On the use of a Nash cascade to improve the lag parameter transferability at different time-step

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On the use of a Nash cascade to improve the lag parameter transferability at different time-step

L. Santos, Guillaume Thirel, C. Perrin

To cite this version:

L. Santos, Guillaume Thirel, C. Perrin. On the use of a Nash cascade to improve the lag parameter transferability at different time-step. EGU General Assembly 2018, Apr 2018, Vienna, Austria. pp.1, 2018. �hal-02608246�

On the use of a Nash cascade to improve the lag parameter transferability at different time-step

Léonard Santos, Guillaume Thirel and Charles Perrin

Irstea, HYDRO research group (HYCAR), Antony, France

EGU2018-14396

Contact:

! PhD student: Léonard Santos  leonard.santos@irstea.fr

 +331-40-96-61-97

https://webgr.irstea.fr/en

On the use of a Nash cascade to improve the lag parameter transferability at different time-step

Léonard Santos, Guillaume Thirel and Charles Perrin

Irstea, HYDRO research group (HYCAR), Antony, France

EGU2018-14396

Contact:

! PhD student: Léonard Santos  leonard.santos@irstea.fr

 +331-40-96-61-97

https://webgr.irstea.fr/en

Objectives

Context

Conceptual bucket-type hydrological models often use lag functions. The lag parameters that govern these functions are dependent on the mod- elling time-step and are difficult to transpose between them. It is an issue because daily flow data are more easily available than hourly data.

Objectives

% To avoid that the lag parameter depends on the time-step

% To more easily transpose parameters from daily to hourly time-step Method

% Substitution of the lag function with a “Nash cascade” (Nash, 1957)

% Use of a near-continuous resolution to solve the model equations

1. Structural modifications

Model used: GR4 (Per- rin et al., 2003)

% Lag function re- placed by stores to obtain a strict state-space repre- sentation

E P

S

Q x₁

R F(x₂)

x₃

Q₉

Q_r

Q_uh Perc

P_n

P_n-P_s

P_r P_s

E_n E_s

Q₁

F(x₂) Q_d

E P

S

Q x₁

R F(x₂)

x₃

Q₉

Q_r UH₂

Q_uh 2x₄ Perc

P_n

P_n-P_s

P_r P_s

E_n E_s

Q₁

F(x₂) Q_d

...

Nash Cascade

S_h,1 S_h,2

S_h,nres

Q_Sh,1

Q_Sh,2 Q_Sh,nres-1

Production store Interception

Routing store Unit

hydrograph

Fig. 1: Substitution of the unit hydrograph of the original GR4J model (left) by a “Nash cascade” to form a state-space model (right)

2. Parametrisation of the Nash cascade

Two parameters in the Nash cascade:

number of stores and outflow coefficient

% Number of stores fixed at nres = 11

% Outflow coefficient linked to the x⁴ pa- rameter of GR4 with the relation: k =

nres−1 x₄

Nash cascade and GR4 unit hydrograph responses have the same timing

0 1 2 3 4 5 6

0.00.20.40.60.81.0

Time (t)

Outflow q(t)

Unit hydrograph response with x₄ = 2 Nash cascade response with x₄ = 2

Fig. 2: Shapes of the unit hydrograph and Nash cascade responses

3. Robust numerical temporal integration

Initial integration technique

% Inputs added at the beginning of the time-step

% Water balance equations solved using a sequential technique Modified integration technique

% Inputs considered as uniform over the time-step

% Implementation of an Euler-implicit method using adaptive sub-step (approaches a continuous time model)

4. Evaluation methodology

% 240 French catchments to get general conclusions

% Calibration of the models using the KGE^′ (Kling et al., 2012) with square root transformed flows at daily and hourly time-steps

% Comparison of perfor- mances and parameter values between daily and hourly time-steps

Fig. 3: Locations of the 240 test catchments in France

5. Results

Impact of the substitution with a Nash cascade

% Performances are similar to the reference GR4 at daily and hourly time steps

% No improvement of param- eter temporal stability

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Reference St−Sp disc

St−Sp cont

Daily performances

KGE'(Q)

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0.00.20.40.60.81.0

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Reference St−Sp disc

St−Sp cont

Hourly performances

KGE'(Q)

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0.00.20.40.60.81.0

Fig. 4: Performances distribution of the different models, in validation, on the 240 catchments

References

Kling, H., Fuchs, M., and Paulin, M.: Runoff conditions in the upper Danube basin under ensemble of climate change scenarios, Journal of Hydrology, 424425, 264–277, doi:10.1016/j.jhydrol.2012.01.011, 2012.

Nash, J. E.: The form of the instantaneous unit hydrograph, Int. Assoc. Sci. Hydrol. Publ., 45, 114–121, 1957.

Perrin, C., Michel, C., and Andréassian, V.: Improvement of a parsimonious model for streamflow simulation, Journal of Hy- drology, 279, 275289, doi:10.1016/s0022-1694(03)00225-7, 2003.

Santos, L., Thirel, G., and Perrin, C.: State-space representation of a bucket-type rainfall-runoff model: a case study with State- Space GR4 (version 1.0), Geosci. Model Dev. Disscuss., doi:10.5194/gmd-2017-264, in Press, 2018.

Impact of the integration technique modification

% Performances also remains similar

% Increase of the lag parameter temporal stability

0 1 2 3 4 5 6

0123456

GR4 with UH

Daily discrete x4 [d]

Hourly discrete x4 [d]

0 1 2 3 4 5 6

0123456

GR4 with Nash casc

Daily state−sp disc x4 [d]

Hourly state−sp disc x4 [d]

0 1 2 3 4 5 6

0123456

Continuous GR4 with Nash casc

Daily state−sp x4 [d]

Hourly state−sp x4 [d]

Fig. 5: Lag parameter values obtained at hourly time-step compared to those obtained at daily time-step

Example: River Sauldre flood, June 2016

River Sauldre at Romorantin ( c⃝ La nouvelle République)

% Calibration at daily time-step

% Test on the hourly flood hydrograph

% Better timing for the continuous in- tegration

+ +

+ +

+ +

+ +

+

+ + + + + + +

Date Flow [mm.h−1 ]

27 May 2016 31 May 2016 04 Jun 2016 08 Jun 2016

0.00.51.01.5 150100500 Rainfall [mm.d−1 ]

+ Observations

Reference GR4J outputs Continuous state−space outputs

River Sauldre at Menetreol

Fig. 7: Daily hydrograph during the flood

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + + ++++

+++++++++++++++++++

+++++++++++++++++

+++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Date Flow [mm.h−1 ]

2016−05−27 01:00:00 2016−06−02 01:00:00 2016−06−08 01:00:00

0.00.51.01.5 150100500 Rainfall [mm.h−1 ]

+ Observations

Reference GR4J outputs Continuous state−space outputs

River Sauldre at Menetreol

Fig. 8: Hourly hydrograph during the flood

Conclusion

% Sequentially integrated model leads to different lag parame- ters at different time-steps

% Continuity in integration technique improves the stability of the lag parameter

% More information in Santos et al. (2018) Application

% For catchments with only daily flow measurements avail- able

% To set-up an adaptive time-step model

EGU General Assembly - April 2018 - Vienna