andy moore, ucsc hernan arango, rutgers gregoire broquet, cnrs chris edwards & milena veneziani,...

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)