Joint state and parameter estimation with an iterative ensemble Kalman smoother Marc Bocquet 1 , 2 , Pavel Sakov 3 1 Universit´ e Paris-Est, CEREA, joint lab ´ Ecole des Ponts ParisTech and EdF R&D, France 2 INRIA, Paris-Rocquencourt Research center, France 3 Bureau of Meteorology, Australia (bocquet@cerea.enpc.fr) 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 1 / 22
Context ◮ New methods called ensemble variational methods that mix variational and ensemble approaches (see Lorenc, 2013 for an almost perfect definition): Hybrid methods, 4D-Var-Ben, 4D-En-Var, Ensemble of data assimilation (EDA) and IEnKF/IEnKS. Lorenc A. 2013. Recommended nomenclature for EnVar data assimilation methods. In Research Activities in Atmospheric and Oceanic Modelling , WGNE. ◮ The IEnKF/IEnKS differ from the other ones in that they are more natural (simple?), regardless of the numerical cost. ◮ The IEnKS has a great potential for parameter estimation, as it is variational but avoids the derivation of the adjoint. 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 2 / 22
Context The IEnKS: at the crossroad between the EnKF and 4D-Var ◮ The IEnKS follows the scheme of the EnKF: Analysis in ensemble space → Posterior ensemble generation → Ensemble forecast ◮ Except that The analysis in ensemble space is variational [e.g. Zupanski, 2005] over a finite time windows. It may require several iterations in strongly nonlinear conditions [Gu & Oliver, 2007; Sakov et al., 2012; Bocquet and Sakov, 2012-2014] . The gradient of the 4D cost function is computed with the ensemble [Gu & Oliver, 2007;Liu et al., 2008] : no need for the tangent linear/adjoint. ◮ It generalises the iterative extended Kalman filter/smoother [Wishner et al., 1969; Jazwinski, 1970; Bell, 1994] to ensemble methods. ◮ It is a unified/straightforward scheme (no hybridization so to speak). 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 3 / 22
The iterative ensemble Kalman smoother The IEnKS: the cycling ◮ L : length of the data assimilation window, ◮ S : shift of the data assimilation window in between two updates. y y L−3 L−2 t L−1 t t L−3 L−2 y L y L−1 S ∆ t t L t t t t L−1 0 1 L y y L+1 L+2 S ∆ t t L+1 t t L+1 L+2 L ∆ t 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 4 / 22
The iterative ensemble Kalman smoother The IEnKS: a variational standpoint ◮ Analysis IEnKS cost function in state space p ( x 0 | y L ) ∝ exp( − J ( x 0 )): L 1 2 ( y k − H k ◦ M k ← 0 ( x 0 )) T β k R − 1 ∑ J ( x 0 ) = k ( y k − H k ◦ M k ← 0 ( x 0 )) k =1 + 1 2 ( x 0 − x 0 ) P − 1 0 ( x 0 − x 0 ) . (1) { β 0 , β 1 ,..., β L } weight the observations impact within the window. ◮ Reduced scheme in ensemble space, x 0 = x 0 + A 0 w , where A 0 is the ensemble anomaly matrix: � J ( w ) = J ( x 0 + A 0 w ) . (2) ◮ IEnKS cost function in ensemble space [Hunt et al., 2007; Bocquet and Sakov, 2012] : L J ( w ) =1 ( y k − H k ◦ M k ← 0 ( x 0 + A 0 w )) T β k R − 1 � ∑ k ( y k − H k ◦ M k ← 0 ( x 0 + A 0 w )) 2 k =1 + 1 2( N − 1) w T w . (3) 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 5 / 22
The iterative ensemble Kalman smoother The IEnKS: minimisation scheme ◮ As a variational reduced method, one can use Gauss-Newton [Sakov et al., 2012] , Levenberg-Marquardt [Bocquet and Sakov, 2012; Chen and Oliver, 2013] , quasi-Newton, etc., minimisation schemes. ◮ Gauss-Newton scheme (the Hessian is approximate): w ( p +1) = w ( p ) − � H − 1 ( p ) ∇ � J ( p ) ( w ( p ) ) , + A 0 w ( p ) , x ( p ) = x (0) 0 0 � � L y k − H k ◦ M k ← 0 ( x ( p ) +( N − 1) w ( p ) , ∇ � k , ( p ) β k R − 1 ∑ Y T J ( p ) = − 0 ) k k =1 L � k , ( p ) β k R − 1 ∑ Y T H ( p ) = ( N − 1) I N + L Y ( p ) , k =1 Y k , ( p ) = [ H k ◦ M k ← 0 ] ′ 0 A 0 . (4) | x ( p ) ◮ One solution to compute the 4D sensitivities: the bundle scheme. It simply mimics the action of the tangent linear by finite difference: �� � � I N − 11 T Y k , ( p ) ≈ 1 x ( p ) 1 T + ε A 0 ε H k ◦ M k ← 0 . (5) N 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 6 / 22
The iterative ensemble Kalman smoother The IEnKS: ensemble update and the forecast step ◮ Generate an updated ensemble from the previous analysis: √ 0 1 T + H − 1 / 2 E ⋆ 0 = x ⋆ N − 1 A 0 � where U1 = 1 . (6) U ⋆ ◮ Just propagate the updated ensemble from t 0 to t S : E S = M S ← 0 ( E 0 ) . (7) In the quasi-static case: S = 1. 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 7 / 22
The iterative ensemble Kalman smoother IEnKS: single vs multiple data assimilation β β β L−1 L 0 y y y L−3 L−2 −2 t L−1 t t L−3 L−2 β β β 0 L−1 L y L y y L−1 S ∆ t 0 t L t t t t L−1 0 1 L β β L−1 L β y y 0 y L+1 L+2 S ∆ t 2 t L+1 t t L+1 L+2 L ∆ t ◮ SDA IEnKS: The observation vector are assimilated once and for all. Exact scheme. ◮ MDA IEnKS: The observation vector are assimilated several times and poundered with weights β k within the window. Exact scheme in the linear/Gaussian limit. More stable for long windows than the SDA scheme. 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 8 / 22
Numerical experiments Application to the Lorenz-95 model ◮ Weakly nonlinear case: Lorenz-95, M = 40, N = 20, ∆ t = 0 . 05, R = I . ◮ Comparison of 4D-Var S = 1, EnKS S = 1, SDA IEnKS S = 1, SDA IEnKS S = L , and MDA IEnKS S = 1. 0.22 0.20 4D-Var S=1 0.220 0.18 4D-Var S=1 EnKS-N S=1 EnKS-N S=1 SDA IEnKS-N S=1 0.16 SDA IEnKS-N S=1 0.210 SDA IEnKS-N S=L SDA IEnKS-N S=L Smoothing analysis RMSE MDA IEnKS-N S=1 0.14 Filtering analysis RMSE MDA IEnKS-N S=1 0.200 0.12 0.195 0.10 0.190 0.09 0.185 0.08 0.180 0.175 0.07 0.170 0.06 0.165 0.05 0.160 0.155 0.04 1 5 10 15 20 25 30 35 40 45 50 1 5 10 15 20 25 30 35 40 45 50 DAW length L DAW length L 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 9 / 22
Numerical experiments Application to the Lorenz-95 model ◮ Strongly nonlinear case: Lorenz-95, M = 40, N = 20, ∆ t = 0 . 20, R = I . ◮ Comparison of 4D-Var S = 1, EnKS S = 1, SDA IEnKS S = 1, SDA IEnKS S = L , and MDA IEnKS S = 1. 0.46 0.40 0.38 0.36 0.44 0.34 0.32 0.42 0.30 4D-Var S=1 EnKS-N S=1 0.28 Smoothing analysis RMSE 0.40 Filtering analysis RMSE 0.26 SDA IEnKS-N S=1 0.39 SDA IEnKS-N S=L 0.24 0.38 MDA IEnKS-N S=1 0.22 0.37 0.20 0.36 0.35 0.18 0.34 0.16 0.33 0.14 0.32 0.31 4D-Var S=1 0.12 EnKS-N S=1 0.30 SDA IEnKS-N S=1 0.29 0.10 SDA IEnKS-N S=L MDA IEnKS-N S=1 0.28 0.27 0.08 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 DAW length L DAW length L 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 10 / 22
Localisation IEnKF/IEnKS: Localisation ◮ Localisation in an EnVar context is non-trivial because localisation and the evolution model do not commute: � � M k ← 0 ( C ◦ B 0 ) M T M k ← 0 B 0 M T k ← 0 � = C ◦ . (8) k ← 0 ◮ Local analysis of IEnKF, and comparison with a non-scalable optimal approach. 0.7 0.6 0.5 Analysis rmse 0.4 0.3 CL IEnKF (bundle, opt. infl., c=10, non-scalable) N=10 IEnKF-N (bundle) N=20 0.2 LA IEnKF-N (bundle, c=10, ε N =1) N=10 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 Time interval between updates 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 11 / 22
Localisation IEnKF/IEnKS: Localisation ◮ Local analysis of IEnKS, and comparison with a non-scalable optimal approach (filtering performance). 0.20 Analysis RMSE 0.18 SDA IEnKS-N filtering N=20 MDA IEnKS-N filtering N=20 0.16 LA EnKS-N filtering N=10 l=10 LA MDA IEnKS-N filtering N=10 l=10 NSCL SDA IEnKS opt.infl. filtering N=10 l=10 LA SDA IEnKS-N filtering N=10 l=10 EnKS-N filtering N=20 1 2 3 4 5 6 7 8 9 10 15 20 30 40 50 DAW length L 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 12 / 22
Augmented state formalism IEnKF/IEnKS: Augmented state formalism ◮ IEnKS treats parameters the way both 4D-Var and EnKF treat them. ◮ The state space is augmented from x ∈ R M to a vector � x � ∈ R M + P , z = (9) θ Technically, there is nothing more to the joint state and parameter IEnKS than in the state IEnKS. ◮ A forward model needs to be introduced for the parameters: For instance, the persistence model ( θ k +1 = θ k ), or some jittering such as a Brownian motion ( θ k +1 = θ k + ε k ). 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 13 / 22
Augmented state formalism Estimation of the Lorenz-95 forcing parameter F ◮ F is static but unknown. EnKF-N EnKS-N L=50 IEnKF-N 8.10 IEnKS-N L=5 IEnKS-N L=10 IEnKS-N L=30 Analysis of parameter F 8.05 8 7.95 7.90 0 1000 2000 3000 4000 5000 Time ◮ Augmented state vector ∈ R 41 , N = 20. The forcing of the true model is F = 8. 9 th EnKF workshop, Bergen, Norway, 22-24 June 2014 M. Bocquet 14 / 22
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