andy moore, ucsc hernan arango, rutgers gregoire broquet, cnrs chris edwards & milena veneziani,...
TRANSCRIPT
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
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
-
“Trinity”(The Matrix, 1999)
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
ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
, Q
Posterior
4D-VarI4-Var, R4D-Var,
4D-PSAS
ROMS 4D-Var
Primal & dualformulations
,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
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
( (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:
( (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
( (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
zPrimalSpace
yObservation
vector
zDual
Space
Primal vs Dual Formulation
Vector ofincrements
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
Observations (y)
CalCOFI &GLOBEC
SST &SSH
Argo
TOPP Elephant Seals
Ingleby andHuddleston (2007)
Data from Dan Costa
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
Observations
4D-VarAnalysis
Posterior
Observations
4D-VarAnalysis
Posterior
Observations
4D-VarAnalysis
Posterior
prior prior prior
Sequential 4D-Var
7-14 days
Forecast Forecast Forecast
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?
What is most important?
i.c.wind stressheat fluxfreshwater fluxopen b.c.
ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
, Q
Posterior
4D-Var Adjoint4D-Var
observation impactobservation sensitivity
ROMS 4D-Var
10km ROMS
I4D-Var, 1 outer, 10 innerStrong constraint
Initial log10 J
Final log10 J
The California Current
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
rms
Analysis Cycle 500m Isobath Transport
Offshore
Onshore
Prior I(xb)
Increment I = I(xa) – I(xb)
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)
• Correlations• Balance
• Advection• Baroclinic waves • Barotropic waves
Physical processes:
Statistics:
Average impactof satellite SST
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)
Impact of ArgoSalinity Obs
ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
, Q
Posterior
4D-Var Adjoint4D-Var
observation impactobservation sensitivity
ROMS 4D-Var
Forecast
Observations
4D-VarAnalysis
Posterior
Observations
4D-VarAnalysis
Posterior
Observations
4D-VarAnalysis
Posterior
prior prior prior
Sequential 4D-Var
7-14 days
Forecast Forecast Forecast
t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
Forecast Error
14 day forecast of 500m isobath transport at t0+14,starting at t0
14fJ
t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
14fJ
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
t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
7fJ
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
t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
aJ
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
t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
14e
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
t0 t0+7 t0+14
x f7x
f14x
AnalysisCycle
ForecastCycle
Overlapping Forecast Cycles
axNext Analysis
Cycle
7e
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]
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
500m Isobath Transport Forecast Error
0e
0e
0e
0e
SST Obs that reduce
forecast errorSST Obs that increase
forecast error
RMS Impact of SST Obs on 500m Isobath Transport Forecast Error
0e 0e
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
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
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
-
Summary and Conclusions
Assimilation impacts on CC
No assim
StrongConstraint
4D-Var
Time meanalongshoreflow (37N)
(10km, 42 lev)
Broquet et al (2009)
Time series of a-b
I
I
(Sv)
(Sv)
rms
rms
• Correlations• Balance
• Advection• Baroclinic waves • Barotropic waves
Physical processes:
Statistics:
Average impactof satellite SST I
(Sv)
Average impactof satellite SST
AdvectionHorizon
BaroclinicWave
Horizon
(based onc1~2 ms-1
Chelton et al,1994)
IGW
IGW
IGW
CTW
AdjointCTW
I(Sv)
Average impactCalCOFI & GLOBEC
Salinity Obs
AdvectionHorizon
BaroclinicWave
Horizon
I(Sv)