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A state-space representation of the GR4J rainfall-runoff model
L. Santos, Guillaume Thirel, C. Perrin
To cite this version:
L. Santos, Guillaume Thirel, C. Perrin. A state-space representation of the GR4J rainfall-runoff model. EGU General Assembly 2017, Apr 2017, Vienna, Austria. pp.1, 2017. �hal-02606378�
A state-space representation of the GR4J rainfall-runoff model
Léonard Santos, Guillaume Thirel and Charles Perrin
Irstea, HBAN - Hydrosystems and Bioprocesses, Antony, France
EGU2017-4851
Contact:
! PhD student: Léonard Santos leonard.santos@irstea.fr +331-40-96-61-97
http://webgr.irstea.fr
A state-space representation of the GR4J rainfall-runoff model
Léonard Santos, Guillaume Thirel and Charles Perrin
Irstea, HBAN - Hydrosystems and Bioprocesses, Antony, France
EGU2017-4851
Contact:
! PhD student: Léonard Santos leonard.santos@irstea.fr +331-40-96-61-97
http://webgr.irstea.fr
Objectives
% Harmonize the mathematical equations of the GR4J model (Perrin et al. , 2003) within a state-space representation
% Replace the sequential resolution of equations by a global resolution with adaptative time-stepping
% Represent the unit hydrographs as a state variables model Constraints
% Keep similar performances
% Keep a lumped model with four free parameters
1. Structural modifications
The two unit hydrographs cannot be written as state space variable model.
The following options were chosen:
% Substitute the two unit hy- drographs by a
“Nash Cascade”
(Nash, 1957)
% Place it before the split of flow components
E P
S
Q x1
R F(x2) x3
Q9
Qr
Interception
Production store
Routing store
HU2
QNC
2x4
Perc Pn
Pn-Ps
Pr Ps
En Es
Q1
F(x2) Qd
... Nash
Cascade
E P
S
Q R F(x2) x3
Q9
Qr
Interception
Production store
Routing store
Perc Pn
Pn-Ps
Pr Ps
En Es
Q1
F(x2) Qd
Unit
Hydrographs
x4 2x4
UH2
UH1
Sh1
Sh2
Shn
Fig. 1: Substitution of the unit hydrographs of the original GR4J model (left) by a
“Nash cascade” for the state-space model (right)
2. Mathematical modifications
% All the state variables are expressed as v˙ = f (v, u) (v is the states vector and u the inputs) and merged in one system
% The “Nash cascade” is formulated with eleven stores and an outflow coef- ficient dependent on the GR4J time base (x4) parameter (to easily compare it with U H2)
The state-space formulation can be written as:
S˙ S˙h1 S˙h2
...
S˙hn R˙
=
−Ux1−β1
t(β−1)νβ−1S(t)β + (En − Pn)
S(t) x1
α
− 2EnS(t)x
1 + Pn
Pn
S(t) x1
α
+ x
1−β 1
(β−1)Utνβ−1S(t)β − n−1x
4 Sh,1(t)
n−1
x4 Sh,1(t) − n−1x
4 Sh,2(t) ...
n−1
x4 Sh,n−1(t) − n−1x
4 Sh,n(t) Φn−1x
4 Sh,n(t) − (γx−1)U1−γ3
tR(t)γ + xxω2
3R(t)ω
Greek letters are fixed parameters, latin letters are model states and inputs, xn are free parameters
3. Evaluation methodology
% 650 French catchments to get general conclu- sions
% Calibration of the mod- els using the KGE0 as an objective function
% Comparison of perfor- mances, output hydro- graphs, parameter val- ues and internal fluxes
% Tests at daily and hourly time steps
River Dives at Saint-Lambert-sur-Dive
Fig. 2: Locations of the 650 test catchments
4. Daily models results
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Original GR4J model State−space GR4J model
Performances comparison (calibration on high flows)
KGE'(Q) 0.00.20.40.60.81.0
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x1 parameter
State−space GR4J model x1 0500100015002000
0 500 1000 1500 2000
Original GR4J model x1
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x4 parameter
State−space GR4J model x4 012345
0 1 2 3 4 5
Original GR4J model x4
Fig. 3: KGE0 and parameter values of the state-space model compared to the original GR4J
% Very similar performances and parameter values (except x4)
% On all the catchments hydrographs, the peak flows are lower for the state-space representation
% Discrepancies in the internal fluxes (see figure 6)
+ + + + + + + + + + + + + + ++
+ + + + + + + + + +
+ ++ + + + + + ++ +
++ ++ + +
+ +
+ ++ + +
+ +
++ + ++ + +
+ + + + + + + + + ++ + + + ++ + +
+
+
++
++ + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Date Flow[m3 /s]
janv. 2002 févr. 2002 mars 2002 avril 2002 mai 2002
0246810 100806040200 Rainfall [mm/j]
+ Observations
Original GR4J outputs State−space outputs
River Dives at Saint−Lambert−sur−Dive
Fig. 4: Simulated hydrographs of the River Dives in winter and spring 2002
References
Ficchì, Andrea, Perrin, Charles, & Andréassian, Vazken. 2016. Impact of temporal resolution of inputs on hydrological model performance: An analysis based on 2400 flood events. Journal of Hydrology, 538(Jul), 454–470.
Nash, J. E. 1957. The form of the instantaneous unit hydrograph. Int. Assoc. Sci. Hydrol. Publ., 45(3), 114–
121.
Perrin, Charles, Michel, Claude, & Andréassian, Vazken. 2003. Improvement of a parsimonious model for streamflow simulation. Journal of Hydrology, 279(1-4), 275–289.
5. Temporal consistency
% For the original GR4J model, time step changes are made possible by the changes in parameters values (Ficchì et al. , 2016)
% State space model parameter values are stable, time step changes are managed by the integration of the differential equations
0 500 1000 1500 2000
0500100015002000
x1 parameter
Daily st−space x1 [mm]
Hourly st−space x1 [mm]
−10 −8 −6 −4 −2 0 2
−10−8−6−4−202
x2 parameter
Daily st−space x2 [mm/day]
Hourly st−space x2 [mm/day]
0 50 100 150 200 250 300
050100150200250300
x3 parameter
Daily st−space x3 [mm]
Hourly st−space x3 [mm]
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00.51.01.52.02.53.0
x4 parameter
Daily st−space x4 [day]
Hourly st−space x4 [day]
0 500 1000 1500 2000
0500100015002000
x1 parameter
Daily original x1 [mm]
Hourly original x1 [mm]
−10 −8 −6 −4 −2 0 2
−10−8−6−4−202
x2 parameter
Daily original x2 [mm/day]
Hourly original x2 [mm/hour]
0 50 100 150 200 250 300
050100150200250300
x3 parameter
Daily original x3 [mm]
Hourly original x3 [mm]
0 1 2 3 4 5
012345
x4 parameter
Daily original x4 [day]
Hourly original x4 [day]
Fig. 5: Scatter plots of the four daily and hourly parameters of the original GR4J (left) and the state-space representation (right)
6. Discrepancies in internal fluxes
% High exchange values occur sooner after rainfall in the state-space model
% Calibrated x4 parameters create faster and higher response from the
“Nash cascade” than with the unit hydrograph in the original GR4J
% These two patterns seem related
% No real consequences on performances
Date
Water exchange [mm/j]
Jan 2002 Feb 2002 Mar 2002 Apr 2002 May 2002
−3−2−10123 50403020100 Rainfall [mm/j]
Classic GR4J exchanges State−space exchanges
River Dives at Saint−Lambert−sur−Dive
Fig. 6: Simulated water exchanges of the River Dives in winter and spring 2002
0 1 2 3 4 5 6
0.00.20.40.60.81.01.2
Time (t)
Outflow q(t)
Unit hydrograph response with x4 = 2 Nash cascade response with x4 = 2
Nash cascade response with a calibrated x4 value
Fig. 7: Comparison of the impulse responses of U H2 and a 11-stores “Nash
cascade”
Conclusion
% Mathematically more uniform and continuous version of the GR4J model
% This version should not substitute the original GR4J model, as it does not outperform the original model and have a higher computational time
% It could be useful for specific applications like time variable modelling, data assimilation or multimodel approaches (see poster EGU2017-4093 Friday, 28 Apr, 17:30–19:00)
EGU General Assembly - April 2017 - Vienna
