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. 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 Goals

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

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

5. REP-ROMS: def:

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

7. TL-ROMS: AD-ROMS: REP-ROMS: Small Errors 1)model missing dynamics 2)boundary conditions errors 3)Initial conditions errors (STEP 2)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

23. ASSIMILATION (3)Cost Function

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

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

26. ASSIMILATION (3)Cost Function Minimize J

27. 4DVAR inversion Hessian Matrix def:

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

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

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

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

32. Representer Matrix TL-ROMSAD-ROMS

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. 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 most general form model to model covariance

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

42. … back to the system to invert ….

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

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

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

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

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 STEP 3) Compute model-data misfit STEP 4) Solve for Representer Coeficients STEP 5) Compute corrections STEP 6) Update model state using REP-ROMS outer loop

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 STEP 3) Compute model-data misfit STEP 4) Solve for Representer Coeficients STEP 5) Compute corrections STEP 6) Update model state using REP-ROMS outer loop inner loop

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

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

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

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

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

54. ROMS Coastal Upwelling Model Topography 10km res, Baroclinic, Periodic NS, CCS Topographic Slope with Canyons and Seamounts, full options! ShelfOpen Ocean 200 km Seasonal Wind Stress

55. ROMS Coastal Upwelling Final Time Initial Condition Temperature Sea Surface Heigth 10km res, Baroclinic, Periodic NS, CCS Topographic Slope with Canyons and Seamounts, full options! = 1 JAN ‘06 = 4 JAN ‘06

56. The ASSIMILATION Setup Final Time Initial Condition Problem: Assimilate a passive tracer, which was perfectly measured. Passive T racer

57. Model Dynamics are Correct True  Corrections only to initial state

58. True Diagonal Covariance Model Dynamics are Correct

59. True Diagonal CovarianceGaussian Covariance How about the initial condition? Model Dynamics are Correct Explained Variance 92%Explained Variance 89%

60. True True Initial Condition Model Dynamics are Correct Diagonal Covariance Explained Variance 75% Explained Variance 92% Gaussian Covariance Explained Variance 83% Explained Variance 89% Strong Constraint

61. What if we allow for error in the model dynamics? True True Initial Condition  Time Dependent corrections to model dynamics and boundary conditions

62. True Gaussian Covariance Explained Variance 24% Explained Variance 99% True Initial Condition Weak Constraint

63. True Gaussian Covariance Explained Variance 24%Explained Variance 83% Explained Variance 99%Explained Variance 89% True Initial Condition Weak Constraint Strong Constraint

64. Data Misfit (comparison) Weak Constraint Strong Constraint

65. What if we really have substantial model errors, or bias?

66. Assimilation of data at time True True Initial Condition

67. Model 1Model 2 Assimilation of data at time True True Initial Condition Which is the model with the correct dynamics?

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

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

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

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

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

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

74. Model 1Model 2 True True Initial Condition Wrong Model Good Model What if we apply more smoothing?

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

76. Assimilating SST Prior (First Guess)

77. Assimilating SST Wrong assumption in Prior (First Guess)After Assimilation

78. Assimilating SST No smoothing Prior (First Guess)After Assimilation

79. Assimilating SST Gaussian Covariance Smoothing Prior (First Guess)After Assimilation

80. RMS difference from TRUE Observations Days RMS Diagonal CASE Gaussian

81. Assimilation of Mesoscale Eddies in the Southern California Bight TRUE 1 st GUESS IOM solution Free SurfaceSurface NS VelocitySST Assimilated data: TS 0-500m Free surface Currents 0-150m

82. Normalized Misfit Datum Assimilated data: TS 0-500m Free surface Currents 0-150m (a) T S VU  Assimilation of Mesoscale Eddies in the Southern California Bight

83. Happy data assimilation and forecasting! Di Lorenzo, E., Moore, A., H. Arango, Chua, B. D. Cornuelle, A. J. Miller and Bennett A. (2005) The Inverse Regional Ocean Modeling System (IROMS): development and application to data assimilation of coastal mesoscale eddies. Ocean Modeling (in preparation)

85. … end

93. True Assimilation EXP 1Assimilation EXP 2 Assimilation of data at time Gaussian Diagonal

94. True GaussianDIagonal Data Misfit (comparison) Weak Constraint Strong Constraint Model Dynamics are InCorrect

95. True Assimilation EXP 1Assimilation EXP 2 Assimilation of data at time Wrong Gaussian

96. True Assimilation EXP 1Assimilation EXP 2 Assimilation of data at time Wrong Gaussian

100. True Gaussian Covariance Explained Variance 24%Explained Variance 91% Explained Variance 99% True Initial Condition Model Dynamics are Incorrect

101. Coastal Upwelling Baroclinic - Passive Sources