A comparison of the model error and state formulations of weak-constraint 4D-Var

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1 A comparison of the model error and state formulations of weak-constraint 4D-Var A. El-Said, A.S. Lawless and N.K. Nichols School of Mathematical and Physical Sciences University of Reading Work supported by NERC

2 Outline 1 Weak constraint 4D-Var 2 Experimental design 3 Numerical results 4 Conclusions

3 Four-dimensional variational assimilation (4D-Var) The 4D-Var data assimilation problem can be expressed as the minimization of J[x 0 ] = 1 2 (x 0 x b ) T B 1 (x 0 x b )+ 1 2 N i=0 (H i [x i ] y i ) T R 1 i (H i [x i ] y i ) subject to the dynamical system x i+1 = M i (x i ) where x b y i A priori (background) estimate Observation B Background error covariance matrix R i Observation error covariance matrix Observation operator H i

4 We consider the model as a weak constraint x i+1 = M i (x i )+η i, η i N(0,Q i )

5 We consider the model as a weak constraint State formulation x i+1 = M i (x i )+η i, η i N(0,Q i ) J(x 0,x 1,...,x N ) = J b +J o + 1 N 1 (x i+1 M i (x i )) T Q 1 i (x i+1 M i (x i )) 2 i=0

6 We consider the model as a weak constraint State formulation x i+1 = M i (x i )+η i, η i N(0,Q i ) J(x 0,x 1,...,x N ) = J b +J o + 1 N 1 (x i+1 M i (x i )) T Q 1 i (x i+1 M i (x i )) 2 Error formulation i=0 J(x 0,η 0,...,η N 1 ) = J b +J o N 1 i=0 η T i Q 1 i η i

7 Condition number The inner loop is solved using a gradient minimization method. The expected accuracy of the numerical solution and the speed of convergence are both determined by the condition number of the Hessian. Condition number κ(a) = A A 1 In the matrix 2-norm, for a symm. pos. def. matrix A, we have κ(a) = λ max (A)/λ min (A) In 4D-Var the condition number of the Hessian matrix determines convergence properties.

8 Notation Define D = diag{b,q 1,...,Q n } R = diag{r 1,...,R n } H = diag{h 1,...,H n } I M 1 I 0 0 L = M n I

9 Hessian - Error formulation S p = D 1 +L T H T R 1 HL 1 S p = B 1 0 Q Qn 1 H T 0 (H 1M 1) T (H 2M 2M 1) T... (H nm n...m 1) T H T 1 (H 2M 2) T... (H nm n...m 2) T H T (H nm n) T H T n R 1 H 0 H 1M 1 H 1 H 2M 2M 1 H 2M 2 H H nm n...m 1... H nm n H n

10 Hessian - State formulation S x = L T D 1 L+H T R 1 H S x = B 1 0 +M T 1 Q 1 1 M 1 M T 1 Q 1 1 Q 1 1 M 1 Q 1 1 +M2 TQ 1 2 M 2 M2 TQ 1 2 Q 1 2 M 2... H0 TR 1 0 H 0 H1 T + R 1 1 H 1... H T n R 1 n H n Mn 1 T Q 1 n 1 Q 1 n 1 M n 1 Q 1 n 1 +MT n Qn 1 M n Mn T Qn 1 Q 1 n M n Q 1 n

11 Aims How does the condition number of the Hessian change with different input parameters? How does this affect convergence of the minimisation? Note: Theoretical bounds on the condition number have been obtained. Here we just illustrate the effects with numerical results.

12 Previous results Previously we have shown for the strong constraint case (preconditioned by B 1/2 ) that the bounds on the condition number will increase as

13 Previous results Previously we have shown for the strong constraint case (preconditioned by B 1/2 ) that the bounds on the condition number will increase as the observations become more accurate; the observations spacing decreases; the background becomes less accurate; the background error correlation lengthscales increase. (Haben et al. (2014) Tellus, Haben et al. (2014) Comput. Fluids)

14 Numerical model We consider results in the context of a simple system, the 1D advection equation with periodic boundary conditions: with u t +a u x = 0 u(x,0) = be (x c)2 2d 2. The model is discretized using an upwind numerical scheme with N = 50 grid points, a = 1.

15 Assimilation scheme Background error covariance matrix B = σ 2 b C SOAR, with σ b = 0.1,L(C) = 2 x. Model error covariance matrix Q i = σ 2 qc LAP, with σ q = 0.05,L(C) = x,. Observation error covariance matrix R i = σ 2 oi, with σ o = Observations every 2 grid points and every 5 time steps in 50 time step window. x = 0.01, t = 0.02.

16 Effect of observation accuracy Large observation error variance Matrix Condition Number No. of iterations S p S x D Small observation error variance Matrix Condition Number No. of iterations S p S x D 838 -

17 Effect of observation accuracy Condition number as σ q /σ o varies: Condition Number κ(s p ) κ(s x ) σ q = σ o σ q / σ o Figure : κ(s p ) (solid line) and κ(s x ) (dashed line) as a function of ratio σ q /σ o. Condition number minimum point at σ q = σ o (dotted line).

18 Effect of assimilation window length - Error formulation κ(s p ) Assimilation window length Number of spatial observa per assimilation step Figure : κ(s p ) as a function of assimilation window length, n, and number of spatial observations, q.

19 Effect of assimilation window length - State formulation x 10 4 κ(s x ) x Assimilation window length Number of spatial observations per assimilation step 50 Figure : κ(s x ) as a function of assimilation window length, n, and number of spatial observations, q.

20 Reduction of model error variance We now set σ b /σ q = 200. Matrix Condition Number No. of iterations S p S x D

21 Change in model error variance - Condition numbers Condition Number 2 x κ(s x ) κ(s p ) σ b / σ q σ b / σ q Figure : κ(s p ) (solid line) and κ(s x ) (dashed line) as a function of ratio σ b /σ q. Condition number minimum point at σ b = σ q (dotted line).

22 Correlation length-scales x 10 4 x κ(s p ) κ(s x ) L(C Q ) L(C B ) L(C Q ) L(C ) B Figure : Condition number of κ(s p ) (left) and κ(s x ) (right) as a function of L(C B ) and L(C Q ).

23 Conclusions The two different formulations of the WC problem have Hessians with different structures.

24 Conclusions The two different formulations of the WC problem have Hessians with different structures. The condition number of both Hessians is sensitive to input parameters, with the state formulation generally being more sensitive.

25 Conclusions The two different formulations of the WC problem have Hessians with different structures. The condition number of both Hessians is sensitive to input parameters, with the state formulation generally being more sensitive. Sensitivities backed up by theory (not shown).

26 Conclusions In particular we find the following sensitivities: For increasing observation accuracy the error formulation is more sensitive, while for small observation accuracy the state formulation is badly conditioned.

27 Conclusions In particular we find the following sensitivities: For increasing observation accuracy the error formulation is more sensitive, while for small observation accuracy the state formulation is badly conditioned. The error formulation is more sensitive to an increase in window length.

28 Conclusions In particular we find the following sensitivities: For increasing observation accuracy the error formulation is more sensitive, while for small observation accuracy the state formulation is badly conditioned. The error formulation is more sensitive to an increase in window length. The state formulation is more sensitive to having fewer observations.

29 Conclusions In particular we find the following sensitivities: For increasing observation accuracy the error formulation is more sensitive, while for small observation accuracy the state formulation is badly conditioned. The error formulation is more sensitive to an increase in window length. The state formulation is more sensitive to having fewer observations. For larger model error the state formulation becomes more ill conditioned than the error formulation.

30 Conclusions In particular we find the following sensitivities: For increasing observation accuracy the error formulation is more sensitive, while for small observation accuracy the state formulation is badly conditioned. The error formulation is more sensitive to an increase in window length. The state formulation is more sensitive to having fewer observations. For larger model error the state formulation becomes more ill conditioned than the error formulation. The state formulation is more sensitive to changes in the condition number of D.

31 Outlook Error formulation seems more stable than state formulation, but not good for longer windows.

32 Outlook Error formulation seems more stable than state formulation, but not good for longer windows. Preconditioning of each formulation could be considered.

33 Outlook Error formulation seems more stable than state formulation, but not good for longer windows. Preconditioning of each formulation could be considered. Effect of correlated observation errors?

34 Outlook Error formulation seems more stable than state formulation, but not good for longer windows. Preconditioning of each formulation could be considered. Effect of correlated observation errors? Saddle-point formulation.

Background and observation error covariances Data Assimilation & Inverse Problems from Weather Forecasting to Neuroscience

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