ROMS 4D-Var: The Complete Story

ROMS 4D-Var: The Complete Story

Andy MooreOcean Sciences Department

University of California Santa Cruz&

Hernan ArangoIMCS, Rutgers University

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

Outline

• What is data assimilation?

• Review 4-dimensional variational methods

• Illustrative examples for California Current

What is data assimilation?

Best Linear Unbiased Estimate (BLUE)

1y 2y

Prior hypothesis: random, unbiased, uncorrelated errors

1 2, Error std:

Find: A linear, minimum variance, unbiased estimate

1 1 2 2ax a y a y 1 22

a a truex x so that is minimised


2 22 1

1 22 2 2 21 2 1 2

ax y y

2 2 2 1

1 1 1 2 2 1( ) ( )y y y

2 2

1 21 22 2 2 2

1 2 1 2

ax y y

2 2 1 2

1 1 2 2 2 1( ) ( )y y y

OR


2 2 2 1( ) ( )a b b b y bx x y x

2 2 1 2( ) ( )a b b y y bx x y x

OR

Let 1 2, by x y y

2 2 1 2( )a b y

Posterior error:

ROMS

,y R

Data Assimilation

bb(t), Bb

fb(t), Bf

xb(0), B

time

x(t)

Obs, y

Model solutions depends on xb(0), fb(t), bb(t), (t)

xb(t)

Data Assimilation

Find ( (0), ( ), ( ), ( ))T T T Tt t t b fz x ε ε η

that minimizes the variance given by:

initialconditionincrement

boundaryconditionincrement

forcingincrement

corrections for model

error

1 11 1

2 2TTJ z D z Gz d R Gz d

diag( , , , ) b fD B B B Q

Background error covariance

TangentLinearModel

ObsErrorCov.

Innovation

bd y Hx

4D-Variational Data Assimilation (4D-Var)

At the minimum of J we have :

( ) ( )T Ta

1 1 1 1b bz z D G R G G R y Hx

( ) ( )T Ta

1b bz z DG GDG R y Hx

OR

time

x(t)

Obs, y

xb(t)

xa(t)

J z 0

Matrix-less Operations

TGDG δThere are no matrix multiplications!

Zonal shear flow


There are no matrix multiplications!

Adjoint ROMS

TGDG δ

Zonal shear flow



Covariance

TGDG δ

Zonal shear flow



Tangent Linear ROMS

TGDG δ

Zonal shear flow

Representers

TGDG δ

A covariance

= A representer

Green’s Function

Zonal shear flow

A Tale of Two Spaces

( ) ( )T Ta


( ) ( )T Ta


K = Kalman Gain Matrix

Solve linear system of equations!


( ) ( )T Ta


( ) ( )T Ta



model t model t(N N ) (N N )

obs obs(N N )obs modelN N


( ) ( )T Ta


( ) ( )T Ta


Model space searches: Incremental 4D-Var (I4D-Var)

Observation space searches: Physical-space Statistical Analysis System (4D-PSAS)

An alternative approach in observation space:The Method of Representers

(t) (t) (t) a bx x β

matrix of representers

vector of representercoefficients

(t)bx : solution of finite-amplitude linearization of ROMS (RPROMS)

R4D-Var

(Bennett, 2002)

Representers

TGDG δ

A covariance

= A representer

Green’s Function

Zonal shear flow

4D-Var: Two Flavours

Strong constraint: Model is error free ( ) 0t η

Weak constraint: Model has errors ( ) 0t η

Only practical in observation space

4D-Var Summary

Model space: I4D-Var, strong only (IS4D-Var)

Observation space: 4D-PSAS, R4D-Var strong or weak

Former Secretary of DefenseDonald Rumsfeld

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.

The California Current (CCS)

The California Current System (CCS)

30km grid 10km grid

Veneziani et al (2009)Broquet et al (2009)

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)

fb(t) bb(t)

3km grid

ChrisEdwards

Observations (y)

CalCOFI &GLOBEC

SST &SSH

ARGO

TOPP Elephant Seals

Ingleby andHuddleston (2007)

Strong Constraint 4D-Var


( ) ( )T Ta


( ) ( )T Ta



5model modelN N ~ 10

4obs obs(N N ) ~ 10

obs modelN <N

CCS 4D-Var

( ) ( )T Ta


T( (0), (t), (t), (t))b b b bz x b f 0

From previouscycle

ECCO COAMPS

( ) ( )T Ta


( )T 1 1 1D G R G

Model space (~105):

Observation space (~104):

( )T 1GDG R

Both matrices are conditioned the samewith respect to inversion(Courtier, 1997)

Jmin

July 2000: 4 day assimilation windowSTRONG CONSTRAINT

# iterations # iterations(1 outer, 50 inner,Lh=50 km, Lv=30m)

Model Space vs Observation Space(I4D-Var vs 4D-PSAS vs R4D-Var)

J J

SST Increments x(0)

I4D-Var 4D-PSAS R4D-VarModel Space

Inner-loop 50

Observation Space

Observation Space

Initial conditions vs surface forcingvs boundary conditions

No

ass

imil

atio

n

i.c.only

i.c. + f i.c.+ f + b.c.

J

IS4D-Var, 1 outer, 50 inner4 day window, July 2000

Model Skill

No assim.Assim.14d frcst

I4D-Var

RMS error in temperature

(1 outer, 20 inner, 14d cyclesLh=50 km, Lv=30m)

Broquet et al (2009)

Surface Flux Corrections, (I4D-Var)

Wind stress increments(Spring, 2000-2004)

Heat flux increments(Spring, 2000-2004)

fε

Broquet

Weak Constraint 4D-Var

Model Error (t)

Model error priorstd in SST


( ) ( )T Ta


( ) ( )T Ta



8model t model t(N N ) (N N ) ~ 10

4obs obs(N N ) ~ 10

obs modelN N

( )T 1 1 1D G R G

( )T 1GDG R

Jmin

# iterations # iterations(1 outer, 50 inner,Lh=50 km, Lv=30m)

Model Space vs Observation Space(I4D-Var vs 4D-PSAS vs R4D-Var)

July 2000: 4 day assimilation windowSTRONG vs WEAK CONSTRAINT

J J

Model space (~108):

Observation space (~104):

4D-Var Post-Processing

• Observation sensitivity• Representer functions• Posterior errors

Assimilation impacts on CC

No assim

IS4D-Var

Time meanalongshore

flow across 37N,2000-2004

(30km)

(Broquet et al,2009)

0 127

500 122

(37N,day 7)W

W

I v d dz

Observation Sensitivity

0.3 SvI

I y

What is the sensitivity of the transport I tovariations in the observations?

What is ?

Observations (y)

CalCOFI &GLOBEC

SST &SSH

ARGO

TOPP Elephant Seals


Sensitivity of upper-ocean alongshoretransport across 37N, 0-500m, on day 7to SST & SSH observations on day 4(July 2000)

SSH day 4 SST day 4

Sverdrups per degree CSverdrups per metre

Observation Sensitivity

Applications: predictability, quality control, array design

CalCOFI GLOBEC

Sv/deg C Sv/psu Sv/deg C Sv/psu

dep

th

I T I T I S I S

Applications: predictability, quality control, array design

Observations (y)

CalCOFI &GLOBEC

SST &SSH

ARGO

TOPP Elephant Seals


The Method of Representers

(t) (t) (t) a bx x β

matrix of representers

vector of representer

coeffiecients

(t)bx : solution of finite-amplitude linearization of ROMS (RPROMS)


Representers

TGDG δ

A covariance

= A representer

Green’s Function

Representer Functions

, day 0T T , day 14T T

, day 0T S , day 0T

70

80

90

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

ROMSROMS

ROMS 4D-Var: The Complete Story

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