The transfer of parameter sets over different temporal and spatial
resolutions is common practice in many large-domain hydrological modelling
studies. The degree to which parameters are transferable across temporal and
spatial resolutions is an indicator of how well spatial and temporal
variability is represented in the models. A large degree of transferability
may well indicate a poor representation of such variability in the employed
models. To investigate parameter transferability over resolution in time and
space we have set up a study in which the Variable Infiltration
Capacity (VIC) model for the Thur basin in Switzerland was run with four
different spatial resolutions (1 km

The history of modern hydrological modelling dates back to halfway through
the nineteenth century, starting with empirical models to predict peak flows

Because the parameters in hydrological models often represent a different
spatial scale than the observation scale, or because conceptual parameters
have no directly measurable physical meaning, calibration of hydrological
models is almost always inevitable

Both to decrease calculation time of the optimization procedure and to be
able to apply the model in ungauged or poorly gauged basins and areas, many
studies have focused on the transferability of parameter values over time,
space, and spatial and temporal resolution
(e.g.

Less intuitive and less common is to transfer parameters across different
grid resolutions.

Although the ambition of GHMs is to move towards hyper-resolution
(

The impact of transferring parameters across spatial and/or temporal resolutions on model performance is thus ambiguous, but relevant in the light of hydrological model development, especially for GHMs which are at the upper boundary of computational power and data availability. Calibration on a coarse temporal or spatial resolution and subsequently transferring to a higher resolution could potentially reduce computation time, and it is therefore relevant to investigate the opportunities. But parameter transferability across spatial and temporal resolutions is also interesting for another reason: it is an indicator of the degree to which spatial variability and temporal variability are represented in the model. Ideally, in a model that describes all relevant hydrological processes correctly, parameters should to a large extent be transferable over time because longer time steps are simply an integration of the shorter time steps. On the other hand, parameters should not be or be hardly transferable over space, because the physical characteristics which they represent are different from place to place. Investigating parameter transferability across spatial and temporal resolutions can thus provide insight into the model's representation of spatial and temporal variability.

In this study, we employ the Variable Infiltration Capacity (VIC) model

Overview of the spatial and temporal resolutions employed in this
study. Top panels from left to right: DEM (digital elevation model) grid
cells for 1 km

Several studies already investigated scale effects in the VIC model, for
instance

The Thur basin (1703 km

The Thur basin and the nine sub-basins for which discharge data were available.

Upper panels: the precipitation sum in the Thur catchment over the
full model period (1 August 2002–31 August 2003) shown for different
resolutions (f.l.t.r.: 1 km

For the station at the Thur outlet (Andelfingen) and eight sub-basins, hourly discharge measurements for the period 1974–2012 were made available by the Swiss Federal Office for the Environment (FOEN). All discharge measurements have been obtained using a stage–discharge relation, based on several measurements conducted by FOEN throughout the years, amongst others, with an ADCP. The discharge measurements for the Rietholzbach catchment were made available by ETH Zürich.

Forcing data for this study were made available by the Swiss Federal Office
for Meteorology and Climatology (MeteoSwiss). These data have previously been
used for numerous applications of hydrological models in the Thur

Land use, hydraulic conductivity, elevation, and soil water storage capacity
maps, all with a spatial resolution of 200 m

The VIC model (version 4.1.2.i) was run at an hourly time step in the energy
balance mode, which implies that both the water and energy balances are
solved. The default routing developed for VIC by

The VIC model

For the upper two soil layers, the Xinanjiang formulation

Baseflow is determined based on the moisture level of layer 3. Baseflow
generation follows the conceptualization of the Arno model

Since the grid size of the VIC model is often larger than the characteristic
scale of snow processes, sub-grid variability is accounted for by means of
elevation bands. For each grid cell the percentage of area within certain
altitude ranges is provided. The snow model is applied for each elevation
band and land-use type separately; the weighted average provides the output
per grid cell. This output consists of the snow water equivalent (SWE) and
the snow depth. The snow model is a two-layer accumulation–ablation model,
which solves both the energy and the mass balance. At the top layer of the
snow cover the energy exchange takes place. A zero energy flux boundary is
assumed at the snow–ground interface. A complete description of the model
can be found in

The mizuRoute routine

With the linearized St. Venant equation,

In the Thur basin, the routing is applied to subcatchments of the order of
1 km

Daily discharge characteristics for the Thur basin. Left panel: the
daily discharge in the Thur for the selected model period. The black lines
show three model runs with the same parameter set but with different initial
conditions (

We have constructed four VIC models with different spatial resolutions:
1 km

Four VIC implementations with different spatial resolutions (0.0109

The models are run at an hourly time step, implying that they solve both the
energy and the water balance. The hourly output of the routing model is
aggregated to daily and monthly time steps for further evaluation; see
Fig.

The four models are run for a period of 1 year and 4 months. The first 3
months are used as a spin-up period and are not used for further analysis.
Tests with the same parameter set and different initial conditions revealed
that 3 months are sufficient to eliminate the effect of initial conditions
(see Fig.

Sampled model parameters.

The analysed period is 1 August 2002–31 August 2003 (see
Fig.

The VIC model has a large number of parameters, divided over three sections:
soil parameters, vegetation parameters, and snow parameters. To determine
which parameters should be sampled in this study, a sensitivity analysis was
conducted on a broad selection of parameters (see Table S1 in the
Supplement). The parameter selection was made such that the main hydrological
processes were represented and included 28 VIC parameters from the three
different sections. Sensitivity analysis was conducted using the distributed
evaluation of local sensitivity analysis (DELSA) method

A base set of 100 parameter samples was created. For each parameter

The analysis showed that parameter sensitivity did not notably change over
the assessed scales: the same parameters were found to be most sensitive, but
in a slightly different order (see Fig. S1 in the Supplement). There are four
parameters which, for all scales and for all objective functions, proved to
be highly sensitive: the parameter describing variable
infiltration (

Parameter sampling as applied in this study.

In comparison with traditional sampling methods, the number of parameter
samples needed to cover the full parameter space can decrease significantly
by selecting only the most sensitive parameters (see
Fig.

For each model run, several objective functions were evaluated. The three
objective functions are

the Kling–Gupta efficiency (KGE) to describe the overall capability of
the model to simulate the discharge

where

the Nash–Sutcliffe efficiency (NSE) of the discharge to describe the model
performance for the higher discharge regions

in which

the Nash–Sutcliffe efficiency of the logarithm of the discharge NSE(

Model performance of the behavioural sets for different temporal
resolutions and different spatial resolutions. The left panel shows the
KGE(

After running the VIC model with 3150 parameter sets, a selection is made of
the best parameter sets, the so-called behavioural runs

We define parameter transferability

First, the impact of temporal and spatial resolution on model performance is
discussed for both uniform and distributed forcing, followed by a discussion
of the impact of the temporal and spatial resolution on parameter
distribution. For these analyses, the temporal and spatial resolution are
assumed to be independent. Subsequently, the parameter transferability across
temporal and spatial resolution is assessed by determining the overlap in
behavioural sets as defined by Eq. (

Figure

With uniform forcing, the lumped model outperforms the distributed models for
all three objective functions and time steps. The monthly time step shows for
all three objective functions an increasing model performance with decreasing
spatial resolution. It is remarkable that the model with the monthly time
step outperforms the models with daily and hourly time steps when the
NSE(

The model performance for the three separate components of the
Kling–Gupta efficiency of the behavioural sets for different temporal and
spatial resolutions. The left panel shows the correlation

Distribution of the sampled parameters for the behavioural sets,
fitted with a kernel density. The width of the line indicates the variation
in distribution between the different spatial resolutions. The left column is
based on KGE(

Distributed forcing leads in all cases except one (1 km

Figure

Figure

The difference in the parameter distribution when comparing distributed and
uniform forcing is limited. The clearest difference can be found for the

With an ANOVA analysis, the significance of temporal and spatial resolutions
in the parameter distribution of the behavioural sets was tested.
Figure

The main research question of this study is to what extent parameters are
transferable across temporal and spatial resolutions, and we will use that as
an indicator of the representation of spatial and temporal variability in the
model. We have defined parameter transferability

Transferability of parameters across spatial resolution, expressed as percentage agreement in detected behavioural runs for different spatial resolutions (in km) at different time steps.

Transferability of parameters across temporal resolution, expressed as percentage agreement in detected behavioural runs for different temporal resolutions at different spatial resolutions.

The effect of spatial and temporal resolutions on parameter
distribution. The

Tables

Parameter transferability as a function of the ratio in temporal and
spatial resolution. The ratio of temporal resolutions is defined as follows:
transfer from hourly to daily time steps is a ratio of 24, whereas transfer
from hourly to monthly is a ratio of 732 (732 h in 1 month of 30.5 days).
The ratio of spatial resolutions is defined as the square root of the number
of cells that would fit in the other cell: from 1 km

Transferability of parameters from the Thur to the nine subbasins,
expressed as percentage agreement (%) in detected behavioural runs. The
forcing was applied uniformly and the KGE(

The advantage of distributed hydrological models over lumped models is that
distributed models can incorporate spatially varying parameters, including
those reflecting land-use and soil characteristics

It seems counter-intuitive that model performance is significantly affected by both the temporal and spatial resolution, while the parameter distribution is mainly impacted by the temporal resolution. This can be explained, however. Model performance can still be significantly impacted by temporal and spatial resolution, even if the same parameters are selected for different spatial resolutions. This implies that the model performance is mainly limited by the model structure or set-up, and much less by the parameter values. This is confirmed by comparing the uniform and distributed forcing. Although the distribution of the behavioural parameters was not very different for the two forcing types, the model performance for distributed forcing was in almost all cases better than the model performance for the uniform forcing.

The conclusion that parameters cannot be transferred across temporal
resolution seems to contradict the results of

Distribution of bulk density over the grid cells for the four different spatial resolutions.

Impact of parameter transfer on model performance. The panels show
the distribution of the NSE(

Figure

Our results show that parameter transferability is more sensitive to temporal
than to spatial resolution. A key question is to what extent this result
stems from the model representation of spatial variability. Spatial
variability can be reflected in three domains of the model: the routing, the
forcing, and the soil and land-use parameters. In this study we excluded the
effect of routing by using a high-resolution drainage network based on
sub-basins with a size of

We investigated the effect of forcing by comparing the results for
distributed and uniformly applied forcing, and we tested the effect of
spatially distributed soil and land-use parameters by aggregating them for
lower resolutions (Fig.

The models in this study are configured in a similar way to many current-day
large-domain hydrological models, using common data like the Harmonized World
Soil Database and uniform application of the most sensitive parameters. As
such, this study is likely representative of many large-domain studies. The
limited sensitivity for spatial resolution is arguable because our
implementation of VIC substantially underestimates the spatial variability in
nature, and, importantly, that similar issues in representing spatial
variability are a common problem in large-domain hydrological modelling
(e.g. see the model configuration in

The results in our study are based on a limited number of model
configurations for a single basin, so the results presented here are only
intended to provide an example of the behaviour in the current generation of
land-surface models. Our results show a low sensitivity for the spatial
resolution, whether applied with distributed forcing or not. The observed
impact of spatial resolution can therefore almost completely be attributed to
the effect of spatially distributed soil and land-use parameters (including
the calibrated ones), which could be substantially underestimated. The impact
of temporal resolution on parameter transferability is large. We employed the
temporal resolutions for which most hydrological observations are available;
thus, our results are relevant for practical applications. Based on the work
of

A VIC model for the Thur basin was run with four different spatial
resolutions (1 km

Both the spatial resolution and the temporal resolution of the VIC model
had a significant impact on the model performance, either expressed in terms
of KGE(

The spatial resolution of the model had little impact on the parameter distribution of the behavioural sets. On the other hand, the temporal resolution significantly impacted the distribution of at least four out of seven parameters, both when applied with uniform and distributed forcing.

Parameters could to a large extent be transferred across the spatial
resolutions, while parameter transferability over the temporal resolutions
was less trivial. Parameter transferability between the hourly and daily time
steps was found to be feasible, but the monthly time step led to
substantially different parameter values. This is crucial information,
because many studies tend to calibrate the VIC model on the monthly time step

We also investigated whether parameters could be transferred across both the
spatial and the temporal resolutions simultaneously. Parameter
transferability decreases when the ratio between the original and the
intended spatial and/or temporal resolution increases. The ratio of temporal
resolutions has a larger negative effect on parameter transferability than
the ratio of spatial resolutions. It was also shown that parameter
transferability depends on the objective function. When the NSE(

The authors would like to thank Kevin Sampson for the preparation of GIS files for the routing, Oldrich Rakovec for providing and helping with DELSA, and Miroslav Vor̂echovský for the provided Hierarchical Latin Hypercube Sample. The Swiss Federal Office for the Environment (FOEN) and Martin Hirschi and Dominic Michel from ETH Zürich are thanked for kindly providing the discharge data. We would like to thank MeteoSwiss for providing the forcing data. Lieke Melsen would like to acknowledge Niko Wanders, Wouter Greuell, Pablo Mendoza, Rohini Kumar, Stephan Tober and Oldrich Rakovec for fruitful discussions that led to the basis of this paper. The data in this study are available from the first author upon request. Edited by: M. Weiler