In many conceptual rainfall–runoff models, the water balance differential equations are not explicitly formulated. These differential equations are solved sequentially by splitting the equations into terms that can be solved analytically with a technique called “operator splitting”. As a result, only the solutions of the split equations are used to present the different models. This article provides a methodology to make the governing water balance equations of a bucket-type rainfall–runoff model explicit and to solve them continuously. This is done by setting up a comprehensive state-space representation of the model. By representing it in this way, the operator splitting, which makes the structural analysis of the model more complex, could be removed. In this state-space representation, the lag functions (unit hydrographs), which are frequent in rainfall–runoff models and make the resolution of the representation difficult, are first replaced by a so-called “Nash cascade” and then solved with a robust numerical integration technique. To illustrate this methodology, the GR4J model is taken as an example. The substitution of the unit hydrographs with a Nash cascade, even if it modifies the model behaviour when solved using operator splitting, does not modify it when the state-space representation is solved using an implicit integration technique. Indeed, the flow time series simulated by the new representation of the model are very similar to those simulated by the classic model. The use of a robust numerical technique that approximates a continuous-time model also improves the lag parameter consistency across time steps and provides a more time-consistent model with time-independent parameters.

Hydrological modelling is a widely used tool to manage rivers at the
catchment scale. It is used to predict floods and droughts as well as
groundwater recharge and water quality. In a review on the different existing
hydrological models,

establish a conceptual representation of reality,

represent this conceptualization in a mathematical model,

set up a computational model to be used on a computer.

In terms of conceptual representation, many models exist and conceptualize
the hydrological processes in the catchment differently, resulting in models
with various levels of complexity. In this study, we will focus on the
bucket-type models, which are among the simplest. These models, such as
Variable Infiltration Capacity (VIC)

In the context of this study, bucket-type models are advantageous
because, even if the concepts are often well documented, this is not
the case of the mathematical and the computational models. In the
models' documentations, the water balance equations that would govern
the models are rarely explicitly formulated

However, several studies in the last decade

Another numerical approximation is commonly applied for bucket-type
models: the operator splitting (OS) technique

According to different studies, an inadequate numerical treatment like
OS can lead to inconsistencies in flux simulations

For these reasons, it is important to use a robust numerical treatment to better estimate the other uncertainties (for example, parameter uncertainty).

The first step to improve the numerical treatment of rainfall–runoff models
is to properly separate the mathematical model from the computational
model

Schemes of the reference GR4 model (

In addition to a clearer mathematical model, we hope that the
state-space representation will gain stability due to the direct
implementation of the time step in the numerical resolution. We thus
hope to obtain more stable parameter values across time steps

To illustrate the methodology proposed, the widely used GR4J
model

Hereafter, the notation GR4 will be used to refer to the structure of the GR4J
model

GR4

The version of GR4 used here is slightly different from the one presented
by

Details of the equations of the GR4 model and discrete and continuous
formulations. The discrete formulations are the continuous equations
integrated individually over the modelling time step using the operator
splitting technique while continuous equations correspond to the terms of the
water balance differential equation of each store. * The values of

The equations of the model are given by

The water balance operators evaluate effective rainfall (i.e. the part of
rainfall that will reach the catchment outlet) by estimating several
quantities: actual evaporation, storage within the catchment and groundwater
exchange. It involves an interception function and a production (soil
moisture accounting) store (

The routing function of the model is fed with the rainfall that does not feed
the production store (

The simulated flow at the catchment outlet (

Four free parameters (called

As mentioned in the introduction, the governing water balance
equations of the model are solved using OS. By
considering that inputs to the store are added at the beginning of the
time step as Dirac functions

Meaning of the free and fixed parameters

To create this state-space representation, it is important to identify
the different model state variables. In the GR4 model, two obvious
states are the levels of the production and routing stores. The main
challenge to describe the state-space formulation is to deal with the
unit hydrograph. The discrete form used in GR4 corresponds to
a convolution product in the state space as implemented in SUPERFLEX

Different combinations of linear stores were tested and the choice was
made to replace the unit hydrograph with a Nash cascade

As introduced in the previous section, the Nash cascade has two parameters, namely the number of stores and the outflow coefficient. The number of stores can only take integer values, which is an issue for automatic calibration because it introduces threshold effects. As a consequence, the number of stores was not optimized automatically and the outflow coefficient is the preferential parameter to calibrate.

To obtain a response that is equivalent to the GR4 unit hydrograph
response, we attempted to determine whether a relationship exists
between the Nash cascade parameters and the GR4

The impulse response of the Nash cascade is

The impulse response of the GR4 symmetrical unit hydrograph
is

The Nash cascade parameters are calculated depending on

Impulse response with

Using this formula, the

Fixing the number of stores in the Nash cascade also provides another
advantage. Indeed, one of the potential issues that arises when
replacing the unit hydrograph with a Nash cascade was the equifinality
with the routing store. Given that the recession curve of the cascade
is theoretically infinite, it could have the same function as the
routing store. Calculating the parameters of the cascade regarding the

Once the model is only represented by stores, a differential equation can be
written for each store (details are provided in Table

The resulting model is composed of the differential equations governing the
states' evolution (here represented as a vector in the Eq. (

Temporal transformations of the GR4 parameters

The output equation to calculate the instantaneous output flow (

The input, state variable and output values are as follows.

The GR4 model was first designed for daily time step modelling and it
was adapted for the hourly time step

The adaptation to the time step is handled by a change in the parameter
values, which depend on time.

The continuous state-space GR4 model used for the hourly time step is exactly
the same as the one used at the daily time step, with no change in the
percolation coefficient. The time step change is not managed by a change in
parameter values but by the numerical integration. For the daily time step,
the model is integrated on

The integration of Eq. (

Following the recommendation in

The choice of using an adaptive sub-step rather than single-step implicit
method

For both hourly and daily time steps, the inputs are considered as constant during the time step. Even if this assumption is a simplification of the truth, we chose to keep it constant to simplify the calculation and not to introduce treatment differences between hourly and daily time step models.

Location of the 240 flow gauging stations used for the tests and
their associated catchments. The River Azergues at Châtillon is used as
an example for the results (Sect.

To compare the performance and behaviour of the reference and the
discrete and continuous state-space GR4 model versions, a large data
set of 240 catchments across France was set up
(Fig.

The data set was built by

Hourly observed flows are available at each catchment outlet and come from
the Banque HYDRO (

The catchments were selected to have less than 10

Three versions of the model were assessed on the 240 catchments
following a split-sample test

The objective function used for calibration is the Kling–Gupta
efficiency

Performance comparisons obtained in validation among the reference
(with unit hydrograph), the discrete state-space (with Nash cascade) and the
continuous state-space daily GR4 on 240 catchments, focusing on high

Simulated hydrograph of the River Azergues in the first half of 2012
during the validation period. The reference GR4 model (output in blue), the
GR4 discrete state-space solution (output in green) and the continuous
state-space solution (output in red) were calibrated with

The results of the calibrations were also analysed in terms of
performance in validation on the three evaluation criteria
(i.e.

A second test was carried out in order to analyse the time step
dependency of the models. The split-sample test was performed at the
hourly time step and the parameter values were compared to those
obtained at the daily time step. With the reference model, the
calibrated parameter values were compared to those theoretically
obtained using the equations in Table

Figure

Scatter plots of the four free parameters of the different versions
of the models obtained by calibration with

Daily inputs in the routing store of the River Azergues in the first
half of 2012. The models are calibrated with the

Daily routing store filling of the River Azergues in the first half
of 2012. The reference GR4 (blue line) and the continuous state-space
representation (red line) are calibrated with the

The study of the hydrographs provides complementary information. The
reference GR4 model and the continuous state-space solution are very
similar while the discrete state-space solution simulates lower peak
flows (see example hydrograph in Fig.

To extend the analysis on the similarity of the models, we compared
the parameter values obtained by calibration. As shown in
Fig.

Last, to understand the internal impact of the state-space formulation
on the model, we analysed state variables and internal fluxes. Two
differences are induced by the model's state-space formulation. First,
the discrete Nash cascade output peaks are lower than the peaks of the
unit hydrograph (Fig.

Moreover, by analysing the differences between the two models, it is also important to take into account the computational time. Indeed, running the original model version is on average 3 times faster than the continuous state-space version due to the adaptive sub-step method. This is important to consider for some applications.

This computational time rise is essentially due to the adaptive
sub-step algorithm. For example, in the River Azergues at
Châtillon catchment, the mean number of sub-steps is

To conclude with these results, we can argue that the modifications brought by the continuous state-space representation, although they modify the model's internal fluxes, do not degrade the model's performance, but only slightly modify the model's internal fluxes. It is important to underline that the OS solving of a Nash cascade creates more errors than a discrete unit hydrograph. To be equivalent to the reference model, the state-space representation of GR4 needs to be solved with a robust numerical technique.

The analysis of temporal consistency provides the most valuable result
produced by the continuous state-space representation. The work of

Scatter plots representing the four parameters of the reference
(daily and hourly) GR4 models obtained by calibration with

In Fig.

The values of

In the continuous state-space model, the time step is taken into account in
the temporal numerical integration of the model. For this reason, in theory
there is no need to adapt the values of the parameters. This is confirmed in
Fig.

Scatter plots representing the four parameters of the continuous
state-space (daily and hourly) GR4 models obtained by calibration with

Performance comparisons obtained in validation among the reference
(with unit hydrograph), the discrete state-space (with Nash cascade) and the
continuous state-space hourly GR4 on 240 catchments, focusing on high

This result is useful in building a model that can adapt its time step
resolution depending on the given conditions. The results are particularly
interesting for the case of

The outliers in

Finally, to verify stability, we also need to compare the performance
of the models at the hourly time step. Figure

Thus, the continuous state-space representation shows better temporal
stability in the

The objective of this study was to present a version of a bucket-type
rainfall–runoff model with a robust numerical resolution of the governing
water balance equations by setting up a continuous state-space
representation. The methodology is based on (i) identifying the state
variables, (ii) writing their differential equations, (iii) replacing certain
components of the model with more easily described components in terms of
differential equations (namely replacing the unit hydrograph with a Nash
cascade here), and (iv) solving these equations with a robust numerical integration
technique. Finally, all the fluxes that form the water balance equation
governing a state are solved simultaneously while they are solved
sequentially in OS models. As stated by

This work was presented using the example of the GR4 model. The new version was created to be as close as possible to the initial model but a single modification was implemented: a Nash cascade substitutes the model's unit hydrograph.

When analysing the results and the output flows, it was shown that the new formulation, when solved with a robust numerical technique, has a limited impact on performance. However, the analysis of the parameter values and of the internal fluxes of the model shows that some discrepancies occur when running the model. The peak flow of the Nash cascade occurs sooner than the peak flow of the unit hydrograph. The amount of water in the routing store and exchanged by the groundwater exchange function is also higher for the state-space representation, particularly during high-flow periods.

Nonetheless, the continuous state-space representation simulates flows that are very similar to the flows simulated by the original GR4 version and performs equally well. It also seems to provide greater stability in the parameter values, particularly regarding different modelling time steps. Moreover, the use of the Nash cascade rather than the unit hydrograph improves (when solved with implicit Euler) the lag parameter value stability with time steps. This improved stability can make it easier to calibrate the model with a given data set and to apply it at a finer time step for which no discharge data are available. It can also allow the use of a model that runs at a finer time step in high-flow periods and a larger time step in low-flow periods.

Furthermore, the comparison between the discrete and continuous state-space model shows that the benefits provided by the continuous state-space representation are a result of the use of a robust numerical integration technique. Indeed, solving the state-space representation using OS introduces errors that impact the simulated flow values and do not result in parameter stability. Thus, the real benefit of the use of the Nash cascade is to simplify the numerical solving application.

The performance obtained with the continuous state-space model is not better than that of the original model. In addition, because the number of sub-steps sometimes needs to be high, the computational time is longer with the continuous state-space representation of the model. Consequently, the use of this representation would be helpful for particular applications such as time-variable modelling. It might also be useful for certain data assimilation techniques (typically variational methods) because all the components are represented as states and the governing equations are clearly defined.

In addition, it could also be advantageous to find a way to adapt the number of stores of the Nash cascade to the catchment studied.

Although it is necessary to adapt the Nash cascade to different unit hydrograph shapes, this article suggests a sufficiently general methodology to erase OS in hydrological bucket-type modelling and can be transposed to other models.

The Fortran code used in this article can be freely
downloaded from GitHub at

This work is part of LS' PhD work; he made the technical development and the analysis and wrote the paper. GT and CP are the PhD supervisors; they supervised this work and the paper writing.

The authors declare that they have no conflicts of interest.

The first author's PhD grant was provided by Irstea. We thank Météo France for providing the SAFRAN climatic data used in this work. We also would like to thank Martyn Clark for his advice in setting-up the differential equations, Nicolas Le Moine for sharing his ideas to replace the unit hydrographs and on the numerical integration, Fabrizio Fenicia for his advice on numerical integration and Paul-Henry Cournède for his analysis of the mathematical adequacy of the model. Finally, we give special thanks to Andrea Ficchí for his work on the database and for the discussions on the temporal stability of the GR4 model.

We thank the topical editor, Jeffrey Neal, for his monitoring of the review process and his relevant reviewers choice. We also acknowledge the two reviewers, Barry Croke, and the anonymous reviewer for their very interesting and complementary remarks. Edited by: Jeffrey Neal Reviewed by: Barry Croke and one anonymous referee