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.
IntroductionOn the need for an adequate mathematical and computational hydrological model
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, determined that all the existing
models follow three modelling steps:
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) , Hydrologiska
Bryåns Vattenbalansavdelning (HBV) and Sacramento
, describe various conceptualizations of the hydrological
processes at the catchment scale. Their parsimony (they usually need few
input data and use few parameters) make them very useful for research as well
as in operational applications thanks to their robustness and good
performance .
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 . The
authors of the models often specify the discrete time equations,
i.e. the result of the analytical or numerical temporal integration of
the governing water balance equations. The problem is that the
temporal resolution of the differential governing equations is part of
the computational model. As a consequence, when the discrete time
equations are the only ones available, the real mathematical model
does not appear clearly. In addition, the descriptions of the
numerical method used to solve the water balance equations and to
obtain these discrete equations are rarely detailed.
However, several studies in the last decade see for
example point out that the
numerical solutions implemented to solve the differential equations
that govern the models are sometimes poorly adapted.
showed that the use of the explicit Euler scheme (which is frequent
for this type of model) can introduce significant errors in the
simulated variables compared to more stable numerical
schemes. Moreover, other studies prove that poorly adapted numerical
treatment causes discontinuities and local optima in the parameter
hyperspace . This results
in problems efficiently calibrating the models and in uncertainty on
parameter values.
Another numerical approximation is commonly applied for bucket-type
models: the operator splitting (OS) technique
. The aim is to split a differential equation into
more simple equations that can be solved analytically in order to
reduce inaccuracies in the numerical treatment. In the case of
hydrological modelling, operator splitting results from the sequential
calculation of processes such as runoff, evaporation and percolation
. , and
identified several widely used models in which the
differential equations are solved using this type of treatment,
e.g. VIC , Sacramento and
GR4J . However, even if OS may reduce numerical
errors, cite several limitations to its use in
hydrology. Indeed, it is physically unsatisfying to separate the
different processes in time because, in reality, they are
concomitant. In addition, it creates numerical splitting errors that
are difficult to identify.
According to different studies, an inadequate numerical treatment like
OS can lead to inconsistencies in flux simulations see for
example the study conducted byon an exponential
store. It may also create inconsistencies in the model
state variables . This results in the
model inaccurately simulating flows.
For these reasons, it is important to use a robust numerical treatment
to better estimate the other uncertainties (for example, parameter
uncertainty).
Scope of this study
The first step to improve the numerical treatment of rainfall–runoff models
is to properly separate the mathematical model from the computational
model . This article proposes a method to do
this by setting up a continuous state-space representation of
a rainfall–runoff model. A state-space representation is a matricial function
of a system that depends on input, output and state variables. At all times,
the system is described by the values of its state variables (referred to as
“states” in this article). In the case of rainfall–runoff models, inputs
can be potential evapotranspiration and precipitation and output can be the
flow at the outlet of the catchment. The soil water content or the amount of
water in the hydrographic network are physical examples of possible state
variables. The level of the bucket-type model stores is a conceptual example
of possible state variables. This state-space representation will give the
governing equations to be solved over time. This resolution will be proceeded
by using an OS technique to be used as a comparison point and
by using a more robust numerical technique, i.e. implicit Euler with an
adaptive sub-step number. The model solved by implicit Euler will be called
continuous state-space because it approximates a continuous model. By
opposition, the OS state-space representation will be named as
discrete.
Schemes of the reference GR4 model (a,
) and the state-space (b) structures. P:
rainfall; E: potential evapotranspiration; Q: streamflow; xi: model
parameter; other letters are model state variables or fluxes. A Nash cascade
replaces the unit hydrograph in the state-space representation.
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 will be taken as an example. Indeed, this model is
currently implemented using the OS technique. A state-space representation
will be set up, following the GR4J's conceptualization of the hydrological
processes as much as possible. Its
behaviour, both with a discrete and a continuous solving, will be compared to
the current formulation of the GR4J model on a wide range of French
catchments with different time steps (day and hour), in terms of performance
and parameters.
GR4 and its new state-space representation
Hereafter, the notation GR4 will be used to refer to the structure of the GR4J
model J stands for journalier, i.e. daily;,
which is transformed and used at different time steps. This is a lumped
bucket-type model described in its current form (Sect. ) and in
its state-space form (Sect. ). A discussion on the Nash cascade
introduced in the GR4 state-space form is given in Sect. . The
continuous differential equations of the state-space form are described in
Sect. . The adaptations needed to change the model time step will
be described in Sect. .
Reference GR4 model
GR4 is a lumped bucket-type daily rainfall–runoff model
with four free parameters. It is widely used for various hydrological
applications in France and in other
countries . It has shown good performances
on a wide range of catchments . The equations of the
reference GR4J model are the result of the integration of
the water balance equations at a discrete time step (here the daily or hourly
time step).
The version of GR4 used here is slightly different from the one presented
by because the two unit hydrographs were replaced by
a single one placed before the flow separation Fig. a,
. This simplification of the model does not
substantially change the resulting simulated flows.
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
UH2 are calculated using Eq. (17) in . Please
note that the two discrete formulations use either the unit hydrograph
equations or the Nash cascade formulation.
Model component nameNotationFlux nameDiscrete formulationsContinuous formulationProduction storeSPrecipitation in the storePs=x11-Sx1αtanhPnx11+Sx1tanhPnx1Ps=Pn1-Sx1αEvaporation from the storeEs=2S-Sαx1tanhEnx11+1-Sx1tanhEnx1Es=En2Sx1-Sx1αPercolationPerc=S1-1+νSx1β-111-βPerc=x11-β(β-1)Utνβ-1SβUnit hydrographUH2UH inflowPr=Pn-Ps+Perc–UH outflowQuh=Pr*UH2(*) (convolution product)Nash cascadeSh,1Precipitation inflow in store 1Pr=Pn-Ps+PercPr=Pn-Ps+PercStore 1 outflowQSh,1=Sh,11-exp1-nresx4QSh,1=nres-1x4Sh,1Sh,2Store 2 inflowQSh,1QSh,1Store 2 outflowQSh,2=Sh,21-exp1-nresx4QSh,2=nres-1x4Sh,2⋮⋮⋮⋮Sh,nStore nres inflowQSh,nres-1Qsh,nres-1=nres-1x4Sh,nres-1Store nres outflowQuh=Sh,nres1-exp1-nresx4Quh=nres-1x4Sh,nresRouting storeRRouting store inflowQ9=ΦQuhQ9=ΦQuhInter-catchment exchangesF=x2x3ωRωF=x2x3ωRωRouting store outflowQr=R1-1+Rx3γ-111-γQr=x31-γ(γ-1)UtRγOutput flowQ=Qr+QdRouting store outflowQrQrDirect flowQd=max(0;(1-Φ)Quh-F)Qd=max(0;(1-Φ)Quh-F)
The equations of the model are given by and listed
in Table . GR4 represents the rainfall–runoff relationship
at the catchment scale using an interception function, two stores,
a unit hydrograph and an exchange function (see
Fig. a). The model structure can be split into water
balance and routing operators.
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 (S in Fig. a). The interception
corresponds to a neutralization of rainfall by potential evapotranspiration.
The remaining rainfall (Pn), if any, is split into a part going
into the production store (Ps in Fig. a) and
a complementary part (Pn-Ps in Fig. a) that
is directed to the routing component of the model. The quantity of rainfall
that feeds the production store depends on the level of water in the store at
the beginning of the time step. In case there is remaining energy for
evapotranspiration after interception (En in Fig. a),
some water is evaporated from the production store at an actual rate
depending on the level of the production store (Es in
Fig. a). The higher the level is at the beginning of the time
step, the closer Es is to En. Thus, the production
store represents the evolution of the catchment moisture content at each time
step. The last water balance operator is a groundwater exchange term (F in
Fig. a, positive or negative), which acts on the routing part of
the model.
The routing function of the model is fed with the rainfall that does not feed
the production store (Ps-Pn) plus a percolation term
(Perc in Fig. a) from the production store, which
generally represents a small amount of water. The total amount
(Pr in Fig. a) is lagged by a symmetric unit
hydrograph and then split into two flow components. The main component
(90 % of Pr, Q9 in Fig. a) is routed by
a nonlinear routing store (R in Fig. a). The complementary
component (10 % of Pr, Q1 in Fig. a)
directly reaches the outlet. The groundwater exchange term (F) is added or
removed from the routing store and from the Q1 component.
The simulated flow at the catchment outlet (Q in Fig. a) is the
sum of the outputs of the two flow components (Qr and
Qd in Fig. a).
Four free parameters (called x1, x2, x3 and x4) are used
to adapt the model to the variety of catchments. Their meanings are
given in Table .
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 , it becomes possible
to find analytical expressions of the model processes when equations
are integrated over the time step. Consequently, the model processes
are treated sequentially.
Meaning of the free and fixed parameters fromexcept for
Ut and nres.
TypeNameSignificationValueUnitFreex1Max capacity of the production store–mmx2Inter-catchment exchange coefficient–mm t-1x3Max capacity of the routing store–mmx4Base time of the unit hydrograph–tFixedαProduction precipitation exponent2–βPercolation exponent5–γRouting outflow exponent5–ωExchange exponent3.5–ϵUnit hydrograph coefficient1.5–ΦPartition between routing store and direct flow0.9–νPercolation coefficient49–UtOne time step length1tnresNumber of stores in Nash cascade11–A state-space formulation for the GR4 model
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
. This convolution product makes the
mathematical resolution of the model that is necessary for the
continuous version that will be introduced in Sect. more complex. Here
we chose to replace this unit hydrograph with a series of linear
stores in order to simplify this resolution. The use of stores is also
convenient because it creates a model that is only composed of stores.
Different combinations of linear stores were tested and the choice was
made to replace the unit hydrograph with a Nash cascade
. It is implemented at the same location in the model
structure as the unit hydrograph (Fig. b). The Nash
cascade is a chain of linear stores that empty into each other. It
has two parameters to govern the shape of the outflow response, namely
the number of stores and the outflow coefficient, which is identical
for all stores. In our case, we decided to fix the number of stores
and to only consider the outflow coefficient as a free parameter. This
choice will be discussed in the following section
(Sect. ). With this type of model, the outflow of the last
store has a similar shape to a unit hydrograph.
Parameterization of the 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 x4 parameter. To
manage this, the determination of the Nash cascade parameter is based
on the comparison of the impulse response of the Nash cascade and the
response of the unit hydrograph.
The impulse response of the Nash cascade is hNash(t)=kΓ(nres)ktnres-1exp(-kt),
where hNash(t) is the impulse response of the Nash cascade at
time t, nres is the number of stores, k is the outflow
coefficient (t-1) and Γ(nres) corresponds to the gamma
function of nres.
The impulse response of the GR4 symmetrical unit hydrograph
is hUH(t)=2.52x4tx41.5, for 0≤t≤x42.52x42-tx41.5, for x4<t≤2x40, for t>2x4,
where hUH(t) is the impulse response of the unit
hydrograph at time t and x4 is the time to peak of the hydrograph.
The Nash cascade parameters are calculated depending on x4 in such
a way that the time to peak and the peak flow would be the same for
the two impulse responses. According to , the time
to peak of the Nash cascade is equal to
tp=nres-1k
and the peak flow is equal to
qp=kΓ(nres)(nres-1)nres-1exp(1-nres).
Using Eq. (), the time to peak of the GR4 unit hydrograph is
equal to
tp=x4
and the peak flow to
qp=1.25x4.
So, from these values the following system can be deduced:
x4=nres-1k1.25x4=kΓ(nres)(nres-1)nres-1exp(1-nres),
which can be transformed into
k=nres-1x41.25=(nres-1)nresΓ(nres)exp(1-nres).
A nres=11 is the best integer approximation
to solve the second equation of Eq. (). The outflow
coefficient is deduced from this number of stores and from x4. By
fixing the parameters in this way, only the x4 parameter has to be
calibrated. This method allows a direct comparison between the
parameters of the Nash cascade and the parameter of the unit
hydrograph. For a given x4 parameter, the unit hydrograph and the
Nash cascade impulse responses have the same time to peak and the same
peak flow (see the dotted and the dashed curve in Fig. ).
Impulse response with x4=2 time steps for the unit hydrograph
of GR4 (dotted line) and the Nash cascade with nres=11 stores and
k=11-1x4 (dashed line).
Using this formula, the x4 parameters of the two models are
equivalent and it can be argued that their meaning is nearly
identical.
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
x4 parameter makes it possible to reduce the possibility of an
infinite impulse response.
Continuous differential equations of the state-space model
Once the model is only represented by stores, a differential equation can be
written for each store (details are provided in Table ). For the
production and routing stores, the equations were built by adding all the
processes that affect the stores. For example, the differential equation for
the production store is the sum of the differential equations of evaporation,
rainfall and the percolation (respectively, Es, Ps
and Perc in Fig. ). This means that all the processes
that are a function of this state are treated simultaneously, unlike the
initial model version in which the processes are treated sequentially. The
state-space representation of the Nash cascade is the same as the one
proposed by .
The resulting model is composed of the differential equations governing the
states' evolution (here represented as a vector in the Eq. (),
taking into account nres stores in the Nash cascade):
S˙S˙h,1S˙h,2⋮S˙h,nresR˙=Ps-Es-PercPr-QSh,1QSh,1-QSh,2⋮QSh,nres-1-QuhQ9+F-Qr.
The notation S˙ stands for
dSdt, the derivative of S against
time t and the different elements of this equation are specified in
Table .
Temporal transformations of the GR4 parameters .
GR4 modelTheoretical transformation from theSource of time step dependencyparameterdaily (Δtd) to the hourly (Δth) time stepννΔth=νΔtdΔtdΔth14Integration of the percolation power 5 function from the production storex1x1(Δth)=x1(Δtd)–x2x2(Δth)=x2(Δtd)ΔtdΔth-18Integration of the exchange flux formulation (dependent on the routing store level)x3x3(Δth)=x3(Δtd)ΔtdΔth14Integration of the fuelling power 5 function of the routing storex4x4(Δth)=x4(Δtd)ΔtdΔthDiscrete concentration time in time step units of the unit hydrographs
The output equation to calculate the instantaneous output flow (q(t)
in Eq. ) completes the model:
q(t)=Qr+Qd.
The different elements in Eqs. () and () are shown
in Table .
The input, state variable and output values are as follows.
Inputs. En and Pn are the potential evapotranspiration
(after the interception) and the precipitation amounts after the
interception phase (mm t-1). We decided to keep the
interception out of the state-space representation because it is not
represented by a store in the reference GR4J and we wanted to avoid
introducing an additional difference between the state-space and the
reference models.
Output. Q is the output flow; it corresponds to the integration of q(t) (Eq. ) over the time step.
State variables. S, R and Sh,k are respectively the levels of the production store, the routing store and
the Nash cascade store number k (with k∈{1,⋯,nres}) in millimetres.
Fluxes. Ps and Es are, respectively, the rainfall added to
the production store and the evapotranspiration extracted from the
production store. Perc is the outflow from the production
store. Pr is the amount of water that reaches the model routing
operators. QSh,k is the outflow of the Nash cascade store number
k (with k∈{1,⋯,nres-1}). Quh is
the outflow of the Nash cascade store number nres (this
notation is used to be consistent with the discrete model). Q9 and
Qr are, respectively, the inflow and the outflow of the routing
store and F is the inter-catchment groundwater exchange. Qd is
the outflow of the complementary flow component.
The parameter meanings are explained in Table . The model
is constructed to ensure that the parameters (x1,⋯,x4 in
the equations) have the same meaning in the continuous model and in
the discrete GR4. The state-space formulation was sought to be as
close as possible to the original model's formulation, to keep the
same general model structure. We expect similar results to be obtained
by the different tested model versions.
Hourly model
The GR4 model was first designed for daily time step modelling and it
was adapted for the hourly time step
GR4H,. The structure and the
equations are similar in GR4H (hourly) and in GR4J (daily). The hourly
versions of the GR4 models used here are the same as the ones showed
in Fig. .
The adaptation to the time step is handled by a change in the parameter
values, which depend on time. gave the theoretical
relationships to transform the GR4 free parameter values as a function of the
time step length (Table ). The fixed percolation coefficient
(ν in Table ) is also time dependent.
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 Δt=1day while; for the hourly time
step, it is integrated on Δt=1h.
Implementation and testing methodologyNumerical integration of the model
The integration of Eq. () (necessary to adapt the model to
discrete input data) cannot be made analytically. It is therefore
necessary to implement a numerical method to solve this integration.
Following the recommendation in , an implicit Euler
algorithm is used to perform this numerical integration. Our choice
was to set up an adaptive sub-step algorithm to
avoid the majority of numerical errors. The implicit equation is
solved using a secant method when necessary.
The choice of using an adaptive sub-step rather than single-step implicit
method as recommended by is a result of several
tests that are not shown here. We compared the modelling results with a
single-step integration to those obtained with the adaptive sub-step
algorithms and found some differences in resulting flows (in
particular for high flows). The differences found this way were not
negligible. In this case, we can say that the stability of the
implicit single-step integration is not sufficient to sufficiently
reduce the integration errors.
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. ).
Catchment set and data
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. ). Testing the models on many catchments will help
obtain general conclusions .
The data set was built by to test GR4 at different
time steps. In this article, we only used daily and hourly data. The
climate data of the SAFRAN daily reanalysis
are used as input data (precipitation
and temperature). Precipitation and temperature are spatially
aggregated on each catchment since the GR4 models are lumped. The
hourly precipitation data were obtained by disaggregating the daily
SAFRAN precipitation using the subdaily distribution of rain gauge
measurements. Potential evapotranspiration at the daily time step was
calculated from the SAFRAN temperature using the Oudin formula
and hourly spread with a Gaussian distribution. Full
details on this data set are available in .
Hourly observed flows are available at each catchment outlet and come from
the Banque HYDRO (http://www.hydro.eaufrance.fr/, last access: 18
April 2018, French Ministry of the Environment). For daily modelling, hourly
measurements are aggregated at the daily time step. Their availability covers
the 2003–2013 period.
The catchments were selected to have less than 10 %
precipitation falling as snow, to avoid requiring a snow model.
Testing methodology
Three versions of the model were assessed on the 240 catchments
following a split-sample test . These three versions
are the reference model, a discrete state-space model (with a Nash
Cascade but solved using OS) and a continuous
state-space model. Comparing the reference and discrete state-space
models allows us to measure the impact of replacing the unit hydrograph
with a Nash cascade. Comparing the discrete and continuous state-space
models allows us to measure the impact of a nearly continuous numerical
integration. For every catchment, the observed flow data period was
divided into a calibration period (the first half) and a validation
period (the second half). A 2-year warm-up period was used for each
catchment, before both the calibration and validation periods. The
calibration was made automatically with an algorithm used in
and based on the work of .
The objective function used for calibration is the Kling–Gupta
efficiency KGE′;. This objective function is
often used in hydrology and assesses different components of the error
made by the model (mean bias, variance bias, correlation). In
addition, to target different flow levels, mathematical
transformations are applied . The logarithm is
applied to analyse the errors in low-flow conditions
(KGE′(log(Q))); no transformation is applied
to preferentially analyse the error on high flows
(KGE′(Q)) and the root square of the flow is used as
a compromise representing the error on intermediate flows
(KGE′(Q)). In the case of logarithm
transformation, following the recommendations made
by , a small quantity which corresponds to one-hundredth of the catchment mean flow is added to avoid troubles with
null flows. These three transformations represent three distinct
objective functions. The models were calibrated separately and
successively on the three objective functions. To avoid strongly
negative values of the KGE′ criterion, we used the C2M
formulation, which restricts the variation range into
[-1;1]see.
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
(a), intermediate (b) and low (c) flows after
calibration with the C2M(Q) (i.e. focusing on intermediate flow).
The large points represent the mean performance and the smaller ones
represent the outliers. The 5, 25, 50, 75 and 95 percentiles are represented
by the box plots.
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 C2M(Q)
as the objective function.
The results of the calibrations were also analysed in terms of
performance in validation on the three evaluation criteria
(i.e. C2M(Q), C2M(log(Q))
and C2M(Q)). Given the large number of
catchments, it is possible to draw a conclusion on the global
difference in performance among the three studied model
versions. This avoids a discrepancy due to specific catchment
conditions. In addition to the performance analysis, the simulated
hydrographs were visually analysed to detect discrepancies in the flow
simulation. An analysis of the time series of internal fluxes and
state variables also provided further insights to interpret the
difference among the model versions. Last, the differences in
parameter values among the models was analysed. It is important to
verify that the parameter values are similar and do not take outlier
values that would compensate for model inconsistencies.
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 . With the continuous
state-space model, we verified the stability of the parameters. This
stability is very important for designing a model that is not
dependent on its time step.
Results and discussionComparison of tested models at the daily time step
Figure shows that performances are globally similar
among the different versions of the model with a calibration using
the C2M on square-rooted flows. The performances of the reference
model and the continuous state-space solution are also similar after
calibration with the two other transformations of the flow in the
objective function (not shown). In the case of the discrete
state-space solution, the model does not seem to be able to
reproduce high flows well but performs better on low flows than the two
other models when the objective function used is the C2M with
logarithmic transformation.
Scatter plots of the four free parameters of the different versions
of the models obtained by calibration with C2M(Q) as an objective
function on the basins of the data set. Parameter comparison between unit
hydrograph and Nash cascade is in black and parameter comparison between
discrete and continuous state-space parameters is in red. The values of
x1, x2 and x3 are similar for the models (the line represents the
y=x line). The x4 values are higher in the discrete state-space model
than for the other model versions.
Daily inputs in the routing store of the River Azergues in the first
half of 2012. The models are calibrated with the C2M(Q) as the
objective function. The peaks are lower with the discrete state-space GR4
(green lines) and occur sooner with the continuous state-space GR4 (red
lines).
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
C2MQ as the objective function.
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. ). This behaviour can
be explained because solving the 11 linear stores introduces
errors that propagate and amplify across the Nash cascade.
To extend the analysis on the similarity of the models, we compared
the parameter values obtained by calibration. As shown in
Fig. , the parameters have the same range of values. We
can still note differences in the values of the x4 parameter, which
are systematically higher for the discrete state-space model. These
differences in the values are probably due to the differences in
response shape between the Nash cascade and the unit hydrograph (see
Sect. ) and to the errors produced by OS
solving of the Nash cascade. The assumption that the differences in
x4 values are due to errors caused by unsuitable solving is
confirmed by the fact that the x4 parameter values are similar for
the three models at an hourly time step (not shown here).
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. ). The peaks of the continuous
state-space representation are more similar with the reference but the
peaks occur sooner. The second difference between the models concerns
the levels of the routing store (Fig. ). Here we only
compared the reference GR4 to the continuous state-space solution
because the inputs in the routing store are too different for the
discrete state-space solution. The peak levels are higher in the
continuous state-space representation, even sometimes higher than the
maximum capacity of the routing store. The reason for this is that we
shifted from the discrete model in which the processes are treated
sequentially to a continuous model in which all the processes are
solved simultaneously. In the discrete model, the exchanges are first
calculated based on the routing level at the beginning of the time
step, then the output of the unit hydrograph is added and last the
outflow of the routing store is calculated. Due to this sequential
treatment, in high-flow conditions, the quantity of exchanged water
and the outflow of the routing store in the discrete model are lower
than those of the continuous state-space representation. Given that
most of the time the exchange parameter is negative, the lower outflow
of the routing store is compensated for by less water loss with the
groundwater exchange in the complementary flow branch. This can
explain why the simulated flows are similar despite these internal
differences.
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 22 and it
can reach 100 during some days. However, in Sect. we
argue that the adaptive sub-step method seems necessary to avoid
numerical errors.
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.
Consistency of the state-space representation through time steps
The analysis of temporal consistency provides the most valuable result
produced by the continuous state-space representation. The work of
resulted in a GR4 model that is nearly consistent
across time steps. However, to adapt the model, they chose to include
the time step variations in a theoretical transformation between the
free parameter values and the percolation fixed coefficient
(Table ) at different time steps. In this section, we only
compare the reference GR4 with the continuous solution of the
state-space representation. The parameters of the state-space
representation discrete solution show the same behaviour as the
reference GR4 ones so it was chosen not to show them. This proves that
all the improvements shown in this section are only due to the
continuous resolution of the state-space model.
Scatter plots representing the four parameters of the reference
(daily and hourly) GR4 models obtained by calibration with C2M(Q)
as the objective function. The solid line represents the y=x regression and the
dashed lines the transformation relations of Table .
In Fig. , the free parameter values obtained by calibration
at the hourly time step are compared to those obtained at the daily
time step using the reference GR4 version. The dashed lines represent
the regression obtained by the theoretical relations reported in
Table . One can note that the calibrated parameters (the
dots in Fig. ) are quite different between the two time
steps but it is important to note that the values of the x3
parameter follow the relations proposed by (the
dashed lines). The high values of x1 are underestimated compared to
the theoretical relation as are the low values of the x2
parameter. There is also an issue with the unit hydrograph parameter
(x4 in Fig. ) for which calibrated hourly parameter
values are systematically lower than the values it would have by
following the transformation. and
encountered the same issue with the lag
parameter of their models.
The values of x1, x2 and x4 are inconsistent compared to the
values expected using the theoretical transformations. Regarding the
work of , we can argue that the changes in the high
values of x1 and the low values of x2 are due to temporal
inconsistencies in the interception calculation. The case of the x4
parameter is more problematic. The differences in the x4 values
probably stem from the discretization of the unit hydrograph at
different time steps.
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. , where the values of calibrated parameters remain
approximately constant despite the time step change. Only the high values of
x1 and the values of x2 slightly diverge from the x=y line.
Scatter plots representing the four parameters of the continuous
state-space (daily and hourly) GR4 models obtained by calibration with
C2M(Q) as the objective function. The solid line represents the
y=x line.
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
(a), intermediate (b) and low (c) flows after
calibration with the C2M(Q) (i.e. focusing on intermediate flow).
The points represent the mean performance.
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 x4 values because the x4 values are
constant between the two time steps, resolving the issue encountered
by , and with lag
parameters. As explained in the work of , this
improvement can be explained by the fact that the adaptive sub-step
integration approximates a continuous time input in the Nash cascade. The
results obtained with the x4 parameter here tend to confirm this earlier work on a wide
range of catchments. However, in addition to the input
errors, the lack of x4 time consistency can also be explained by the
integration errors produced by the OS at a daily time step.
The outliers in x3 values that occur in Fig. are also
present in Fig. . No explanations relating to physical
characteristics of these catchments or simulation performance were
found. We assume that these outlier values are due to the non-sensitivity of the x3 parameter for these catchments.
Finally, to verify stability, we also need to compare the performance
of the models at the hourly time step. Figure shows that,
as at the daily time step, the performance is similar for the
different versions.
Thus, the continuous state-space representation shows better temporal
stability in the x4 parameter values with similar performance.
Conclusions and perspectives
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 is more physically satisfying.
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.
Code and data availability
The Fortran code used in this article can be freely
downloaded from GitHub at
https://github.com/HYDRO-group-Irstea-Antony/GR4-State-space-version-1.0
(last access: 18 April 2018). The state-space model can be tested on an
example catchment data set with already calibrated model parameters. The full
reference for this code can be found in the
references ; it is referenced with the following DOI:
https://doi.org/10.5281/zenodo.1118183.
Author contributions
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.
Competing interests
The authors declare that they have no conflicts of interest.
Acknowledgements
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
References
Andréassian, V., Hall, A., Chahinian, N., and Schaake, J.: Large
sample basin experiments for hydrological model parametrization,
chap. Introduction and Synthesis: Why should hydrologists work on a
large number of basin data sets?, IAHS-AISH P., 307, 1–5, 2006.
Bergström, S. and Forsman, A.: Development of a conceptual and
deterministic rainfall-runoff model, Nord. Hydrol., 4, 147–170, 1973.
Burnash, R. J. C.: The NWS river forecast system – catchment modeling, chap.
10, in: Computer Model of Watershed Hydrology, Water Resources Publications,
Highlands Ranch, Colorado, USA, 311–366, 1995.Clark, M. P. and Kavetski, D.: Ancient numerical daemons of conceptual
hydrological modeling: 1. Fidelity and efficiency of time stepping schemes,
Water Resour. Res., 46, W10510, 10.1029/2009wr008894, 2010.Coron, L., Andréassian, V., Perrin, C., Lerat, J., Vaze, J., Bourqui, M.,
and Hendrickx, F.: Crash testing hydrological models in contrasted climate
conditions: An experiment on 216 Australian catchments, Water Resour. Res.,
48, W05552, 10.1029/2011WR011721, 2012.Coron, L., Thirel, G., Delaigue, O., Perrin, C., and Andréassian, V.: The
suite of lumped GR hydrological models in an R package, Environ. Modell.
Softw., 94, 166–177, 10.1016/j.envsoft.2017.05.002, 2017.Dakhlaoui, H., Ruelland, D., Tramblay, Y., and Bargaoui, Z.: Evaluating the
robustness of conceptual rainfall-runoff models under climate variability in
northern Tunisia, J. Hydrol., 550, 201–217,
10.1016/j.jhydrol.2017.04.032, 2017.Fenicia, F., Kavetski, D., and Savenije, H. H. G.: Elements of a flexible
approach for conceptual hydrological modeling: 1. Motivation and theoretical
development, Water Resour. Res., 47, W11510, 10.1029/2010wr010174,
2011.
Ficchí, A.: An adaptive hydrological model for multiple time-steps:
Diagnostics and improvements based on fluxes consistency, PhD thesis,
Université Pierre et Marie Curie, Université Pierre et Marie Curie,
Paris, France, 2017.Ficchí, A., Perrin, C., and Andréassian, V.: Impact of temporal
resolution of inputs on hydrological model performance: An analysis based on
2400 flood events, J. Hydrol., 538, 454–470,
10.1016/j.jhydrol.2016.04.016, 2016.Grouillet, B., Ruelland, D., Vaittinada Ayar, P., and Vrac, M.: Sensitivity
analysis of runoff modeling to statistical downscaling models in the western
Mediterranean, Hydrol. Earth Syst. Sci., 20, 1031–1047,
10.5194/hess-20-1031-2016, 2016.Gupta, H. V., Clark, M. P., Vrugt, J. A., Abramowitz, G., and Ye, M.: Towards
a comprehensive assessment of model structural adequacy, Water Resour. Res.,
48, W08301, 10.1029/2011wr011044, 2012.Kavetski, D. and Clark, M. P.: Numerical troubles in conceptual hydrology:
Approximations, absurdities and impact on hypothesis testing, Hydrol.
Process., 25, 661–670, 10.1002/hyp.7899, 2010.Kavetski, D. and Fenicia, F.: Elements of a flexible approach for conceptual
hydrological modeling: 2. Application and experimental insights, Water
Resour. Res., 47, W11511, 10.1029/2011wr010748, 2011.Kavetski, D. and Kuczera, G.: Model smoothing strategies to remove microscale
discontinuities and spurious secondary optima in objective functions in
hydrological calibration, Water Resour. Res., 43, W03411,
10.1029/2006wr005195, 2007.Kavetski, D., Kuczera, G., and Franks, S. W.: Semidistributed hydrological
modeling: A “saturation path” perspective on TOPMODEL and VIC, Water
Resour. Res., 39, 1246, 10.1029/2003wr002122, 2003.Kavetski, D., Fenicia, F., and Clark, M. P.: Impact of temporal data
resolution on parameter inference and model identification in conceptual
hydrological modeling: Insights from an experimental catchment, Water Resour.
Res., 47, W05501, 10.1029/2010wr009525, 2011.Klemeš, V.: Operational testing of hydrological simulation models,
Hydrolog. Sci. J., 31, 13–24, 10.1080/02626668609491024, 1986.Kling, H., Fuchs, M., and Paulin, M.: Runoff conditions in the upper Danube
basin under ensemble of climate change scenarios, J. Hydrol., 424–425,
264–277, 10.1016/j.jhydrol.2012.01.011, 2012.Littlewood, I. G. and Croke, B. F. W.: Data time-step dependency of
conceptual rainfall streamflow model parameters: an empirical study with
implications for regionalisation, Hydrolog. Sci. J., 53, 685–695,
10.1623/hysj.53.4.685, 2008.Littlewood, I. G. and Croke, B. F. W.: Effects of data time-step on the
accuracy of calibrated rainfall–streamflow model parameters: practical
aspects of uncertainty reduction, Hydrol. Res., 44, 430–440,
10.2166/nh.2012.099, 2013.
Mathevet, T.: Quels modèles pluie-débit globaux au pas de temps
horaire? Développements empiriques et comparaison de modèles sur un
large échantillon de bassins versants, PhD thesis, Ecole Nationale du
Génie Rural, des Eaux et des Forêts, Paris, France, 2005 (in French).
Mathevet, T., Michel, C., Andréassian, V., and Perrin, C.: A bounded
version of the Nash-Sutcliffe criterion for better model assessment on large
sets of basins, IAHS-AISH P., 307, 211–219, 2006.
Michel, C.: Hydrologie appliquée aux petits bassins versants
ruraux, Tech. rep., Cemagref, Antony, 320 pp., 1991 (in French).Michel, C., Perrin, C., and Andreassian, V.: The exponential store: a correct
formulation for rainfall runoff modelling, Hydrolog. Sci. J., 48, 109–124,
10.1623/hysj.48.1.109.43484, 2003.
Michel, C., Perrin, C., Andréassian, V., Oudin, L., and Mathevet, T.: Has
basin-scale modelling advanced beyond empiricism?, IAHS-AISH P., 307,
108–116, 2006.
Nash, J. E.: The form of the instantaneous unit hydrograph, Int. Assoc. Sci.
Hydrol. Publ., 45, 114–121, 1957.Oudin, L., Hervieu, F., Michel, C., Perrin, C., Andréassian, V.,
Anctil, F., and Loumagne, C.: Which potential evapotranspiration input for a
lumped rainfall–runoff model?, J. Hydrol., 303, 290–306,
10.1016/j.jhydrol.2004.08.026, 2005.Perrin, C., Michel, C., and Andréassian, V.: Improvement of a
parsimonious model for streamflow simulation, J. Hydrol., 279, 275–289,
10.1016/s0022-1694(03)00225-7, 2003.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P.:
Numerical recipes in C, 2nd Edn., Press Syndicate of the University of
Cambridge, 1992.Pushpalatha, R., Perrin, C., Moine, N. L., and Andréassian, V.: A review
of efficiency criteria suitable for evaluating low-flow simulations, J.
Hydrol., 420–421, 171–182, 10.1016/j.jhydrol.2011.11.055, 2012.Quintana Seguì, P., Le Moigne, P., Durand, Y., Martin, E., Habets, F.,
Baillon, M., Canellas, C., Franchisteguy, L., and Morel, S.: Analysis of
Near-Surface Atmospheric Variables: Validation of the SAFRAN Analysis over
France, J. Appl. Meteorol. Clim., 47, 92–107, 10.1175/2007JAMC1636.1,
2008.Santos, L.: HYDRO-group-Irstea-Antony/GR4-State-space-version-1.0: First
release of GR4-State-space-version-1.0, Zenodo, available at:
10.5281/zenodo.1118183 (last access: 18 April 2018), 2017.Schoups, G., Vrugt, J. A., Fenicia, F., and van de Giesen, N. C.: Corruption
of accuracy and efficiency of Markov chain Monte Carlo simulation by
inaccurate numerical implementation of conceptual hydrologic models, Water
Resour. Res., 46, W10530, 10.1029/2009wr008648, 2010.Seiller, G., Roy, R., and Anctil, F.: Influence of three common calibration
metrics on the diagnosis of climate change impacts on water resources, J.
Hydrol., 547, 280–295, 10.1016/j.jhydrol.2017.02.004, 2017.
Szöllösi-Nagy, A.: The discretization of the continuous linear
cascade by means of state space analysis, J. Hydrol., 58, 223–236,
10.1016/0022-1694(82)90036-1, 1982.van Esse, W. R., Perrin, C., Booij, M. J., Augustijn, D. C. M., Fenicia, F.,
Kavetski, D., and Lobligeois, F.: The influence of conceptual model structure
on model performance: a comparative study for 237 French catchments, Hydrol.
Earth Syst. Sci., 17, 4227–4239, 10.5194/hess-17-4227-2013, 2013.Vidal, J.-P., Martin, E., Franchisteguy, L., Baillon, M., and
Soubeyroux, J.-M.: A 50-year and high-resolution atmospheric reanalysis over
and France with the Safran system, Int. J. Climatol., 30, 1627–1644,
10.1002/joc.2003, 2010.Wood, E. F., Lettenmaier, D. P., and Zartarian, V. G.: A landsurface
hydrology parameterization with subgrid variability for general circulation
models, J. Geophys. Res., 97, 2717–2728, 10.1029/91JD01786, 1992.Young, P. and Garnier, H.: Identification and estimation of continuous-time,
data-based mechanistic (DBM) models for environmental systems, Environ.
Modell. Softw., 21, 1055–1072, 10.1016/j.envsoft.2005.05.007, 2006.