1. Weak and Strong Constraint 4DVAR in the R egional O cean M odeling S ystem ( ROMS ): Development and Applications Di Lorenzo, E. Georgia Institute of Technology Arango, H. Rutgers University Moore, A. and B. Powell UC Santa Cruz Cornuelle, B and A.J. Miller Scripps Institution of Oceanography

2.  Short review of development and theory (an alternative derivation of the representer method)  Current applications (some pending issues)

3. Inverse Regional Ocean Modeling System (ROMS) Chua and Bennett (2001) Inverse Ocean Modeling System (IOMs) Moore et al. (2004) NL-ROMS, TL-ROMS, REP-ROMS, AD-ROMS To implement a representer-based generalized inverse method to solve weak constraint data assimilation problems a representer-based 4D-variational data assimilation system for high-resolution basin-wide and coastal oceanic flows Di Lorenzo et al. (2007)

4. Non Linear Model Tangent Linear Model Representer Model Adjoint Model Sensitivity Analysis Data Assimilation 1) Incremental 4DVAR Strong Constrain 2) Indirect Representer Weak and Strong Constrain 3) PSAS Ensemble Ocean Prediction Stability Analysis Modules ROMS Block Diagram NEW Developments Arango et al. 2003 Moore et al. 2004 Di Lorenzo et al. 2007

5. Download: ROMS components http://myroms.org Arango H. IOM components http://iom.asu.edu Muccino, J. et al. Inverse Ocean Modeling Portal

6. STRONG Constraint WEAK Constraint (A)(B) …we want to find the corrections e Best Model Estimate (consistent with observations) Initial Guess ASSIMILATION Goal

7. Quadratic Linear Cost Function for residuals

8. 2) corrections should not exceed our assumptions about the errors in model initial condition. 1) corrections should reduce misfit within observational error Quadratic Linear Cost Function for residuals

9. ASSIMILATIONCost Function

10. ASSIMILATIONCost Function def: is a mapping matrix of dimensions observations X model space

11. ASSIMILATIONCost Function def: is a mapping matrix of dimensions observations X model space

12. ASSIMILATIONCost Function

13. 4DVAR inversion Hessian Matrix def:

14. 4DVAR inversion IOM representer-based inversion Hessian Matrix def:

15. 4DVAR inversion IOM representer-based inversion Hessian Matrix Stabilized Representer Matrix Representer Coefficients Representer Matrix def:

16. 4DVAR inversion IOM representer-based inversion Hessian Matrix Stabilized Representer Matrix Representer Coefficients Representer Matrix def:

17. An example of Representer Functions for the Upwelling System Computed using the TL-ROMS and AD-ROMS

18. An example of Representer Functions for the Upwelling System Computed using the TL-ROMS and AD-ROMS

19. Applications of inverse ROMS:  Baroclinic coastal upwelling: synthetic model experiment to test the development  CalCOFI Reanalysis: produce ocean estimates for the CalCOFI cruises from 1984- 2006. Di Lorenzo, Miller, Cornuelle and Moisan  Intra-Americas Seas Real-Time DA Powell, Moore, Arango, Di Lorenzo, Milliff et al.

20. Coastal Baroclinic Upwelling System Model Setup and Sampling Array section

21. 1) The representer system is able to initialize the forecast extracting dynamical information from the observations. 2) Forecast skill beats persistence Applications of inverse ROMS:  Baroclinic coastal upwelling: synthetic model experiment to test the development 10 day assimilation window 10 day forecast

22. SKILL of assimilation solution in Coastal Upwelling Comparison with independent observations SKIL L DAYS Climatology Weak Strong Persistence Assimilation Forecast  Di Lorenzo et al. 2007; Ocean Modeling

23. Day= 0 Day= 2 Day= 6 Day= 10

24. Day= 0 Day= 2 Day= 6 Day= 10 Assimilation solutions

25. Day= 14 Day= 18 Day= 22 Day= 26

26. Day= 14 Day= 18 Day= 22 Day= 26

27. Forecast Day= 14 Day= 18 Day= 22 Day= 26 Day= 14 Day= 18 Day= 22 Day= 26

28. April 3, 2007 Intra-Americas Seas Real-Time DA Powell, Moore, Arango, Di Lorenzo, Milliff et al. www.myroms.org/ias

29. CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. Di Lorenzo, Miller, Cornuelle and Moisan

30. Things we struggle with …  Tangent Linear Dynamics can be very unstable in realistic settings.  Background and Model Error COVARIANCE functions are Gaussian and implemented through the use of the diffusion operator.  Fitting data vs. improving the dynamical trajectory of the model.

31. Assimilation of surface Salinity True True Initial Condition

32. True True Initial Condition Which model has correct dynamics? Assimilation of surface Salinity Model 1Model 2

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

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

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

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

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

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

39. RMS difference from TRUE Observations Days RMS Less constraint More constraint

40. Applications of inverse ROMS (cont.)  Improve model seasonal statistics using surface and open boundary conditions as the only controls.  Predictability of mesoscale flows in the CCS: explore dynamics that control the timescales of predictability. Mosca et al. – (Georgia Tech)

41. inverse machinery of ROMS can be applied to regional ocean climate studies …

42. EXAMPLE: Decadal changes in the CCS upwelling cells Chhak and Di Lorenzo, 2007; GRL

43. SSTa Composites 1 2 3 4 Observed PDO index Model PDO index Warm PhaseCold Phase Chhak and Di Lorenzo, 2007; GRL

44. -50 -100 -150 -250 -200 -350 -300 -450 -400 -500 -140W -130W -120W 30N 40N 50N -50 -100 -150 -250 -200 -350 -300 -450 -400 -500 -140W -130W -120W 30N 40N 50N COLD PHASE ensemble average WARM PHASE ensemble average April Upwelling Site Pt. Conception Chhak and Di Lorenzo, 2007; GRL Pt. Conception depth [m] Tracking Changes of CCS Upwelling Source Waters during the PDO using adjoint passive tracers enembles

45. Concentration Anomaly Model PDO PDO lowpassed Surface 0-50 meters (-) 50-100 meters (-) 150-250 meters year Changes in depth of Upwelling Cell (Central California) and PDO Index Timeseries Chhak and Di Lorenzo, 2007; GRL Adjoint Tracer

46. Arango, H., A. M. Moore, E. Di Lorenzo, B. D. Cornuelle, A. J. Miller, and D. J. Neilson, 2003: The ROMS tangent linear and adjoint models: A comprehensive ocean prediction and analysis system. IMCS, Rutgers Tech. Reports. Moore, A. M., H. G. Arango, E. Di Lorenzo, B. D. Cornuelle, A. J. Miller, and D. J. Neilson, 2004: A comprehensive ocean prediction and analysis system based on the tangent linear and adjoint of a regional ocean model. Ocean Modeling, 7, 227-258. Di Lorenzo, E., Moore, A., H. Arango, Chua, B. D. Cornuelle, A. J. Miller, B. Powell and Bennett A., 2007: Weak and strong constraint data assimilation in the inverse Regional Ocean Modeling System (ROMS): development and application for a baroclinic coastal upwelling system. Ocean Modeling, doi:10.1016/j.ocemod.2006.08.002. References

47. Adjoint passive tracers ensembles physical circulation independent of

48. Australia Asia USA Canada Pacific Model Grid SSHa (Feb. 1998) Regional Ocean Modeling System (ROMS)

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

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

51. COLD PHASE ensemble average WARM PHASE ensemble average April Upwelling Site Pt. Conception Chhak and Di Lorenzo, 2007; GRL

52. What if we really have substantial model errors?

53. Current application of inverse ROMS in the California Current System (CCS): 1)CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. NASA - Di Lorenzo, Miller, Cornuelle and Moisan 2)Predictability of mesoscale flow in the CCS: explore dynamics that control the timescales of predictability. Mosca and Di Lorenzo 3)Improve model seasonal statistics using surface and open boundary conditions as the only controls.

54. Comparison of SKILL score of IOM assimilation solutions with independent observations HIRES: High resolution sampling array COARSE: Spatially and temporally aliased sampling array

55. RP-ROMS with CLIMATOLOGY as BASIC STATE RP-ROMS with TRUE as BASIC STATE RP-ROMS WEAK constraint solution Instability of the Representer Tangent Linear Model (RP-ROMS) SKILL SCORE

56. TRUE Mesoscale Structure SSH [m] SST [C] ASSIMILATION Setup California Current Sampling: (from CalCOFI program) 5 day cruise 80 km stations spacing Observations: T,S CTD cast 0-500m Currents 0-150m SSH Model Configuration: Open boundary cond. nested in CCS grid 20 km horiz. Resolution 20 vertical layers Forcing NCEP fluxes Climatology initial cond.

57. SSH [m] WEAK day=5 STRONG day=5 TRUE day=5 ASSIMILATION Results 1 st GUESS day=5

58. WEAK day=5 STRONG day=5 ASSIMILATION Results ERROR or RESIDUALS SSH [m] 1 st GUESS day=5

59. WEAK day=0 STRONG day=0 TRUE day=0 Reconstructed Initial Conditions 1 st GUESS day=0

60. Normalized Observation-Model Misfit Assimilated data: TS 0-500m Free surface Currents 0-150m T S V U  observation number Error Variance Reduction STRONG Case = 92% WEAK Case = 98%