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4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Optimal state estimation for numerical weather prediction using reduced order models A.S. Lawless 1 , C. Boess 1 , N.K. Nichols 1 and A. Bunse-Gerstner 2 1 School of Mathematical and Physical Sciences, University of Reading, U.K. 2 ZeTeM, Universtaet Bremen, Germany. A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Outline 1 4D-Var and Gauss-Newton 2 Model reduction in data assimilation 3 Extension to unstable systems 4 Numerical results 5 Conclusions A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions 4-Dimensional Variational Data Assimilation (4D-Var) Aim To find the best estimate of the true state of a system consistent with observations distributed in time and with the system dynamics. x x b t N t 0 Time A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions 4-Dimensional Variational Data Assimilation (4D-Var) Aim To find the best estimate of the true state of a system consistent with observations distributed in time and with the system dynamics. x x a x b t 0 t N Time A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Nonlinear least squares problem We consider a dynamical system x i +1 = M i ( x i ) H ( x i ) + η i y i = with M i Model dynamics State vector O (10 7 − 10 8 ) x i Observations at time t i O (10 6 − 10 7 ) y i Observational noise η i H i Observation operator A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Nonlinear least squares problem Then the data assimilation problem is to minimize N J [ x 0 ] = 1 0 ( x 0 − x b )+1 2( x 0 − x b ) T B − 1 � ( H i [ x i ] − y i ) T R − 1 ( H i [ x i ] − y i ) i 2 i =0 subject to the dynamical system, where x b A priori (background) estimate B 0 Background error covariance matrix Observation error covariance matrix R i A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions This can be written as a standard nonlinear least squares problem J [ x ] = 1 2 || f ( x ) || 2 2 with 1   0 ( x 0 − x b ) B 2 1   R 0 ( H 0 [ x 0 ] − y 0 ) 2   f ( x 0 ) =   .  .  .     1 R N ( H N [ x N ] − y N ) 2 A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Gauss-Newton method (Incremental 4D-Var) For the nonlinear least squares problem J [ x ] = 1 2 || f ( x ) || 2 2 the Gauss-Newton method is equivalent to the following iteration Start with iterate x (0) For k = 0 , . . . , K − 1 Solve min δ x || J ( x ( k ) ) δ x + f ( x ( k ) ) || 2 2 Update x ( k +1) = x ( k ) + δ x A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions For the data assimilation problem the inner cost function is 1 − [ x b − x 0( k ) ]) T B − 1 − [ x b − x 0( k ) ]) J ( k ) [ δ x ( k ) 2( δ x ( k ) 0 ( δ x ( k ) ˜ 0 ] = 0 0 N 1 ( H i δ x ( k ) − d ( k ) ( H i δ x ( k ) − d ( k ) � ) T R − 1 + ) i i i i i 2 i =0 with d i = y i − H i [ x i ], subject to the linear dynamical system δ x i +1 = M i δ x i d i = H i δ x i . A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions In practice the inner problem can only be solved approximately. Common approximations include 1 Truncate iterative solution, so that exact minimum not found. 2 Use approximate linear model. 3 Solve the problem in a restricted space (usually a lower spatial resolution). Gratton, Lawless and Nichols ( SIOPT , 2007) proved conditions for convergence of Gauss-Newton under approximations 1 & 2. In this work we consider approximation 3. A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Low resolution G-N in data assimilation i ∈ R r × n such that We introduce linear restriction operators U T x i = U T i δ x i ∈ R r , r < n . δ ˆ We also define linear prolongation operators V i ∈ R n × r , with U T i V i = I r and V i U T a projection. i A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Low resolution G-N in data assimilation i ∈ R r × n such that We introduce linear restriction operators U T x i = U T i δ x i ∈ R r , r < n . δ ˆ We also define linear prolongation operators V i ∈ R n × r , with U T i V i = I r and V i U T a projection. i We can then define a restricted dynamical system in R r ˆ δ ˆ x i +1 = M i δ ˆ x i , ˆ ˆ H i δ ˆ d i = x i , where V i ˆ M i U T approximates M i i and ˆ H i U T approximates H i . i A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions We solve the minimization problem in R r 1 0 [ x b − x 0( k ) ]) T x ( k ) x ( k ) J ( k ) [ δ ˆ ˆ − U T 0 ] = 2( δ ˆ 0 0 [ x b − x 0( k ) ]) x ( k ) × ˆ B − 1 − U T 0 ( δ ˆ 0 N 1 x ( k ) − d ( k ) x ( k ) − d ( k ) � ( ˆ ) T R − 1 ( ˆ + H i δ ˆ H i δ ˆ ) , i i i i i 2 i =0 subject to the restricted system, and then update the G-N iterate using δ x ( k ) x ( k ) = V 0 δ ˆ 0 0 A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions In practice the restricted system is chosen to be a low resolution version of the full system. This does not take into account the dynamics or observations. Can we do better using ideas of model reduction? A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Model reduction Given a time-invariant linear system δ x i +1 = M δ x i + Gu i d i = H δ x i where u i ∼ N ( 0 , W ), find projection matrices U , V with U T V = I r and r << n , such that the reduced order system U T MV i δ ˆ x i + U T Gu i δ ˆ = x i +1 ˆ d i = HV i δ ˆ x i approximates the full system with δ x i ≈ V δ ˆ x i . A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions We seek matrices U , V such that we minimize �� �� � T R − 1 � ˆ ˆ i →∞ E lim d i − d i d i − d i over all inputs of normalized unit length. A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions We seek matrices U , V such that we minimize �� �� � T R − 1 � ˆ ˆ i →∞ E lim d i − d i d i − d i over all inputs of normalized unit length. Bernstein et al. (1986) derive necessary conditions for a minimum, but it is not practicable to obtain the optimal matrices. The method of balanced truncation finds an approximation solution, with error bounded in terms of the Hankel singular values. A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Balanced truncation Balanced truncation removes states which are least affected by inputs and which have least effect on outputs. There are 2 steps 1 Balancing - Transform the system into one in which these states are the same. 2 Truncation - Truncate states related to the smallest singular values of the transformed covariance matrices (Hankel singular values). A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Step 1: The balancing transformation simultaneously diagonalizes the state covariance matrices P and Q associated with the inputs and outputs. This requires the solution of the Stein equations MPM T + GG T , = P M T QM + H T R − 1 H . Q = The transformation Ψ , given by the matrix of eigenvectors of PQ , is used to transform the system into balanced form. A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions Step 1: The balancing transformation simultaneously diagonalizes the state covariance matrices P and Q associated with the inputs and outputs. This requires the solution of the Stein equations MPM T + GG T , = P M T QM + H T R − 1 H . Q = The transformation Ψ , given by the matrix of eigenvectors of PQ , is used to transform the system into balanced form. Step 2: The balanced system is truncated using � I r � U T = [ I r , 0 ] Ψ − 1 , V = Ψ . 0 A.S. Lawless et al. Optimal state estimation using reduced order models