Optimal state estimation for numerical weather prediction using reduced order models

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1 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.

2 Outline 1 4D-Var and Gauss-Newton 2 Model reduction in data assimilation 3 Extension to unstable systems 4 Numerical results 5 Conclusions

3 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 0 Time t N

4 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 Time t N

5 Nonlinear least squares problem We consider a dynamical system x i+1 = M i (x i ) y i = H(x i ) + η i with M i Model dynamics x i State vector O( ) y i Observations at time t i O( ) η i Observational noise Observation operator H i

6 Nonlinear least squares problem Then the data assimilation problem is to minimize J [x 0 ] = 1 2 (x 0 x b ) T B 1 0 (x 0 x b )+ 1 2 N 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 (H i [x i ] y i ) T R 1 i (H i [x i ] y i )

7 This can be written as a standard nonlinear least squares problem J [x] = 1 2 f(x) 2 2 with f(x 0 ) = B (x 0 x b ) R (H 0 [x 0 ] y 0 ). R 1 2 N (H N [x N ] y N )

8 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 δx J(x (k) )δx + f(x (k) ) 2 2 Update x (k+1) = x (k) + δx

9 For the data assimilation problem the inner cost function is J (k) [δx (k) 0 ] = 1 2 (δx(k) 0 [x b x 0 (k) ]) T B 1 0 (δx(k) 0 [x b x 0 (k) ]) N i=0 (H i δx (k) i d (k) i ) T R 1 (H i δx (k) i i d (k) i ) with d i = y i H i [x i ], subject to the linear dynamical system δx i+1 = M i δx i d i = H i δx i.

10 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.

11 Low resolution G-N in data assimilation We introduce linear restriction operators U T i R r n such that δˆx i = U T i δx i R r, r < n. We also define linear prolongation operators V i R n r, with U T i V i = I r and V i U T i a projection.

12 Low resolution G-N in data assimilation We introduce linear restriction operators U T i R r n such that δˆx i = U T i δx i R r, r < n. We also define linear prolongation operators V i R n r, with U T i V i = I r and V i U T i a projection. We can then define a restricted dynamical system in R r δˆx i+1 = ˆM i δˆx i, ˆd i = Ĥ i δˆx i, where and V i ˆM i U T i approximates M i Ĥ i U T i approximates H i.

13 We solve the minimization problem in R r ˆ J (k) [δˆx (k) 0 ] = 1 2 (δˆx(k) 0 U T 0 [x b x 0 (k) ]) T ˆB 1 0 (δˆx(k) 0 U T 0 [x b x (k) 0 ]) + 1 N (Ĥ i δˆx (k) i d (k) i ) T R 1 i (Ĥ i δˆx (k) i 2 i=0 d (k) i ), subject to the restricted system, and then update the G-N iterate using δx (k) 0 = V 0 δˆx (k) 0

14 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?

15 Model reduction Given a time-invariant linear system δx i+1 d i = Mδx i + Gu i = Hδx 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 δˆx i+1 ˆd i = U T MV i δˆx i + U T Gu i = HV i δˆx i approximates the full system with δx i Vδˆx i.

16 We seek matrices U,V such that we minimize { ] lim [ˆd [ˆd T ] } E i d i R 1 i d i i over all inputs of normalized unit length.

17 We seek matrices U,V such that we minimize { ] lim [ˆd [ˆd T ] } E i d i R 1 i d i 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.

18 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).

19 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 P = MPM T + GG T, Q = M T QM + H T R 1 H. The transformation Ψ, given by the matrix of eigenvectors of PQ, is used to transform the system into balanced form.

20 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 P = MPM T + GG T, Q = M T QM + H T R 1 H. 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 U T = [I r,0]ψ 1, V = Ψ [ Ir 0 ].

21 Application to data assimilation In data assimilation we have the (time-invariant) stochastic dynamical system δx i+1 d i = Mδx i = Hδx i with stochastic initial condition δx 0 N(0,B 0 ). To apply model reduction we need to write this in the form δx i+1 d i = Mδx i + Gu i = Hδx i

22 We write our original system in the form δx i+1 d i = Mδx i + Gũ i = Hδx i with and For 4D-Var we have G = 0. G = [G I n ], ũ i = [ ui [ W 0 ũ i N(0, W), W = 0 B 0 v i ] ].

23 Reduced order cost function The reduced order minimization problem is then min ˆ J (k) [δˆx (k) 0 ] = 1 2 (δˆx(k) 0 U T [x b x 0 (k) ]) T (U T B 0 U) 1 (δˆx (k) 0 U T [x b x (k) 0 ]) + 1 N (HVδˆx (k) i d (k) i ) T R 1 (HVδˆx (k) i d (k) i ), 2 i=0 subject to and set δˆx i+1 = U T MVδˆx i δx (k) 0 = Vδˆx (k) 0.

24 Extension to unstable systems Previously we have shown the benefit of using reduced order models in data assimilation when the system matrix is stable (Lawless et. al, 2008, Mon. Wea. Rev.; Lawless et. al, 2008, Int. J. Numer. Methods in Fluids.) But How can we choose the projection operators U T and V when the system matrix is unstable?

25 We consider 3 methods: 1 Choose U T as a low resolution spatial restriction. 2 Balanced truncation, with standard splitting for unstable systems. 3 New α-bounded balanced truncation for unstable systems.

26 Standard extension of balanced truncation Divide the system S into its stable and unstable parts S = S + + S Apply the usual balanced truncation method to the stable part S + (S + ) red Combine the reduced stable part with the unchanged unstable part Ŝ = (S + ) red + S

27 Standard extension of balanced truncation Divide the system S into its stable and unstable parts S = S + + S Apply the usual balanced truncation method to the stable part S + (S + ) red Combine the reduced stable part with the unchanged unstable part Ŝ = (S + ) red + S Problem: Only works if the number of unstable poles is small.

28 New approach for unstable systems I Boess, Bunse-Gerstner and Nichols (2010) introduce an extension of balanced truncation for α-bounded systems, i.e. systems whose eigenvalues lie in a circle of radius α > 1. 1 Find α R + such that all eigenvalues of the system matrix M lie inside an α-circle around the origin. 2 Choose a reduction order r n. 3 Apply model reduction to the α-scaled system δx (α) i+1 = M α δx (α) i + G α u i d (α) i = H α δx (α) i with M α := M/α, G α := G/ α, H α := H/ α.

29 New approach for unstable systems II This yields projection matrices U α,v α R n r and a low order system: δˆx (α) i+1 = ˆM α δˆx (α) i + G α u i ˆd (α) i = Ĥ α δˆx (α) i with ˆM α := U T αm α V α, Ĝ α := U T α G α and Ĥ α := H α V α. 4 The scaling α is then reversed to give the reduced system δˆx i+1 = ˆMδˆx i + Gu i ˆd i = Ĥδˆx i with ˆM = α ˆM α, G = α G α and Ĥ = αĥ α.

30 Reduced order cost function The reduced order minimization problem is now min ˆ J (k) [δˆx (k) 0 ] = 1 2 (δˆx(k) 0 U T α [x b x 0 (k) ]) T (U T αb 0 U α ) 1 (δˆx (k) 0 U T α [x b x (k) 0 ]) + 1 N (HV α δˆx (k) i d (k) i ) T R 1 (HV α δˆx (k) i d (k) i ), 2 i=0 subject to the reduced system. We then set δx (k) 0 = V α δˆx (k) 0.

31 Numerical results - 2D Eady model The Eady model is a 2D x z model describing the evolution of a buoyancy b on the boundaries and a potential vorticity q in the interior. We have the evolution equations ( t + z x ) b = ψ x, on z = ±1, x [0, X], 2 and ( t + z ) q = 0, in z [ 1 x 2, 1 ], x [0, X], 2 where the streamfunction ψ satisfies 2 ψ x ψ z 2 = q, with boundary conditions ψ z = b, in z [ 1 2, 1 ], x [0, X], 2 on z = ±1, x [0, X]. 2

32 Methodology Define the true solution at the initial time. Run the model to generate synthetic observations at times t 0,...,t 5 (12h window). Define a linear least squares problem using these observations and a background field. Solve the reduced order least squares problem using low resolution model; reduced order system using standard balanced truncation; reduced order system using new method. Lift solution to full resolution and compare with solution to full linear least squares problem.

33 Experiment 1: Zero PV When the interior potential voriticity is zero, the only evolution is on the upper and lower boundaries. We consider 20 horizontal grid points (system size 40). True state is the eigenvector associated with the largest eigenvalue of the model. Zero background state with inverse Laplacian covariance matrix of variance 1. Observation error covariance matrix is identity. Observations taken of buoyancy on lower boundary only. Reduced problem solved with size 20.

34 0.2 Eady40, lower buoyancy, α= 1.12: Solution plot 0.15 x 0 x 0 (low), order x 0 (std), order 20 x 0 (alpha), order solution vector components 10 0 Eady40, lower buoyancy, α= 1.12: Error plot error e 1 =x 0 x 0 (low), order 20 e 2 =x 0 x 0 (std), order 20 e 3 =x 0 x 0 (alpha), order vector components Figure: Lower boundary solution (top) and error (bottom).

35 0.2 x 0 Eady40, upper buoyancy, α= 1.12: Solution plot x 0 (low), order 20 x 0 (std), order 20 x 0 (alpha), order solution vector components 10 0 Eady40, upper buoyancy, α= 1.12: Error plot error e 1 =x 0 x 0 (low), order 20 e 2 =x 0 x 0 (std), order 20 e 3 =x 0 x 0 (alpha), order vector components Figure: Upper boundary solution (top) and error (bottom).

36 Eigenvalues of α reduced M, α = 1.12 Eigenvalues of low resolution M eigs M eigs M 0.6 eigs M eigs M α 0.6 low unit circle unit circle imaginary axis imaginary axis real axis real axis Eigenvalues of standard reduced M eigs M eigs M std unit circle 0.4 imaginary axis real axis Figure: Comaprison of eigenvalues with full system for α-reduced (top left), standard reduced (top right) and low resolution (bottom).

37 Experiment 2: Non-zero PV We now consider a case when there is evolution of the PV in the interior. The full system size is now 1040 (80 horizontal grid points). True solution is non-modal growing mode. Observations are taken at every second grid point. Background error variances are 10 5 on the boundaries and 1 in the interior. Reduced problem solved with size 520.

38 2e 006 2e Contour plot of true solution x e e 006 6e 006 8e 006 2e 006 4e e 006 6e Contour plot of lifted low res solution, k=520 x e e 006 4e 006 6e e e 006 6e 006 4e Figure: True solution (top) and low resolution (bottom).

39 2e 006 2e 007 2e Contour plot of true solution x e e 006 6e 006 8e 006 2e 006 4e e 006 6e Contour plot of lifted mod red solution using standard BT, k=520 x e 007 2e 007 4e 007 6e 007 4e 007 2e 007 4e 007 2e 007 2e Figure: True solution (top) and standard BT (bottom).

40 2e 006 2e 006 2e 006 2e Contour plot of true solution x e e 006 6e 006 8e 006 2e 006 4e e 006 6e Contour plot of lifted mod red solution using α BT, α = 1.13, k=520 x e e 006 6e 006 8e 006 2e 006 4e e 006 6e Figure: True solution (top) and α BT (bottom).

41 Conclusions In many applications dynamical systems may be unstable over finite time intervals.

42 Conclusions In many applications dynamical systems may be unstable over finite time intervals. A new method of α-bounded balanced truncation has been incorporated into 4D-Var.

43 Conclusions In many applications dynamical systems may be unstable over finite time intervals. A new method of α-bounded balanced truncation has been incorporated into 4D-Var. Numerical results show a more accurate solution than using low resolution models or standard model reduction.

44 Conclusions In many applications dynamical systems may be unstable over finite time intervals. A new method of α-bounded balanced truncation has been incorporated into 4D-Var. Numerical results show a more accurate solution than using low resolution models or standard model reduction. In part this can be explained by preservation of the model eigenvalues.

45 Remaining issues In order to be implemented in real systems the method still requires some further work:

46 Remaining issues In order to be implemented in real systems the method still requires some further work: Extension to time-varying systems.

47 Remaining issues In order to be implemented in real systems the method still requires some further work: Extension to time-varying systems. Efficient reduction methods for large-scale systems.

48 Remaining issues In order to be implemented in real systems the method still requires some further work: Extension to time-varying systems. Efficient reduction methods for large-scale systems. Update of reduced order system within iterative process.

49 Remaining issues In order to be implemented in real systems the method still requires some further work: Extension to time-varying systems. Efficient reduction methods for large-scale systems. Update of reduced order system within iterative process. Nevertheless results indicate that this is research that is worth pursuing.

50 References I Boess, C., Nichols, N.K. and Bunse-Gerstner, A. (2010), Model reduction for discrete unstable control systems using a balanced truncation approach. Preprint MPS , Department of Mathematics, University of Reading. Gratton, S., Lawless, A.S. and Nichols, N.K. (2007), Approximate Gauss-Newton methods for nonlinear least squares problems. SIAM J. on Optimization, 18, Lawless, A.S., Nichols, N.K., Boess, C. and Bunse-Gerstner, A. (2008), Using model reduction methods within incremental 4D-Var. Mon. Wea. Rev., 136,

51 References II Lawless, A.S., Nichols, N.K., Boess, C. and Bunse-Gerstner, A. (2008), Approximate Gauss-Newton methods for optimal state estimation using reduced order models. Int. J. Numer. Methods in Fluids, 56, Boess, C., Lawless, A.S., Nichols, N.K. and Bunse-Gerstner, A. (2011), State estimation using model order reduction for unstable systems. Comput. Fluids, 46,

Using Model Reduction Methods within Incremental Four-Dimensional Variational Data Assimilation

APRIL 2008 L A W L E S S E T A L. 1511 Using Model Reduction Methods within Incremental Four-Dimensional Variational Data Assimilation A. S. LAWLESS AND N. K. NICHOLS Department of Mathematics, University