Prediction of Ocean Circulation in the Prediction of Ocean Circulation in the Gulf of Mexico and Caribbean SeaGulf of Mexico and Caribbean Sea
An application of the ROMS/TOMS An application of the ROMS/TOMS Data Assimilation ModelsData Assimilation Models
Hernan G. Arango (IMCS, Rutgers University)Hernan G. Arango (IMCS, Rutgers University)
Emanuele Di Lorenzo (Georgia Institute of Technology)Emanuele Di Lorenzo (Georgia Institute of Technology)
Arthur J. Miller, Bruce D. CornuelleArthur J. Miller, Bruce D. Cornuelle (Scripps Institute of Oceanography, UCSD)(Scripps Institute of Oceanography, UCSD)
Andrew M. Moore (PAOS, Colorado University)Andrew M. Moore (PAOS, Colorado University)
Gulf of Mexico and Caribbean Seas
plus satellite data (SSH, SST) and radar
Ocean Observations
OCEAN INIT IALIZE
FINALIZE
RUN
S4DVAR_OCEAN
IS4DVAR_OCEAN
W4DVAR_OCEAN
ENSEMBLE_OCEAN
NL_OCEAN
TL_OCEAN
AD_OCEAN
PROPAGATOR
KERNELNLM, TLM, RPM, ADM
physicsbiogeochemicalsedimentsea ice
Optimal pertubations
ADM eigenmodes
TLM eigenmodes
Forcing singular vectors
Stochastic optimals
Pseudospectra
ADSEN_OCEAN
SANITY CHECK S
PERT_OCEAN
PICARD_OCEAN
GRAD_OCEAN
TLCHECK _OCEAN
RP_OCEAN
ESMF
AIR_OCEAN
MASTER
ROMS/TOMS
cean M odel
earch C o mm
Ocean Modeling Framework
Gulf of Mexico Ocean Model Grid
Ocean Modeling of North Atlantic
Ocean Model Surface Currents and Sea LevelOcean Model Surface Currents and Sea Level
Ocean Modeling Applications in
• Develop a real-time Develop a real-time data assimilationdata assimilation and and prediction prediction systemsystem for the Gulf of Mexico and Caribbean Seas for the Gulf of Mexico and Caribbean Seas based on a continuous upper ocean monitoring systembased on a continuous upper ocean monitoring system
• Demonstrate the utility of variational data assimilation Demonstrate the utility of variational data assimilation in a real-time, sea-going environmentin a real-time, sea-going environment
• Demonstrate the value of collecting routine ocean Demonstrate the value of collecting routine ocean observations from specially equipped ocean vessels observations from specially equipped ocean vessels ((Explorer of the SeasExplorer of the Seas))
• Develop much needed experience in both the Develop much needed experience in both the assimilation of disparate ocean data and ocean assimilation of disparate ocean data and ocean prediction in regional ocean models.prediction in regional ocean models.
• Add platform oceanic measurements (a possibility)Add platform oceanic measurements (a possibility)
Gulf of Mexico and Caribbean Seas
Ensemble PredictionEnsemble Prediction
t
s
HighSpread
U npredic table
timet
sLow
Spread
P redic table
time
For an appropriate forecast skill measure, For an appropriate forecast skill measure, ss
Time=tN
Kelvin Wave Pattern
Maximum transport
Time=t0SSH SSHTime=tN
Kelvin Wave Pattern
Maximum transport
Time=t0SSH SSH
Example from the Caribbean model run, of sensitivity of the transport through the Yucatan Strait given a particular realization of the circulation. In this case the maximum transport at time tN, indicated by the strong
gradients in sea surface height (SSH), is sensitive to a pattern of Kelvin waves at previous time t0. These types of sensitivity, computed using the
non-linear and Adjoint models of ROMS, will be applied for the Florida Strait to explore how different topographic shapes affect the transport during different circulation regimes.
Ocean Adjoint Modeling Applications
4D Variational Data Assimilation Platforms 4D Variational Data Assimilation Platforms (4DVAR)(4DVAR)
• Strong Constraint (S4DVAR) drivers:Strong Constraint (S4DVAR) drivers: Conventional S4DVAR: outer loop, Conventional S4DVAR: outer loop, NLMNLM, , ADMADM Incremental S4DVAR: inner and outer loops, Incremental S4DVAR: inner and outer loops, NLMNLM, ,
TLMTLM, , ADMADM (Courtier et al., 1994) (Courtier et al., 1994) Efficient Incremental S4DVAR (Weaver et al., 2003)Efficient Incremental S4DVAR (Weaver et al., 2003)
• Weak Constraint (W4DVAR) - IOMWeak Constraint (W4DVAR) - IOM Indirect Representer Method: inner and outer loops, Indirect Representer Method: inner and outer loops,
NLMNLM, , TLMTLM, , RPMRPM, , ADM ADM (Egbert et al., 1994; Bennett (Egbert et al., 1994; Bennett et al, 1997)et al, 1997)
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misfit variance reduced 62%
1st guessIOM solution
2000 4000 60000 8000
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TRUE
1 st
GUESS
IOM solution
Free Surfac e Surface NS Veloc ity SST
No
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lize
d m
isfi
t
T S U V
Strong Constraint 4DVAR from IOM
(Di Lorenzo et al., 2005)
NormalizedMisfit
Datum
Assimilated data:TS 0-500m Free surface Currents 0-150m
Strong Constraint
1st Guess
True Synthetic Data
SST
SST SST
SST
TS
V U
Weak Constraint
Strong and Weak Constraint 4DVAR(Southern California Bight)
0-500 mdata
CalCOFISampling
grid
AnnualClimatology
• Given the model state vector:Given the model state vector:
• Consider a Yucatan Strait transport index, , Consider a Yucatan Strait transport index, , defined in terms of space and/or time integrals of defined in terms of space and/or time integrals of : :
• Small changes in will lead to changes in Small changes in will lead to changes in where: where:
• We will define sensitivity as etc.We will define sensitivity as etc.
J
dJ
J J J J JdJ du dv dT dS d
u v T S
, , ,J J J
u v T
dJ
Adjoint Sensitivity
+ …
PublicationsPublications
Arango, H.G., Moore, A.M., E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2003: The ROMS Tangent Linear and Arango, H.G., Moore, A.M., E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2003: The ROMS Tangent Linear and
Adjoint Models: A comprehensive ocean prediction and analysis system, Adjoint Models: A comprehensive ocean prediction and analysis system, Rutgers Tech. ReportRutgers Tech. Report..
http://marine.rutgers.edu/po/Papers/roms_adjoint.pdfhttp://marine.rutgers.edu/po/Papers/roms_adjoint.pdf
Di Lorenzo, E., A.M. Moore, H.G. Arango, B. Chua, B.D. Cornuelle, A.J. Miller and A. Bennett, 2005: The Inverse Regional Di Lorenzo, E., A.M. Moore, H.G. Arango, B. Chua, B.D. Cornuelle, A.J. Miller and A. Bennett, 2005: The Inverse Regional
Ocean Modeling System: Development and Application to Data Assimilation of Coastal Mesoscale Eddies, Ocean Modeling System: Development and Application to Data Assimilation of Coastal Mesoscale Eddies, Ocean Ocean
ModellingModelling, In preparation., In preparation.
Moore, A.M., H.G Arango, E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2004: A comprehensive ocean prediction Moore, A.M., H.G Arango, E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2004: A comprehensive ocean prediction
and analysis system based on the tangent linear and adjoint of a regional ocean model, and analysis system based on the tangent linear and adjoint of a regional ocean model, Ocean Modelling,Ocean Modelling, 7, 227-258. 7, 227-258.
http://marine.rutgers.edu/po/Papers/Moore_2004_om.pdfhttp://marine.rutgers.edu/po/Papers/Moore_2004_om.pdf
Moore, A.M., E. Di Lorenzo, H.G. Arango, C.V. Lewis, T.M. Powell, A.J. Miller and B.D. Cornuelle, 2005: An Adjoint Moore, A.M., E. Di Lorenzo, H.G. Arango, C.V. Lewis, T.M. Powell, A.J. Miller and B.D. Cornuelle, 2005: An Adjoint
Sensitivity Analysis of the Southern California Current Circulation and Ecosystem, Sensitivity Analysis of the Southern California Current Circulation and Ecosystem, J. Phys.J. Phys. Oceanogr.Oceanogr., In preparation., In preparation.
Wilkin, J.L., H.G. Arango, D.B. Haidvogel, C.S. Lichtenwalner, S.M.Durski, and K.S. Hedstrom, 2005: A Regional Modeling Wilkin, J.L., H.G. Arango, D.B. Haidvogel, C.S. Lichtenwalner, S.M.Durski, and K.S. Hedstrom, 2005: A Regional Modeling
System for the Long-term Ecosystem Observatory, System for the Long-term Ecosystem Observatory, J. Geophys. Res.J. Geophys. Res., 110, C06S91, doi:10.1029/2003JCC002218., 110, C06S91, doi:10.1029/2003JCC002218.
http://marine.rutgers.edu/po/Papers/Wilkin_2005_jgr.pdfhttp://marine.rutgers.edu/po/Papers/Wilkin_2005_jgr.pdf
Warner, J.C., C.R. Sherwood, H.G. Arango, and R.P. Signell, 2005: Performance of Four Turbulence Closure Methods Warner, J.C., C.R. Sherwood, H.G. Arango, and R.P. Signell, 2005: Performance of Four Turbulence Closure Methods
Implemented Using a Generic Length Scale Method, Implemented Using a Generic Length Scale Method, Ocean ModellingOcean Modelling, 8, 81-113., 8, 81-113.
http://marine.rutgers.edu/po/Papers/Warner_2004_om.pdfhttp://marine.rutgers.edu/po/Papers/Warner_2004_om.pdf
Background Material
OverviewOverview•Let’s represent Let’s represent NLMNLM ROMS as: ROMS as:
•The The TLMTLM ROMS is derived by considering a small perturbation ROMS is derived by considering a small perturbation ss to to SS. A first-order Taylor expansion yields:. A first-order Taylor expansion yields:
A is real, non-symmetricA is real, non-symmetric Propagator MatrixPropagator Matrix
•The The ADMADM ROMS is derived by taking the inner-product with an ROMS is derived by taking the inner-product with an
arbitrary vector , where the inner-product defines an arbitrary vector , where the inner-product defines an
appropriate norm (L2-norm):appropriate norm (L2-norm):
Tangent Linear and Adjoint Based GST Tangent Linear and Adjoint Based GST DriversDrivers
• Singular vectors:Singular vectors:
• Forcing Singular vectors:Forcing Singular vectors:
• Stochastic optimals:Stochastic optimals:
• Pseudospectra:Pseudospectra: 1HI A I A
( ,0) (0, )TR t XR t
andand• Eigenmodes ofEigenmodes of (0, )R t ( ,0)TR t
0 0
( , ) ( , )
T
R t dt X R t dt
| '|/ '
0 0
( , ) ( , ) 'ct t t Te R t XR t dt dt
Two InterpretationsTwo Interpretations
• Dynamics/sensitivity/stability of flow to Dynamics/sensitivity/stability of flow to
naturally occurring perturbationsnaturally occurring perturbations
• Dynamics/sensitivity/stability due to Dynamics/sensitivity/stability due to error error
or uncertainties in the forecast systemor uncertainties in the forecast system
• Practical applications:Practical applications:
Ensemble predictionEnsemble prediction
Adaptive observationsAdaptive observations
Array design ...Array design ...
GSA on the Southern California Bight (SCB)GSA on the Southern California Bight (SCB)
Free-SurfaceSST and Surfacecurrents
EigenmodesEigenmodes
SCB coastally trapped wavesSCB coastally trapped waves
• TLM TLM eigenvectors ( eigenvectors (AA): normal modes): normal modes• ADMADM eigenvectors ( eigenvectors (AATT): optimal excitations ): optimal excitations
Real Part Imag Part
diffluencediffluence
Optimal PerturbationsOptimal Perturbations
• A measurement of the fastest growing of all possible A measurement of the fastest growing of all possible perturbations over a given time intervalperturbations over a given time interval
SCB maximum growth of perturbation energy over 5 daysSCB maximum growth of perturbation energy over 5 days
confluenceconfluence
Stochastic OptimalsStochastic OptimalsProvide information about the influence of stochasticvariations (biases) in ocean forcing
SCB patterns of stochastic forcing that maximizes theperturbation energy variance for 5 days
Open Boundary Sensitivity: Open Boundary Sensitivity: errorserrors growth quickly and appear to growth quickly and appear to propagate through the model domain as coastally trapped waves.propagate through the model domain as coastally trapped waves.
Singular VectorsSingular Vectors
Ensemble PredictionEnsemble Prediction
• Optimal perturbations / singular vectors and Optimal perturbations / singular vectors and stochastic optimal can also be used to generate stochastic optimal can also be used to generate ensemble forecasts.ensemble forecasts.
• Perturbing the system along the most unstable Perturbing the system along the most unstable directions of the state space yields information directions of the state space yields information about the about the firstfirst and and secondsecond moments of the moments of the probability density function (PDF):probability density function (PDF):
ensemble meanensemble mean
ensemble spreadensemble spread
• Adjoint based perturbations excite the full spectrumAdjoint based perturbations excite the full spectrum
Data Assimilation OverviewData Assimilation Overview
•Cost Function:Cost Function:
wherewhere model,model, background,background, observations,observations,
inverse background error covariance,background error covariance,
inverse observations error covarianceinverse observations error covariance
•Model solution depends on initial conditions ( ), Model solution depends on initial conditions ( ), boundary conditions, and model parametersboundary conditions, and model parameters
•Minimize JMinimize J to produce a best fit between model and to produce a best fit between model and observations by adjusting initial conditions, and/or observations by adjusting initial conditions, and/or boundary conditions, and/or model parameters.boundary conditions, and/or model parameters.
MinimizationMinimization
• Perfect model constrained minimization (Lagrange Perfect model constrained minimization (Lagrange function):function):
We require the minimum of at which:We require the minimum of at which:
, , ,, , ,
yieldingyielding
• AATT is the transpose of is the transpose of AA, often called the adjoint , often called the adjoint operator. It can be shown that: operator. It can be shown that:
The adjoint equation solutionThe adjoint equation solutionprovides gradient informationprovides gradient information
4D Variational Data Assimilation Platforms 4D Variational Data Assimilation Platforms (4DVAR)(4DVAR)
• Strong Constraint (S4DVAR) drivers:Strong Constraint (S4DVAR) drivers: Conventional S4DVAR: outer loop, Conventional S4DVAR: outer loop, NLMNLM, , ADMADM Incremental S4DVAR: inner and outer loops, Incremental S4DVAR: inner and outer loops, NLMNLM, ,
TLMTLM, , ADMADM (Courtier et al., 1994) (Courtier et al., 1994) Efficient Incremental S4DVAR (Weaver et al., 2003)Efficient Incremental S4DVAR (Weaver et al., 2003)
• Weak Constraint (W4DVAR) - IOMWeak Constraint (W4DVAR) - IOM Indirect Representer Method: inner and outer loops, Indirect Representer Method: inner and outer loops,
NLMNLM, , TLMTLM, , RPMRPM, , ADM ADM (Egbert et al., 1994; Bennett (Egbert et al., 1994; Bennett et al, 1997)et al, 1997)
RP:RP:
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Forward and Adjoint MPI Communications
Forward
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