1. H.L. Tanaka and K. Kondo University of Tsukuba
Comparative study on the error covariance matrices for KF and EnKF using the barotropic S-model
H.L. Tanaka and K. Kondo
University of Tsukuba

2. Introduction
KF (Kalman Filter) (Kalman 1960) implementation needs to calculate inverse of a matrix with the dimension of model variables.
We can’t directly implement KF in recent numerical models.
EnKF (Ensemble Kalman Filter) approximates KF(Evensen 1994).

3. Introduction EKF vs EnKF
Dimension of the barotropic S-model is low (Tanaka 2003).
We can directly implement EKF (Extended Kalman Filter) in the barotropic S-model.
EKF vs EnKF

4. Methods Barotropic S-Model EnKF
Local Ensemble Transform Kalman Filter: LETKF (Hunt 2005)
Ensemble member: 51

5. Kalman Filter Ensemble Kalman Filter

6. Methods
EKF and EnKF are implemented in perfect model experiments with the barotropic S-model.
Observation data ＝ Ture data + noise
EKF and EnKF assimilated observation data to forecast data in every 6 hours.

7. Results

8. EKF
EnKF
Observational Error
Initial 1990/01/01/00z

9. EKF vs EnKF Pf norm of EnKF Pa norm of EnKF Pf norm of EKF
Pa norm of EKF 25 1
3.5

10. 1day(24hr)
Pf of EKF
Pa of EKF
260

11. 1day(24hr)
Pf of EnKF
Pa of EnKF
Pf of EKF
Pa of EKF

12. 3.5day(84hr)
Pf of EKF
Pa of EKF
110

13. 3.5day(84hr)
Pf of EnKF
Pa of EnKF
Pf of EKF
Pa of EKF

14. 25day(600hr)
Pf of EKF
Pa of EKF 50

15. 25day(600hr)
Pf of EnKF
Pa of EnKF
Pf of EKF
Pa of EKF

16. 1day(24hr)

17. 1day(24hr)

18. 3.5day(84hr)

19. 3.5day(84hr)

20. 25day(600hr)

21. 25day(600hr)

22. Conclusions Performance of EnKF is as good as that of EKF.
The assimilation cycle leads to degenerate dimensions of the error covariance matrices.
Eigenvalue of the error covariance matrices of EKF is about 50% of that of EnKF.
Eigenvector of the error covariance matrices of EnKF is similar to that of EKF.