:,,, PACS: a. , (3D-Var) (4D-Var) [7 9].,. 4D-Var 3D-Var,.,4D-Var,., ,.,, ;,,,,. ( ), . (PF). PF EnKF,,,

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1 * (, ) ( ; ) (PDF),... :,,, PACS: a 1,.,, ;,,,. ( ), [1]. (KF) [2 6]. KF,. (EKF) KF,. (EnKF), (UKF).,,., (3D-Var) (4D-Var) [7 9].,. 4D-Var 3D-Var,.,4D-Var,.,. (PF). PF EnKF,,, [10]. PF, Pitt (auxiliary particle filter) [11], Musso (regularized particle filter) [12], Arulampalam (likelihood particle filter) [13], Kunihama MCMC (improved MCMC particle filter) [14], [15], [16],. Kotecha, Djuric Van der Merwe [17] * ( : , ). hzleng@nudt.edu.cn c 2012 Chinese Physical Society

2 [18] σ (gaussian mixture sigma-point particle filter) [19],. Eyink (maximum entropy particle filtering) [20],,., [21],,,,,,. 2 (dynamic state space, DSS) [22]. {ψ k,k N}, {d k,k N},, k, ψ k R n d k R m k. Markov,,. ψ k = f(ψ k 1,μ k ), (1) d k = h(ψ k,υ k ),, μ k R n υ k R m., Bayesian p(ψ k d k )= p(d k ψ k ) p(ψ k ), (2) p(d k ψ k ) p(ψ k )dψ k, p(d k ψ k ), p(ψ k ) (PDF), p(ψ k d k ) PDF., PDF,, Bayesian (minimum mean square error, MMSE) [23], (KF)., p(ψ k d k ),,, [24], EnKF. p(ψ), N {ψ i : i =1, 2,,N}, p(ψ) = 1 N δ(ψ ψ i ), (3) N i=1 ψ f(ψ) E(f(ψ)) = f(ψ)p(ψ)dψ E P (f(ψ)) = 1 N f(ψ i ). N i=1 (4) PDF, q,, w k i = w k 1 i i ) p(dk ψ k i )p(ψk i ψk 1 q(ψ k i ψk 1 i,d n ), (5), p(ψi k ψk 1 i ), q(ψi k ψk 1 i,d n ),, p(ψ k d k ). q. q(ψi k ψk 1 i,d n ) = p(ψ k ψ k 1 ) [24], w k i = w k 1 i p(d k ψ k i ). (6),, [21],,,. 3, PDF,, PDF,.,,,. ψ k = f(ψ k 1 )+μ k, d k = h(ψ k )+υ k, (7)

3 , k, f(ψ k 1 ) h(ψ k ), μ k υ k ( ),, Q R. 3.1, ( ),,,,., dt =0.01s, n dt = n 0.01s(n =10 100),,,. d ln+j = d ln + j n (d(l+1)n d ln ), (8), n, l {0, 1, 2, }, j {0, 1,,n}, d ln,d (l+1)n. j (j 0,n),,,, j (j 0,n) R j = R (1 + n/2 e n/2 j 2 ), j=1, 2,,n 1, (9), R j.,, j =1, 2,,n 1, ψ k,old = f(ψ k 1 )+μ k, (10) ψ k,new = ψ k,old + K(d k + γ k h(ψ k,old )), (11), k = ln + j, γ k = γ ln+j R j, K, (EKF), K = PH T (HPH T + R j ) 1, (12), H h, P. 3.2,, ,,,,.,,,,.,. (auxiliary particle filter) [11],,,,,,,.,, Z = 1 trace(r)/trace(hqht ) M 1 N, (13), M, N, 1/N, trace( ). Z, ;. Z M,,,, ;, Z trace(r)/trace(hqh T ),

4 ,,,,., ln 1 ln,,.,,, ln ψ k = f(ψ k 1,new )+K(d k h(f(ψ k 1,new ))) + μ k, (14) ψ k 1,new, K, μ k, μ k,, Q. K(d k h(f(ψ k 1,new ))), K., K. EnKF,,.,. ψ = w i ψ i, (15) i P f = w i (ψ i ψ)(ψ i ψ) T. (16) i.,, K,. K i = α i P f H T (HP f H T + R) 1, (17) / 1, α i = w i N,, α i,,,., w i p(dk ψi k)p(ψk ψk 1 i i ) q(ψ k i ψk 1 i,d n ) exp[ 1 2 (dk ψi k )T R 1 (d k ψi k ) 1 2 (ψk i f(ψ k 1 i )) T Q 1 (ψi k f(ψ k 1 i )) μt Q μ], (18),,, 1/N. 3.3, 0,.,,. 1 (l 1)n + j(j {1, 2,,n 1}). 2 d (l 1)n+j R j. 3 (l 1)n + j. 4 j = n 1, ln;, w i. 5 w i,. 6, w i. 7 w i ψ, P f K, ψ ln. 8 w i, w i. 4,. [20,25] Lorenz63 Lorenz63,

5 dx = a(y x), dt dy =(b z)x y, dt dz = xy cz. dt (19) a =10,b =28,c = 8/3 [20]. Runge-Kutta, dt =0.01, T = 1000., m =20, x, H = I 1, R =2. x 0 = , y 0 = , z 0 = , , , Q = , , , , , Σ ini = 2I 3, I 1,I 3., P = 1 (ψ i N ψ)(ψ i ψ) T. i 1 (,, * ) (a) SPF; (b) EnKF; (c) N =20. Lorenz63,. 1, 1(a) (SPF), 1(b) (EnKF), 1(c). 1, SPF,, [26] ;EnKF SPF,,

6 ,. 1,, R, Q. 1,., 2,, EnKF,,,. 1 (100, m=20, N=20, SPF, EnKF, newpf ) R /R Q /Q SPF EnKF newpf ,, EnKF, SPF, 3 4, SPF,. 5,,,. 6 7,,. 2 (100, m=10, N=3, 1) R /R Q /Q SPF EnKF newpf m =10 N =3. 2,, SPF, 3.,,, 6 7.,, 4,, 4D-Var. 4.2 Lorenz96, [21],. Lorenz96, dx j dt =(x j+1 x j 2 )x j 1 x j +F, j =1, 2,,J (20), J. Runge-Kutta, dt =0.005, F =8, J =40, j =20, F = , ( [25] ). 10, j =2i(i =1, 2,,J/2). σ obs =1, σ ini =1, Q 1/2 =0.05C 1/2, C, 1, 0.5, , T = 1000, N =20., P =0.05C., EnKF. 2,, 10 ( ),. 2,,,,., EnKF 300,,,,,

7 ,,., EnKF,, EnKF,. diff(t) = 1 J (xs t j J xtt j )2, xs t j j=1 xt t j t j, diff(t) t. 4 diff(t) t., diff(t) 1 4.5, EnKF. 2 (,, ) (a) ;(b)enkf 3 (a) ; (b)enkf 4 (40D, σ obs =1) (a) ; (b)enkf

8 50, RMS newpf(diff t ) 1.51, EnKF RMS EnKF 2.86.,,, σ obs =0.5σ obs,, RMS newpf = 0.95, RMS EnKF = 2.27,. 400D,, EnKF, RMS newpf = 2.15, RMS EnKF = 4.53, 5. 5 (400D, σ obs =0.5) (a) ; (b)enkf 5, KF.,.,,,,.,,, PDF,., EnKF K, K. KF,,,.,,,,. [1] Zou X L 2009 Data Assimilation-Theory and Application (Vol.1)(Beijing China Meteorological Press) p43 (in Chinese) [ ( )( ) 43 ] [2] Evensen G 1994 J. Geophys. Res [3] Evensen G, Van Leeuwen P J 2000 Mon Wea. Rev [4] Houtekamer P L, Mitchell H L 2001 Mon Wea. Rev [5] Lorenc A C 2003 Q. J. R. Meteorol. Soc [6] Julier S, Uhlmann J, Durrant W 2000 IEEE Transactions on Automated Control, Technical Notes and Correspondence [7] Liu C S, Xiao Q N, Wang B 2008 Mon Wea. Rev [8] Navon I M, Zou X, Derber J 1992 Mon Wea. Rev [9] Cao X Q, Huang S X, Zhang W M 2008 Acta Phys. sin (in Chinese) [,, ] [10] Michail D V, Dan C, Manfred O 2011 Physica D (in Press) [11] Pitt M, Shephard N 1999 J. Amer. Statist. Assoc [12] Musso C, Oudjane N, LeGland F 2001 Sequential Monte Carlo Methods in Practice (New York: Springer-Verlag) pp [13] Arulampalam M S, Maskell S, Gordon N, Clapp T 2002 IEEE Transactions On Signal Processing

9 [14] Kunihama T, Omori Y, Zhang Z J 2011 Journal of Time Series Analysis dx.doi.org/ / j x [15] Cheng S Y, Zhang J Y 2008 Acta Electronica Sinica (in Chinese) [, ] [16] Du Z C, Tang B, Li K 2006 Acta Phys. Sin (in Chinese) [,, ] [17] Kotceha J H, Djuric P M 2003 IEEE Transactions On Signal Processing [18] Kotceha J H, Djuric P M 2003 IEEE Transactions On Signal Processing [19] VanderMerwe R, Wan E 2003 Proceedings of the IEEE International Conference on Acoustics Speech and Signal Processing(ICASSP) [20] Eyink G L, Kim S 2006 J. Stat. Phys [21] VanLeeuwen P J 2009 Mon Wea. Rev [22] Li X R, Jilkov V P 2003 IEEE Trans. AES [23] Bar S. Y, Li X R, Kirubarajan T 2001 Estimation with Applications to Tracking and Navigation: Theory, Algorithms, and Software(New York: John Wiley & Sons) pp [24] Doucet A, Freitas N D, Gordon N 2001 Sequential Mente Carlo Methods in Practice(New York Springer-Verlag) pp25 26 [25] VanLeeuwen P J 2010 Q. J. R. Meteorol. Soc [26] Nakano S, Ueno G, Higuchi T 2007 Nonlin. Processes Geophys Improved particle filter in data assimilation Leng Hong-Ze Song Jun-Qiang Cao Xiao-Qun Yang Jin-Hui ( College of Computer, National University of Defense Technology, Changsha , China ) ( Received 25 June 2011; revised manuscript received 11 August 2011, China ) Abstract Owing to the fact that standard particle filter and ensemble Kalman filter can not efficiently represent the posterior probability density function (PDF), an improved particle filter is proposed. In this algorithm, an innovation step is introduced after the prediction step, and the analyses of non-observation time and observation time are treated separately. The numerical simulations of a low- and a high-dimensional systems show that this new particle filter can follow the true state of a highly nonlinear non-gaussian system very well. Keywords: assimilation, nonlinear, particle filter, ensemble Kalman filter PACS: a * Project supported by the National Natural Science Foundation of China (Grant Nos , ). hzleng@nudt.edu.cn