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ROMS 4D-Var: Past, Present & Future
Andy MooreUC Santa Cruz
Overview
• Past: A review of the current system.• Present: New features coming soon.• Future: Planned new features and developments.
The Past….
Acknowledgements
• Hernan Arango – Rutgers University• Art Miller – Scripps• Bruce Cornuelle – Scripps• Emanuelle Di Lorenzo – GA Tech• Brian Powell – University of Hawaii• Javier Zavala-Garay - Rutgers University• Julia Levin - Rutgers University• John Wilkin - Rutgers University• Chris Edwards – UC Santa Cruz• Hajoon Song – MIT• Anthony Weaver – CERFACS• Selime Gürol – CERFACS/ECMWF• Polly Smith – University of Reading• Emilie Neveu – Savoie University
Acknowledgements
• Hernan Arango – Rutgers University• Art Miller – Scripps• Bruce Cornuelle – Scripps• Emanuelle Di Lorenzo – GA Tech• Doug Nielson - Scripps
“In the beginning…” Genesis 1.1
No grey hair!!!
“In the beginning…” Genesis 1.1
Regions where ROMS 4D-Var has been used
Data Assimilation
bb(t), Bb
fb(t), Bf
xb(0), Bx
ROMS
Model Observations
Incomplete picture ofthe real ocean
A complete picture butsubject to errors and
uncertainties
Prior +
Posterior
Bayes’ Theorem Data Assimilation
Data Assimilation
bb(t), Bb
fb(t), Bf
xb(0), Bx
ROMS
Model Observations
Prior +
(0)
x
z f
b
The control vector:
x
f
b
B
B B
B
Prior error covariance:
Maximum Likelihood Estimate & 4D-Var
expP J z y
azz
P z yProbability
1 1( ) ( )T T
H HJ b b yz z z yRBz z z
Prior Priorerrorcov.
Obserrorcov.
Obs Obsoperator
The cost function:
Maximize P(z|y) byminimizing J usingvariational calculus
T -1k b k b
T -1k k ( ) ( )
NLJ
H H
z z B z z
y z R y z
4D-Var Cost Function
Cost function minimum identified using truncatedGauss-Newton method via inner- and outer-loops:
TT -1 -1k k k k k-1 k k k-1J z B z G z d R G z d
k k-1 b z z z
k G Tangent linear ROMS sampled at obs points(generalized observation operator)
k-1 k-1( )H d y z
Control vector
initial conditions
surface forcing
open boundary conditions
corrections for model error
z
1
N
y
y
y
Observation vector
Solution
k b k k z z K d
-1-1 T -1 T -1k k k k K B G R G G R
-1T T Tk k k kK BG G BG R
Optimal estimate:
Gain matrix – primal form:
Gain matrix – dual form:
Okay for strong constraint, prohibitive for weak constraint.
Okay for strong constraint and weak constraint.
Solution
-1 T -1 T -1k k k k k B G R G x G R d
T Tk k k k G BG R λ d
Traditionally, primal form used by solving:
The dual form is appropriate for strong and weakconstraint:
Okay for strong constraint, prohibitive for weak constraint.
Tk k k x BG λ
The Lanczos Formulation of CG
ROMS offers both primal and dual options
In both J is minimized using Lanczos formulation of CG
Au b -1 Tu VT V bTT V AV T V V I
Generalform:
Approxsolution:
Tridiagonalmatrix:
Orthonormalmatrix:
iV v iv Lanczos vectors: one per inner-loop
-1 T T -1p p pK V T V G RPrimal
T -1 Td d dK BG V T VDual
-1 T -1 A B G R G
T A GBG R
Primal:
Dual:
• Incremental (linearized about a prior) (Courtier et al, 1994)• Primal & dual formulations (Courtier 1997) • Primal – Incremental 4-Var (I4D-Var) • Dual – PSAS (4D-PSAS) & indirect representer (R4D-Var) (Da Silva et al, 1995; Egbert et al, 1994)• Strong and weak (dual only) constraint• Preconditioned, Lanczos formulation of conjugate gradient
(Lorenc, 2003; Tshimanga et al, 2008; Fisher, 1997)• 2nd-level preconditioning for multiple outer-loops• Diffusion operator model for prior covariances
(Derber & Bouttier, 1999; Weaver & Courtier, 2001)• Multivariate balance for prior covariance (Weaver et al, 2005)• Physical and ecosystem components • Parallel (MPI)• Moore et al (2011a,b,c, PiO); www.myroms.org
ROMS 4D-Var
• Observation impact (Langland and Baker, 2004)
• Observation sensitivity – adjoint of 4D-Var (OSSE) (Gelaro et al, 2004)
• Singular value decomposition (Barkmeijer et al, 1998)
• Expected errors (Moore et al., 2012)
ROMS 4D-Var Diagnostic Tools
Observation Impacts
The impact of individual obs on the analysis orforecast can be quantified using:
T -1 -1 Tp p pK R GV T V
T -1 Td d dK V T V GB
Primal
Dual
Conveniently computed from 4D-Var output
Observation Sensitivity
Treat 4D-Var as a function:
k b k x x dK
T
k dK Quantifies sensitivity ofanalysis to changes in obs
Adjoint of 4D-Var
Adjoint of 4D-Var also yields estimates of expectederrors in functions of state.
Impact of the Observations on AlongshoreTransport
Total number of obs
Observation Impact
March 2012 Dec 2012
March 2012 Dec 2012Ann Kristen Sperrevik (NMO)
Impact of HF radar on 37N transport
Impact of HF radar on 37N transport
Impact of MODIS SST on 37N transport
The Present….
New stuff not in the svn yet
• Augmented B-Lanczos formulation
New stuff not in the svn yet
4D-Var Convergence Issues
Primal preconditioned by B has good convergenceproperties: T -1I G R GB Preconditioned Hessian
Dual preconditioned by R-1 has poor convergenceproperties: -1 T R GBG I Preconditioned stabilized
representer matrix
Restricted preconditioned CG ensures that dual4D-Var converges at same rate as B-preconditionedPrimal 4D-Var (Gratton and Tschimanga, 2009)
Can be partly alleviated using the Minimum ResidualMethod (El Akkraoui et al, 2008; El Akkraoui and Gauthier, 2010)
Restricted Preconditioned Conjugate Gradient
Strong Constraint Weak Constraint
(Gürol et al, 2013, QJRMS)
Augmented Restricted B-Lanczos
For multiple outer-loops:
• Augmented B-Lanczos formulation• Background quality control
New stuff not in the svn yet
2 2 2 21o bi i b o by y
ˆ 2ln[ / max( )]f f f f
Background Quality Control(Andersson and Järvinen, 1999)
PDF of in situ T innovations Transformed PDF of in situ T innovations
16
16
• Augmented B-Lanczos formulation• Background quality control• Biogeochemical modules: - TL and AD of NEMURO - log-normal 4D-Var
New stuff not in the svn yet
Hajoon Song
Ocean Tracers: Log-normal or otherwise?
Campbell (1995) – in situ ocean Chlorophyll, northern hemisphere
Assimilation of biological variables
• Differs from physical variables in statistics. – Gaussian vs skewed
non-Gaussian
• We use lognormal transformation
• Maintains positive definite variables and reduces rms errors over Gaussian approach
Song et al. (2013)
NPZ model
Lognormal 4DVAR (L4DVAR) Example• PDF of biological variables is often closer to lognormal than Gaussian.• Positive-definite property is preserved in L4DVAR.
Model twin experiment. Initial surface phytoplankton concentration (log scale).Negative values in black.
Truth Prior L4DVARPosterior
G4DVARPosterior
Biological Assimilation, an example• 1 year (2000) SeaWiFS ocean color assimilation• NPZD model• Being implemented in near-realtime system
Gray color indicates cloud cover
Song et al. (in prep)
1-Day SeaWiFS
8-Day SeaWiFS
Model –No Assimilation
Model –With Assimilation
• Augmented B-Lanczos formulation• Background quality control• Biogeochemical modules: - TL and AD of NEMURO - log-normal 4D-Var• Correlations on z-levels• Improved mixed layer formulation in balance operator• Time correlations in Q
New stuff not in the svn yet
Recent Bug Fixes
• Normalization coefficients for B
• Open boundary adjustments in 4D-Var
T T T b bΛ ΛB K Σ L Σ K
The Future….
Planned Developments
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
Planned Developments
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R
Planned Developments
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)
Planned Developments
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B
Planned Developments
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B• Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)
Planned Developments
0m
100m
200m
0m
100m
200m EQ 15S 15N
SEC NECNECC
EUCNEC=N. Eq. Curr.SEC=S. Eq. CurrNECC=N. Eq. Counter Curr.EUC=Eq. Under Curr.
Equatorial PacificTemperature
Observation
Weaver and Courtier (2001)(3D-Var & 4D-Var)
Diffusion eqn with adiffusion tensor.
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B• Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)
Planned Developments
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B• Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)• TL and AD of parameters
Planned Developments
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B• Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)• TL and AD of parameters• Nested 4D-Var
Planned Developments
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B• Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)• TL and AD of parameters• Nested 4D-Var• POD for biogeochemistry
Planned Developments
22
2H V P P
P PP A P A Q S
t z
u
Biogeochemical Tracer Equation
Sources of P Sinks of P
Replace with an EOF decompositionP PQ S
(Following Pelc, 2013)
• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B• Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)• TL and AD of parameters• Nested 4D-Var• POD for biogeochemistry• TL and AD of sea-ice model
Planned Developments
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