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Transcript of Andy Moore, UCSC Hernan Arango, Rutgers Gregoire Broquet, CNRS Chris Edwards & Milena Veneziani,...
![Page 1: Andy Moore, UCSC Hernan Arango, Rutgers Gregoire Broquet, CNRS Chris Edwards & Milena Veneziani, UCSC Brian Powell, U Hawaii Jim Doyle, NRL Monterey Dave.](https://reader036.fdocuments.net/reader036/viewer/2022062500/56649e855503460f94b87858/html5/thumbnails/1.jpg)
Andy Moore, UCSCHernan Arango, RutgersGregoire Broquet, CNRS
Chris Edwards & Milena Veneziani, UCSCBrian Powell, U Hawaii
Jim Doyle, NRL MontereyDave Foley, NOAA Pacific Grove
A Comprehensive 4D-Var Data Assimilationand Analysis System Applied to the
California Current System using ROMS
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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• NOPP
• Dan Costa• Patrick Robinson
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“Trinity”(The Matrix, 1999)
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ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
, Q
Posterior
4D-Var
Priors &Hypotheses
ClippedAnalyses
Ensemble(SV, SO)
HypothesisTests
Forecast
dof Adjoint4D-Var
impact
Term balance,eigenmodes
UncertaintyAnalysis
error
ROMS 4D-VarEnsemble
4D-Var
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ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
, Q
Posterior
4D-VarI4-Var, R4D-Var,
4D-PSAS
ROMS 4D-Var
Primal & dualformulations
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,y R
Data Assimilation
bb(t), Bb
fb(t), Bf
xb(0), B
Model solutions depends on xb(0), fb(t), bb(t), (t)
time
x(t)
Obs, y
Prior
Posterior
ROMS
Prior
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Notation & Nomenclature
T
S
x u
v
ζ
(0)
( )
( )
( )
t
t
t
x
fz
b
η
1
N
y
y
y
bd y Hx
Statevector
Controlvector
Observationvector
Innovationvector
×
HObservation
matrix
Prior
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( (0), ( ), ( ), ( ))T T T Tt t t b fz x ε ε η
initialconditionincrement
boundaryconditionincrement
forcingincrement
corrections for model
error
bb(t), Bb
fb(t), Bf
xb(0), B
Prior
Incremental Formulation & Bayes Theorem
Thomas Bayes(1702-1761)
( | ) JP e z d
Posterior distribution of z:
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( (0), ( ), ( ), ( ))T T T Tt t t b fz x ε ε η
initialconditionincrement
boundaryconditionincrement
forcingincrement
corrections for model
error
bb(t), Bb
fb(t), Bf
xb(0), B
1 11 1
2 2TTJ z D z G z d R G z d
diag( , , , ) b fD B B B Q
Prior (background) error covariance
TangentLinear Modelsampled atobs points
ObsErrorCov.
Innovation
bd y Hx
Prior
Incremental Formulation & Bayes Theorem
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( (0), ( ), ( ), ( ))T T T Tt t t b fz x ε ε η
initialconditionincrement
boundaryconditionincrement
forcingincrement
corrections for model
error
bb(t), Bb
fb(t), Bf
xb(0), B
1 11 1
2 2TTJ z D z G z d R G z d
Prior
The minimum of J is identified iteratively by searchingfor ∂J/∂z=0
Incremental Formulation & Bayes Theorem
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zPrimalSpace
yObservation
vector
zDual
Space
Primal vs Dual Formulation
Vector ofincrements
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The Priors for ROMS CCS
30km, 10 km & 3 km grids, 30- 42 levels
Veneziani et al (2009)Broquet et al (2009)
COAMPSforcing
ECCO openboundaryconditions
fb(t), Bf
bb(t), Bb
xb(0), B
Previous assimilationcycle
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Observations (y)
CalCOFI &GLOBEC
SST &SSH
Argo
TOPP Elephant Seals
Ingleby andHuddleston (2007)
Data from Dan Costa
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1 11 1
2 2TTJ z D z G z d R G z d
Recall the Cost Function
The aim of 4D-Var is to find the increments zcorresponding to the minimum variance (maximum likelihood) estimate:
( (0), ( ), ( ), ( ))T T T Tt t t b fz x ε ε η
initialconditionincrement
boundaryconditionincrement
forcingincrement
corrections for model
error
The minimum of J is identified iteratively by searchingfor ∂J/∂z=0
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Observations
4D-VarAnalysis
Posterior
Observations
4D-VarAnalysis
Posterior
Observations
4D-VarAnalysis
Posterior
prior prior prior
Sequential 4D-Var
7-14 days
Forecast Forecast Forecast
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I4D-Var (primal)
R4D-Var (dual)
4D-PSAS (dual)
Jinitial
Jfinal
30km, 1X50, strong
Which elements of the control vectorexert the largest influence on J?
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What is most important?
i.c.wind stressheat fluxfreshwater fluxopen b.c.
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ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
, Q
Posterior
4D-Var Adjoint4D-Var
observation impactobservation sensitivity
ROMS 4D-Var
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10km ROMS
I4D-Var, 1 outer, 10 innerStrong constraint
Initial log10 J
Final log10 J
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The California Current
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Adjoint
4D-Var
Observation impacts
7day average transport across 500m isobath upper 14m(Veneziani et al, 2009)
I 500m Transport Increment = (Posterior-Prior)
1
platform
pp
I
(Langland & Baker, 2004; Gelaro et al., 2007)
I
I I I x f bI
1
obsN
ii
I
initialcondition
surfaceforcing
boundaryconditions
Controlvectorimpact
Obsimpact
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rms
Analysis Cycle 500m Isobath Transport
Offshore
Onshore
Prior I(xb)
Increment I = I(xa) – I(xb)
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Analysis Cycle 500m Isobath Transport
Satellite SSH
Satellite SST T XBT
T CTD
T Argo
T TOPP
S CTD
S Argo
Increment I = I(xa) – I(xb)
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• Correlations• Balance
• Advection• Baroclinic waves • Barotropic waves
Physical processes:
Statistics:
Average impactof satellite SST
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Average impactof satellite SST
AdvectionHorizon
BaroclinicWave
Horizon
(based onc1~2 ms-1
Chelton et al,1994)
IGW
IGW
IGW
CTW
AdjointCTW
I
(based on v~0.1 ms-1)
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Impact of ArgoSalinity Obs
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ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
, Q
Posterior
4D-Var Adjoint4D-Var
observation impactobservation sensitivity
ROMS 4D-Var
Forecast
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Observations
4D-VarAnalysis
Posterior
Observations
4D-VarAnalysis
Posterior
Observations
4D-VarAnalysis
Posterior
prior prior prior
Sequential 4D-Var
7-14 days
Forecast Forecast Forecast
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t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
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Forecast Error
14 day forecast of 500m isobath transport at t0+14,starting at t0
14fJ
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t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
14fJ
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Forecast Error
14 day forecast of 500m isobath transport at t0+14,starting at t0
14fJ
7fJ 7 day forecast of 500m isobath transport at t0+14,
starting at t0+7
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t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
7fJ
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Forecast Error
14 day forecast of 500m isobath transport at t0+14,starting at t0
14fJ
7fJ 7 day forecast of 500m isobath transport at t0+14,
starting at t0+7aJ Verifying analysis of 500m isobath transport at t0+14
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t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
aJ
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Forecast Error
14 day forecast of 500m isobath transport at t0+14,starting at t0
14fJ
7fJ 7 day forecast of 500m isobath transport at t0+14,
starting at t0+7aJ Verifying analysis of 500m isobath transport at t0+14
14 142( )f ae J J 14 day forecast error
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t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
14e
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Forecast Error
14 day forecast of 500m isobath transport at t0+14,starting at t0
14fJ
7fJ 7 day forecast of 500m isobath transport at t0+14,
starting at t0+7aJ Verifying analysis of 500m isobath transport at t0+14
14 142( )f ae J J 14 day forecast error
7 72( )f ae J J 7 day forecast error
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t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
7e
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Forecast Error
14 day forecast of 500m isobath transport at t0+14,starting at t0
14fJ
7fJ 7 day forecast of 500m isobath transport at t0+14,
starting at t0+7aJ Verifying analysis of 500m isobath transport at t0+14
14 142( )f ae J J 14 day forecast error
7 72( )f ae J J 7 day forecast error
7 14e e e Change in e due to assimilationof observations over [t0+7,t0+14]
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Forecast Error
7 14 0e e e 7 day forecast betterthan 14 day forecast
7 14 0e e e 7 day forecast worsethan 14 day forecast
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500m Isobath Transport Forecast Error
0e
0e
0e
0e
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SST Obs that reduce
forecast errorSST Obs that increase
forecast error
RMS Impact of SST Obs on 500m Isobath Transport Forecast Error
0e 0e
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ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
, Q
Posterior
4D-Var
Priors &Hypotheses
HypothesisTests
ROMS 4D-Var
degrees of freedomdegrees of reachabilityarray modes
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Degrees of Freedom
1 11 1
2 2TTJ z D z G z d R G z d
Recall that the optimal increments minimize:
No. of dof in obs
No. of dof in prior
“dof” – degrees of freedom
min obs / 2J NTheoretical min:
min( ) Tr( ) / 2bJ KG
min obs( ) ( - Tr( )) / 2oJ N KG
(Bennett et al, 1993; Cardinali et al, 2004; Desroziers et al., 2009)
bJ oJ
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Assimilation cycle (2002-2004)
Lo
g10
(J)
dofof
obs
(30km, 30 level, dual, strong, sequential, 7 day, 200 inner-loops)
• Less than 10% of all observations provide independent info• LOTS OF REDUNDANCY!• Jb>(Jb)min and indicates over fitting to the obs• J≠Jmin and indicates that prior hypotheses are incorrect
2obsN
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Summary and Conclusions
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Assimilation impacts on CC
No assim
StrongConstraint
4D-Var
Time meanalongshoreflow (37N)
(10km, 42 lev)
Broquet et al (2009)
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Time series of a-b
I
I
(Sv)
(Sv)
rms
rms
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• Correlations• Balance
• Advection• Baroclinic waves • Barotropic waves
Physical processes:
Statistics:
Average impactof satellite SST I
(Sv)
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Average impactof satellite SST
AdvectionHorizon
BaroclinicWave
Horizon
(based onc1~2 ms-1
Chelton et al,1994)
IGW
IGW
IGW
CTW
AdjointCTW
I(Sv)
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Average impactCalCOFI & GLOBEC
Salinity Obs
AdvectionHorizon
BaroclinicWave
Horizon
I(Sv)