1. ROMS 4D-Var: The Complete Story Andy Moore Ocean Sciences Department University of California Santa Cruz & Hernan Arango IMCS, Rutgers University

2. Acknowledgements Chris Edwards, UCSC Jerome Fiechter, UCSC Gregoire Broquet, UCSC Milena Veneziani, UCSC Javier Zavala, Rutgers Gordon Zhang, Rutgers Julia Levin, Rutgers John Wilkin, Rutgers Brian Powell, U Hawaii Bruce Cornuelle, Scripps Art Miller, Scripps Emanuele Di Lorenzo, Georgia Tech Anthony Weaver, CERFACS Mike Fisher, ECMWF ONR NSF

3. Outline What is data assimilation? Review 4-dimensional variational methods Illustrative examples for California Current

4. What is data assimilation?

5. Best Linear Unbiased Estimate (BLUE) Prior hypothesis: random, unbiased, uncorrelated errors Error std: Find: A linear, minimum variance, unbiased estimate so that is minimised

6. Best Linear Unbiased Estimate (BLUE) OR

7. Best Linear Unbiased Estimate (BLUE) OR Let Posterior error:

8. ROMS Data Assimilation b b (t), B b f b (t), B f x b (0), B time x(t) Obs, y Model solutions depends on x b (0), f b (t), b b (t), h(t) x b (t)

9. Data Assimilation Find that minimizes the variance given by: initial condition increment boundary condition increment forcing increment corrections for model error Background error covariance Tangent Linear Model Obs Error Cov. Innovation

10. 4D-Variational Data Assimilation (4D-Var) At the minimum of J we have : OR time x(t) Obs, y x b (t) x a (t)

11. Matrix-less Operations There are no matrix multiplications! Zonal shear flow

12. Matrix-less Operations There are no matrix multiplications! Adjoint ROMS Zonal shear flow

13. Matrix-less Operations There are no matrix multiplications! Adjoint ROMS Zonal shear flow

14. Matrix-less Operations There are no matrix multiplications! Covariance Zonal shear flow

15. Matrix-less Operations There are no matrix multiplications! Covariance Zonal shear flow

16. Matrix-less Operations There are no matrix multiplications! Tangent Linear ROMS Zonal shear flow

17. Matrix-less Operations There are no matrix multiplications! Tangent Linear ROMS Zonal shear flow

18. Representers A covariance = A representer Green’s Function Zonal shear flow

19. A Tale of Two Spaces K = Kalman Gain Matrix Solve linear system of equations!

20. A Tale of Two Spaces Solve linear system of equations!

21. A Tale of Two Spaces Model space searches: Incremental 4D-Var (I4D-Var) Observation space searches: Physical-space Statistical Analysis System (4D-PSAS)

22. An alternative approach in observation space: The Method of Representers matrix of representers vector of representer coefficients : solution of finite-amplitude linearization of ROMS (RPROMS) R4D-Var (Bennett, 2002)

23. Representers A covariance = A representer Green’s Function Zonal shear flow

24. 4D-Var: Two Flavours Strong constraint: Model is error free Weak constraint: Model has errors Only practical in observation space

25. 4D-Var Summary Model space: I4D-Var, strong only (IS4D-Var) Observation space: 4D-PSAS, R4D-Var strong or weak

26. Former Secretary of Defense Donald Rumsfeld

27. Why 3 4D-Var Systems? I4D-Var: traditional NWP, lots of experience, strong only (will phase out). R4D-Var: formally most correct, mathematically rigorous, problems with high Ro. 4D-PSAS: an excellent compromise, more robust for high Ro, formally suboptimal.

28. The California Current (CCS)

29. The California Current System (CCS) 30km grid 10km grid Veneziani et al (2009) Broquet et al (2009)

30. The California Current System (CCS) COAMPS 10km winds; ECCO open boundary conditions 30km grid 10km grid Veneziani et al (2009); Broquet et al (2009) June mean SST (2000-2004) f b (t)b b (t)

31. 3km grid Chris Edwards

32. Observations (y) CalCOFI & GLOBEC SST & SSH ARGO TOPP Elephant Seals Ingleby and Huddleston (2007)

33. Strong Constraint 4D-Var

34. A Tale of Two Spaces Solve linear system of equations!

35. CCS 4D-Var From previous cycle ECCOCOAMPS

36. Model space (~10 5 ): Observation space (~10 4 ): Both matrices are conditioned the same with respect to inversion (Courtier, 1997) J min July 2000: 4 day assimilation window STRONG CONSTRAINT # iterations (1 outer, 50 inner, L h =50 km, L v =30m) Model Space vs Observation Space (I4D-Var vs 4D-PSAS vs R4D-Var) J J

37. SST Increments  x(0) I4D-Var 4D-PSAS R4D-Var Model Space Inner-loop 50 Observation Space Observation Space

38. Initial conditions vs surface forcing vs boundary conditions No assimilationi.c. only i.c. + f + b.c. J IS4D-Var, 1 outer, 50 inner 4 day window, July 2000

39. Model Skill No assim. Assim. 14d frcst I4D-Var RMS error in temperature (1 outer, 20 inner, 14d cycles L h =50 km, L v =30m) Broquet et al (2009)

40. Surface Flux Corrections, (I4D-Var) Wind stress increments (Spring, 2000-2004) Heat flux increments (Spring, 2000-2004) Broquet

41. Weak Constraint 4D-Var

42. Model Error  (t) Model error prior std in SST

43. A Tale of Two Spaces Solve linear system of equations!

44. J min # iterations (1 outer, 50 inner, L h =50 km, L v =30m) Model Space vs Observation Space (I4D-Var vs 4D-PSAS vs R4D-Var) July 2000: 4 day assimilation window STRONG vs WEAK CONSTRAINT J J Model space (~10 8 ): Observation space (~10 4 ):

45. 4D-Var Post-Processing Observation sensitivity Representer functions Posterior errors

46. Assimilation impacts on CC No assim IS4D-Var Time mean alongshore flow across 37N, 2000-2004 (30km) (Broquet et al, 2009)

47. Observation Sensitivity What is the sensitivity of the transport I to variations in the observations? What is ?

48. Observations (y) CalCOFI & GLOBEC SST & SSH ARGO TOPP Elephant Seals Ingleby and Huddleston (2007)

49. Sensitivity of upper-ocean alongshore transport across 37N, 0-500m, on day 7 to SST & SSH observations on day 4(July 2000) SSH day 4SST day 4 Sverdrups per degree CSverdrups per metre Observation Sensitivity Applications: predictability, quality control, array design

50. CalCOFI GLOBEC Sv/deg CSv/psuSv/deg CSv/psu depth Applications: predictability, quality control, array design

51. Observations (y) CalCOFI & GLOBEC SST & SSH ARGO TOPP Elephant Seals Ingleby and Huddleston (2007)

52. The Method of Representers matrix of representers vector of representer coeffiecients : solution of finite-amplitude linearization of ROMS (RPROMS)

53. There are no matrix multiplications! Representers A covariance = A representer Green’s Function

54. Representer Functions 70 80 90

55. Summary ROMS 4D-Var system is unique Powerful post-processing tools All parallel 4D-Var rounds out the adjoint sensitivity and generalized stability tool kits in ROMS CCS, CGOA, IAS, EAC, PhilEX Biological assimilation Outstanding issues: - multivariate refinements for coastal regions - non-isotropic, non-homogeneous cov. - multiple grids - posterior errors

56. ROMS