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 assimilation system 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: REP-ROMS: TL-ROMS: AD-ROMS: Representer Tangent Linear Model
Perturbation Tangent Linear Model
TL-ROMS:
Adjoint Model
AD-ROMS:

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

8. REP-ROMS: TL-ROMS: AD-ROMS: Small Errors model missing dynamics
(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 TL-ROMS: 1) Set of observations
2) Model trajectory
3) Find that minimizes
Sampling functional
TL-ROMS:

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

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

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

20. Modeling the Corrections
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. Modeling the Corrections
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. STRONG CONSTRAINT 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. Modeling the Corrections
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. Modeling the Corrections
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. observations X model space
is a mapping matrix of dimensions
observations X model space
def:
ASSIMILATION (3)
Cost Function

27. observations X model space
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. IOM representer-based inversion
def:
4DVAR inversion
Hessian Matrix
IOM representer-based inversion
Representer Coefficients
Stabilized Representer Matrix
Representer Matrix

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

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

34. IOM representer-based inversion
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. IOM representer-based inversion
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. IOM representer-based inversion
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 and take 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 and take 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. model to model covariance
Representer Matrix
model to model covariance
if we sample at observation locations through
data to data covariance

46. model to model covariance
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
preconditioned data to data covariance

47. … back to the system to invert ….

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

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

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

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

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

53. 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

54. 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

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

56. Baroclinic Jet Example -

57. … end