Ideally, semi-distributed hydrologic models should provide better streamflow simulations than lumped models, along with spatially-relevant water resources management solutions. However, the spatial distribution of model parameters raises issues related to the calibration strategy and to the identifiability of the parameters. To analyse these issues, we propose to base the evaluation of a semi-distributed model not only on its performance at streamflow gauging stations, but also on the spatial and temporal pattern of the optimised value of its parameters. We implemented calibration over 21 rolling periods and 64 catchments, and we analysed how well each parameter is identified in time and space. Performance and parameter identifiability are analysed comparatively to the calibration of the lumped version of the same model. We show that the semi-distributed model faces more difficulties to identify stable optimal parameter sets. The main difficulty lies in the identification of the parameters responsible for the closure of the water balance (i.e. for the particular model investigated, the intercatchment groundwater flow parameter).

Developing modelling tools that help to understand the spatial distribution
of water resources is a key issue for better management. The dynamics of
streamflow depends on (i) the spatial variability of precipitation (which,
a priori, should be better handled by a semi-distributed hydrological model),
(ii) the heterogeneity of catchment behavior (which can be dealt explicitly
with by spatially-variable model parameters), and, increasingly,
(iii) localized human regulations (for instance, water reservoirs). Since
calibration is generally based on discharge measurements at the outlet of the
catchment only, and gauging stations are not available everywhere,
semi-distributed hydrological models are often difficult to parameterize. As
argued by

This raises the need to better understand how well parameters are identified
in a semi-distributed model compared to a lumped model. The variability of
catchment model parameters calibrated over different periods (“time
variability”) is one way of approaching this question. Indeed, as reminded
by

However, literature provides many examples showing that this assumption is
hardly satisfied.

Secondly, conditions of the catchment itself may change over time, which
consequently, and understandably, shifts optimal model parameters, and
justifies rigourous evaluations of the model robustness

This paper investigates the procedure of parameter identifiability in a semi-distributed model by comparing model calibration schemes and results with a lumped model on which it is based. From this comparison, we address two main questions: (1) Does spatial distribution of parameters interfere with parameters identifiability? Indeed, one could hope that applying parameters to a more geographically-limited area tends to facilitate their identification. (2) What are the parameters that are the most variable in the lumped and in the semi-distributed models? In this way, we aim to diagnose which components of the model are the least robust, in the sense that their parameterisation is difficult to transpose in time and space.

The model is implemented in Eastern France, close to the border with Germany,
over 64 sub-catchments of French tributaries of the River Rhine
(Fig.

The two hydrological models implemented (lumped and semi-distributed) require
daily time series of

Location of the study area (left), and average annual precipitation (centre) and annual potential evapotranspiration (right) for each sub-catchment (climate data are estimated from the 1971–2000 SAFRAN database).

Schematic representation of the GR5J and GRSD semi-distributed model
(from

The GRSD semi-distributed rainfall–runoff model was developed by

List of the parameters for the semi-distributed conceptual rainfall–runoff model GRSD.

The semi-distributed model is applied on sub-catchments. The delineation of
sub-catchments is performed only at gauging stations, which means that
discharge measurements are available for every hydrological units of the
model. The lumped GR5J model is applied on hydrological units composed of
upstream catchments (headwater catchments) or intermediate sub-catchments
(drained area between downstream and upstream stations). In that way, each
hydrological unit receives its own meteorological inputs (

The outflow of each GR5J model is finally routed to its downstream catchment
using a linear lag propagation model

In order to quantify the agreement between simulations (

Following

Sequential calibration is a common strategy for semi-distributed models

Similarly to

Following the work of

Illustration of the rolling calibration period methodology: 21
parameter sets

The comparison of the goodness-of-fit between the lumped GR5J model and the
semi-distributed GRSD model shows slightly better results of the lumped model
during calibration and identical results during validation (Fig.

As expected, performance on upstream catchments are similar between the
lumped and the semi-distributed catchments. Indeed, for those catchments,
models are strictly identical (Sect.

We did not detect any significant performance trends in time. Calibration
performances are rather stable, whereas validation performances are subject
to more fluctuations. These are similar between the lumped and the
semi-distributed models. Results illustrate that both models are potentially
able to produce stable efficiency (KGE

Quantile values of the goodness-of-fit (KGE

Quantile values of optimised parameter values according to the calibration periods for the 64 catchments. Upstream catchments are all headwater hydrological units, and downstream catchments are all the others.

Here, we compare the temporal trends and the variability of parameters, and
differences between the two models. The distribution of parameter values
according to the 21 calibration periods is given in Fig.

Variability of parameters values among calibration periods within
one catchment (64 catchments summarized by boxplots,

As expected from the structure of the models (Sect.

The smaller production store in GRSD appears to be compensated by X2 and X5.
Indeed, those parameters aim to quantify intercatchment groundwater flows
(IGF), which is the amount of water that daily gets out/in of the catchment
to fill/empty the routing store and the direct flow component. The X2
parameter quantifies IGF according to a linear relation with the routing
store rate (

By looking at parameter values according to the calibration periods, a relative stability of the median value appears among catchments. Only the parameters X2 and X5 in GRSD tend to slowly decrease. These trends are not observed in the lumped model, whose parameters appear more stable for downstream catchments.

From the relative stability in Fig.

First, results clearly show a higher temporal variability of the parameters
of the semi-distributed model, comparatively to the variability observed with
the lumped model (Fig.

Second, it is shown that the most important temporal instability of parameter
values is related to the parameter X2, followed by X3 and X5 parameters. All
three parameters are used to quantify IGF (Eq.

The most stable parameter appears to be X4 (time base of unit hydrograph),
which is consistent with previous works

We also analysed the spatial variability of the parameter values, considering variability between parameters and between models. To this end, we estimated the coefficient of variation of the parameter values among catchments (one performed by calibration period). The aim is to quantify how much parameters can be different between catchments.

Similarly to the temporal variability of parameters, spatial variability appears to be higher with the semi-distributed model than with the lumped model. However, contrary to the temporal variability, spatial variability is more expected here, as one of the objectives of a semi-distributed model is precisely to consider those spatial heterogeneities of the hydrological response. However, we noticed again that parameters X1 and X2 are the most variable parameters for the semi-distributed model. Therefore they appear among the most variable parameters for both analyses, the time variability and the space variability analyses. These two parameters control water balance. Similarly to the temporal variability, they are expected to be highly variable in space in order to get along with sequential observations at each downstream station during calibration.

Spatial variabilities are not constant over time (as observed by the boxplot
widths on Fig.

In this paper, we compared the spatio-temporal variability of the parameters of a semi-distributed model (GRSD) and a lumped model (GR5J) on which it is based. We applied a rolling calibration strategy over 21 periods and 64 French catchments.

A classical evaluation of discharge simulations using goodness-of-fit criteria was applied to the outputs of both models. It illustrates a slightly better performance of the lumped model during calibration, and similar performance of the models during validation. However, further investigation on parameter identifiability highlighted much higher temporal variabilities of the semi-distributed model. This study also showed that it is more difficult to identify catchment's specific parameter sets with the semi-distributed model than with the lumped model.

The methodology applied also enabled to identify the more unstable parameters. Results showed that the parameters related to the quantification of intercatchment groundwater flows (IGF) are the most unstable. We conclude that further modelling efforts should focus on the model structure in order to better quantify IGF.

This work also emphasizes the fact that the calibration strategy and the evaluation approach of a semi-distributed model should not focus only on goodness-of-fit performance, but also on parameter identifiability, especially if the model aims to be used to explore future scenarios in a changing world. Such an approach would also facilitate the application of the model at ungauged locations, since parameters that depict high variability in time and space might be more difficult to regionalize.

The streamflow data used in this study are freely available at:

The first author was partially funded by the French Waterboard “Agence de l'Eau Rhin-Meuse” within the MOSARH21 project (project #15C92002). Météo-France and SCHAPI are thanked for making climatic and hydrological data available for this study. The authors thank the anonymous reviewer for his comments on the manuscript.