1. Page 1 The Construction and Use of Linear Models in Large-scale Data Assimilation Tim Payne Large-Scale Inverse Problems and Applications in the Earth Sciences October 24th 2011

2. Page 2 © Crown copyright 2011 Part I. The Construction of Linear Models in Data Assimilation

3. Page 3 © Crown copyright 2011 Notation

4. Page 4 © Crown copyright 2011 Update-Prediction Cycle

5. Page 5 © Crown copyright 2011 First Strategy – exact evolution of covariances

6. Page 6 © Crown copyright 2011 Second strategy – EKF using tangent-linear

7. Page 7 © Crown copyright 2011 Third strategy – EKF using best linear approximation

8. Page 8 © Crown copyright 2011 Explicit formula for best linear approximation

9. Page 9 © Crown copyright 2011 Basic properties of best linear approximation

10. Page 10 © Crown copyright 2011 Cloud function

11. Page 11 © Crown copyright 2011 Smith Cloud Scheme with ‘ad hoc’ Regularisation

12. Page 12 © Crown copyright 2011 Smith Cloud Scheme with ‘Optimal’ Regularisation

13. Page 13 © Crown copyright 2011 Incremental 4D-Var

14. Page 14 © Crown copyright 2011 Options for linearisation step required for incremental 4D-Var

15. Page 15 © Crown copyright 2011 Gain matrix implied by each option

16. Page 16 © Crown copyright 2011 Advantage of BLA over TL in incremental 4D-Var

17. Page 17 © Crown copyright 2011 Pseudo Chain-rule for best linear approximation

18. Page 18 © Crown copyright 2011 Use of best linear approximation in EKF

19. Page 19 © Crown copyright 2011 The prior covariance implied by different approximations

20. Page 20 © Crown copyright 2011 Prior covariance using best linear estimate

21. Page 21 © Crown copyright 2011 Prior covariance using the best linear estimate always underestimates the true prior

22. Page 22 © Crown copyright 2011 The Duffing Map

23. Page 23 © Crown copyright 2011 100,000 iterates of Duffing Map

24. Page 24 © Crown copyright 2011 Reminder of EKF algorithm

25. Page 25 © Crown copyright 2011 Prior covariance for Duffing Map

26. Page 26 © Crown copyright 2011 Mean square analysis error in Duffing map: TL and best linear estimate compared

27. Page 27 © Crown copyright 2011 Part II. The Use of Linear Models in Data Assimilation

28. Page 28 © Crown copyright 2011 Linearisation error in 4D-Var as used in real numerical weather prediction models

29. Page 29 © Crown copyright 2011

30. Page 30 © Crown copyright 2011 Linear model for evolution of increments

31. Page 31 © Crown copyright 2011 Linearisation error as a stochastic error

32. Page 32 © Crown copyright 2011 Issues in forming EKF

33. Page 33 © Crown copyright 2011 Signal model for system with time correlated linearisation error

34. Page 34 © Crown copyright 2011 EKF with time correlated linearisation error

35. Page 35 © Crown copyright 2011 Parameters for filter including linearistion error

36. Page 36 © Crown copyright 2011 Example: L95, nearly perfect full model, persistence for linear model

37. Page 37 © Crown copyright 2011 Example: L95, nearly perfect full model, persistence for linear model, results

38. Page 38 © Crown copyright 2011 Variational version: weak constraint 4D-Var allowing for time correlated linearisation error

39. Page 39 © Crown copyright 2011 Remarks on variational form

40. Page 40 © Crown copyright 2011 Long window weak constraint 4D-Var allowing for linearisation error, same example

41. Page 41 © Crown copyright 2011 Summary to Part I

42. Page 42 © Crown copyright 2011 Summary to Part II

43. Page 43 © Crown copyright 2011 The End