Comparative study on the error covariance matrices for KF and EnKF using the barotropic S-model

1. 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 • 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. Observational Error EKF EnKF 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. Pf of EKF Pa of EKF 1day(24hr) 260

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

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

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

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