ROMS 4 DVar The Complete Story Andy Moore

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

Acknowledgements • • • • Chris Edwards, UCSC • ONR Jerome Fiechter, UCSC • NSF 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

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

What is data assimilation?

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

Best Linear Unbiased Estimate (BLUE) OR

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

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)

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

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

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

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

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

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

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

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

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

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

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

A Tale of Two Spaces Solve linear system of equations!

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

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) R 4 D-Var

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

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

4 D-Var Summary Model space: I 4 D-Var, strong only (IS 4 D-Var) Observation space: 4 D-PSAS, R 4 D-Var strong or weak

Former Secretary of Defense Donald Rumsfeld

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

The California Current (CCS)

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

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

3 km grid Chris Edwards

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

Strong Constraint 4 D-Var

A Tale of Two Spaces Solve linear system of equations!

CCS 4 D-Var From previous cycle ECCO COAMPS

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

SST Increments dx(0) Inner-loop 50 I 4 D-Var 4 D-PSAS Model Space Observation Space R 4 D-Var Observation Space

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

Model Skill RMS error in temperature (1 outer, 20 inner, 14 d cycles Lh=50 km, Lv=30 m) No assim. Assim. 14 d frcst I 4 D-Var Broquet et al (2009)

Surface Flux Corrections, (I 4 D-Var) Wind stress increments (Spring, 2000 -2004) Heat flux increments (Spring, 2000 -2004) Broquet

Weak Constraint 4 D-Var

Model Error h(t) Model error prior std in SST

A Tale of Two Spaces Solve linear system of equations!

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

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

Assimilation impacts on CC No assim Time mean alongshore flow across 37 N, 2000 -2004 (30 km) IS 4 D-Var (Broquet et al, 2009)

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

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

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

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

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

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

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

Representer Functions 70 80 90

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

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