ROMS 4D-Var: Past, Present & Future Andy Moore UC Santa Cruz.
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Transcript of ROMS 4D-Var: Past, Present & Future Andy Moore UC Santa Cruz.
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ROMS 4D-Var: Past, Present & Future
Andy MooreUC Santa Cruz
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Overview
• Past: A review of the current system.• Present: New features coming soon.• Future: Planned new features and developments.
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The Past….
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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
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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
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No grey hair!!!
“In the beginning…” Genesis 1.1
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Regions where ROMS 4D-Var has been used
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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
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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:
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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
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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
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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.
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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
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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:
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• 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
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• 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
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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
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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.
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Impact of the Observations on AlongshoreTransport
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Total number of obs
Observation Impact
March 2012 Dec 2012
March 2012 Dec 2012Ann Kristen Sperrevik (NMO)
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Impact of HF radar on 37N transport
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Impact of HF radar on 37N transport
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Impact of MODIS SST on 37N transport
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The Present….
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New stuff not in the svn yet
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• Augmented B-Lanczos formulation
New stuff not in the svn yet
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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)
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Restricted Preconditioned Conjugate Gradient
Strong Constraint Weak Constraint
(Gürol et al, 2013, QJRMS)
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Augmented Restricted B-Lanczos
For multiple outer-loops:
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• Augmented B-Lanczos formulation• Background quality control
New stuff not in the svn yet
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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
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16
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• 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
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Ocean Tracers: Log-normal or otherwise?
Campbell (1995) – in situ ocean Chlorophyll, northern hemisphere
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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
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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
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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
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• 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
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Recent Bug Fixes
• Normalization coefficients for B
• Open boundary adjustments in 4D-Var
T T T b bΛ ΛB K Σ L Σ K
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The Future….
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Planned Developments
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• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
Planned Developments
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• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R
Planned Developments
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• Digital filter – Jc to suppress initialization shock (Gauthier & Thépaut, 2001)
• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)
Planned Developments
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• 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
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• 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
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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.
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• 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
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• 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
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• 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
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• 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
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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)
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• 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