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The Inverse Regional Ocean Modeling System:

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. Goals. A brief overview of the Inverse Regional Ocean Modeling System

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The Inverse Regional Ocean Modeling System:

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  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

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