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 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  large  h 大 ) Large R (large  ) Which one should be chosen?

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

5 Kalman gain Simulation Filtered dist. at t-1Predicted dist. at t Filtered dist. at

6 EnKF and PF Resampling Approx. Kalman gain EnKFPF KF

Likelihood Which is the most likely distribution that produces observation y obs ? Likelihood L(  ) = p(y obs |θ) In this example,  3 is most likely. y obs

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

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

10 Estimation of covariance matrix Minimizing innovation [predicted error] Bayes estimation Naive Ensemble mean and covariance of state Adjustment according to cost function Matcing with innovation covariance 1.With assumption of Gaussian dist. of state Maximum likelihood Ensemble mean of likelihood 2.Without assumption of Gaussian dist. of state This study Covariance matching Ueno et al., Q. J. R. Met. Soc. (2010)

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

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

13 Maximum likelihood

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

15 Filtered estimates with different θ Large Q  large  h 大 ) Large R (large  ) 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 magnitude = (5.95cm) 2, correlation lengths= (2.38, 2.52deg) Filtered estimate Smoothed estimate year longitude

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: … Tractable for just four parameters? Ueno et al., Q. J. R. Met. Soc. (2010)

22 Motivation for the second half The output of DA (i.e. “analysis”) varies with prescribed parameter θ, where θ = (B, Q 1:T, R 1:T ) B: covariance matrix of the initial state (i.e. V 0|0 ) Q t : covariance matrix of system noise R t : 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 R 1:T (dim y t ) 2 /2 elements at maximum

23 Likelihood of Rt Current assumption Log-likelihood

24 Estimation design Use ℓ t (R 1:t ) for estimating R t only It is of course that R 1:t-1 are parameters of ℓ t (R 1:t ) But they are assumed to have been estimated with former log-likelihood, ℓ 1 (R 1 ), …, ℓ t-1 (R 1:t-1 ), and to be fixed at current time step t. R t 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 R t

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

27 Estimate of R t (Temporal mean) var cov Case  t  similar output for  Case diagonal: large variance near equator, small variance for off- equator Case  t  uniform variance with intermediate value

28 Estimate of R t (Spatial mean) var Case  t  : small variance for first half, large for second half Case diagonal: large variance around 1998 Case  t  : similar for the diagonal case year -2002

29 Filtered estimates Case  t  : false positive anomalies in the east Case  t  : 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 (  or I).