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 • Overview of the Inverse Regional Ocean Modeling System • Implementation - How do we assimilate data using the ROMS set of models • Examples, (a) Coastal upwelling (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: REP-ROMS:

6. def: REP-ROMS: TL-ROMS: AD-ROMS:

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

8. REP-ROMS: TL-ROMS: AD-ROMS:

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

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

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

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

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

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

15. 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:

16. 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

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

18. 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

19. 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

20. 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

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

22. 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.

23. ASSIMILATION (3) Cost Function

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

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

26. Minimize J ASSIMILATION (3) Cost Function

27. def: 4DVAR inversion Hessian Matrix

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

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

30. def: 4DVAR inversion Hessian Matrix IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix Data to Data Covariance

31. def: How to understand the physical meaning of the Representer Matrix? IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

32. Representer Matrix TL-ROMS AD-ROMS Representer Matrix

33. Representer Matrix Assume a special assimilation case:  Observations = Full model state at time T  Diagonal Covariance with unit variance

34. Representer Matrix Assume a special assimilation case:  Observations = Full model state at time T  Diagonal Covariance with unit variance

35. Representer Matrix

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

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

38. Representer Matrix model to model covariance Temperature Temperature Covariance for grid point n

39. Representer Matrix model to model covariance Temperature Temperature Covariance for grid point n Temperature Velocity Covariance for grid point n

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

41. Representer Matrix model to model covariance model to model covariance most general form if we sample at observation locations through data to data covariance

42. … back to the system to invert ….

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

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

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

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

47. Method of solution in IROMS STEP 1) Produce background state using nonlinear model starting from initial guess. STEP 2) Run REP-ROMS linearized around background state to generate first estimate of model trajectory outer loop STEP 3) Compute model-data misfit STEP 4) Solve for Representer Coeficients STEP 5) Compute corrections STEP 6) Update model state using REP-ROMS

48. Method of solution in IROMS STEP 1) Produce background state using nonlinear model starting from initial guess. STEP 2) Run REP-ROMS linearized around background state to generate first estimate of model trajectory outer loop STEP 3) Compute model-data misfit inner loop STEP 4) Solve for Representer Coeficients STEP 5) Compute corrections STEP 6) Update model state using REP-ROMS

49. How to evaluate the action of the stabilized Representer Matrix

50. How to evaluate the action of the stabilized Representer Matrix