The most common parameterization techniques found in the literature are (1) look-up tables (LUTs), (2) manual or automatic calibration, (3) hydrologic response units (HRUs), (4) representative elementary watersheds (REWs), (5) a priori regularization functions, (6) simultaneous regionalization/regularization functions, and (7) dissimilarity-based metrics to transfer model parameters.

The simplest technique to assign a parameter value to a modeling unit (e.g., grid cell, HRU, or subcatchment) is based on a LUT. In this case, a categorical index associated with a modeling unit links it with information taken from an external reference file (i.e., the LUT) which maps this index with parameter values that are usually taken from the literature. This technique is commonly used in most of the (operational) LSMs such as CABLE, CHTESSEL, CLM, JULES, and Noah-MP . A disadvantage of this method is the difficulty to perform sensitivity analysis . Moreover, the number of classes defined in LUT is often limited to a few (e.g., 13 soil classes in Noah-MP) resulting in non-seamless parameter fields that are not continuous.

Manual or automatic calibration is a commonly used technique to parameterize spatially lumped hydrologic models e.g., and semi-distributed hydrologic models e.g.,. The aim is to minimize the disagreement between model simulations and observations. In the majority of the cases, the target variable is streamflow. The main drawback of this parameterization technique is that the parameter fields, which are obtained by colocating lumped model parameters from sub-basins, are doubtful because they exhibit sharp discontinuities along individually calibrated sub-basin boundaries despite having spatial continuity in basin physical attributes like soil, vegetation, and geological properties that govern spatial dynamics of hydrological processes . In addition, the “patchwork quilt” parameter fields shown in these references exhibit significant sensitivity to the calibration conditions as demonstrated by . Thus, models that are parameterized with this technique may exhibit (1) poor predictability of state variables and fluxes at locations and periods not considered in calibration and (2) sharp discontinuities along sub-basin boundaries in state, flux, and parameter fields e.g.,. Parameter fields derived from basin-wise “calibrated” lumped models lack spatial seamlessness and thus are “inadequate representations of real-world systems” . Moreover, excessive reliance on parameter calibration leads to deficient performance at interior points of the basin or at other locations at which the model was not calibrated .

There have been many attempts to improve the parameterization of lumped and semi-distributed models by further discretizing the sub-basins into a given number of regions that exhibit nearly similar hydrologic behavior, i.e., the so-called HRU concept initially proposed by and further developed by others e.g.,. Unfortunately, results obtained in these parameterization attempts have not been very successful in realistically representing the spatial variability of model parameters, states, and fluxes because of the lack of regionalized parameters and the unabridged reliance on parameter calibration to improve model performance . Commonly, the effective parameters estimated for the HRUs are found by automatic calibration. Efforts have been made to enforce continuity on parameter fields but with somewhat limited success during the transferability of parameters across scales and locations. In addition, models parameterized using HRUs do not lead to mass conservation of water fluxes (i.e., flux-matching) when applied to scales other than those used for calibration . Recent attempts have been made to improve the HRU concept to increase the seamless representation of parameters, states, and fluxes . However, this concept has not been tested for scalability and seamlessness of the estimated fields at coarse resolutions. Lately, a thermodynamic reinterpretation of the HRU concept was proposed by , but to date, the implementation of this approach has not found its way into meso-scale to macro-scale LSMs/HMs.

The representative elementary watershed approach is an interesting theoretical concept, which scales mass and momentum balance equations. Unfortunately, to the best of our knowledge, it has not been used to estimate effective parameters at meso- and regional scales.

A priori regularization functions (e.g., pedotransfer functions) were introduced by to ensure the “inappropriate randomness in the spatial patterns of model parameters”, i.e., the lack of seamlessness. Unfortunately, in this case, the parameters (or coefficients) of regularization functions were not subject to parameter estimation or to the verification of their ability to predict fluxes and states across various scales. The use of empirical point-scale-based relationships to link geophysical characteristics with LSM/HM parameters and the assumption that their coefficients are universally applicable with certainty (e.g., the coefficients in the pedotransfer functions) are the major reasons for the proliferation of hidden parameters in LSM/HM code . It is of pivotal importance to understand that these point-scale relationships should not be applied beyond the scale at which they were derived.

Many types of regionalization (or regularization) approaches have been tested for semi-distributed and distributed models. According to , these approaches can be broadly classified into post-regionalization and simultaneous regionalization approaches, depending on if the regionalization function parameters (or global parameters) are estimated after or during the model calibration . None of these procedures consider the subgrid variability of the model parameters or geophysical characteristics. noted that most of these regionalization procedures exhibit limited transferability because of the use of discrete soil texture classes as predictors, and very likely discontinuous parameter fields.

Recently, a dissimilarity-based regionalization technique was used by to generate an ensemble of global parameters of the Hydrologiska Byråns Vattenbalansavdelning (HBV) model at a 0.5∘ resolution for global-scale hydrological modeling. A shortcoming of this approach is the use of ad hoc nearest-neighbor interpolation of parameter fields to fill gaps where no donor basins are available in (geographically) surrounding regions. Following a similar concept of that of , the parameterization method proposed by for the contiguous United States (CONUS) will likely lead to discontinuous parameter fields for reasons similar to those mentioned above.

Many attempts have been made in the land surface modeling community to address Dooge's challenges, especially with respect to the transferability of model parameters across locations and scales, and to obtain seamless parameter fields. One of the earliest prominent experiments was conducted in the Project for Intercomparison of Land-surface Parameterizations (PILPS) . In this project, calibrated LSM parameters were transferred from small catchments to their nearest computational grid cells. The results indicated that LSMs exhibited poor transferability across space, leading to significant differences in the partitioning of water and energy fluxes. For instance, used calibrated variable infiltration capacity (VIC) model parameters from small basins to generate parameter fields for continental-scale land surface modeling by “linearly interpolating to fill in those grid cell not calibrated” on a sparse grid. As noted by , this type of regionalization is inadequate because of the nonlinearity of soil and geological formations. The spatial patterns of model parameters that would be obtained by ad hoc extrapolations based on calibrated parameters from small basins or grid cells would most likely lead to unrealistic parameter fields with spatial discontinuities circumscribing river basins, as shown in recent studies by and for the VIC model parameters.

Recent community-driven efforts, such as the Protocol for the Analysis of Land Surface Models (PALS) and the Land Surface Model Benchmarking Evaluation Project (PLUMBER) , indicate that the hurdles noted in PILPS have not been overcome. Thus, it is required to gain understanding on whether the inferior predictability of many LSMs evaluated with empirical benchmarks in the PLUMBER project (e.g., CABLE, CHTESSEL, JULES, Noah) may be the result of deficient parameterizations, among other factors.

The above-mentioned challenges that we face in estimating key physical parameters in LSMs/HMs have been intensively discussed in many studies . To further visualize the problems and to understand the deficiencies of current parameterization techniques, we selected a representative sample of LSMs/HMs used for research and/or operational purposes, namely CABLE, CLM, JULES, LISFLOOD, Noah-MP, mHM, PCR-GLOBWB, WaterGAP2 (30 arcmin), WaterGAP3 (5 arcmin), CHTESSEL, and HBV. These models vary in process complexity and spatial resolution.

We selected soil porosity as an example to visualize existing shortcomings because it is one of the most common parameters in many LSMs/HMs. This parameter controls the dynamic of several state variables and fluxes such as soil moisture, latent heat, and soil temperature, and its sensitivity has been demonstrated in various studies . A representation of the porosity of the top 2 m soil column in these models over the Pan-European domain (Pan-EU) is shown in Fig. . The Pan-EU domain was selected for depiction, but we note that the problem is general and persistent across other domains . For cases in which a HM does not use this parameter, the “available water capacity” (WaterGAP) or the “field capacity” (HBV) were selected as a surrogate due to their similarity with porosity. Both surrogate fields are normalized (in space) to ease their comparison with the porosity fields. Soil porosity is expressed in m3 m-3 to ease the comparison among different models.

The following lessons can be learned from Figure :

There is a large variability in the parameterization of this key physical parameter because none of the analyzed models have comparable spatial patterns or comparable estimates at a given location. It should be noted that the definition of the selected parameter is rather simple: it represents the ratio of the volume of voids to the total volume in the soil column. One can now wonder how large the uncertainty of other parameters would be (e.g., hydraulic conductivity) whose relationship with soil properties is very nonlinear.

The key postulation aiming at obtaining scalable (global) parameters that are transferable across locations and scales was proposed by and further tested in and . We hypothesize that flux matching across scales leads to quasi-scale-invariant global parameters γ^; thus, ∑i∑t|Wi(γ^,t)ai-∑k∈iwk(γ^,t)ak|→0,∀i∈Ω.

Here, k denotes the subgrid elements constituting a given modeling cell i with area ak. i denotes a modeling grid cell i with area ai. Wi and wk denote fluxes at two modeling scales ℓ1 and ℓ1′, respectively, with ℓ1ℓ1′2=aiak. Ω denotes the modeling domain, e.g., a river basin, and t a point in time. It should be noted that the topology of the cells at either level is not specified. Normally, rectangular grid cells are used for convenience, but this is not a necessary condition. This strong flux-matching condition can be used as a penalty function or as an additional test to discriminate parameter sets obtained with conventional parameter estimation approaches.

MPR, proposed by , aims to estimate model parameters that are seamless across scales, satisfy the flux-matching conditions (see Sect. ), and enable the transferability of global or transfer-function parameters across scales and locations . The development of MPR is ongoing. Regionalization functions used in MPR for the mHM model (www.ufz.de/mhm) by were further improved by . More recently, a model-agnostic implementation of MPR has been proposed by and tested in the VIC model in over 500+ basins in the CONUS. The study of , in contrast to the present study, does not include flux-matching tests nor the evaluation of model skill across different spatial scales.

The scaling problem in MPR is addressed by using process-specific representative elementary areas (REAs) that determine the minimum computational grid size ℓ1 at which the continuum assumptions can be used without explicit knowledge of the actual patterns of the topography, soil, or rainfall fields . The REA of a specific process, such as streamflow, can be determined by conducting a careful sensitivity analysis as shown by . To estimate an “effective” model parameter (e.g., total soil porosity) at the selected modeling scale, it is first necessary to estimate its variability at a much finer scale ℓ0≪ℓ1 such that the effects of its spatial heterogeneity can be adequately represented. In other words, the parameter at the fine scale ℓ0 represents the minimum support at which the proposed equations are still valid. indicated that a suitable estimate of ℓ0 for a given parameter could be near its correlation length. The subgrid variability of a parameter β0 depends, in turn, on the spatial heterogeneity of geophysical and biophysical characteristics (u0), such as terrain elevation, slope and aspect, soil texture, geological formation, and land cover, which are now available at hyper-resolution for the entire globe. The mathematical relationships that link model parameters with these characteristics at the finer resolution are called pedotransfer, regionalization, or regularization functions f . The constants required in these functions are usually denoted as global parameters γ^; thus, β0=fu0,γ^. Note that the fields β0 and u0 are dependent on space and time, but the vector γ^ is not.

Regularization functions are commonly used in mathematics and statistics to solve ill-posed problems (which is the case when the parameters of a distributed LSM/HM are determined by calibration) and/or to prevent overfitting. The direct consequence of the regularization is the substantial decrease in degrees of freedom of the optimization problem because the cardinality of the gridded parameter fields #{β0} is orders of magnitude larger than that of the vector of the global parameters #{γ^}. Hence, MPR is a parsimonious parameterization technique that offers spatially continuous parameter fields and removes spatial discontinuities in water fluxes and states, as observed by and discussed by . From the Bayesian point of view, the regularization functions impose a prior distribution on the model parameters. Consequently, greater care should be taken in their selection.

The second step of the MPR approach consists of upscaling the subgrid distribution of a regionalized parameter to the modeling scale. In other words, β1=〈β0〉. Here, the symbol 〈⋅〉 represents an averaging or scaling operator that is parameter specific, and thus β1 denotes the upscaled effective parameter field. It is important to note that this scaling operator is not necessarily the arithmetic mean.

A schematic representation of the MPR procedure can be seen in Fig. . In short, the motto of MPR is “estimate first, then average”, whereas other existing regionalization methods follow the opposite approach of “average first, then estimate.” Because the processes in LSMs/HMs are highly nonlinear, this sequence of operations does not commute. The consequences can be dramatic (to be shown in the results section). The latter, which is the standard approach, does not preserve fluxes/states across scales, whereas MPR does to a considerable extent. The key question here is in finding the right scaling rule for the model parameters such that the fluxes/states are preserved across a range of spatial scales.

Model parameters at the ℓ1 scale (i.e., 1 to 100 km) are called “effective” parameters because they cannot be measured by physical means at this resolution and can only be inferred by heuristic relationships f(⋅). Thus, it is essential that the inequality ℓ0≪ℓ1 is fulfilled so that the law of large numbers leads to stable estimates of the effective parameter β1 having low uncertainty. Since every LSM/HM (e.g., those mentioned in Sect. ) contains “effective” model parameters, depending on heuristic relationships (that are hidden in the source code in many cases; ), it is logical that existing LSMs/HMs are subject to parameter uncertainty. These models can be treated as stochastic models, even though their governing equations are deterministic in nature and based on physical principles such as the conservation of mass and energy . Effective parameters should not be the pure result of a blind calibration algorithm. MPR varies from other regionalization approaches in that the introduced relationships may lead to seamless parameter fields and model simulations fulfilling the flux-matching condition.

Currently, MPR is the only method that consistently and simultaneously addresses the scale, nonlinearity, and overparameterization issues if global parameters are estimated simultaneously at multiple locations (i.e., basins). The MPR approach also addresses the principle of scale-dependent subgrid parameterization (i.e., “net fluxes must satisfy the conservation of mass” proposed by ) but does not adhere to Beven's other principles, such as that subgrid parameterizations may be data and scale dependent (principles 3 and 4 in ), because exhaustive tests reported in the above-mentioned references carried out over hundreds of river basins do not appear to support them. We find MPR to be a robust technique that has the ability to provide “effective parameters” and is capable of addressing the scaling problem; in this sense, it diverges from the Beven's view p. 507 that these “effective parameters” are an “inadequate approach to the scale problem”. Furthermore, MPR differs on the regionalization and aggregation scheme (i.e., patch model areal weighting) proposed by p. 520.

The selection of regionalization functions and scaling operators is fundamental to ensuring the transferability of global parameters across scales and to guarantee the seamlessness of parameter fields across scales, e.g., from ℓ1 to 2ℓ1 and so on. proposed that the key to determining them is the flux-matching condition mentioned above. A seamless parameter field β1 can be interpreted as the corollary of the flux-matching condition. Moreover, MPR employs geophysical properties at ℓ0 that allow for a representative sample at the hyper-resolution promoted by and .

The development of LSMs/HMs and their parameterizations should be guided by a strict hypothesis-driven framework that aims at finding parsimonious and robust parameter sets that fulfill the flux-matching condition and a number of efficiency metrics that are not used during the parameter estimation phase. A multivariate, multiscale evaluation assessing the reliability of model simulations should follow the scheme presented in . Based on our previous experiences, we synthesize a formalized scheme (i.e., protocol) for systematically implementing the MPR technique in other LSMs/HMs with the aim to obtain a robust and seamless parameterization. A graphical depiction of the estimation procedure at multiple scales is shown in Fig. .

Retrofit the source code of an LSM/HM so that all model parameters are exposed to analysis algorithms. Parameters are the values of a model that can be considered random variables, i.e., those that are subject to various outcomes and can be fully defined by a probability density function. Parameters should not be confused with numerical or physical constants.

It should be noted that any of the steps above can be tested within a sequential hypothesis-testing framework . A substantial difference from a standard model optimization exercise is that the transfer function f(⋅) (step 3) and the upscaling operator (step 4) can also be modified in the modeling protocol.

Failure to satisfy the imposed condition, such as the flux-matching test, after exhaustively testing the options in steps 3 to 6 may indicate deficits in process understanding and/or poor data. Consequently, the evaluation step should also provide guidance on detecting and separating the errors stemming from process conceptualization (modeling) and input data.

In Sect. , it was postulated that the MPR technique aims at estimating seamless parameter fields across scales which minimize the occurrence of artificial discontinuities and ease the transferability of model parameters across scales and locations. The latter has been tested and reported in many studies in Europe, USA, and other basins worldwide . In this study, we provide evidence in favor of the former postulation.

To achieve this goal, the mHM model is parameterized using MPR with hyper-resolution fields of geophysical characteristics at ℓ0=500 m resolution as input. Among them, the land cover data were obtained from the Corine datasets (http://land.copernicus.eu/pan-european/corine-land-cover), and the soil texture information was derived from SoilGrids (soilgrids.org). These very detailed and homogenized soil texture fields provide the fractions of clay and sand, mineral bulk density, and fraction of organic matter for six soil horizons up to 2 m deep. A hyper-resolution digital elevation model (DEM) over Europe (approximately 30 m) from the GMES RDA project (EU-DEM; www.eea.europa.eu/data-and-maps/data/eu-dem) was used to derive terrain characteristics such as slope, aspect, and flow direction. The underlying hydrogeological characteristics are based on the International Hydrogeological Map of Europe (IHME; www.bgr.bund.de/ihme1500), available at a 1 : 1 500 000 scale. Details on the pedotransfer function used for these simulations can be found in . mHM global parameters were obtained by closing the water balance over selected river basins in Europe .

Based on these settings, which constitute the basis for the EDgE project (edge.climate.copernicus.eu), we estimated porosity fields at three modeling resolutions of ℓ1= 5, 10, and 25 km, based on the same ℓ0 support information. Following the MPR procedure depicted in Fig. , the parameter fields for the mHM model at these three resolutions can be estimated. Results are shown in Fig. .

The results illustrate that the MPR approach can preserve the spatial pattern of the porosity fields (see Fig. a, b, and c) and the first and second moments of its probability density function shown in Fig. e–g. Two-sample Kolmogorov–Smirnov tests indicate that there is insufficient evidence to reject the null hypothesis that any of the three possible pairs of empirical distributions were drawn from the same unknown distribution. This highlights that the MPR approach leads to consistent parameter fields across scales. In this case, the mean porosity is estimated to be 0.42 m3 m-3 independent of the scale.

The MPR approach, as any method, has some limitations. One of the crucial aspects of MPR is the selection of transfer functions and upscaling operators. Existing theories could be the first guess, but in the event that nothing is available, the protocol proposed in Sect.  could be used to guide the search of robust transfer functions. Testing the model parameterization for flux-matching conditions across a range of basin and spatial scales may help to identify adequate upscaling operators. This procedure, although tedious, is the only solution for the moment.

In the event that some state variables change over time (e.g., land cover/use), or during parameter estimation, the MPR algorithm has to be linked to the model because every time a global parameter (γ^) is re-estimated, all related model parameters (β1) have to be updated as illustrated in Fig. . The computational cost of performing MPR is therefore larger than other parameterization method discussed before.

Another limitation of the applicability of the MPR technique until recently was its availability only as an intrinsic module of the mHM model (www.ufz.de/mhm). This implies that tailored algorithms (i.e., source code) to perform the regionalization and upscaling of parameters for a target LSM/HM have to be developed from scratch, as it is demonstrated here as a case study for the PCR-GLOBWB model. This activity is of course time-consuming and not pleasing due to its complexity. For this reason, have started a community effort to develop a model-agnostic MPR implementation (MPR-flex), which has been so far evaluated for the VIC model.

The availability of high-resolution biophysical characteristics at the spatial scale ℓ0 constitutes another limitation of the applicability of MPR. Since the subgrid variability is fundamental to estimating robust effective parameter values at coarser scales, the minimum scale at which a model can be applied (ℓ1) is strongly determined by the data availability. For example, if the soil data are available for the Pan-EU domain at ℓ0= 250 m, the ℓ1 should not be lower than 1000 m, so that each modeling cell (ℓ1) has a representative number of underlying subgrid cells (ℓ0).

MPR has been mainly developed for a hydrologic model representing the water cycle. However, land surface models also include the energy and carbon cycles and thus have greater complexity. In particular, they have more detailed representation of vegetation. It is a topic for future research to develop a MPR approach (i.e., transfer functions and upscaling operators) for plant functional-type-specific parameters such as carboxylation rate and the slope of the Ball–Berry equation for stomatal conductance , which are required for a successful implementation of MPR in LSMs.

Finally, the computational effort for MPR is also considerably larger in comparison with other methods, because of its requirement to estimate model parameters (β0) at the highest resolution at which the biophysical characteristics are available. The computational time, however, could be substantially reduced by using a restart file (i.e., a dataset containing a copy of all parameters, state variables, and fluxes of a model at a given point in time). If this capability is available, the MPR estimation can be greatly reduced for operational simulations because the effective parameter fields and past modeled states do not need to be estimated often.

As noted in the introduction, on-site (basin-specific) parameter estimation based on HRU or similar techniques (such as clustering grid cells or sub-basins into regions that exhibit quasi-similar hydrological behavior) leads to non-seamless parameter fields such as those reported in . Here, we go one step further to show the consequences of this common practice on state variables such as soil moisture. Our postulation is that an on-site calibration of global parameters γ^ leads to biased state variables even with regularization techniques such as MPR. To falsify this postulation, we performed two model simulations denoted “on-site” and “multisite” calibration schemes. In both cases, we used the mHM setup described in over the Pan-EU domain at a 0.25∘ resolution.

In the first simulation, we perform on-site calibrations at 400 river basins in the Pan-European domain. Subsequently, the respective optimized parameter sets are used in each corresponding basin to generate the target variable, in this case, the daily soil moisture of the top 1 m soil column. Lastly, daily soil moisture fields are assembled using the independent basin simulations for the entire Pan-EU domain. The results of this experiment are shown in Fig. a for a day in August 2005. In the second simulation, the global parameters γ^ are estimated simultaneously for a set of 13 basins covering various hydroclimatic regimes in the Pan-EU domain. The corresponding soil moisture field for the same point in time is depicted in Fig. b.

The first simulation shows clear evidence of strong spatial imprint in the soil moisture fields that is easily identifiable because the shapes of the constituent river basins (Fig. a) are apparent. Another interesting feature is a strong wet bias in a basin located in center of the Iberian Peninsula compared to its neighboring regions. Wet soils during this period are very unlikely because the entire region was enduring a prolonged and extreme drought. Moderate dry bias is apparent in basins in southwest Germany, and a strong dry bias was detected in basins in west Croatia, south Lithuania, south Hungary, and north Bosnia and Herzegovina. Conversely, the soil moisture field obtained with the multi-basin parameter estimation does not exhibit these nuisances and thus can be regarded as a spatially seamless field. In this case, parameter estimation with a large sample of geophysical characteristics and many streamflow time series to estimate efficiency measures leads to a well-posed parameter estimation problem.

Based on these results, it can be concluded that parameter sets obtained using the on-site parameter estimation technique do not lead to seamless parameter fields or state variables. Moreover, automatic optimization algorithms, such as SCE or DDS, tend to overlearn from time series with large observational errors, which in turn leads to poor identifiability of parameters and biased simulations, as demonstrated above. Consequently, parameter estimation should be performed with a representative sample of basins that adequately cover the variability of hydrological regimes and geophysical properties (e.g., soil types) . It is worth noting that if the parameters of a model are estimated in a small basin with very few soil types, a single geological formation, or very flat terrain, then it is very likely that some parameters cannot be constrained during calibration. The obtained parameter set is biased to the specific basin in which it has been estimated, and hence it is not skillful for seamless and continental-scale simulations.

The effects of the commonly used parameterization techniques to generate the porosity fields of LSMs (such as CHTESSEL and Noah-MP depicted in Fig. ) are important to investigate. These fields are obtained by combining the majority (or dominant) upscaling operator and a look-up table containing categorical values of model parameters tabulated for a limited set of dominant soil types (e.g., p. 20., p. 137). The majority-based operator is mostly used for estimating grid-specific vegetation classes in LSMs .

The porosity field, based on a majority upscaling for the Noah-MP model used in EURO-CORDEX (www.euro-cordex.net) at an approximately 12 km resolution, is depicted in Fig. . Compared with the other model-derived porosity fields, the Noah-MP field appears to be most homogeneously distributed in space. It is very likely that the spatial heterogeneity is underrepresented in this case as the default soil LUT contains only 13 soil classes. It should be noted that a model such as CABLE that uses a porosity field with an approximately 100 km resolution has a larger variability than that of Noah-MP at 12 km.

The following experiment is carried out to evaluate whether the variability of the soil map or the upscaling operator has a larger effect on the derived porosity field. The highest resolution soil map available for Europe is used and applied in the same manner to derive porosity fields as described above. The texture field is provided by the SoilGrids dataset (http://soilgrids.org) at 1000 m resolution (level-0). The upscaled porosity field is generated at 5 km for the EDgE project. The soil characteristics for Noah-MP are estimated using the same look-up table as in the EURO-CORDEX–Noah-MP case. The comparison of both parameter fields (i.e., EDgE–Noah-MP and EURO-CORDEX–Noah-MP) and the main statistical moments describing the spatial variability of the porosity fields are shown in Fig. . The results clearly indicate the inappropriateness of the majority-based upscaling operator for this parameter in both cases. It leads to reduction of the variance of the porosity field and thus can be considered the least sensitive operator. This means that the informational content of the hyper-resolution soil maps, commonly available globally, is almost lost.

Notably, although the overall mean of the porosity estimated using MPR over the Pan-EU domain for mHM (Fig. a) is only 6.6 % lower than that calculated using the majority-based approach for Noah-MP (Fig. a), the spatial patterns obtained by both models are very different. The evidence of this remarkable dissimilarity can also be visualized by comparing the empirical density functions shown in Figs. d and c, both corresponding to a field at ℓ1=5 km and with the same input data. A detailed evaluation conducted by in Germany showed that large porosity values estimated with the majority-based approach could overestimate those obtained with MPR by up to 40 %, whereas in other locations, underestimation up to 15 % from those estimated by MPR can be found.

Other upscaling operators, such as the weighted arithmetic mean, are commonly used in LSMs in combination with the mosaic approach. For example, in CLM see p. 160, the texture class of the subunits of the cell, called tiles, are provided in a look-up table. The upscaled porosity field obtained using this approach is shown in Fig.  at a 1∘ (100 km) resolution. Methods based on the majority and weighted arithmetic mean operators exhibit some similarity and lack spatial variability. In both cases, the spatial mean is approximately 0.43 m3 m-3.

Hydrologic models that do not use soil porosity tend to use a similar conceptualization and values denoted as the total available water capacity (TAWC; WaterGAP versions 2 and 3) and field capacity (FC; HBV). For these types of conceptual models, normalized values of these parameters are used as surrogates for soil porosity. The consistency of the spatial patterns of TAWC and FC are compared here instead of their actual values. A distinctive difference in the patterns can be observed. For example, WaterGAP3 exhibits lower values than WaterGAP2, whereas the pattern of the normalized FC in HBV is the opposite in many locations (e.g., Spain, Germany, and Scandinavia).

Details of the parameterization schemes used to estimate TAWC and FC are beyond the scope of this study. Interested readers may refer to or , respectively. However, the TAWC in WaterGAP is obtained by linking the soil type provided by the FAO soil map with available water capacity values estimated by . Thus, no scaling rule or form of regularization is used in this case. The field capacity parameters used in HBV were determined using an ad hoc nearest-neighbor interpolation technique that relies on calibrated parameters from nearby similar donor basins that might exhibit very different geophysical characteristics. The parameter fields obtained for two versions of WaterGAP (30 and 5 arcmin) and HBV are depicted in Fig. . It can be concluded that the parameterization technique employed is not scale invariant as revealed by distinct parameter sets from WaterGAP model versions, which are operated at different resolutions. The regionalization proposed by leads to a patchwork-quilt field that does not resemble to any other field presented. Evident from Fig. , the HBV field lacks seamlessness that may result in non-seamless fields of water fluxes and states.

There is a complex interplay between soil moisture (SM) and latent heat (LH) in LSMs/HMs. Improving our understanding of soil–land–atmosphere feedback is fundamental for making reliable predictions of water and energy fluxes. In this context, we carry out a sensitivity experiment to investigate the effects of soil-related parameterizations (e.g., soil porosity) on latent heat and soil moisture. Two contrasting modeling paradigms (Noah-MP and mHM) are employed.

The WRF/Noah-MP system is forced with ERA-Interim at the boundaries of the rotated CORDEX grid (www.meteo.unican.es/wiki/cordexwrf) at a spatial resolution of 0.11∘ covering Europe from 1989 to 2009. To ease the comparison, the process-based hydrological model mHM (www.ufz.de/mhm) is driven with daily precipitation and temperature fields generated by the WRF/Noah-MP system during the same period. The spatial resolution of mHM is fixed at 5 × 5 km2. The main geophysical characteristics in WRF/Noah-MP of land cover and soil texture are represented with a 1 × 1 km2 MODIS and a single-horizon, coarse-resolution FAO soil map with 16 soil texture classes, respectively. The porosity field of Noah-MP is estimated by applying a majority-based operator to values for different soil classes, as shown in Fig. b.

The settings of the mHM model used in this experiment are described in Sect. . In contrast to those of Noah-MP, the global parameters of mHM estimated using the MPR technique are obtained by closing the water balance over selected river basins in Europe . The porosity fields obtained for mHM over the Pan-EU are depicted in Fig. .

The phase diagrams of the monthly fraction of soil water saturation fSM=θθs (i.e., plots of monthly fSM(t) vs. fSM(t+1)) are subsequently investigated to understand the effect of differences in porosity estimates of the top 2 m soil column on the soil moisture dynamics (Fig. ). Two locations in Germany are selected in which Noah-MP systematically over- or underestimated the latent heat fluxes with respect to mHM (the latitude and longitude coordinates of the center of the selected Noah-MP grids are A (54∘ N, 10∘ E) and B (51∘ N, 7∘ E), respectively). At location A, the majority-based approach underestimates the MPR soil porosity by -10 %, whereas in location B, it overestimates it by 40 %. This experiment unambiguously shows that, at locations where Noah-MP overestimates latent heat with respect to mHM, the temporal variance (i.e., dynamic) of the monthly SM time series simulated by Noah-MP is almost doubled compared to that of mHM, leading to much lower soil moisture values (Fig. a). Conversely, underestimation of latent heat greatly reduces the variance of the soil moisture dynamics (Fig. b).

In Sect. , we postulated that ad hoc parameterization schemes do not necessarily fulfill the flux-matching test performed with a flux simulated by a given model at two modeling resolutions (ℓ1= 5 and 30 arcmin). A detailed description of how to perform this test is provided in . The following experiment is conducted with three models (mHM, PCR-GLOBWB, and WaterGAP) in an attempt to falsify the above postulation. All models use the same forcings and geophysical information. The simulations are conducted in the Rhine River upstream of the Lobith gauging station. All three models are driven by daily forcing with a spatial resolution of 5 km, which was kindly provided by the EFAS team at JRC (www.eea.europa.eu). Additional details of the modeling settings of this experiment are provided in and at www.hyperhydro.org/. The KGE and bias values of these three models obtained for both scales at the Lobith station during 2003 are reported in Table . The daily streamflow time series during this year is selected for evaluation because it exhibits strong temporal dynamics, with wet conditions in the beginning of the year followed by a drought during the summer and fall seasons. The performances obtained for the three models are satisfactorily, but the results shown in Table  indicate that mHM is the only model that can have higher KGE values regardless of the spatial modeling resolution.

The flux-matching test presented in Sect.  is performed with simulated evapotranspiration (ET) because it is the largest flux in the water cycle besides precipitation, and is prone to the largest predictive uncertainties . To ease the comparison, collocated grids are employed for every model such that every coarser scale grid cell has exactly the same number of underlying cells at finer resolution (5 arcmin). The results of this test are shown in Fig. a, b. They reveal that mHM exhibits the best flux-matching between these two scales. This experiment also shows that the MPR technique implemented in mHM leads to ET fields that are of similar magnitude at both scales, indicating a close conservation of mass leading to the lowest relative errors (Fig. c) among the three models.

The PCR-GLOBWB and WaterGAP models reveal large inconsistencies in preserving the spatial pattern of annual ET across two modeling scales, although the streamflow performance at the outlet is good (greater than 0.83 in both cases). PCR-GLOBWB at coarse resolution tends to underestimate ET (up to 50 %) compared with those at finer resolution (Fig. f). Conversely, the coarser version of WaterGAP tends to overestimate ET (up to 60 %) compared with those at the finer resolution (Fig. i). Interestingly, it can be observed that changes in model resolution affect the dynamic of water fluxes in those models that do not use any consistent scaling rules for model parameterization. These results also confirm the postulation that “streamflow-related metrics are a necessary but not sufficient condition to warrant the proper partitioning of incoming precipitation P into various spatially distributed water storage components (e.g., SM) and fluxes (e.g., ET)” . Because all models are forced with the same forcings, share the same geophysical information, and have almost similar hydrological process descriptions, it can be safely concluded that the parameterization method used in the models caused the ET mismatch. To falsify this postulation, the MPR parameterization protocol proposed in Sect.  is next applied to PCR-GLOBWB.