Asynchronous data assimilation

Transcription

1 Ensemble Kalman Filter, lecture 2 Asynchronous data assimilation Pavel Sakov Nansen Environmental and Remote Sensing Center, Norway This talk has been prepared in the course of evita-enkf project funded by RCN Summer data assimilation school, Sibiu 27 July - 7 August 2009 NERSC

2 Outline Asynchronous assimilation Reasoning Historical overview Asynchronous EnKF: formulation EnKF update and ensemble observations Evolution of corrections Parallel assimilation of asynchronous data with the AEnKF Other schemes EnKS 4DEnKF Asynchronous EnKF: resume

3 Synchronous and asynchronous assimilation Synchronous, or 3D assimilation = observations are assumed to be taken at the assimilation time Asynchronous, or 4D assimilation = observations can be taken at time different than the assimilation time

4 Synchronous and asynchronous assimilation Synchronous, or 3D assimilation = observations are assumed to be taken at the assimilation time Asynchronous, or 4D assimilation = observations can be taken at time different than the assimilation time observations updates (a) (b) (c) t 0 t 1 t2 t 0 t 1 t2 t 0 t 1 t2 (a) Synchronous assimilation at each observation time (b) Synchronous assimilation; asynchronous observations are assumed to be synchronous (c) Asynchronous assimilation

5 Reasoning 0-order correction: x 2 d δy 0 δy 1 Hx f 1 Hx f 0 x 1

6 Reasoning 0-order correction: x 2 d 1 δy 1 d 0 δy 0 Hx f 1 Hx f 0 x 1

7 Reasoning 0-order correction: 1-order correction: x 2 d 1 x 2 Hx t 1 δy 1 δy 1 d 0 δy 0 Hx f 1 Hx t 0 δy 0 Hx f 1 Hx f 0 Hx f 0 x 1 x 1

8 Reasoning 0-order correction: 1-order correction: x 2 d 1 δy 1 x 2 Hx t 1 d 1 δy 1 d 0 δy 0 Hx f 1 Hx t 0 d 0 δy 0 Hx f 1 Hx f 0 Hx f 0 x 1 x 1

9 Reasoning 0-order correction: 1-order correction: x 2 d 1 δy 1 x 2 x a 1 δx 1 d 0 δy 0 Hx f 1 x a 0 δx 0 x f 1 Hx f 0 x f 0 x 1 x 1

10 Historical overview Ensemble smoother (ES) (van Leeuwen and Evensen, 1996) Ensemble Kalman smoother (EnKS) (Evensen and van Leeuwen, 2000) 4DEnKF (Hunt et al., 2004) 4D-LETKF (Hunt et al., 2007)

11 Historical overview Ensemble smoother (ES) (van Leeuwen and Evensen, 1996) Ensemble Kalman smoother (EnKS) (Evensen and van Leeuwen, 2000) 4DEnKF (Hunt et al., 2004) 4D-LETKF (Hunt et al., 2007)

12 Historical overview Ensemble smoother (ES) (van Leeuwen and Evensen, 1996) Ensemble Kalman smoother (EnKS) (Evensen and van Leeuwen, 2000) 4DEnKF (Hunt et al., 2004) 4D-LETKF (Hunt et al., 2007)

13 Historical overview Ensemble smoother (ES) (van Leeuwen and Evensen, 1996) Ensemble Kalman smoother (EnKS) (Evensen and van Leeuwen, 2000) 4DEnKF (Hunt et al., 2004) 4D-LETKF (Hunt et al., 2007)

14 EnKF update and ensemble observations The linear update equations: x a x f δx = A f Gs, A a A f δa = A f T where s = R 1/2 (y Hx f )/ m 1 The gain writes as G = (I + S T S) 1 S T = S T (I + SS T ) 1 where S R 1/2 HA f / m 1. EnKF: T = G(D S) ETKF: T = (I + S T S) 1/2 I DEnKF: T = 1 2 GS HE f R y HA f Hx f S G T s

15 EnKF update and ensemble observations The linear update equations: x a x f δx = A f Gs, A a A f δa = A f T where s = R 1/2 (y Hx f )/ m 1 The gain writes as G = (I + S T S) 1 S T = S T (I + SS T ) 1 where S R 1/2 HA f / m 1. EnKF: T = G(D S) ETKF: T = (I + S T S) 1/2 I DEnKF: T = 1 2 GS HE f R y HA f Hx f S G T s

16 EnKF update and ensemble observations The linear update equations: x a x f δx = A f Gs, A a A f δa = A f T where s = R 1/2 (y Hx f )/ m 1 The gain writes as G = (I + S T S) 1 S T = S T (I + SS T ) 1 where S R 1/2 HA f / m 1. EnKF: T = G(D S) ETKF: T = (I + S T S) 1/2 I DEnKF: T = 1 2 GS HE f R y HA f Hx f S G T s

17 EnKF update and ensemble observations The linear update equations: x a x f δx = A f Gs, A a A f δa = A f T where s = R 1/2 (y Hx f )/ m 1 The gain writes as G = (I + S T S) 1 S T = S T (I + SS T ) 1 where S R 1/2 HA f / m 1. EnKF: T = G(D S) ETKF: T = (I + S T S) 1/2 I DEnKF: T = 1 2 GS HE f R y HA f Hx f S G T s

18 Evolution of corrections Let us assimilate observations at t 0 δx 0 = A f 0 G 0 s 0 δa 0 = A f 0 T 0 Let M 01 be the tangent linear propagator along the forecast system trajectory between t 0 and t 1 : δx 1 = M 01 δx 0 + O ( δx 0 2) At t 1 the corrections become: δx 1 M 01 δx 0 = M 01 A f 0G 0 s 0 A f 1 G 0 s 0 δa 1 M 01 δa 0 = M 01 A f 0T 0 A f 1 T 0

19 Evolution of corrections Let us assimilate observations at t 0 δx 0 = A f 0 G 0 s 0 δa 0 = A f 0 T 0 Let M 01 be the tangent linear propagator along the forecast system trajectory between t 0 and t 1 : δx 1 = M 01 δx 0 + O ( δx 0 2) At t 1 the corrections become: δx 1 M 01 δx 0 = M 01 A f 0G 0 s 0 A f 1 G 0 s 0 δa 1 M 01 δa 0 = M 01 A f 0T 0 A f 1 T 0

20 Evolution of corrections Let us assimilate observations at t 0 δx 0 = A f 0 G 0 s 0 δa 0 = A f 0 T 0 Let M 01 be the tangent linear propagator along the forecast system trajectory between t 0 and t 1 : δx 1 = M 01 δx 0 + O ( δx 0 2) At t 1 the corrections become: δx 1 M 01 δx 0 = M 01 A f 0G 0 s 0 A f 1 G 0 s 0 δa 1 M 01 δa 0 = M 01 A f 0T 0 A f 1 T 0

21 Parallel assimilation of asynchronous data with the EnKF s = [s T 1... st k ]T S = [S T 1...ST k ]T observations update E f HE f

22 EnKS k E a = E f X 5 (t i ), i=1 X 5 = 1 m 11T + (I 1 m 11T )(G δy1 T + T) observations t 1 t 2 t 3... t i... t k 1 t k update E f HE f experiments

23 EnKS k E a = E f X 5 (t i ), i=1 X 5 = 1 m 11T + (I 1 m 11T )(G δy1 T + T) observations t 1 t 2 t 3... t i... t k 1 t k update E f HE f experiments

24 4DEnKF 4DEnKF: Hx 0 HE 0 (E T 1 E 1 ) 1 E T 1 x 1, HA 0 HE 0 (E T 1 E 1 ) 1 E T 1 A 1, What is (E T E) 1 Ex doing? - It is a vector of coefficients of projection of x onto the range of E (E T 1 E 1) 1 E T 1 x 1 = 1/m E 0 (E T 1 E 1) 1 E T 1 x 1 = x 0 (E T 1 E 1) 1 E T 1 A 1 = I 11 T /m E 0 (E T 1 E 1) 1 E T 1 A 1 = A 0

25 4DEnKF 4DEnKF: Hx 0 HE 0 (E T 1 E 1 ) 1 E T 1 x 1, HA 0 HE 0 (E T 1 E 1 ) 1 E T 1 A 1, What is (E T E) 1 Ex doing? - It is a vector of coefficients of projection of x onto the range of E (E T 1 E 1) 1 E T 1 x 1 = 1/m E 0 (E T 1 E 1) 1 E T 1 x 1 = x 0 (E T 1 E 1) 1 E T 1 A 1 = I 11 T /m E 0 (E T 1 E 1) 1 E T 1 A 1 = A 0

26 4DEnKF 4DEnKF: Hx 0 HE 0 (E T 1 E 1 ) 1 E T 1 x 1, HA 0 HE 0 (E T 1 E 1 ) 1 E T 1 A 1, What is (E T E) 1 Ex doing? - It is a vector of coefficients of projection of x onto the range of E (E T 1 E 1) 1 E T 1 x 1 = 1/m E 0 (E T 1 E 1) 1 E T 1 x 1 = x 0 (E T 1 E 1) 1 E T 1 A 1 = I 11 T /m E 0 (E T 1 E 1) 1 E T 1 A 1 = A 0

27 Asynchronous assimilation: resume Recipe: use ensemble observations stored at observation times, as in 4D-LETKF description (Hunt et al., 2007) This method is scheme-independent It is essentially equivalent to the EnKS solution (Evensen and van Leeuwen, 2000) It is well suited for using with local analysis but not with covariance localisation No tangent linear or adjoint model required Do not use 4DEnKF (Hunt et al., 2004)

28 Asynchronous assimilation: resume Recipe: use ensemble observations stored at observation times, as in 4D-LETKF description (Hunt et al., 2007) This method is scheme-independent It is essentially equivalent to the EnKS solution (Evensen and van Leeuwen, 2000) It is well suited for using with local analysis but not with covariance localisation No tangent linear or adjoint model required Do not use 4DEnKF (Hunt et al., 2004)

29 Asynchronous assimilation: resume Recipe: use ensemble observations stored at observation times, as in 4D-LETKF description (Hunt et al., 2007) This method is scheme-independent It is essentially equivalent to the EnKS solution (Evensen and van Leeuwen, 2000) It is well suited for using with local analysis but not with covariance localisation No tangent linear or adjoint model required Do not use 4DEnKF (Hunt et al., 2004)

30 Asynchronous assimilation: resume Recipe: use ensemble observations stored at observation times, as in 4D-LETKF description (Hunt et al., 2007) This method is scheme-independent It is essentially equivalent to the EnKS solution (Evensen and van Leeuwen, 2000) It is well suited for using with local analysis but not with covariance localisation No tangent linear or adjoint model required Do not use 4DEnKF (Hunt et al., 2004)

31 Asynchronous assimilation: resume Recipe: use ensemble observations stored at observation times, as in 4D-LETKF description (Hunt et al., 2007) This method is scheme-independent It is essentially equivalent to the EnKS solution (Evensen and van Leeuwen, 2000) It is well suited for using with local analysis but not with covariance localisation No tangent linear or adjoint model required Do not use 4DEnKF (Hunt et al., 2004)

32 Asynchronous assimilation: resume Recipe: use ensemble observations stored at observation times, as in 4D-LETKF description (Hunt et al., 2007) This method is scheme-independent It is essentially equivalent to the EnKS solution (Evensen and van Leeuwen, 2000) It is well suited for using with local analysis but not with covariance localisation No tangent linear or adjoint model required Do not use 4DEnKF (Hunt et al., 2004)

33 References Evensen, G. and P. J. van Leeuwen, 2000: An ensemble Kalman smoother for nonlinear dynamics. Mon. Wea. Rev., 128, Hunt, B. R., E. Kalnay, E. J. Kostelich, E. Ott, D. J. Patil, T. Sauer, I. Szunyogh, J. A. Yorke, and A. V. Zimin, 2004: Four-dimensional ensemble Kalman filtering. Tellus, 56A, Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, Sakov, P., G. Evensen, and L. Bertino, 2009, submitted: Asynchronous data assimilation with the EnKF. Tellus A. van Leeuwen, P. J. and G. Evensen, 1996: Data assimilation and inverse methods in terms of a probabilistic formulation. Mon. Wea. Rev., 124,