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ROMS 4D-Var: The Complete Story

ROMS 4D-Var: The Complete Story. Andy Moore Ocean Sciences Department University of California Santa Cruz & Hernan Arango IMCS, Rutgers University. Acknowledgements. ONR NSF. Chris Edwards, UCSC Jerome Fiechter, UCSC Gregoire Broquet, UCSC Milena Veneziani, UCSC

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ROMS 4D-Var: The Complete Story

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

  2. Acknowledgements • ONR • NSF • 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

  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 is minimised so that

  6. Best Linear Unbiased Estimate (BLUE) OR

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

  8. Data Assimilation fb(t), Bf ROMS bb(t), Bb xb(0), B Obs, y xb(t) x(t) time Model solutions depends on xb(0), fb(t), bb(t), h(t)

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

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

  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 representer Green’s Function A covariance Zonal shear flow

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

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

  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 vector of representer coefficients matrix of representers (Bennett, 2002) : solution of finite-amplitude linearization of ROMS (RPROMS) R4D-Var

  23. Representers = A representer Green’s Function A covariance 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) 10km grid 30km grid Veneziani et al (2009) Broquet et al (2009)

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

  31. 3km grid Chris Edwards

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

  33. Strong Constraint 4D-Var

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

  35. CCS 4D-Var From previous cycle ECCO COAMPS

  36. Model Space vs Observation Space (I4D-Var vs 4D-PSAS vs R4D-Var) Model space (~105): Observation space (~104): J J Both matrices are conditioned the same with respect to inversion (Courtier, 1997) Jmin # iterations # iterations (1 outer, 50 inner, Lh=50 km, Lv=30m) July 2000: 4 day assimilation window STRONG CONSTRAINT

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

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

  39. Model Skill RMS error in temperature No assim. Assim. 14d frcst I4D-Var (1 outer, 20 inner, 14d cycles Lh=50 km, Lv=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 h(t) Model error prior std in SST

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

  44. Model Space vs Observation Space (I4D-Var vs 4D-PSAS vs R4D-Var) Model space (~108): Observation space (~104): J J Jmin # iterations # iterations (1 outer, 50 inner, Lh=50 km, Lv=30m) July 2000: 4 day assimilation window STRONG vs WEAK CONSTRAINT

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

  46. Assimilation impacts on CC No assim Time mean alongshore flow across 37N, 2000-2004 (30km) IS4D-Var (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 Ingleby and Huddleston (2007) TOPP Elephant Seals ARGO

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

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

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