1. Tbilisi, 10.07.2014 GGSWBS'14 Optimization for inverse modelling Ketevan Kasradze 1 Hendrik Elbern 1,2 kk@riu.uni-koeln.de he@riu.uni-koeln.dekk@riu.uni-koeln.dehe@riu.uni-koeln.de and the Chemical Data Assimilation group of RIU 1 Rhenish Institute for Environmental Research at the University of Cologne, Germany 2 Institute for Energy and Climate Research -Troposphere, Germany

2. Tbilisi, 10.07.2014 GGSWBS'14 Atmospheric layers 3/18

3. Tbilisi, 10.07.2014 GGSWBS'14 Atmospheric layers 3/18 SACADA

4. Tbilisi, 10.07.2014 GGSWBS'14 SACADA assimilation-system Background Meteorological ECMWF analyses Trace gas observations Analysis SACADA PREP DWD GME CTM CTMad Diffusion L-BFGS

5. Tbilisi, 10.07.2014 GGSWBS'14 Horizontal GME Grid 9/18 ~147km between the grid points 23 042 grid points per Model layer

6. Tbilisi, 10.07.2014 GGSWBS'14 Additional refinement troposphere/lower stratosphere SACADA Vertical Grid 54 layer CRISTA-NF MLS

7. Tbilisi, 10.07.2014 GGSWBS'14 HN O 3 4.11.2005 ~137hPa 12 Uhr UTC MLS 15

8. Tbilisi, 10.07.2014 GGSWBS'14 SCOUT-O3 campaign Stratospheric-Climate Links with Emphasis on the UTLS - O3 November-December 2005 AMMA- campaign African Monsoon Multidisciplinary Analyses 29.07.2006 -17.08.2006 12/18

9. Tbilisi, 10.07.2014 GGSWBS'14 Cost function Vector of observations Observation error covariance matrix Projection operator Background Model operator SACADA assimilation-system 4D-Var Background error covariance matrix BECM ~ 10 12 ~ 80 Terrabyte

10. Tbilisi, 10.07.2014 GGSWBS'14 Gradient Adjoint Model SACADA assimilation-system 4D-Var

11. Tbilisi, 10.07.2014 GGSWBS'14 Quasi-Newton method L-BFGS SACADA assimilation-system 4D-Var

12. Tbilisi, 10.07.2014 GGSWBS'14 Quasi-Newton method L-BFGS Background error covariance matrix BECM ~ 10 12 ~ 80 Terrabyte SACADA assimilation-system 4D-Var

13. Tbilisi, 10.07.2014 GGSWBS'14 Radius of Influence ((de-)correlation length): Extending the information from an observation location Textbook: horizontal influence radius L around a measurement site, to be based on a priori statistical assessments L vertical cut L Horizontal structure function, to be stored as a column of the forecast error covariance matrix diffusion operator construction For atmospheric chemistry covariance modelling the diffusion approach is advocated: localisation intrinsically performed sharp gradients easily feasible matrix square roots for preconditioning straightforward to calculate; no inversion needed Background error covariance matrix formulation

14. Tbilisi, 10.07.2014 GGSWBS'14 Isopleths of the cost function and transformed cost function and minimisation steps Minimisation by mere gradients, quasi-Newon method L-BFGS (Large dimensional Broyden Fletcher Goldfarb Shanno), and preconditioned (transformed) L-BFGS application concentration species 1 transformed species 1 concentration species 2 transformed species 2

15. Tbilisi, 10.07.2014 GGSWBS'14 Solution: Diffusion Approach Transformation of cost-function: => Inverse of B and B -1/2 are not needed, if x b = 1. guess. 2 outstanding problems: 1.With linear estimation: How to treat the background error covariance matrix B (O(10 12 ))? 2.How can this be treated for preconditioning? (need B -1, B 1/2, B -1/2 ) With variational methods: minimisation procedure Background error covariance matrix formulation

16. Tbilisi, 10.07.2014 GGSWBS'14 Background error covariance matrix formulation Background

17. Tbilisi, 10.07.2014 GGSWBS'14 Background error covariance matrix formulation Background Observation: 3 ppm Ozone

18. Tbilisi, 10.07.2014 GGSWBS'14 Analysis (B diagonal) Background error covariance matrix formulation Background Observation: 3 ppm Ozone

19. Tbilisi, 10.07.2014 GGSWBS'14 Background error covariance matrix formulation Background Observation: 3 ppm Ozone

20. Tbilisi, 10.07.2014 GGSWBS'14 Background error covariance matrix formulation Observation: 3 ppm Ozone Analysis increment isotropic correlation The increment in initial values is spread out to neighbouring grid-points depending on the correlations that are known / assumed. Background

21. Tbilisi, 10.07.2014 GGSWBS'14 Assumption: Strong correlation along isolines of Potential Vorticity  Enhancement of diffusion  flow-dependent BECM Diffusion can be generalised to account for inhomogeneous and anisotropic correlations: Stratospheric case Background error covariance matrix formulation use PV field for anisotropic correlation modelling

22. Tbilisi, 10.07.2014 GGSWBS'14 Background Observation: 3 ppm Ozone Background error covariance matrix formulation

23. Tbilisi, 10.07.2014 GGSWBS'14 Background Observation: 3 ppm Ozone Analysis increment Background error covariance matrix formulation

24. Tbilisi, 10.07.2014 GGSWBS'14 Quasi-Newton method L-BFGS Adjoint Model SACADA assimilation-system 4D-Var

25. Tbilisi, 10.07.2014 GGSWBS'14 Construction of the adjoint code (3 different possible pathways) forward model (forward differential equation) algorithm (solver) code backward model (backward differential equation) adjoint algorithm (adjoint solver) adjoint code

26. Tbilisi, 10.07.2014 GGSWBS'14 Adjoint model A numerical model integration over a time interval [t 0 ; t i ] Accordingly, the tangent linear of this sequence of model operators is given by Thus, the adjoint model operator Mi propagates the gradient of the cost function with respect to xi backwards in time, to deliver the gradient of the cost function with respect to x0.

27. Tbilisi, 10.07.2014 GGSWBS'14 Adjoint model example

28. Tbilisi, 10.07.2014 GGSWBS'14 Adjoint model example

29. Tbilisi, 10.07.2014 GGSWBS'14 Quasi-Newton method L-BFGS Limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm SACADA assimilation-system 4D-Var

30. Tbilisi, 10.07.2014 GGSWBS'14 Gradient of the cost function h Hessian of the cost function

31. Tbilisi, 10.07.2014 GGSWBS'14 BFGS algorithm (2) From an initial guess x 0 a nd an approximate Hessian matrix H 0 the following steps are repeated as x k converges to the solution. 1.Obtain a direction s k by solving: 2.Perform a line search to find an acceptable step size in the direction found in the first step, then update 3.Set 4. 5. Convergence can be checked by observing the norm of the gradient,.

32. Tbilisi, 10.07.2014 GGSWBS'14 BFGS example with MATLAB

33. Tbilisi, 10.07.2014 GGSWBS'14 BFGS example with MATLAB

34. Tbilisi, 10.07.2014 GGSWBS'14 BFGS example with MATLAB

35. Tbilisi, 10.07.2014 GGSWBS'14 BFGS example with MATLAB

36. Tbilisi, 10.07.2014 GGSWBS'14 BFGS example with MATLAB

37. Tbilisi, 10.07.2014 GGSWBS'14 BFGS example with MATLAB

38. Tbilisi, 10.07.2014 GGSWBS'14 BFGS example with MATLAB

39. Tbilisi, 10.07.2014 GGSWBS'14 BFGS example with MATLAB it= 40 f=1.497581e-13 ||g||=1.726061e-05 sig=1.200 step=BFGS it= 41 f=5.990317e-15 ||g||=3.452127e-06 Successful termination with ||g||<1.000000e-08*max(1,||g0||):

40. Tbilisi, 10.07.2014 GGSWBS'14 Thank you for your attention! გმადლობთ ყურადღებისათვის ! Vielen Dank für Ihre Aufmerksamkeit!