アンサンブルカルマンフィルターによる大気海洋結合モデルへのデータ同化

1. アンサンブルカルマンフィルターによる大気海洋結合モデルへのデータ同化アンサンブルカルマンフィルターによる大気海洋結合モデルへのデータ同化 On-line estimation of observation error covariance for ensemble-based filters Genta Ueno The Institute of Statistical Mathematics

2. Covariance matrix in DA State space model Cost function

3. Filtered estimates with different θ Large Q (largesh大) Large R (large a) Which one should be chosen?

4. Ensemble approx. of distribution Gaussian dist. Non-Gaussian dist. Ensemble approx. / Particle approx. Exactly represented Ensemble Kalman filter(EnKF), Particle filter(PF) Kalman filter(KF)

5. Kalman filter (KF) Filtered dist. at Filtered dist. at t-1 Predicted dist. at t Kalman gain Simulation

6. EnKF andPF Resampling Approx. Kalman gain EnKF PF KF

7. Likelihood Which is the most likely distribution that produces observation yobs ? yobs yobs yobs Likelihood L(t) = p(yobs|θ) In this example, q3 is most likely.

8. Likelihood of time series Find θ that maximizes L(θ). In practice, log-likelihood is easy to handle:

9. Likelihood of time series likelihood Predicted dist. Observation model Non-Gaussian dist. [due to nonlinear model] If it were Gaussian,

10. Estimation of covariance matrix Minimizing innovation[predicted error] Maximum likelihood With assumption of Gaussian dist. of state • Naive • Ensemble mean and covariance of state • Adjustment according to cost function • Matcing with innovation covariance • Without assumption of Gaussian dist. of state • Ensemble mean of likelihood This study Bayes estimation Ueno et al., Q. J. R. Met. Soc. (2010) Covariance matching

11. Ensemble approx. of likelihood Observation model Ensemble mean of likelihood of each member xt|t-1(n) • Find θ that maximizes the ensemble approx. log-likelihood.

12. Regularization of Rt Regularization with Gaussian graphical model 12 neighborhood Sample covariance (singular due to n<<p) 12

13. Maximum likelihood

14. Data and Model year longitude The color shows SSH anomalies.

15. Filtered estimates with different θ Large Q (largesh大) Large R (large a) Which one should be chosen?

16. System noise: magnitude

17. System noise: zonal correlation length

18. System noise: meridional correlation length

19. Observation noise: magnitude

20. Estimates with MLE Smoothed estimate Filtered estimate year longitude magnitude= (5.95cm)2, correlation lengths= (2.38, 2.52deg)

21. Summary for the first half • Maximum likelihood estimation can be carried out even for non-Gaussian state distribution with ensemble approximation • Applicable for ensemble-based filters such as EnKF and PF • Estimated parameters: Ueno et al., Q. J. R. Met. Soc. (2010) • … Tractable for just four parameters?

22. Motivationfor the second half • The output of DA (i.e. “analysis”) varies with prescribed parameter θ, where • θ = (B, Q1:T, R1:T) • B: covariance matrix of the initial state(i.e. V0|0) • Qt: covariance matrix of system noise • Rt: covariance matrix of observation noise • My interest is how to construct optimal θ for a fixed dynamic model • Only four parameters so far … • We allow more degree of freedom on R1:T • (dim yt)2/2 elements at maximum

23. Likelihood of Rt Current assumption Log-likelihood

24. Estimation design • Use ℓt(R1:t) for estimatingRt only • It is of course that R1:t-1 are parameters of ℓt(R1:t) • But they are assumed to have been estimated with former log-likelihood, ℓ1(R1), …, ℓt-1(R1:t-1) , and to be fixed at current time step t. • Rt is estimated at each time step t. • Bad news: • The estimated Rt may vary significantly between different time steps. • A time-constant R cannot be estimated within the present framework.

25. Experiment • Assumed structure of Rt

26. Data and Model year longitude The color shows SSH anomalies.

27. Estimate of Rt (Temporal mean) var cov • Case atS: similar output for 20S. • Case diagonal: large variance near equator, small variance for off-equator • Case atI: uniform variance with intermediate value

28. Estimate of Rt (Spatial mean) var 1992- year-2002 • Case atS: small variance for first half, large for second half • Case diagonal: large variance around 1998 • Case atI: similar for the diagonal case

29. Filtered estimates • Case atS: false positive anomalies in the east • Case atI: negative anomalies in the east, but the equatorial Kelvin waves unclear • Case diagonal: negative anomalies and equatorial Kelvin reproduced

30. Iteration times • Only 2-4 times • Small number of parameters requires large iteration numbers

31. Summary of the second half • An on-line and iterative algorithm for estimating observation error covariance matrix Rt. • The optimality condition of Rt leads a condition of Rt in a closed form. • Application to a coupled atmosphere-ocean model • Only 4-5 iterations are necessary • A diagonal matrix with independent elements produces more likely estimates • than those of scalar multiplication of fixed matrices (S or I).