An iterative ensemble Kalman filter in presence of additive model error Marc Bocquet 1 , Pavel Sakov 2 , Jean-Matthieu Haussaire 1 (1) CEREA, joint lab ´ Ecole des Ponts ParisTech and EdF R&D, Universit´ e Paris-Est, France Institut Pierre-Simon Laplace (2) Environment and Research Division, Bureau of Meteorology, Melbourne, Australia (marc.bocquet@enpc.fr) 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 1 / 32
What this talk is about . . . ◮ Iterative ensemble Kalman smoother (IEnKS): exemplar of nonlinear four-dimensional EnVar methods. ◮ It propagates the error statistics from one cycle to the next with the ensemble (errors of the day). ◮ It performs a 4D-Var analysis at each cycle (within the ensemble subspace). ◮ Typical cycling ( L = 6, S = 2): y L − 3 y L − 2 t L − 2 t L − 3 t L − 2 y L − 1 y L S ∆ t t L t L − 1 t L y L + 1 y L + 2 S ∆ t t L + 2 t L + 1 t L + 2 L ∆ t Variational analysis in ens. space → Posterior ens. generation → Ens. forecast 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 2 / 32
Cost functions ◮ General cost function over [ t 1 ,..., t L ]; weak-constraint formalism: L L 1 � 2 � y i − H i ( x i ) � 2 � x i − M i ( x i − 1 ) � 2 J L ( x 1 ,..., x L ) = � x 1 − x f ∑ ∑ 1 ) − 1 + + i . R − 1 Q − 1 ( P f i i =1 i =2 ◮ Configurations addressed in this talk: ◮ The case L = 1, S = 1, the so-called iterative ensemble Kalman filter → IEnKF: J L ( x 1 , x 2 ) = � x 1 − x f 1 � 2 1 ) − 1 + � y 2 − H 2 ◦ M 2 ( x 1 ) � 2 2 . R − 1 ( P f ◮ The IEnKF but, now, with additive model error → IEnKF-Q : J L ( x 1 , x 2 ) = � x 1 − x f 1 � 2 1 ) − 1 + � y 2 − H 2 ( x 2 ) � 2 2 + � x 2 − M 2 ( x 1 ) � 2 2 . ( P f R − 1 Q − 1 ◮ The linearized case L +1 = S with additive model error, called the asynchronous ensemble Kalman filter → AEnKF. P. Sakov, J.-M. Haussaire, and M. Bocquet , An iterative ensemble Kalman filter in presence of additive model error , Q. J. R. Meteorol. Soc., 0 (2017), pp. 0–0. Submitted 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 3 / 32
The iterative ensemble Kalman filter (IEnKF) Outline The iterative ensemble Kalman filter (IEnKF) 1 Theory of the IEnKF-Q 2 Formulation Decoupling Base algorithm Numerics for the IEnKF-Q 3 Asynchronous EnKF with additive model error 4 Conclusions 5 References 6 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 4 / 32
The iterative ensemble Kalman filter (IEnKF) Iterative ensemble Kalman filter: a Bayesian standpoint ◮ Gaussian assumption for the prior: p ( x 1 ) = n ( x 1 | x 1 , P 1 ) . ◮ Forecast under perfect model assumption: p ( x 2 | x 1 ) ∝ δ { x 2 − M 2 ( x 1 ) } . ◮ Likelihood used in the analysis: p ( y 2 | x 2 ) = n ( y 2 − H 2 ( x 2 ) | 0 , R 2 ) . ◮ (Full cycle) analysis of the initial condition x 1 : p ( x 1 | y 2 ) ∝ p ( y 2 | x 1 ) p ( x 1 ) ∝ p ( y 2 | x 2 = M 2 ( x 1 )) p ( x 1 ) . ◮ Analysis (forecast!) of the filtering distribution: � p ( x 2 | y 2 ) = d x 1 p ( x 2 | x 1 , y 2 ) p ( x 1 | y 2 ) � = d x 1 δ { x 2 − M 2 ( x 1 ) } p ( x 1 | y 2 ) . M. Bocquet and P. Sakov , An iterative ensemble Kalman smoother , Q. J. R. Meteorol. Soc., 140 (2014), pp. 1521–1535 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 5 / 32
The iterative ensemble Kalman filter (IEnKF) Iterative ensemble Kalman filter: a variational standpoint ◮ Analysis IEnKF cost function in state space p ( x 1 | y 2 ) ∝ exp( − J ( x 1 )): J ( x 1 ) = 1 2 + 1 2 � y 2 − H 2 ◦ M 2 ( x 1 )) � 2 2 � x 1 − x 1 � 2 1 . R − 1 P − 1 ◮ Reduced scheme in ensemble subspace, x 1 = x 1 + A 1 w 1 , where A 1 is the normalized ensemble anomaly matrix: � J ( w 1 ) = J ( x 1 + A 1 w 1 ) . ◮ IEnKF cost function in ensemble space: J ( w 1 ) = 1 2 + 1 � 2 � y 2 − H 2 ◦ M 2 ( x 1 + A 1 w 1 ) � 2 2 � w 1 � 2 . R − 1 P. Sakov, D. S. Oliver, and L. Bertino , An iterative EnKF for strongly nonlinear systems , Mon. Wea. Rev., 140 (2012), pp. 1988–2004 M. Bocquet and P. Sakov , Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems , Nonlin. Processes Geophys., 19 (2012), pp. 383–399 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 6 / 32
The iterative ensemble Kalman filter (IEnKF) Iterative ensemble Kalman filter: minimization scheme ◮ As a variational reduced method, one can use Gauss-Newton [Sakov et al., 2012] , Levenberg-Marquardt [Bocquet and Sakov, 2012; Chen and Oliver, 2012] , etc, minimization schemes (not limited to quasi-Newton). ◮ Gauss-Newton scheme: w ( j +1) = w ( j ) J ( j ) ( w ( j ) 1 − � H − 1 ( j ) ∇ � 1 ) , 1 x ( j ) 1 = x 1 + A 1 w ( j ) 1 , � � J ( j ) = w ( j ) y 2 − H 2 ◦ M 2 ( x ( j ) ∇ � 1 − Y T ( j ) R − 1 1 ) , 2 H ( j ) = I N + Y T � ( j ) R − 1 2 Y ( j ) , Y ( j ) = [ H 2 ◦ M 2 ] ′ 1 A 1 . | x ( j ) 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 7 / 32
The iterative ensemble Kalman filter (IEnKF) Iterative ensemble Kalman filter: computing the sensitivities ◮ Sensitivities Y ( p ) computed by ensemble propagation without TLM and adjoint ( [Gu and Oliver, 2007; Liu et al., 2008; Buehner et al., 2010] ) ◮ First alternative [Sakov et al., 2012] : the transform scheme. The ensemble is preconditioned before its propagation using the ensemble transform � � − 1 / 2 ( j ) R − 1 Y ( j ) I N + Y T T ( j ) = , obtained at the previous iteration. The inverse transformation is applied after propagation. ◮ Second alternative [Bocquet and Sakov, 2012] : the bundle scheme. It simply mimics the action of the tangent linear by finite difference: �� � � I N − 11 T Y ( j ) ≈ 1 x ( j ) 1 T + ε A 1 ε H 2 ◦ M 2 . N 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 8 / 32
Theory of the IEnKF-Q Outline The iterative ensemble Kalman filter (IEnKF) 1 Theory of the IEnKF-Q 2 Formulation Decoupling Base algorithm Numerics for the IEnKF-Q 3 Asynchronous EnKF with additive model error 4 Conclusions 5 References 6 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 9 / 32
Theory of the IEnKF-Q Formulation IEnKF-Q: formulation ◮ Analysis cost function: 1 � 2 1 ) − 1 + � y 2 − H ( x 2 ) � 2 R − 1 + � x 2 − M ( x 1 ) � 2 J ( x 1 , x 2 ) = � x 1 − x a Q − 1 . ( P a ◮ Ensemble subspace representation: 1 ) T = P a x 1 = x a 1 + A a A a A a 1 ( A a 1 u , 1 , 1 1 = 0 , x 2 = M ( x 1 )+ A q A q 2 ( A q 2 ) T = Q , A q 2 v , 2 1 = 0 . ◮ Cost function in ensemble subspace: J ( u , v ) = u T u + v T v + � y 2 − H ( x 2 ) � 2 R − 1 . 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 10 / 32
Theory of the IEnKF-Q Formulation IEnKF-Q: formulation ◮ Compactification: � u � J ( w ) = w T w + � y 2 − H ( x 2 ) � 2 w ≡ = ⇒ R − 1 . v ◮ Condition of zero gradient: w − ( HA ) T R − 1 [ y 2 − H ( x 2 )] = 0 , where 1 , A q A ≡ [ MA a 2 ] , H ≡ ∇ H ( x 2 ) , M ≡ ∇ M ( x 1 ) . ◮ The cost function can be minimized using a Gauss-Newton method w i +1 = w i − D i ∇ J ( w i ) , where the inverse Hessian is approximated as � I +( H i A i ) T R − 1 H i A i � − 1 D i ≈ . 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 11 / 32
Theory of the IEnKF-Q Formulation IEnKF-Q: formulation ◮ Posterior anomalies: 1 δ u + A q δ x 1 = A a δ x 2 = MA a 1 δ u , 2 δ v , ◮ Updated perturbations over [ t 1 , t 2 ]: 2 ) T = E [ δ x ⋆ 2 ) T ] = A ⋆ E [ w ⋆ ( w ⋆ ) T ]( A ⋆ ) T = A ⋆ D ⋆ ( A ⋆ ) T , 2 ( δ x ⋆ A a 2 ( A a which implies 2 = A ⋆ ( D ⋆ ) 1 / 2 = A ⋆ � I +( H ⋆ A ⋆ ) T ( R ) − 1 H ⋆ A ⋆ � − 1 / 2 A a . ◮ Updated (smoothed) perturbations at t 1 : 1 ) T = E [ δ x ⋆ 1 ( δ x ⋆ 1 E [ u ⋆ ( u ⋆ ) T ]( A a A s 1 ( A s 1 ) T ] = A a 1 ) T , which implies 1 ( D ⋆ 1: m , 1: m ) 1 / 2 . A s 1 = A a 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 12 / 32
Theory of the IEnKF-Q Decoupling IEnKF-Q: Decoupling ◮ In all generality: J ( x 1 , x 2 ) = − 2ln p ( x 1 , x 2 | y 2 ) = − 2ln p ( x 2 | x 1 , y 2 ) p ( x 1 | y 2 ) . ◮ If the observation operator H is linear: 1 � 2 − 2ln p ( x 1 | y 2 ) = � x 1 − x a 1 ) − 1 + � y 2 − H ◦ M ( x 1 ) � 2 ( R + HQH T ) − 1 + c 1 , ( P a and − 2ln p ( x 2 | x 1 , y 2 ) = � x 2 − M ( x 1 ) − QH T ( R + HQH T ) − 1 [ y 2 − H ◦ M ( x 1 )] � 2 Q − 1 + H T R − 1 H + c 2 , ◮ Then the MAP of J ( x 1 , x 2 ) can be computed in two steps: ◮ Minimize − 2ln p ( x 1 | y 2 ) over x 1 just like the IEnKF in the absence of model error but with R → R + HQH T . ◮ The MAP of − 2ln p ( x 2 | x ⋆ 1 , y 2 ) is then directly given by: 1 )+ QH T � R + HQH T � − 1 x ⋆ 2 = M ( x ⋆ [ y 2 − H ◦ M ( x ⋆ 1 )] . 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 13 / 32
Theory of the IEnKF-Q Decoupling IEnKF-Q: Decoupling ◮ This decoupling also implies the decoupling of ( u , v ): � u i +1 − u i = D i u ) − 1 ( HM i A a 1 ) T ( R i u � � − u i � y 2 − H ◦ M ( x a 1 + A a 1 u i ) × . v ⋆ = D ⋆ v ) − 1 [ y 2 − H ( x ⋆ v ( HA q 2 ) T ( R ⋆ 2 )+ HM ⋆ A a 1 u ⋆ ] . ◮ However, this decoupling does not convey to the perturbations update! ◮ The same decoupling is used in particle filtering [Doucet et al., 2000] to build the optimal importance proposal particle filter. 12 th EnKF workshop, Os, Norway, 12-14 June 2017 M. Bocquet 14 / 32
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