The Regional Ocean Modeling System (ROMS) 4D-Var Assimilation Systems

applied to the California Current System

Andy Moore, Gregoire Broquet, Hernan Arango, Chris Edwards, Brian Powell, Milena Veneziani and Javier Zavala-

Garay

Research Supported by: NSF, ONR, NOPP

The Incremental Control Vector

ROMSbb(t), Bb

fb(t), Bf

xb(0), B

State vector: x=(T,S,u,v,)T Controls: initial conditions, x(0) surface forcing, f(t) boundary conditions, b(t)

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

initialconditionincrement

boundaryconditionincrement

forcingincrement

corrections for model

error

The Incremental Control Vector

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

initialconditionincrement

boundaryconditionincrement

forcingincrement

corrections for model

error

ROMSbb(t), Bb

fb(t), Bf

xb(0), B

1 11 1

2 2TTJ z D z Gz d R Gz d

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

Prior error covariance

TangentLinearModel

ObsErrorCov.

Innovation

bd y Hx

Cost/Penalty:

a bz z Kd

T T -1( ) K DG GDG R

1 T -1 -1 T -1( ) K D G R G G R

Analysis:

Gain (primal):

Gain (dual):

Primal vs Dual Formulation

ROMS has primal and dual 4D-Var options & strong vs weak constraint

a bz z Kd

T T -1( ) K DG GDG R

1 T -1 -1 T -1( ) K D G R G G R

Analysis:

Gain (primal):

Gain (dual):

Primal vs Dual Formulation

state stateN N

obs obsN N

obs stateN N

a bz z KdAnalysis:

Gain (primal):

Gain (dual):

Primal vs Dual Formulation

The Lanczos formulation of the CG algorithm lendsconsiderable utility to 4D-Var:

T 1 T d d dK DG V T V

1 T T -1 p p pK V T V G R

Vp = matrix of Lanczos vectors of the Hessian

Vd = matrix of dual Lanczos vectors

a bz z KdAnalysis:

Gain (primal):

Gain (dual):

Primal vs Dual Formulation

The Lanczos formulation of the CG algorithm lendsconsiderable utility to 4D-Var:

T 1 T d d dK DG V T V

1 T T -1 p p pK V T V G R

• Posterior (analysis) error variance (dual)

• Posterior (analysis) error EOFs (dual)

• Observation impact (primal & dual)

• Observation sensitivity (dual)

ROMS: California Current System (CCS)

30km, 10 km & 3 km grids, 30- 42 levels

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

COAMPSforcing

ECCO openboundaryconditions

fb(t), Bf

bb(t), Bb

xb(0), B

Previous assimilationcycle

Observations (y)

CalCOFI &GLOBEC/LTOP

SST &SSH

ARGO

TOPP Elephant Seals

Ingleby andHuddleston (2007)

Observation Impact(37N transport, 0-500m)

Nobs

T T TI I xd K M(Sv)I

TotalI

ISSH

ISST

IXBT

IT CTD

IT Argo

IS CTD

IS Argo

ITOPP

Total

SSH

SST

XBT

T CTD

T Argo

S CTD

S Argo

TOPP

TransportIncrement

7 day assimcycle

(strong)

Observation Impact(37N transport, 0-500m)

Nobs

T T TI I xd K M(Sv)I

TotalI

ISSH

ISST

IXBT

IT CTD

IT Argo

IS CTD

IS Argo

ITOPP

Total

SSH

SST

XBT

T CTD

T Argo

S CTD

S Argo

TOPP

TransportIncrement

7 day assimCycle

(strong)

Obs Impact vs Obs Sensitivity

T T TI I xd K M(Sv)I

TotalI

ISSH

ISST

IXBT

IT CTD

IT Argo

IS CTD

IS Argo

ITOPP

T T TI I y xd MK(Sv)I

TotalI

ISSH

ISST

IXBT

IT CTD

IT Argo

IS CTD

IS Argo

ITOPP

Impact

Sensitivity:based onadjoint of4D-Var

TransportIncrement

TransportIncrement

7 day assimcycle

(strong)

Obs Impact vs Obs Sensitivity

T T TI I xd K M(Sv)I

TotalI

ISSH

ISST

IXBT

IT CTD

IT Argo

IS CTD

IS Argo

ITOPP

T T TI I y xd MK(Sv)I

TotalI

ISSH

ISST

IXBT

IT CTD

IT Argo

IS CTD

IS Argo

ITOPP

Impact

Sensitivity:based onadjoint of4D-Var

TransportIncrement

TransportIncrement

7 day assimcycle

(strong)

Expected Posterior Error

SST 75mT T( ) ( ) aE I KG D I KG KRK

Expected Posterior Error

SST 75mdiag( )aE D

Posterior variance – Prior variance

Current Applications

NJ Bight

Intra-AmericasSea

CaliforniaCurrent

Gulf of AlaskaPhilippine

Archipelago

EastAustraliaCurrent

Jmin

(1 outer, 75 inner,Lh=50 km, Lv=30m,Obs errors:SSH 2 cmSST 0.4 Chydrographic 0.1 C 0.01psu)

Primal vs Dual Space(I4D-Var vs 4D-PSAS vs R4D-Var)

July 2000: 4 day assimilation windowSTRONG vs WEAK CONSTRAINT

1 - 4 July, 2000

(Slow convergence ofdual form also noted byEl Akkraoui and Gauthier, 2009)

i.c. = initial conditionwind = wind stressQ = heat flux(E-P) = freshwater fluxb.c. = open boundary conditions

Jmin

(R4DVAR: 1 outer-loop;75 inner-loops)

ROMS 4D-Var in the CCS

SSH error

SST error

no assim4D-Varforecast

ROMS 4D-Var DiagnosticsThe Lanczos formulation of the CG algorithm lendsconsiderable utility to 4D-Var:

a bx x Kd

T 1 T d d dK DG V T V

1 T T -1 p p pK V T V G R

Analysis:

Gain (primal):

Gain (dual):

Vp = matrix of Lanczos vectors of the Hessian

Vd = matrix of dual Lanczos vectors

ROMS 4D-Var Diagnostics

• Posterior/Analysis error variance (dual)• Posterior/Analysis error EOFs (dual)• Observation impact (primal & dual)• Observation sensitivity (dual)

The Lanczos formulation of the CG algorithm lendsconsiderable utility to 4D-Var:

a bx x KdAnalysis:

Gain (primal):

Gain (dual):T 1 T d d dK DG V T V

1 T T -1 p p pK V T V G R

Assimilation impacts on CC

No assim

IS4D-Var

Time meanalongshore

flow across 37N,2000-2004

(30km)

(Broquet et al,2009)

Sv

Numberof Obs

5 – 11 April, 2003

Nobs NSST NT NSNSSH NT NS NT NS NT

CalCOFI GLOBEC ARGO

To

tal

SS

T

SS

H T S T S T S TCalCOFI GLOBEC ARGO

980 obs(6%)

0.49Sv(63%)

XBT

XBT

Analysis Increment: upper 500 m Transport at 37N

11 April, 2003