2 4 (four-dimensional variational(4dvar))(talagrand and Courtier(1987), Courtier et al.(1994)) (Ensemble Kalman Filter( EnKF))(Evensen(1994), Evensen(

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1 1,3 2,3 2,3 ; ; ; 1. (Wunsch(1996), Daley(1991), Bennett(2002), (1997)) (JST) (CREST)

2 2 4 (four-dimensional variational(4dvar))(talagrand and Courtier(1987), Courtier et al.(1994)) (Ensemble Kalman Filter( EnKF))(Evensen(1994), Evensen(2003)) (Kalman(1960)) 2

3 3 (Higuchi(2003)) EnKF (Particle Filter( PF))(Kitagawa(1996), Gordon et al.(1993), Liu(2001), Doucet et al.(2001), (1996), (1996)) EnKF PF EnKF 5 EnKF 6 PF EnKF PF EnKF 7 2.

4 4 (Lorenz(1963)) EnKF ( (1983) (1977)) EnKF Evensen(1994)

5 5 EnKF Lisæter et al.(2003) Lisæter et al EnKF EAKF(Ensemble Adjustment Kalman Filter)(Anderson (2001)) EnKF (EnKS) Evensen and van Leeuwen(2000) EnKF PF Pham(2001) 3 Lorenz van Leeuwen(2003) Manda et al.(2003) 3.

7 7 3.2 x n M {1, 2,, M} T i S i 2 (U i, V i ) x x = (T 1, S 1, U 1, V 1, T 2, S 2, U 2, V 2,, T M, S M, U M, V M ) n x x n M i {1, 2,, M} ξ i x x = (ξ1 T, ξ2 T,, ξm) T T n x x n T x n 1 n v n x n x n = f n (x n 1, v n )

8 8 v n v n y n = h n (x n, w n ) w n ( (2005)) :p(x n y 1:n 1 ) = p(x n y 1, y 2,, y n 1 ) :p(x n y 1:n ) = p(x n y 1, y 2,, y n ) :p(x n y 1:T ) = p(x n y 1, y 2,, y T ) ( T > n) y n = h n (x n ) + w n EnKF (3.1) y n = H n x n + w n

9 x n n y 1:n = {y 1,, y n } ˆx n n θ ˆθ p(x n y 1:n ), p(θ y 1:n ) y n p(x n y 1:n ) EnKF PF 4. EnKF 4.1 EnKF (3.1) w n N(0, R n ) n 1 p(x n 1 y 1:n 1 ) N (

10 10 ) {x (i) n 1 n 1 }N i=1 p(x n 1 y 1:n 1 ) = 1 N δ(x n 1 x (i) n 1 n 1 N ) i=1 PF y n p(x n y 1:n ) = 1 N N i=1 δ(x n x (i) n n ) {x (i) n n }N i=1 EnKF x (i) n 1 n 1 {x (i) n n 1 }N i=1 : x (i) n n 1 = f n(x (i) n 1 n 1, v(i) n ), v (i) n N(0, Q n ). PF EnKF PF v (i) n 4.1.2

11 11 {w (i) n } N i=1 N(0, R n ) : x (i) n n 1 = x(i) n n 1 1 N x (j) n n 1 N, ˆV n n 1 = 1 N 1 ˆR n = 1 N 1 N j=1 N j=1 j=1 x (j) n n 1 x (j)t n n 1, w n (j) w n (j)t. : (4.1) ˆKn = ˆV n n 1 H T n (H n ˆVn n 1 H T n + ˆR n ) 1. : (4.2) x (i) n n = x(i) n n 1 + ˆK n (y n + w n (i) H n x (i) n n 1 ). 3 PF (4.2) w (i) n EnKF PF w (i) n (4.2) ˆK n (4.2)

12 w (i) n EnKF PF n = 0 {x (i) 0 0 }N i=1 n = 0 x 0 ( ) N x 0 (Evensen(2003)) 4.2 EnKF (Evensen(2003)) Evensen(2003)

13 : 13 X n = [x (1) n, x(2) n,, x(n) n ], 1 N X n 1 N = [ˆx n, ˆx n,, ˆx n ], X n = X n 1 N X n 1 N, W n = [w n (1), w n (2),, w n (N) ], Y n = [y n,, y n ] (y n N ). 1 N 1 N N ˆx n {x (i) n }N i=1 : (4.2) ˆV n = 1 N 1 X n X T n, ˆR n = 1 N 1 W nw T n. X n n = X n n 1 + ˆV n n 1 H T n (H n ˆVn n 1 H T n + ˆR n ) 1 (Y n + W n H n X n n 1 ) X n n = X n n N 1 X n n 1(I 1 N 1 N)X T n n 1H T n (H n ˆVn n 1 H T n + ˆR n ) 1 (Y n + W n H n X n n 1 ) (4.3) = X n n 1 (I + X T n n 1H T n (H n X n n 1X T n n 1H T n + W n W T n ) 1 (Y n + W n H n X n n 1 )) 1 N X T n n 1 = O

14 14 (4.3) Z n n 1 (4.4) X n n = X n n 1 Z n n 1 Z n n 1 N N 2 1 EnKF PF PF EnKF X n PF (4.4) X n n = X n n 1 Z n n 1 PF Z n n Z n n 1 i 1 x (i) n n (Ensemble Kalman Smoother EnKS) PF (Particle Smoother PS)( (1996) Kitagawa(1996)) L n (n L

15 n n 1) Ĵ n n 1 N N 1 ( i=1 (x (i) 15 n n 1 ˆx n n 1)(x (i) n n 1 ˆx n n 1) T )Hn T (H n ˆVn n 1 Hn T + ˆR n ) 1 x (i) n n x(i) n n 1 + Ĵn n(y n + w n (i) H n x (i) n n 1 ) x (i) n n n (4.5) X n n = X n n 1Z n n 1 Z n n 1 EnKF (4.4) Z n n 1 EnKS PS EnKF PF (4.5) EnKS PS 5. EnKF 5.1 EnKF (Burgers et al.(1998))

16 16 x (i) n n 1 w(i) n 2 x n y 1:n 1 w n N 1 N x (i) n n 1 N x n n 1, ˆVn n 1 V n n 1, 1 N i=1 N i=1 w (i) n 0, ˆRn R n x n n 1, V n n 1 x n n 1 =E[x n y 1:n 1 ], V n n 1 =E[(x n x n n 1 )(x n x n n 1 ) T ] K n N (4.1) ˆK n ˆK n K n ˆx n n ˆx n n x n n 1 + K n (y n H n x n n 1 ) =x n n

17 ˆV n n ˆV n n = 1 N 1 N i=1 17 {(x (i) n n ˆx n n)(x (i) n n ˆx n n) T } = (I ˆK n H n ) ˆV n n N 1 N i=1 { (I ˆK n H n )(x (i) n n 1 ˆx n n 1)w n (i)t ˆK n T + ˆK n w (i) n (x (i) n n 1 ˆx n n 1) T (I ˆK n H n ) T } ( ˆK n = ˆV n n 1 H T n (H n ˆVn n 1 H T n + ˆR n ) 1 ) (I ˆK n H n ) ˆV n n 1 ( 1 N 1 N i=1 w (i) n (x (i) n n 1 ˆx n n 1) T O) (I K n H n )V n n 1 (N ) w n x n ˆV n n V n n (4.1) ˆR n R n 2 Evensen(1994) (4.2) w (i) n ˆV n n ˆV n n (I K n H n )V n n 1 (I K n H n ) T (N ) (Burgers et al.(1998)) EnKF 2

18 18 EnKF N 5.2 PF EnKF h n (3.1) H n (Evensen (2003)) : (5.1) x n = [x T n, h n (x n ) T ] T. x n = [f n (x n 1, v n ) T, h n (f n (x n 1 )) T ] T f n ( x n 1, v n ), y n = [O l k, I l l ] x n + w n H n x n + w n k, l x n, y n I l l l O l k l k

19 19 ε n = y n H n x n n 1 Cov(x n, ε n ) = Cov(x n, H n ( x n x n n 1 ) + w n ) = Cov([I k k, O k l ] x n, H n ( x n x n n 1 ) + w n ) 1 N 1 [I k k, O k l ] X n n 1 X = 1 N 1 X X n n 1 T H n n 1 n T, T H n n 1 n T V ar(ε n ) 1 N 1 ( H T n X n n 1 X H n n 1 n T + W n Wn T ) X n n = X n n 1 + X n n 1 X T n n 1 H T n ( H n X n n 1 X T n n 1 H T n + W n W T n ) 1 (Y n + W n H n Xn n 1 ) EnKF 2 6. PF 6.1 EnKF PF PF

20 20 EnKF 2 2 PF 1 EnKF EnKF 2 EnKF EnKF PF PF 2 EnKF PF

21 21 ( ) EnKF (4.1) PF PF 3 EnKF (4.3) PF ( ) Rejection Control(Liu et al.(1998))

22 22 PF PF 6.2 EnKF EnKS PF PS y n (Carlin et al.(1992), Gordon et al.(1993), Kitagawa and Gersch(1996), Kitagawa(1998)) (6.1) x n = 1x 2 n x n cos(1.2n) + v 1+x 2 n n 1 y n = x2 n 20 + w n x 0 N(0, 5), v n N(0, 1), w n N(0, 10), n = 1,, : (6.1) x n (6.1) v n (Kitagawa (1998))

23 23 EnKF (5.1) y n EnKF EnKS PF PS N 100, 1000, y n x n n=1 (ˆx n x n ) 2 ˆx n EnKS PS L EnKF PF EnKS PS 2 EnKF PF, EnKS PS PF PS PF PS

24 24 EnKF PF x n 100 x n EnKF PF ˆx n PF EnKF EnKF ˆx n x n n = 45 x n x n 0 (Doucet(2001)) EnKF EnKF x n θ n 1 = x 2 n x n cos (1.2n) 1+x 2 n 1 + θ n 1 v n u n x 0 N(0, 5), v n N(0, exp (θ n )), θ 0 U( 2, 6)

25 25 exp (θ n ) v n θ n ˆθ n exp (ˆθ n ) v n ˆθ n = 1 N 1 exp (ˆθ n ) 1 ˆθ n 0 v n 1 N i=1 θ (i) n u n = 0 EnKF u n = 0 PF ˆθ n PF N(0, ) u n = 0 y 1:100 ˆθ y n EnKF u n = 0 Kitagawa(1987) ˆθ 100

26 26 PF 0 n {θ n (j) } N j=1 n = 100 EnKF PF EnKF 0 2 PF 0 PF PF 2 1 EnKF

27 27 2 PF 2 PF EnKF {θ (i) n } N i=1 PF EnKF θ n PF 7. EnKF PF

28 28 PF PF (van Leeuwen(2003)) ( ) EnKF (Evensen (2003)) PF EnKF

29 29 PF EnKF Pham(2001) EnKF smooth bootstrap(efron and Tibshirani(1993), Stavropoulos and Titterington(2001)) (JST/CREST) Anderson, J.L. and Anderson, S.L.(1999). A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Monthly Weather Review, 127, Anderson, J.L.(2001). An Ensemble Kalman Filter for data assimilation, Monthly Weather Review, 129, (1977). Bennett, A.F.(2002). Inverse Modeling of the Ocean and Atmosphere, Cambridge University Press, Cambridge. Burgers, G., van Leeuwen, P.J. and Evensen, G.(1998). Analysis scheme in the ensemble Kalman Filter. Monthly Weather Review, 126, Carlin, B. P., Polson, N. G. and Stoffer, D. S.(1992). A Monte Carlo approach to nonnormal and nonlinear state-space modeling, Journal of the American Statistical

30 30 Association, 87, Courtier, P., Thepaut, T. and Hollingsworth, A.(1994). A strategy for operational implementation of 4DVAR, using an incremental approach, Quarterly Journal of the Royal Meteorological Society, 120, Daley, R.(1991). Atmospheric Data Analysis, Cambridge University Press, Cambridge. Doucet, A., de Freitas, N. and Gordon, N.(2001). An introduction to sequential Monte Carlo methods, Sequential Monte Carlo methods in practice (eds. Doucet, A., de Freitas, N. and Gordon, N.), 3 14, Springer-Verlag, New York. Efron, B. and Tibshirani, R. J.(1993). An Introduction to the Bootstrap, Chapman & Hall, New York. Evensen, G.(1994). Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, Journal of Geophysical Research, 99, 10,143 10,162. Evensen, G. and van Leeuwen, P. J.(2000). An Ensemble Kalman Smoother for nonlinear dynamics, Monthly Weather Review, 128, Evensen, G.(2003). The Ensemble Kalman Filter: theoretical formulation and practical implementation, Ocean Dynamics, 53, Gordon, N. J., Salmond, D. J. and Smith, A. F. M.(1993). Novel approach to nonlinear/non- Gaussian Bayesian state estimation, IEE Proceedings F, 140, (1996) Higuchi, T.(2003). Data assimilation with Monte Carlo mixture Kalman filter toward space weather forecasting, Proceedings of International Symposium on Information Science and Electrical Engineering 2003, Kalman, R. E.(1960). A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82, (1983). (1996) (2005). Kitagawa, G.(1987). Non-Gaussian state-space modeling of nonstationary time series

31 31 (with discussion). Journal of the American Statistical Association, 82, Kitagawa, G.(1996). Monte Carlo filter and smoother for non-gaussian nonlinear state space models, Journal of Computational and Graphical Statistics, 5, Kitagawa, G.(1998). Self-organizing state space model, Journal of the American Statistical Association, 93, Kitagawa, G. and Gersch W.(1996). Smoothness priors analysis of time series, Lecture Notes in Statistics 116, Springer-Verlag, New York. Liu, J. S.(2001). Monte Carlo Strategies in Scientific Computing, Springer-Verlag, New York. Liu, J. S., Chen, R. and Wong, W. H.(1998). Rejection control and sequential importance sampling, Journal of the American Statistical Association, 93, Lorenz, E. N.(1963). Deterministic nonperiodic flow. Quarterly Journal of Atmospheric Science, 20, Manda, A., Hirose, N. and Yanagi, T.(2003). Application of a nonlinear and non-gaussian sequential estimation method for an ocean mixed layer model. Engineering Sciences Reports, Kyushu University, 25, Pham, D. T.(2001). Stochastic methods for sequential data assimilation in strongly nonlinear systems, Monthly Weather Review, 129, Stavropoulos, P., Titterington, D. M.(2001). Improved particle filters and smoothing, Sequential Monte Carlo methods in practice (eds. Doucet, A., de Freitas, N. and Gordon, N.), , Springer-Verlag, New York. (1997) Talagrand, O. and Courtier, P.(1987). Variational assimilation of meteorological observations with the adjoint vorticity equation I: theory. Quarterly Journal of the Royal Meteorological Society, 113, van Leeuwen, P.J.(2003). A variance-minimizing filter for large-scale applications, Monthly Weather Review, 131, Wunsch, C.(1996). The Ocean Circulation Inverse Problem, Cambridge University Press, Cambridge.

32 32 PF EnKF PS EnKS :. 100

33 33 PF EnKF : ˆθ

34 i(i = 1, 2,, M) T i S i (U i, V i ).

35 35 20 observation y n. n y n

36 state EnKF PF x n. x n EnKF PF.

37 integration EnKF PF EnKF PF ˆθ n. EnKF PF ˆθ n exp(ˆθ n ).

38 number of ensemble 100 members time:n EnKF {θ n (j) } N j=1. n θ n

39 number of ensemble 100 members time:n 6. PF {θ n (j) } N j=1. n θ n

40 40 Data Assimilation : Concept and Algorithm Kazuyuki Nakamura (Department of Statistical Science, The Graduate University for Advanced Studies;JST CREST) Genta Ueno and Tomoyuki Higuchi (The Institute of Statistical Mathematics;JST CREST) Data assimilation technique, which is developed in the meteorology and the oceanography, aims at accommodating states of a physical simulation model to observations. It is motivated to provide the good initial condition for the nonlinear simulation model and to realize the online model parameter estimation. To attain these purposes, sequential data assimilation such as the Ensemble Kalman Filter is used. Compared to the Particle Filter, the Ensemble Kalman Filter is similar methodology in view of ensemble-based method, but it has different structures. For example, linear calculation is done instead of resampling at filtering step and approximated probability density does not converge to real probability density even if the number of the ensemble members goes to infinity in almost all cases. In this paper, the concept of data assimilation is reviewed, and is formulated using the nonlinear state space model. On the basis of this formulation, the Ensemble Kalman Filter is reviewed, focusing on the difference and similarity between the Ensemble Kalman Filter and the Particle Filter. After the review, we demonstrate assimilation experiments of nonlinear observations with small scale simulation and the superiority of the Particle Filter to the Ensemble Kalman Filter at these conditions. Applicability to actual system including nonlinear observations is also suggested. Key words: Data Assimilation, Particle Filter, Ensemble Kalman Filter, State Space Model.

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toukei04.dvi 2005 53 2 211 229 c 2005 1,3 2,3 2,3 2005 3 31 2005 8 31 1. Wunsch, 1996; Daley, 1991; Bennett, 2002;, 1997 4 four-dimensional variational 4DVAR Talagrand and Courtier, 1987; Courtier et al., 1994 Ensemble