Weak and Strong Constraint 4D variational data assimilation: Methods and Applications

1. Weak and Strong Constraint 4D variational data assimilation:Methods and Applications Di Lorenzo, E. Georgia Institute of Technology Arango, H. Rutgers University Moore, A. and B. Powell UC Santa Cruz Cornuelle, B and A.J. Miller Scripps Institution of Oceanography Bennet A. and B. Chua Oregon State University

2. Short review of 4DVAR theory (an alternative derivation of the representer method and comparison between different 4DVAR approaches) • Overview of Current applications • Things we struggle with

3. ASSIMILATION Goal Initial Guess Best Model Estimate (consistent with observations) (A) (B) WEAK Constraint STRONG Constraint …we want to find the correctionse

4. Best Model Estimate Corrections Initial Guess ASSIMILATION Goal (A) (B) WEAK Constraint STRONG Constraint …we want to find the correctionse

5. Best Model Estimate Corrections Initial Guess ASSIMILATION Goal

6. Best Model Estimate Corrections Initial Guess ASSIMILATION Goal

7. Best Model Estimate Corrections Initial Guess ASSIMILATION Goal

8. Best Model Estimate Corrections Initial Guess ASSIMILATION Goal Tangent Linear Dynamics

9. ASSIMILATION Goal Best Model Estimate Corrections Initial Guess Integral Solution Tangent Linear Propagator

10. ASSIMILATION Goal Best Model Estimate Corrections Initial Guess

11. ASSIMILATION Goal Best Model Estimate Corrections Initial Guess The Observations

12. ASSIMILATION Goal Best Model Estimate Corrections Initial Guess Data misfit from initial guess

13. ASSIMILATION Goal is a mapping matrix of dimensions observations X model space def: Data misfit from initial guess

14. ASSIMILATION Goal is a mapping matrix of dimensions observations X model space def: Data misfit from initial guess

15. Quadratic Linear Cost Function for residuals is a mapping matrix of dimensions observations X model space

16. Quadratic Linear Cost Function for residuals 2) corrections should not exceed our assumptions about the errors in model initial condition. 1) corrections should reduce misfit within observational error is a mapping matrix of dimensions observations X model space

17. Minimize Linear Cost Function

18. def: 4DVAR inversion Hessian Matrix

19. def: 4DVAR inversion Hessian Matrix Representer-based inversion

20. def: 4DVAR inversion Hessian Matrix Representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

21. def: 4DVAR inversion Hessian Matrix Representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

22. An example of Representer Functionsfor the Upwelling System Computed using the TL-ROMS and AD-ROMS

23. An example of Representer Functionsfor the Upwelling System Computed using the TL-ROMS and AD-ROMS

24. Applications of the ROMS inverse machinery: • Baroclinic coastal upwelling: synthetic model experiment to test the development • CalCOFI Reanalysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. Di Lorenzo, Miller, Cornuelle and Moisan • Intra-Americas Seas Real-Time DAPowell, Moore, Arango, Di Lorenzo, Milliff et al.

25. Coastal Baroclinic Upwelling System Model Setup and Sampling Array section

26. Applications of inverse ROMS: • Baroclinic coastal upwelling: synthetic model experiment to test inverse machinery 1) The representer system is able to initialize the forecast extracting dynamical information from the observations. 2) Forecast skill beats persistence 10 day assimilation window 10 day forecast

27. SKILL of assimilation solution in Coastal UpwellingComparison with independent observations Weak SKILL Strong Climatology Persistence Assimilation Forecast  DAYS Di Lorenzo et al. 2007; Ocean Modeling

28. Day=0 Day=2 Day=6 Day=10

29. Assimilation solutions Day=0 Day=2 Day=6 Day=10

30. Day=14 Day=18 Day=22 Day=26

31. Day=14 Day=18 Day=22 Day=26

32. Forecast Day=14 Day=14 Day=18 Day=18 Day=22 Day=22 Day=26 Day=26

33. Intra-Americas Seas Real-Time DAPowell, Moore, Arango, Di Lorenzo, Milliff et al. www.myroms.org/ias April 3, 2007

34. CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. Di Lorenzo, Miller, Cornuelle and Moisan

35. …careful

36. …careful Data Assimilation is NOT a black box

37. …careful Data Assimilation is NOT a black box • Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems  fitting vs. assimilating data)

38. …careful Data Assimilation is NOT a black box • Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems  fitting vs. assimilating data) • Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics)

39. …careful Data Assimilation is NOT a black box • Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems  fitting vs. assimilating data) • Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics) • Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)

40. …careful Data Assimilation is NOT a black box • Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems  fitting vs. assimilating data) • Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics) • Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)

41. Assimilation of SSTa True Initial Condition True

42. Assimilation of SSTa True Initial Condition Which model has correct dynamics? Model 1 Model 2 True

43. Wrong Model Good Model True Initial Condition Model 1 Model 2 True

44. Time Evolution of solutions after assimilation Wrong Model DAY 0 Good Model

45. Time Evolution of solutions after assimilation Wrong Model DAY 1 Good Model

46. Time Evolution of solutions after assimilation Wrong Model DAY 2 Good Model

47. Time Evolution of solutions after assimilation Wrong Model DAY 3 Good Model

48. Time Evolution of solutions after assimilation Wrong Model DAY 4 Good Model

49. What if we apply more background constraints? Wrong Model Good Model True Initial Condition True Model 1 Model 2

50. Assimilation of data at time True Initial Condition True Model 1 Model 2