulated recharge and the total actual ET, expressed as a percentage of the total precipitation at the four FLUXNET sites (mean and 95 % confidence interval across the behavioural parameterisations). At the Spanish site, we present the results over two different time periods that have very different precipitation values, namely a drier period from 1 January 2006 to 31 December 2008 and a wetter period from 1 January 2009 to 30 December 2011 (see Fig. 2). Figure 5 shows that, apart from extremely wet periods at the Spanish site, in all other cases the fraction of recharge (Qepi) is significantly lower than the fraction of actual ET (ETact). Figure 5b shows the partitioning of ET among its different components (transpiration, soil evaporation and interception). We observe that transpiration (Tact) is the largest component at all sites, while the relative importance of evaporation from canopy interception (Ecact) and soil evaporation (Esact) varies across sites. In particular, at the German site Ecact is remarkably high on average compared to the other sites.

Finally, Fig. 6 reports monthly time series of observed variables, namely precipitation input P, actual ET Eact,cor (in blue) and soil moisture SMobs (in green). It also reports time series of simulated variables, including recharge Qepi, actual ET Eact,sim and soil moisture in the root zone SMsim (non-behavioural simulations are in grey and behavioural simulations in black). We see that the soft rules allow for a significant reduction of the uncertainty in model outputs at all sites. In fact, the width of the behavioural ensemble, i.e. the ensemble of simulations obtained by the application of the rules (black lines), is much narrower than the non-behavioural ensemble (grey lines). Simulated actual ET (Eact,sim) is also closer to the observations (blue line) in the behavioural ensemble compared to the non-behavioural ensemble. This means that the application of the soft rules and a priori information on parameter ranges allows not only for an improvement of the precision of the simulated states and fluxes (reduced uncertainty ranges of the simulations), but also for an improvement of the accuracy of simulated actual ET (simulations close to observations).

From Fig. 6, we also observe that the seasonal variations in model predictions are consistent with our understanding of the sites over the entire simulation horizon and not only over the months for which ET and soil moisture observations are used to estimate the parameters (blue and green areas in the plot). Specifically, at the German site we find a marked seasonality of simulated Eact,sim and SMsim: low Eact,sim and high SMsim in winter and high Eact,sim and low SMsim in spring and summer. In fact, in winter, the energy available for ET is low and the deciduous vegetation is not able to transpire or intercept large amounts of precipitation. On the contrary, in spring and summer more energy is available for ET and the vegetation has a higher LAI value; therefore, losses due to ET can occur during this time and deplete the soil moisture. At the other sites we observe a similar pattern for SMsim, while Eact tends to peak in spring and be lower in summer when the ET fluxes are more water-limited than at the German site.

The sensitivity analysis results are reported in Fig. 7 and refer to the sensitivity of the total simulated recharge (expressed as a percentage of total precipitation) to the 15 parameters of the V2Karst model. For each parameter, the plots in Fig. 7 report the absolute mean of the elementary effects (μ*, total effect of the parameters) on the horizontal axis and their standard deviation (σ, degree of linearity of the effect of the parameters) on the vertical axis. In all plots, we observe that the bootstrap confidence intervals of the sensitivity indices are narrow and show little overlap, which gives confidence that the sensitivity results are robust. Similar to the analysis of the simulated fluxes in Sect. 5.1 (Fig. 5), at the Spanish site we present the results for two different time periods with different precipitation amounts.

We first examine the left panels in Fig. 7, which show the sensitivity results when hveg, rst, LAImin, LAImax, Vr and Vsoi are sampled within the constrained ranges (Table 3), while the ranges for other parameters are taken from Table 1. The objective is to inform model calibration in future model applications, as such parameter ranges capture the uncertainty in parameter values left after considering site-specific information. We first note that μ* and σ take a non-zero value for all parameters at all sites, which means that all parameters are influential and have a non-linear effect on recharge, possibly through interactions with other parameters.

We observe that the spatial variability coefficient a has the largest influence by far, followed by the Vsoi and Vr parameters. In fact, their value of μ* is significantly higher than the other parameters at all sites. The implication for model calibration in future applications of V2Karst is that efforts should primarily seek to reduce the uncertainty in a, Vsoi and Vr parameters. These three parameters also have a significantly large value of σ, which indicates non-linearities in the model response to variations in these parameters. Interestingly, Vcan, which controls evaporation from interception, has an impact on recharge at most sites, and rs,soi, which controls soil evaporation, has an impact on recharge at the Spanish site during wet years. This shows that the processes of evaporation from interception and soil evaporation can be important for recharge simulations.

Moreover, we observe that the LAImin, z0, k and Ve parameters have a very small impact on total recharge at all sites (μ*<3 %). However, Sect. S7 of our Supplement reports additional sensitivity analysis results for other model outputs and shows that the most influential parameters which should be the focus of the calibration strategy vary depending on the output of interest.

We then examine the right panels of Fig. 7, which show the sensitivity indices when sampling parameters within unconstrained ranges (Table 1). This analysis allows one to test the plausibility of the model structure through the assessment of the model sensitivity across a large spectrum of soil and vegetation conditions.

The most apparent difference with respect to the previous results is that vegetation parameters (hveg, rst, LAImin, LAImax and Vr) now have a much higher value of the sensitivity indices (both μ* and σ). More specifically, LAImax has a very high sensitivity index at all sites (μ*>10.5 %), which can be attributed to the fact that this parameter is used to calculate different model components. Interestingly, the seasonality of the LAI appears to play an important role in V2Karst as μ* for LAImin⁡, although always lower than μ* for LAImax⁡, stands out at all sites.

The considerable variations in parameter sensitivities observed across sites can be interpreted by considering their climatic differences. We would expect transpiration to be mainly energy-limited at the German site, given that it has a suboceanic–submountain climate; in contrast, we would expect it to be mainly water-limited at the French sites, which have a Mediterranean climate, and at the Spanish site, which has a semi-arid Mediterranean climate. In this regard, the most influential parameter at the Spanish site is by far the a parameter (high μ*), which has an impact on the water storage in the soil and therefore on the amount of water available to sustain ET between rain events, while at the German site, parameters rst and hveg, which control PET, are more influential compared to the other sites.

Finally, we observe that, the parameters that specifically control the volume of transpiration (rst and Vr) have a significantly higher value of μ* than the parameters that specifically control soil evaporation (z0, rs,soi and Ve) and evaporation from interception (Vcan). Moreover, z0, rs,soi and Ve have a very small impact (μ*<3 %), while parameter Vcan can have an important effect at the German site (μ*=5.7 %). Additionally, we see that parameter fred, which controls transpiration from the third soil layer, has an impact on recharge simulated at the Spanish site.

After showing that the V2Karst model behaves reasonably at the four FLUXNET sites, in this section we use virtual experiments to further learn about the sensitivity of simulated recharge to precipitation characteristics and land cover using virtual sites (see Sect. 4.3). Figure 8 shows the monthly average value of simulated recharge Qepi, for a range of synthetic precipitation inputs with different values of the precipitation monthly amount Pm (x axis) and of the interval between rainy days Hp (y axis) at the virtual forest and shrub sites. We do not report Qepi values in the top right of the plots because this region corresponds to very intense precipitation events (higher than 200 mm d-1) that have a very low probability of occurrence (see Sect. 4.3).

From the top left panel of Fig. 8, we see that winter Qepi is mostly sensitive to Pm, in fact simulated recharge increases along the x axis from left to right but shows little variations along the y axis (when Hp is varied). This result is due to the fact that the actual ET is very limited in winter because of the low energy available. We indeed estimated that the maximum value of total ET across the different precipitation inputs is 13 mm month-1 at the forest site and 35 mm month-1 at the shrub site. Therefore, a large part of precipitation becomes recharge rather independently of its temporal distribution.

From the right panel of Fig. 8, we observe a systematic reduction in summer Qepi compared to winter at both virtual sites. Moreover, summer recharge is generally highly sensitive not only to Pm but also to Hp, as it increases when moving along the y axis from bottom to top, i.e. when the same amount of monthly precipitation falls in less frequent but more intense events. This result can be explained by the fact that in summer potential ET is larger and therefore, if events are less intense, a larger part of the precipitation is lost via ET. In contrast, if events are more intense, the canopy and soil stores reach saturation. In this case, a saturation excess flow to the epikarst is generated – hence, there is more recharge and less ET. Moreover, in summer, Qepi shows a limited sensitivity to Pm and Hp when these quantities take low values (brown and red dots on the left of the plots). This is due to the fact that only a few soil compartments reach saturation under drier conditions and therefore little recharge can be generated. We also note that Qepi is a significant flux (Qepi>5 mm) for smaller values of Pm and Hp at the shrub site compared to the forest site. This may be due to the fact that the soil water capacity (Vsoi) is much smaller at the shrub site, and the soil compartments can therefore reach saturation under drier conditions.

Finally, Fig. 9 reports the results of another virtual experiment which is focused on the impact of land cover change. Specifically, the panels in Fig. 9 show the variation in simulated recharge when land cover is changed from forest to shrub at the virtual forest site (and vice versa at the virtual shrub site), and more specifically, Fig. 9 reports ΔQepi=Qepishrub-Qepiforest. We see that in all plots ΔQepi is positive, which means that recharge is larger and actual ET is lower under shrub land cover compared with forest land cover for both sites. From the left panels of Fig. 9, we observe that ΔQepi is very limited in winter, which is expected as ET fluxes are small in winter as explained above.

Conversely, the right panels of Fig. 9 show that summer ΔQepi is much higher than in winter conditions and is sensitive to both Pm and Hp. The sensitivity of ΔQepi is highly variable across the different precipitation inputs, and more specifically an increase in Hp can have a different effect on ΔQepi depending on the value of Pm (no variation, increase or decrease in ΔQepi). In fact, when Pm is low, ΔQepi is always low and does not vary sensibly when Pm and Hp are varied (brown area in the left end of the plot), since recharge is always low under these precipitation conditions as shown in Fig. 8. For intermediate values of Pm, ΔQepi has a similar pattern at both sites and increases when either Hp or Pm increases. Instead, for high values of Pm, we see that for both sites ΔQepi decreases when Hp increases. ΔQepi is largest when the monthly precipitation Pm is high and the interval between wet days Hp is low (green dots at the virtual forest site and dark blue dots at the virtual shrub site), because under these precipitation conditions the amount of moisture available for ET is maximum.

Importantly, our results also show that the impact of a change in land cover can vary greatly across sites, as at the virtual shrub site summer ΔQepi reaches much higher values and is sensitive to Pm and Hp over a larger range of values of Pm and Hp compared to the virtual forest site.

We tested the model by evaluating its ability to reproduce observations at four carbonate rock FLUXNET sites, which is a standard approach to model testing, used for instance to test the previous version of the model VarKarst (Hartmann et al., 2015) and large-scale ET products (Martens et al., 2017; McCabe et al., 2016; Miralles et al., 2011). We demonstrated that V2Karst is able to produce behavioural simulations consistent with observations and a priori information at FLUXNET sites. In addition, the time series of simulated water balance components are coherent with our understanding of the sites. A different number of behavioural parameterisations was identified at the different sites, with the highest number found at the more humid German site and the lowest number at the semi-arid Spanish site, which is coherent with previous findings that a better fit to observations can be obtained at wetter locations (Atkinson et al., 2002; Bai et al., 2015).

Interestingly, the behavioural parameter sets for the French 1 site corroborate the hypothesis that root water uptake is likely to extend below the physical soil layer, as communicated by Guillaume Simioni (investigator of the site). In fact, we found behavioural values of the Vr parameter above 59 mm, while site-specific information indicates that the physical soil layer has a storage capacity of 49 mm (Table B1). This result further attests to the realism of V2Karst structure.

The results of the global sensitivity analysis of simulated recharge to the model parameters are easily interpreted in light of the different climatic conditions at the four FLUXNET sites. Parameters that control PET and the soil water storage capacity generally have a large impact on simulated recharge. The interception capacity is also very important at the German site. These findings are in line with the few sensitivity analysis studies of large-scale hydrological models (Güntner et al., 2007; Werth et al., 2009), which have examined the sensitivity of continental water storage and river discharge simulated by the WaterGAP model. The importance of the maximum leaf area index and, to a lesser extent, of the seasonality of vegetation in the V2Karst model is also consistent with previous studies. For example, Tesemma et al. (2015) found that assimilating year-to-year monthly LAI in the VIC model can significantly improve runoff simulations compared with using long-term average LAI; furthermore, Rosero et al. (2010) determined that LAI has a large influence on simulated latent heat in the Noah land surface model. Therefore, our sensitivity analysis results generally suggest that the model behaves sensibly and consistently with our understanding of the key vegetation–recharge processes that we aim to reproduce.

Our results also provide valuable insights about possible vulnerabilities of karst systems. For example, the fact that the LAI is a highly influential parameter at all FLUXNET sites suggests that the increasing trend in global LAI documented by Zhu et al. (2016) could be reflected as a significant impact on groundwater recharge and on the partitioning between green and blue water in karst areas.

Variations in simulated recharge were also systematically examined under combined changing land cover type and climate conditions (precipitation overall amount and temporal distribution), as the model should be appropriate for climate and land cover change impact studies. We used virtual experiments with synthetic data because they allow one to unequivocally attribute changes in recharge to changes in climate and land cover. Importantly, we showed that precipitation characteristics and land cover have an interacting effect on simulated recharge; these findings are consistent with the global and observational analysis by Kim and Jackson (2012) of non-karst areas, which revealed interactions among vegetation and climate (in particular the overall amount of precipitation) in producing recharge. We found that simulated recharge is lower (and ET is higher) for forest than for the shorter vegetation type considered, i.e. shrubs, which is in line with expectations and with previous studies undertaken in karst areas (Ford and Williams, 2007; Williams, 1993). We demonstrated that the overall amount and daily intensity of precipitation have a positive effect on simulated recharge, i.e. the higher these factors are the higher the recharge. This is consistent with previous observational studies, which found that heavy precipitation events enhance groundwater recharge in non-karst areas (e.g. Taylor et al., 2013, in a semi-arid tropical region; Owor et al., 2009, in a seasonally humid tropical region). Conversely, the modelling study by Hartmann et al. (2017) showed that heavy precipitation events can have a negative effect on recharge simulated by the VarKarst model in some carbonate rock areas in Europe, the Middle East and northern Africa. However, Hartmann et al. (2017) analysed the sensitivity of simulated recharge using forcing data from global circulation models (GCMs). Given that climate variables from GCMs show complex patterns, it might be difficult to isolate the effect of a specific climate property (e.g. heavy precipitation events) on the simulated fluxes, which could explain the differences between their findings and ours.

Precipitation patterns are more complex than the simple periodic variations we used here, and the steady-state conditions may never be reached in practice. Nevertheless, we believe that performing virtual experiments similar to those proposed in the present study is a complementary approach to the application of climate projections provided by GCMs and future land cover change scenarios (e.g. Holman et al., 2017; Hurtt et al., 2011), to systematically assess the sensitivity of a model to changes in input characteristics, to test the soundness of the model sensitivities and to determine which aspects of a model input would be worth further investigating.

The results of our global sensitivity analyses suggest that all newly introduced processes in V2Karst are relevant for applications over large domains because the parameters that control these processes can affect simulated recharge, depending on the climatic, soil and land cover conditions. Specifically, transpiration and vegetation seasonality are important processes under the climate of the four FLUXNET sites, evaporation from canopy interception is important at all forested sites (German site and two French sites), the contribution of water stored below the root zone is important under the climate of the Spanish site and soil evaporation is important at the semi-arid site with sparse and short vegetation (Spanish site). The significance of representing canopy interception, in particular for forested land covers, has already been mentioned in previous studies (Gerrits, 2010; Savenije, 2004), whilst the importance of separating transpiration and soil evaporation was reported in Maxwell and Condon (2016) and Wang and Dickinson (2012).

Regarding parameter estimation, the use of soft rules similar to this study and to the large-scale study of Hartmann et al. (2015) can be envisaged for applications over large domains. We showed that a priori information on parameter ranges is needed to constrain the simulations. At large scales, a priori information on vegetation parameters can be derived from large-scale databases of vegetation properties (more details in Sect. S1 of our Supplement), and a priori information on parameters that control the subsurface properties (Vsoi) can be derived following Hartmann et al. (2015). It is also particularly important to assess the sensitivity of model parameters across the modelling domain to test the suitability of fixing model parameters, as done in this study at FLUXNET sites. We indeed found that parameters Vcan and rs,soi, which are typically fixed in the other large-scale models detailed in Table A1, do have an impact on simulated recharge at least one FLUXNET site. Moreover, as mentioned in Sect. 2.3, Vcan and rs,soi are understood to vary across land cover types and soil types, respectively, even if no clear ranges of these parameters have been established across either land cover or soil types. Therefore, fixing these two parameters could potentially introduce large uncertainties in V2Karst simulations. Other studies have reported on this issue; Cuntz et al. (2016), in particular, showed that some constant parameters of the Noah-MP land surface model can be highly influential for some model outputs. Finally, parameter estimation for applications over large domains will need to account for the large variability in hydraulic properties and recharge patterns observed across karst systems reported, for example, in Hartmann et al. (2014) and Klimchouk and Ford (2000), for instance using a simplified classification of karst landscapes based on climate and topography as in Hartmann et al. (2015).

We note that although soils may be absent in some karst areas (e.g. karren field in high mountain areas as reported in Hartmann et al., 2014), V2Karst always includes a soil layer. However, this soil layer can be very thin and has a limited impact on recharge. Additionally, when the simulation units have a large extent, such as 0.25∘×0.25∘ in Hartmann et al. (2015), it is reasonable to assume that soil layers can always be found.

The V2Karst model can account for the sub-grid heterogeneity in the vegetation type using a “tile” approach as in Mac-PDM (Gosling and Arnell, 2011) and VIC (Bohn and Vivoni, 2016), which consists of subdividing each model grid cell into a number of independent units (tiles), each of which has a specific land cover (e.g. short or tall vegetation). The model is then evaluated separately over each tile and the overall simulated fluxes are computed as the area weighted average of the fluxes calculated over the tiles. In V2Karst, each “tile” includes a vegetated and non-vegetated fraction as explained in Sect. 2.3.3.

The V2Karst model implements a simplified representation of ET and karst processes, given the limited amount of data available at large scales as discussed in Sect. 2.1. In particular, further information on typical karstic features over large domains would be valuable to refine the representation of karst processes in V2Karst in future developments of the model, to consider additional processes explicitly (the formation of a perched aquifer in the epikarst, which can occur in highly weathered karst systems as described e.g. in Williams (1983, 2008); this process is represented e.g. in a previous karst model developed for applications at the local scale introduced in Hartmann et al., 2012b).