The Inverse Regional Ocean Modeling System:

1. The Inverse Regional Ocean Modeling System: Development and Application to Data Assimilation of Coastal Mesoscale Eddies. Di Lorenzo, E., Moore, A., H. Arango, B. Chua, B. D. Cornuelle , A. J. Miller and Bennett A.

2. Goals • A brief overview of the Inverse Regional Ocean Modeling System • How do we assimilate data using the ROMS set of models • Examples, (a) zonal baroclinic jet and (b) mesoscale eddies in the Southern California Current

3. Inverse Ocean Modeling System (IOMs) Chua and Bennett (2001) To implement a representer-based generalized inverse method to solve weak constraint data assimilation into a non-linear model NL-ROMS, TL-ROMS, REP-ROMS, AD-ROMS Moore et al. (2003) Inverse Regional Ocean Modeling System (IROMS) a 4D-variational data assimilationsystem for high-resolution basin-wide and coastal oceanic flows

4. NL-ROMS: def: REP-ROMS: Approximation of NONLINEAR DYNAMICS (STEP 1) also referred to as Picard Iterations

5. def: Representer Tangent Linear Model REP-ROMS:

6. def: Representer Tangent Linear Model REP-ROMS: Perturbation Tangent Linear Model TL-ROMS: Adjoint Model AD-ROMS:

7. REP-ROMS: TL-ROMS: AD-ROMS:

8. REP-ROMS: (STEP 2) TL-ROMS: • Small Errors • model missing dynamics • boundary conditions errors • Initial conditions errors AD-ROMS:

9. REP-ROMS: TL-ROMS: AD-ROMS:

10. Integral Solutions REP-ROMS: TL-ROMS: ….. AD-ROMS:

11. Integral Solutions REP-ROMS: TL-ROMS: ….. AD-ROMS: Tangent Linear Propagator

12. Integral Solutions REP-ROMS: TL-ROMS: AD-ROMS: Tangent Linear Propagator

13. Integral Solutions REP-ROMS: TL-ROMS: AD-ROMS: Adjoint Propagator

14. Integral Solutions REP-ROMS: Tangent Linear Propagator TL-ROMS: AD-ROMS: Adjoint Propagator

15. How is the tangent linear model useful for assimilation? TL-ROMS:

16. ASSIMILATION (1) Problem Statement 1) Set of observations 2) Model trajectory 3) Find that minimizes Sampling functional TL-ROMS:

17. Best Model Estimate Corrections Initial Guess ASSIMILATION (1) Problem Statement 1) Set of observations 2) Model trajectory 3) Find that minimizes Sampling functional TL-ROMS:

18. ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Model Tangent Linear trajectory 3) Find that minimizes TL-ROMS: Best Model Estimate Corrections Initial Guess

19. ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Model Tangent Linear trajectory 3) Find that minimizes TL-ROMS:

20. ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Model Tangent Linear trajectory 3) Find that minimizes TL-ROMS:  Corrections to initial conditions  Corrections to model dynamics and boundary conditions

21. ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Corrections to Model State 3) Find that minimizes  Corrections to initial conditions  Corrections to model dynamics and boundary conditions

22. ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Correction to Model Initial Guess 3) Find that minimizes Assume we seek to correct only the initial conditions STRONG CONSTRAINT  Corrections to initial conditions  Corrections to model dynamics and boundary conditions

23. ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Correction to Model Initial Guess 3) Find that minimizes ASSIMILATION (3) Cost Function

24. ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Correction to Model Initial Guess 3) Find that minimizes ASSIMILATION (3) Cost Function 1) corrections should reduce misfit within observational error 2) corrections should not exceed our assumptions about the errors in model initial condition.

25. ASSIMILATION (3) Cost Function

26. is a mapping matrix of dimensions observations X model space def: ASSIMILATION (3) Cost Function

27. is a mapping matrix of dimensions observations X model space def: ASSIMILATION (3) Cost Function

28. Minimize J ASSIMILATION (3) Cost Function

29. def: 4DVAR inversion Hessian Matrix

30. def: 4DVAR inversion Hessian Matrix IOM representer-based inversion

31. def: 4DVAR inversion Hessian Matrix IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

32. def: What is the physical meaning of the Representer Matrix? IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

33. Representer Matrix TL-ROMS AD-ROMS IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

34. Representer Matrix Assume a special assimilation case:  Observations = Full model state at time T  Diagonal Covariance with unit variance IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix

35. Representer Matrix Assume a special assimilation case:  Observations = Full model state at time T  Diagonal Covariance with unit variance IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix

36. Representer Matrix IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix

37. Representer Matrix Assume you want to compute the model spatial covariance at time T

38. Representer Matrix Assume you want to compute the model spatial covariance at time T STEP 1) AD-ROMS

39. Representer Matrix Assume you want to compute the model spatial covariance at time T STEP 1) AD-ROMS STEP 2) run TL-ROMS forced with

40. Representer Matrix Assume you want to compute the model spatial covariance at time T STEP 1) AD-ROMS STEP 2) run TL-ROMS forced with STEP 3) multiply by andtake expected value

41. Representer Matrix Assume you want to compute the model spatial covariance at time T STEP 1) AD-ROMS STEP 2) run TL-ROMS forced with STEP 3) multiply by andtake expected value

42. Representer Matrix model to model covariance

43. Representer Matrix model to model covariance model to model covariance most general form

44. Representer Matrix model to model covariance

45. Representer Matrix model to model covariance if we sample at observation locations through data to data covariance

46. Representer Matrix model to model covariance if we sample at observation locations through data to data covariance if we introduce a non-diagonal initial condition covariance preconditioneddata to data covariance

47. … back to the system to invert ….

48. STRONG CONSTRAINT def: 4DVAR inversion Hessian Matrix IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

49. WEAK CONSTRAINT def: 4DVAR inversion Hessian Matrix IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

50. WEAK CONSTRAINT How to solve for corrections ?  Method of solution in IROMS IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix