roms 4d-var: the complete story
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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
Best Linear Unbiased Estimate (BLUE)
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
Best Linear Unbiased Estimate (BLUE)
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
Matrix-less Operations
There are no matrix multiplications!
Adjoint ROMS
TGDG δ
Zonal shear flow
Matrix-less Operations
There are no matrix multiplications!
Adjoint ROMS
TGDG δ
Zonal shear flow
Matrix-less Operations
There are no matrix multiplications!
Covariance
TGDG δ
Zonal shear flow
Matrix-less Operations
There are no matrix multiplications!
Covariance
TGDG δ
Zonal shear flow
Matrix-less Operations
There are no matrix multiplications!
Tangent Linear ROMS
TGDG δ
Zonal shear flow
Matrix-less Operations
There are no matrix multiplications!
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
1 1 1 1b bz z D G R G G R y Hx
( ) ( )T Ta
1b bz z DG GDG R y Hx
K = Kalman Gain Matrix
Solve linear system of equations!
A Tale of Two Spaces
( ) ( )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
Solve linear system of equations!
model t model t(N N ) (N N )
obs obs(N N )obs modelN N
A Tale of Two Spaces
( ) ( )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
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
A Tale of Two Spaces
( ) ( )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
Solve linear system of equations!
5model modelN N ~ 10
4obs obs(N N ) ~ 10
obs modelN <N
CCS 4D-Var
( ) ( )T Ta
1 1 1 1b bz z D G R G G R y Hx
T( (0), (t), (t), (t))b b b bz x b f 0
From previouscycle
ECCO COAMPS
( ) ( )T Ta
1b bz z DG GDG R y Hx
( )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
A Tale of Two Spaces
( ) ( )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
Solve linear system of equations!
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
Ingleby andHuddleston (2007)
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
Ingleby andHuddleston (2007)
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)
There are no matrix multiplications!
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
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