Ensemble data assimilation methods

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Ensemble data assimilation methods

for merging remotely sensed terrestrial water storage changes with hydrological models

Maike Schumacher, Jürgen Kusche

Institute of Geodesy and Geoinformation, University of Bonn, Germany

Workshop on Data Assimilation in Terrestrial Systems

Bonn, September 19-21, 2016

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Canopy Snow

Soil/Vegetation Surface

Water River

Groundwater

Simplification of Reality

2

+ Precipitation - Evapotranspiration

- Runoff

Terrestrial Water Cycle

Large-Scale Hydrological Modelling

1 Motivation 2 Frameworks 3 Applications 4 Conclusions

Limitation of models - parameters

- forcing data (precipitation)

- model structure

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3

Canopy Snow

Surface Water

River

Groundwater

Total Water Storage Anomaly (TWSA)

Altimetry

Groundwater well Soil moisture

missions

GRACE Discharge

Integrating (Geodetic) Data

Soil/Vegetation

Gravity Recovery And

Climate Experiment

since 2003

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4

Time variable satellite gravimetry data

1 Motivation 2 Frameworks 3 Applications 4 Conclusions

Gravity field changes induced by mass changes (atmosphere, oceans, hydrosphere)

Observed by GRACE and represented as monthly potential coefficients - Conversion to total water storage anomalies (TWSA)

- Accounting for mass in atmosphere and oceans

m m

300-600 km

filtering

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Improve hydrological model with GRACE data

5

WaterGAP Global Hydrology Model (WGHM)

GRACE

0.5°x0.5° grid

- daily time step - monthly averaged potential coefficients

calibration parameters

Döll et al. (2003)

300-600 km

DDK filtered TWSA

w at er co m pa rtm en ts

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Uncertainties of GRACE TWSA

6

Standard deviation Error correlations

10 20 30 40

[mm] -1 -0.5 0 0.5 1

[-]

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Research gaps

7

Main challenges:

- resolution mismatch

- disaggregation of TWSA

- uncertainties of model and data

- parameters calibration against TWSA

- data-model fusion methods

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Classical Approach: EnKF

Ensemble Covariance Matrix

Model Prediction GRACE TWSA Observations

) p , u , f(x

x

⁽¹⁾_k ^



⁽¹⁾_k_₁

…

⁽¹⁾_k_₁ ⁽¹⁾_k_₁

N

_e

water state samples at time k forcing

) p , u , f(x

x

^(N_k ^e⁾^



^(N_k_₁ê⁾ ^(N_k_₁ê⁾ ^(N_k_₁ê⁾

parameters

ε , Ax l

_k

 

_k

design matrix errors







 

 

^N^e

1 i

T k (i)

k k (i)

k e

k

e

(x x )(x x )

1 N ) 1 (x C

Σ

ll

Ensemble Kalman Filter (EnKF)

1 ll T

k e T

k

e

(x )A (AC (x )A Σ )

C

K 

^ ^



^

ensemble generation

with gain matrix ) (x C ,

x

_k^ ^e _k^







⁽ⁱ⁾_k

(i) 1

k

x

Evensen (1994) Burgers et al. (1998)

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δl

k

  Ax

⁽¹⁾_k ^

l

k





(1)

x

k

K [ ]





(1)

x

k

( )

) (N k

δl

e

k



 l

) (N k

x

e

K [ ]





) (N k

x

e

( )  Ax

^(N_k ^e⁾^

…

White noise vs.

Correlated errors

1 Motivation 2 Frameworks 3 Applications 4 Conclusions

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Variant 1: SQRA

SQRA

with gain matrix K as in EnKF

EnKF

Avoid additional source of sampling errors

Update

ensemble mean

Re-initialization required for model perturbations

, C

^e

(x

_k^

)



^









_k



_k



_k

k

x K l A x

x

(1)

δl

k

  Ax

⁽¹⁾_k ^

l

k

K [ ]





(1)

x

k

( )

) (N k

δl

e

k



 l

) (N k

x

e

K [ ]





) (N k

x

e

( )  Ax

^(N_k ^e⁾^

… 

^⁽¹⁾_k

x

SQuare Root Analysis Scheme (SQRA)

Evensen (2004)

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Variant 2: SEIK

Pham et al. (1998)

SEIK

1 ll T T

k 1 k 1 ll T T

k 1

k

(G L A Σ AL ) L A Σ

L

K 

^



^ ^ ^

with

SQRA

1 ll T

k e T

k

e

(x )A (AC (x )A Σ )

C

K 

^ ^



^

Update in observation space

Singular Evolutive Interpolated Kalman filter (SEIK) Computationally more efficient:

especially, if number of observations >> ensemble size N

_e

Update in ensemble space

#obs x #obs



^









_k



_k



_k

k

x K l A x

x

N

_e

x N

_e

After some transformations and



^









_k



_k



_k

k

x K l A x

x

1 Motivation 2 Frameworks 3 Applications 4 Conclusions

T k k

k

) L GL

C(x

^



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EnKF results with WGHM

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Soil water Snow water

Synthetic experiment within the Mississippi River Basin (MRB)

USA

MRB

WGHM GRACE

TWSA EnKF

‘Truth ‘

Schumacher et al. (2016b)

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Error correlations of GRACE TWSA

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4 sub-basins 11 sub-basins 16 grid cells: 5° x 5°

10 – 15 mm 15 – 25 mm 20 – 25 mm



σ σ  σ 

up to 50 % up to 90 % > 90 %

ρ ρ ρ

1 Motivation 2 Frameworks 3 Applications 4 Conclusions

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Root mean square error (RMSE)

[mm]

TWSA

C/DA results: exchange error covariance matrix

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Correlation of residual

curves* [-]

EnKF USA

MRB

white noise

or correlated errors

Synthetic experiment within the Mississippi River Basin (MRB)

*

after subtracting linear trend, annual and semi-annual cycles

WGHM EnKF white

EnKF

correlated

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C/DA results: exchange filter algorithm

14

Synthetic experiment within the Mississippi River Basin (MRB)

EnKF or SEIK USA

MRB

correlated errors Root mean

square error (RMSE)

[mm]

Correlation of residual

curves* [-]

*

after subtracting linear trend, annual and semi-annual cycles

TWSA

EnKF SEIK

1 Motivation 2 Frameworks 3 Applications 4 Conclusions

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Murray – Darling River Basin (MDB)

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USA

MRB Transfer Australia

Groundwater loss

WGHM

GRACE EnKF

Schumacher et al. (2016c)

w ater com par tme nts

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Conclusions and Outlook

Insights about data assimilation frameworks:

- EnKF is commonly used due to simple implementation

- Alternative methods show benefit for state / parameter estimation  uncertainties, computational efficiency

- effect of implementing GRACE error correlations visible Recommendations:

- consider GRACE error correlations - alternative methods

 Further investigations will contribute in enlarging our

understanding of hydrological data assimilation and parameter calibration

- parameter calibration

- constraints for individual water states - disaggregation of GRACE TWSA

- C/DA methodology

1 Motivation 2 Frameworks 3 Applications 4 Conclusions

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Contact:

Maike Schumacher Astronomical, Physical and Mathematical Geodesy University of Bonn, Germany email: schumacher@geod.uni-bonn.de homepage: http://www.igg.uni-bonn.de/apmg/index.php?id=schumacher

Thank you for your attention!

Any questions?

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Main References

Ensemble Kalman Filter Approaches

- Evensen (1994) Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J Geophys Res 99(C5):10143-10162. doi:10.1029/94JC00572

- Burger et al. (1998) Analysis scheme in the ensemble Kalman filter. Mon Weather Rev 126:1719-1724. doi:

10.1175/1520-0493(1998)126$<$1719:ASITEK$>$2.0.CO;2

- Evensen (2004) Sampling strategies and square root analysis schemes for the EnKF. Ocean Dynam 54:539-560.

doi:10.1007/s10236-004-0099-2

- Pham et al. (1998) A singular evolutive extended Kalman filter for data assimilation in oceanography. J Marine Syst 16(3-4):323-340. doi:10.1016/S0924-7963(97)00109-7

Calibration and Data Assimilation Approach:

- Schumacher M. (2012): Assimilation of GRACE data into a global hydrological model using an ensemble Kalman filter. Master thesis, University of Bonn.

- Eicker et al. (2014): Calibration/Data Assimilation Approach for Integrating GRACE Data into the WaterGAP Global Hydrology Model (WGHM) Using an Ensemble Kalman Filter: First Results. Surv Geophys, Vol. 35 (6), pp. 1285-1309.

doi:10.1007/s10712-014-9309-8

- Schumacher et al. (2016a): A Systematic Impact Assessment of GRACE Error Correlation on Data Assimilation in Hydrological Models. J Geod, doi:10.1007/s00190-016-0892-y

- Schumacher et al. (2016b) Exploring the Hydrologial Millennium Drought Pattern (2003-2009) in the Murray-Darling Basin by Assimilating GRACE into WGHM (in preparation)

WaterGAP Global Hydrology Model (WGHM):

- Döll et al. (2003): A global hydrological model for deriving water availibility indicators: model tuning and validation. J Hydrol, 207, pp. 105-134. doi:10.1016/S0022-1694(02)00283-4

- Müller Schmied et al. (2014): Sensitivity of simulated global-scale freshwater uxes and storages to input data,

hydrological model structure, human water use and calibration. HESS 18:3511-3538. doi:10.5194/hess-18-3511-2014