ROMS 4D-Var: Past, Present & Future

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)

xz f

b

The control vector:

x

f

b

BB B

B

Prior error covariance:

Maximum Likelihood Estimate & 4D-Var

expP J z y

azz

P z yProbability

1 1( ) ( )T TH 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 conditionssurface forcingopen boundary conditionscorrections 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

Tk dK Quantifies sensitivity of

analysis 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 MODIS SST on 37N transport

The Present….

New stuff not in the svn yet

• Augmented B-Lanczos formulation


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


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

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• Augmented B-Lanczos formulation• Background quality control• Biogeochemical modules: - TL and AD of NEMURO - log-normal 4D-Var


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


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)



• Non-diagonal R



• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)



• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B



• Non-diagonal R• Bias-corrected 4D-Var (Dee, 2005)• Time correlations in B• Correlations rotated along isopycnals using diffusion tensor

(Weaver & Courtier, 2001)


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.



(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)




(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)• TL and AD of parameters




(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)• TL and AD of parameters• Nested 4D-Var




(Weaver & Courtier, 2001)• Combine 4D-Var and EnKF (hybrid B)• TL and AD of parameters• Nested 4D-Var• POD for biogeochemistry


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2H V P PP PP A P A Q St z

u

Biogeochemical Tracer Equation

Sources of P Sinks of P

Replace with an EOF decompositionP PQ S

(Following Pelc, 2013)



(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


ROMS 4D-Var: Past, Present & Future

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