Adaptive ensemble Kalman filtering of nonlinear systems

Transcription

1 Adaptive ensemble Kalman filtering of nonlinear systems Tyrus Berry George Mason University June 12, 213

2 : Problem Setup We consider a system of the form: x k+1 = f (x k ) + ω k+1 ω N (, Q) y k+1 = h(x k+1 ) + ν k+1 ν N (, R) We initially assume Gaussian system and observation noise Our goal is to estimate the covariance matrices Q and R as part of the filter procedure Later we consider Q to be an additive inflation which attempts to compensate for model error

3 : Influence of Q and R 3.2 Simple example with full observation and diagonal noise covariances Red indicates of unfiltered observations Black is of optimal filter (true covariances known) Variance used in R k Variance used in Q k

4 : Influence of Q and R Standard Kalman Update: Pk f = F k 1 Pk 1 a F k 1 T + Q k 1 P y k = H k Pk f HT k + R k 1 K k = Pk f HT k (Py k ) 1 Pk a = (I K k H k )Pk f Variance used in R k ɛ k = y k y f k = y k H k x f k x a k = x f k + K kɛ k Variance used in Q k

5 : Influence of Q and R 3.2 Covariances Q and R effect filter performance Seems better to underestimate observation noise Seems better to overestimate model error Can we estimate these parameters from the data? Variance used in R k Variance used in Q k

6 Adaptive Filter: Estimating Q and R Innovations contain information about Q and R ɛ k = y k yk f = h(x k ) + ν k h(xk f ) = h(f (x k 1 ) + ω k ) h(f (x a k 1 )) + ν k H k F k 1 (x k 1 x a k 1 ) + H kω k + ν k IDEA: Use innovations to produce samples of Q and R : E[ɛ k ɛ T k ] HPf H T + R E[ɛ k+1 ɛ T k ] HFPe H T HFKE[ɛ k ɛ T k ] P e FP a F T + Q In the linear case this is rigorous and was first solved by Mehra in 197

7 Adaptive Filter: Estimating Q and R To find Q and R we estimate H k and F k 1 from the ensemble and invert the equations: E[ɛ k ɛ T k ] HPf H T + R E[ɛ k+1 ɛ T k ] HFPe H T HFKE[ɛ k ɛ T k ] This gives the following empirical estimates of Q k and R k : Pk e = (H k+1 F k ) 1 (ɛ k+1 ɛ T k + H k+1f k K k ɛ k ɛ T k )H T k Qk e = Pk e F k 1Pk 1 a F k 1 T R e k = ɛ k ɛ T k H kp f k HT k Note: Pk e is an empirical estimate of the background covariance

8 An Adaptive Kalman-Type Filter for Nonlinear Problems We combine the estimates of Q and R with a moving average Original Kalman Eqs. P f k = F k 1 P a k 1 F T k 1 + Q k 1 P y k = H k P f k HT k + R k 1 Our Additional Update P e k 1 = F 1 k 1 H 1 k ɛ kɛ T k 1 H T k 1 + K k 1 ɛ k 1 ɛ T k 1 H T k 1 K k = P f k HT k (Py k ) 1 P a k = (I K k H k )P f k Q e k 1 = P e k 1 F k 2P a k 2 F T k 2 R e k 1 = ɛ k 1 ɛ T k 1 H k 1P f k 1 HT k 1 ɛ k = y k y f k x a k = x f k + K kɛ k Q k = Q k 1 + (Q e k 1 Q k 1)/τ R k = R k 1 + (R e k 1 R k 1)/τ

9 How does this compare to inflation? We extend Kalman s equations to estimate Q and R Estimates converge for linear models with Gaussian noise When applied to nonlinear, non-gaussian problems We interpret Q as an additive inflation Q can have complex structure, possibly more effective than multiplicative inflation? Downside: many more parameters than multiplicative inflation Somewhat less ad hoc than other inflation techniques?

10 Adaptive Filter: We will apply the adaptive EnKF to the 4-dimensional Lorenz96 model integrated over a time step t =.5 dx i dt = x i 2 x i 1 + x i 1 x i+1 x i + F We augment the model with Gaussian white noise x k = f (x k 1 ) + ω k ω k = N (, Q) y k = h(x k ) + ν k ν k = N (, R) We will consider full and sparse observations The Adaptive EnKF uses F = 8 We will consider model error where the true F i = N (8, 16)

11 Recovering Q and R, Full Observability True Covariance Initial Guess Final Estimate Difference x 1 3 Q of Q Q k Filter Step k R Filter Steps shown for the initial guess covariances (red) the true Q and R (black) and the adaptive filter (blue)

12 Recovering Q and R, Sparse Observability Observing 1 sites results in divergence with the true Q and R True Covariance Initial Guess Final Estimate Difference 4 Q Filter Steps x R Filter Steps x 1 4 shown for the initial guess covariances (red) the true Q and R (black) and the adaptive filter (blue)

13 Compensating for Model Error The adaptive filter compensates for errors in the forcing F i True Covariance Initial Guess Final Estimate Difference Q R (Q f ) ii Relative Change Site Number F i Model Error (%) Filter Steps x 1 shown for the initial guess covariances (red) an Oracle EnKF (black) and the adaptive filter (blue)

14 Integration with the LETKF Simply find a local Q and R for each region True Covariance Initial Guess Final Estimate Difference 1 Q Relative Variance Site Number R Filter Steps x 1 5 shown for the initial guess covariances (red) the true Q and R (black) and the adaptive filter (blue)

15 This research was partially supported by the National Science Foundation Directorates of Engineering (EFRI ) and Mathematics and Physical Sciences (DMS ). T. Berry, T. Sauer,. To appear in Tellus A. R. Mehra, 197: On the identification of variances and adaptive Kalman filtering. P. R. Bélanger, 1974: Estimation of noise covariance matrices for a linear time-varying stochastic process. J. Anderson, 27: An adaptive covariance inflation error correction algorithm for ensemble filters. H. Li, E. Kalnay, T. Miyoshi, 29: Simultaneous estimation of covariance inflation and observation errors within an ensemble Kalman filter. B. Hunt, E. Kostelich, I. Szunyogh, 27: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. E. Ott, et al. 24: A local ensemble Kalman filter for atmospheric data assimilation.