Land surface and hydrologic models (LSMs/HMs) are used at diverse spatial resolutions ranging from catchment-scale (1–10 km) to global-scale (over 50 km) applications. Applying the same model structure at different spatial scales requires that the model estimates similar fluxes independent of the chosen resolution, i.e., fulfills a flux-matching condition across scales. An analysis of state-of-the-art LSMs and HMs reveals that most do not have consistent hydrologic parameter fields. Multiple experiments with the mHM, Noah-MP, PCR-GLOBWB, and WaterGAP models demonstrate the pitfalls of deficient parameterization practices currently used in most operational models, which are insufficient to satisfy the flux-matching condition. These examples demonstrate that J. Dooge's 1982 statement on the unsolved problem of parameterization in these models remains true. Based on a review of existing parameter regionalization techniques, we postulate that the multiscale parameter regionalization (MPR) technique offers a practical and robust method that provides consistent (seamless) parameter and flux fields across scales. Herein, we develop a general model protocol to describe how MPR can be applied to a particular model and present an example application using the PCR-GLOBWB model. Finally, we discuss potential advantages and limitations of MPR in obtaining the seamless prediction of hydrological fluxes and states across spatial scales.

“If it disagrees with experiment, it's wrong”. Richard P. Feynman

Land surface and hydrologic models (LSMs/HMs) are currently used at
diverse spatial resolutions ranging from 1 to 10 km in catchment-scale
impact analysis and forecasting

A parameterization is a simplified and idealized representation of
subgrid physical phenomenon that is either “too small, too brief, too
complex, or too poorly understood” to be explicitly represented by a
model at a given resolution

Effective parameters of LSMs/HMs are usually obtained by ad hoc
procedures (e.g., automatic calibration) at a given spatial resolution
for a given modeling domain. As a consequence of this standard
practice, parameter fields of LSMs/HMs often exhibit artificial
spatial “discontinuities” such as calibration imprints circumscribing
river basin boundaries, and consequently they are not seamless

Further reasons that have prevented the improvement of
parameterization techniques are

the lack of procedures and theories for linking physical properties (e.g., soil porosity) that can be measured at the field scale with “effective” parameter values that represent the aggregate behavior of the land characteristics at the scale of a grid cell required in LSMs or HMs,

poor understanding of
the scaling of parameters

limited
inclusion of subgrid heterogeneity in hydrological
parameterizations and multiscale modeling of hydrologically relevant
variables as suggested by

lack of significant progress on the
applicability of seminal upscaling theories

lack of
transparency in most of the existing LSM/HM source codes with respect to
the meaning, origin, and uncertainty associated with the hard-coded
numerical values (i.e., parameters) either in the code or in the look-up
tables

There are potential methods available in the literature that may lead
toward coherent parameterizations and prediction of water and energy
fluxes in LSMs/HMs. For example, (1) sidestepping the scaling problem of
key model parameters by assuming scale-independent distribution
functions with regionalized distribution parameters

In contrast to these existing methods, we argue that the multiscale
parameter parameterization (MPR) technique

In this study, we analyze to which extent existing LSM/HM
parameterizations are limited to obtain seamless predictions of water
fluxes and states across multiple spatial resolutions. Through several
modeling experiments addressing

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

Manual or automatic calibration is a commonly used technique to
parameterize spatially lumped hydrologic models

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

The representative elementary watershed approach

A priori regularization functions (e.g., pedotransfer functions) were
introduced by

Many types of regionalization (or regularization) approaches have been
tested for semi-distributed and distributed models. According to

Recently, a dissimilarity-based regionalization technique was used by

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)

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)

The above-mentioned challenges that we face in estimating key physical
parameters in LSMs/HMs have been intensively discussed in many studies

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

Porosity fields (top 2 m) of typical LSM/HM over Pan-EU at
various resolutions: CABLE (1

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 degree of seamlessness strongly depends on the level of aggregation and the upscaling of underlying soil texture fields. For example, the proxy of porosity for WaterGAP is substantially different in spatial pattern and magnitude for 30 arcmin and 5 arcmin simulations. On the contrary, the spatial pattern and magnitude for porosity used in mHM remain almost unchanged for application at 30 and 5 arcmin resolution.

A parameter field becomes highly discontinuous and patchy when, for a given model, the parameter is calibrated in a limited domain (or basins) and then extrapolated to other regions (e.g., as shown in the panel corresponding to the HBV).

These experimental results confirm the postulation of

Why are there such large differences between models in estimating a parameter that has a physical meaning?

What are the consequences of poor parameterizations on the spatiotemporal dynamics of state variables and fluxes?

What are the consequences of model calibration on parameter fields?

Are current model parameterizations scale invariant?

Do the fluxes estimated with these models at various scales satisfy the fundamental mass conservation criterion (hereafter denoted as the flux-matching test)?

The key postulation aiming at obtaining scalable (global) parameters that are
transferable across locations and scales was proposed by

Here,

MPR, proposed by

The scaling problem in MPR is addressed by using process-specific
representative elementary areas (REAs) that determine the minimum
computational grid size

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

The second step of the MPR approach consists of upscaling the subgrid
distribution of a regionalized parameter to the modeling scale. In other
words,

A schematic representation of the MPR procedure can be seen in
Fig.

Schematic representation of the proposed seamless prediction
framework based on

Model parameters at the

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

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

The development of LSMs/HMs and their parameterizations should be guided by a
strict hypothesis-driven framework

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.

Determine a set of the most sensitive model parameters through a
sensitivity analysis (SA). For computationally expensive LSMs such as
CLM or Noah-MP, computationally frugal methods such as the elementary
effects method

Regionalize sensitive model parameters that exhibit marked spatial
variabilities. The selection of the regionalization function

Estimate effective parameter fields

Estimate the global parameters

Perform multi-basin, multiscale, multivariate cross-validation
tests to evaluate the robustness of the regionalization functions,
scaling operators, and global parameters

Evaluate the parameter seamlessness and the preservation of the
statistical moments of fluxes and states across scales (seamless
prediction step in Fig.

If the cross-validation tests provide satisfactory results (e.g.,
Kling–Gupta efficiency (KGE) of the compromise solution

Seamless soil porosity (top 2 m) fields obtained using MPR
at three spatial resolutions

It should be noted that any of the steps above can be tested within a
sequential hypothesis-testing framework

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.

To achieve this goal, the mHM model is parameterized using MPR

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

The results illustrate that the MPR approach can preserve the spatial
pattern of the porosity fields (see Fig.

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.

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
(

Another limitation of the applicability of the MPR technique until
recently was its availability only as an intrinsic module of the mHM
model (

The availability of high-resolution biophysical characteristics at the
spatial scale

mHM simulations of soil moisture as the fraction from
saturation

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

Finally, the computational effort for MPR is also considerably larger in
comparison with other methods, because of its requirement to estimate
model parameters (

In this section, we perform four modeling experiments, inspired by

the effects of the overcalibration of global parameters on the spatial patterns of modeled state variables,

the effects of a parameterization technique on the spatial pattern of effective parameters,

the effects of a parameterization technique on the dynamics of a state variable, and

the effects of not satisfying the flux-matching condition on simulated flux across different spatial scales. In these experiments, four models are employed: mHM, Noah-MP, PCR-GLOBWB, and WaterGAP.

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

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.

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.

Porosity fields obtained using the majority upscale operator
for spatial resolutions of

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

The effects of the commonly used parameterization techniques to generate the
porosity fields of LSMs (such as CHTESSEL and Noah-MP depicted in
Fig.

The porosity field, based on a majority upscaling for the Noah-MP model used
in EURO-CORDEX (

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
(

Notably, although the overall mean of the porosity estimated using MPR over
the Pan-EU domain for mHM (Fig.

Other upscaling operators, such as the weighted arithmetic mean, are commonly
used in LSMs in combination with the mosaic approach. For example, in CLM

Phase diagrams of monthly soil moisture fraction for two
locations in Germany,

Hydrologic models that do not use soil porosity tend to use a similar
conceptualization and values denoted as the total available water capacity
(

Details of the parameterization schemes used to estimate

There is a complex interplay between soil moisture (

Multiscale simulation of annual ET for the Rhine River in
2003 with mHM, PCR-GLOBWB, and WaterGAP (versions 3 and 2) at
spatial resolutions

The WRF/Noah-MP system is forced with ERA-Interim at the boundaries of
the rotated CORDEX grid (

The settings of the mHM model used in this experiment are described in
Sect.

The phase diagrams of the monthly fraction of soil water saturation

In Sect.

Data sources and parameterization methods used by models used in this study.

Efficiency of mHM, PCR-GLOBWB (Ludovicus et al., 2017), and WaterGAP obtained for the Rhine Basin at the Lobith station during 2003 for spatial resolutions of 5 and 30 arcmin.

The flux-matching test presented in Sect.

Porosity fields of PCR-GLOBWB before

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.

To evaluate the consistency of land surface fluxes before and after MPR
implementation, we analyze the impact of MPR on evaporative fluxes and
soil moisture content in PCR-GLOBWB

Annual ET fields in 2003 of PCR-GLOBWB before

The original PCR-GLOBWB parameterization does not include consistency in
upscaling as enforced by MPR, leading to a larger difference in soil
properties. Figure

These differences in soil hydraulic properties influence the derived
hydrological properties, leading to changes in saturated conductivity
and storage capacity in the unsaturated zone. The considerable
differences in ET fluxes are shown in Fig.

From these evaluations, we conclude that MPR implementation leads to significant improvement in the flux-matching and discharge simulations across scales, allowing for more consistency across scales for hydrological model simulations. Notably, additional parameters in PCR-GLOBWB still need to be regionalized within the MPR framework, which could potentially lead to better performance and transferability.

Hyper-resolution modeling initiatives

We revisited a technique called MPR

This study has shown that two models that use ad hoc parameterizations can have reasonable efficiency with respect to simulated streamflow but poor performance with respect to distributed fluxes such as evapotranspiration. The implementation of this protocol in PCR-GLOBWB in this study increased the model efficiency by almost 6 % and improved the consistency of simulated ET fields across scales. For example, the estimation of evapotranspiration without MPR at 5 and 30 arcmin spatial resolutions for the Rhine River basin resulted in a difference of approximately 29 %. Applying MPR reduced this difference to 9 %. For total soil water, the differences without and with MPR are 25 and 7 %, respectively. We have also shown that the PCR-GLOBWB global parameters can be transferred across scales with consistent ET patterns and model efficiency.

In general, it can be concluded that the estimation of global parameters
is feasible with MPR and that these scalars are transferable across
scales and locations. The successful application of MPR implies that the
averaging procedure of geophysical properties matters and that having
the right physics with incorrect “effective” parameters leads to
inconsistent fluxes and states. Consequently, MPR is a step forward to
quasi-scale-invariant parameterizations and is feasible to implement in
existing LSMs/HMs whose goal should be seamless parameter fields across
scales that do not exhibit artificial spatial “discontinuities” such
as calibration imprints, and that lead to consistent predictions across
scales. We consider that this feature is the key for the next generation
of LSM and NWP models such as the model for prediction across scales
(MPAS) (

Finally, we would like to reiterate that a flux obtained from a land surface/hydrologic model should always be evaluated with local observations when available and across scales. If “it disagrees with the experiment, it's wrong.”

All datasets can be obtained by email from the corresponding author.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Observations and modeling of land surface water and energy exchanges across scales: special issue in Honor of Eric F. Wood”. It is a result of the Symposium in Honor of Eric F. Wood: Observations and Modeling across Scales, Princeton, New Jersey, USA, 2–3 June 2016.

We kindly acknowledge our data providers: Noah-MP: Kirsten Warrach-Sagi
(University of Hohenheim), PCR-GLOBWB: Niko Wanders
(Princeton University), WaterGAP: Hannes Müller Schmied (University
of Frankfurt), JULES: Anne Verhoef (The University of Reading),
LISFLOOD: Peter Salamon (JRC), CABLE: Matthias Cuntz (formerly UFZ,
now INRA), CLM: David Lawrence (UCAR) and Edwin H. Sutanudjaja and
Marc F. P. Bierkens, Hannes Müller Schmied, Stephanie Eisner, Oldrich Rakovec
for providing results from the HyperHydro WG1
Workshop 9–12 June 2015 in Utrecht. This study was carried out within
the Helmholtz Association climate initiative REKLIM
(