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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. initial condition - PowerPoint PPT Presentation
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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
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