Lecture 1: Primal 4D-Var · 2 ROMS Obs y, R fb, B f bb, B b xb, B η, Q Posterior 4D-Var Priors &...

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1 Lecture 1: Primal 4D-Var Outline ROMS 4D-Var overview 4D-Var concepts Primal formulation of 4D-Var Incremental approach used in ROMS The ROMS I4D-Var algorithm ROMS Obs y, R f b , B f b b , B b x b , B η, Q Posterior 4D-Var Clipped Analyses Ensemble (SV, SO) Hypothesis Tests Forecast dof Adjoint 4D-Var impact Term balance, eigenmodes Uncertainty Analysis error ROMS 4D-Var Ensemble 4D-Var Priors & Hypotheses Priors

Transcript of Lecture 1: Primal 4D-Var · 2 ROMS Obs y, R fb, B f bb, B b xb, B η, Q Posterior 4D-Var Priors &...

Page 1: Lecture 1: Primal 4D-Var · 2 ROMS Obs y, R fb, B f bb, B b xb, B η, Q Posterior 4D-Var Priors & Hypotheses Clipped Analyses Ensemble (SV, SO) Hypothesis Tests Forecast dof Adjoint

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Lecture 1: Primal 4D-Var

Outline

• ROMS 4D-Var overview• 4D-Var concepts• Primal formulation of 4D-Var• Incremental approach used in ROMS• The ROMS I4D-Var algorithm

ROMS Obsy, R

fb, Bf

bb, Bb

xb, B

η, Q

Posterior

4D-Var

ClippedAnalyses

Ensemble(SV, SO)

HypothesisTests

Forecast

dof Adjoint4D-Var

impact

Term balance,eigenmodes

UncertaintyAnalysiserror

ROMS 4D-VarEnsemble

4D-Var

Priors &Hypotheses

Priors

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ROMS Obsy, R

fb, Bf

bb, Bb

xb, B

η, Q

Posterior

4D-Var

Priors &Hypotheses

ClippedAnalyses

Ensemble(SV, SO)

HypothesisTests

Forecast

dof Adjoint4D-Var

impact

Term balance,eigenmodes

UncertaintyAnalysiserror

ROMS 4D-VarEnsemble

4D-VarPriors

• Broquet, G., C.A. Edwards, A.M. Moore, B.S. Powell, M. Veneziani and J.D. Doyle, 2009: Application of 4D-variational data assimilation to the California Current System. Dyn. Atmos. Oceans, 48, 69-91.

• Broquet, G., A.M. Moore, H.G. Arango, C.A. Edwards and B.S. Powell, 2009: Ocean state and surface forcing correction using the ROMS-IS4DVAR data assimilation system. Mercator Ocean Quarterly Newsletter, #34. 5-13.

• Broquet, G., A.M. Moore, H.G. Arango, and C.A. Edwards, 2010: Corrections to ocean surface forcing in the California Current System using 4D-variational data assimilation. Ocean Modelling, Under review.

• Di Lorenzo, E., A.M. Moore, H.G. Arango, B.D. Cornuelle, A.J. Miller, B. Powell, B.S. Chua and A.F. Bennett, 2007: Weak and strong constraint data assimilation in the inverse Regional Ocean Modeling System (ROMS): development and application for a baroclinic coastal upwelling system. Ocean Modeling, 16, 160-187.

• Powell, B.S., H.G. Arango, A.M. Moore, E. DiLorenzo, R.F. Milliff and D. Foley, 2008: 4DVAR Data Assimilation in the Intra-Americas Sea with the Regional Ocean Modeling System (ROMS). Ocean Modelling, 23, 130-145.

• Powell, B.S., A.M.Moore, H.G. Arango, E. Di Lorenzo, R.F. Milliff and R.R. Leben, 2009: Near real-time assimilation and prediction in the Intra-Americas Sea with the Regional Ocean Modeling System (ROMS). Dyn. Atmos. Oceans, 48, 46-68.

• Zhang, W.G., J.L. Wilkin, and J.C. Levin, 2010: Towards an integrated observation and modeling system in the New York Bight using variational methods, Part I. Ocean Modelling, Under review.

• Zhang, W.G., J.L. Wilkin, and J.C. Levin, 2010: Towards an integrated observation and modeling system in the New York Bight using variational methods, Part II. Ocean Modelling, Under review

ROMS 4D-Var Applications

,y R

Data Assimilation

bb(t), Bb

fb(t), Bf

xb(0), Bx

Model solutions depends on xb(0), fb(t), bb(t), h(t)

time

x(t)

Obs, yPriorPosterior

ROMS

Prior

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Notation & Nomenclature

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

TS

x ζuv

(0)( )( )( )

ttt

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

xf

zbη

1

N

y

y

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

y ( )( )H= − bd y x

Statevector

Controlvector

Observationvector

Innovationvector

×

HObservation

operator

Prior

Thomas Bayes(1702-1761)

Bayes Theorem

( | ) ( | ) ( ) ( )p p p p=z y y z z y

( ) ( )( )T 1 exp 1 2 ( ) ( )c H H−= − − −y x R y x

( ) ( )( )T 1 exp 1 2 −× − − −b bz z D z z

Conditional probability:

Maximum likelihood estimate: identify the minimum ofT T1 1( ) ( ) ( ) ( ( )) ( ( ))

2 2NLJ H H− −= − − + − −1 1b bz z z D z z y x R y x

which maximizes p(z|y).

(Wikle and Berliner, 2007)

Posteriordistribution

Prior MarginalDatadistribution

(“likelihood”)

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Variational Data Assimilation

T T1 1( ) ( ) ( ) ( ( )) ( ( ))2 2NLJ H H− −= − − + − −1 1

b bz z z D z z y x R y x

diag( , , , )= x b fD B B B Q

Background error covariance

( )( | ) exp NLP J∝ −z yConditional Probability:

Problem: Find z=za that minimizes J (i.e. maximizes P)using principles of variational calculus.za is the “maximum likelihood” or “minimum variance”estimate.

JNL is called the “cost” or“penalty” function.

Observation error covariance

( (0), ( ), ( ), ( ))T T T Tt t tδ δ= b fz x ε ε ηinitial

conditionincrement

boundaryconditionincrement

forcingincrement

corrections for model

error

bb(t),Bb

fb(t), Bf

xb(0), Bx

( ) ( )1 11 12 2

TTJ δ δ δ δ− −= + − −z D z G z d R G z d

diag( , , , )= x b fD B B B Q

Prior (background) error covariance

TangentLinear Modelsampled atobs points

ObsErrorCov.

Innovation

( )H= − bd y x

Prior

Incremental Formulation

(Courtier et al., 1994)

(H=linearizedobs operator)

( (0), ( ), ( ), ( ))T T T Tt t tδ δ= b fz x ε ε ηinitial

conditionincrement

boundaryconditionincrement

forcingincrement

corrections for model

error

bb(t),Bb

fb(t), Bf

xb(0), Bx

( ) ( )1 11 12 2

TTJ δ δ δ δ− −= + − −z D z G z d R G z d

Prior

The minimum of J is identified iteratively by searchingfor ∂J/∂δz=0

Incremental Formulation

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Incremental Formulation

bb(t),Bb

fb(t), Bf

xb(0), Bx

( (0), ( ), ( ), ( ))T T T Tt t tδ δ= b fz x ε ε ηinitial

conditionincrement

boundaryconditionincrement

forcingincrement

corrections for model

error

Assumptions:

(i) δ bz z(ii) (t)= (t)+ (t)δbx x x(iii) (t)δ δx M z(iv) (t) =δ δ δ∗ ∗H x H M z G z

Prior

δ= +bz z zTangent Linear ModelTangent Linear H

==

MH

i i-1 i i(t ) ( (t ), (t ), (t ))=b b b bx x f bM

The Tangent Linear Model(TLROMS)

time

x(t)

Obs, yPrior

0 τPrior is solution of model:

Increment: (t) (t); (t) (t); etcδ δb bx x f fi i-1 i-1

i-1

(t )= ( (t ) (t ), ) ( (t ), )

δδ

++

b

b

b x

x x xx M z

MM

Nonlinearmodel

Tangent linear model

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢

•••••••

⎥⎢ ⎥⎢•

⎥⎢ ⎥⎢ ⎥⎣

⎦•

••

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢

•••••••

⎥⎢ ⎥⎢•

⎥⎢ ⎥⎢ ⎥⎣

⎦•

zPrimalSpace

yObservation

vector

zDual

Space

Primal vs Dual FormulationVector of

increments

“Observationspace”

“Modelspace”

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The Solution

= +a bz z Kd

T T -1( )= +K DG GDG R

Analysis:

Gain matrix (dual form):

Gain matrix (primal form):1 T 1 1 T 1( )− − − −= +K D G R G G R

Two Spaces

T T -1( )= +K DG GDG RGain (dual):

Gain (primal):1 T 1 1 T 1( )− − − −= +K D G R G G R

obs obsN N×

model modelN N×obs modelN N

Two Spaces

G maps from model spaceto observation space

GT maps from observation spaceto model space

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Iterative Solution of Primal Formulation(define IS4DVAR, is4dvar_ocean.h)

( ) ( )T1 11 12 2

TJ δ δ δ δ− −= + − −z D z G z d R G z d

Recall the incremental cost function:

At the minimum of J we have J δ∂ ∂ =z 0

( )1 T 1J δ δ δ− −∂ ∂ = + −z D z G R G z d

Given J and ∂J/∂δz, we can identify the δz that minimizes J

bJ oJ

Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Tangent LinearModel

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Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Tangent LinearModel

Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Consider a single Observation

Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Consider a single Observation

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Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Inverse Obs ErrorCovariance

Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Inverse Obs ErrorCovariance

Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Adjoint Model

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Matrix-less Operations

( )T -1 δ −G R G z dThere are no matrix multiplications!

Zonal shear flow

Adjoint Model

Green’s Function

oJ δ∂ ∂ z

Primal 4D-Var Algorithm (I4D-Var)

Choose δz

Run TLROMS

R-1

Run ADROMS

ConjugateGradient

Algorithm

( ), , Jδ δ −G z G z d

( )-1 δ −R G z d

( )T -1 δ −G R G z d1J δ δ−∂ ∂ = +z D z

( )T 1 δ− −G R G z d

δ z

NLROMS, zb xb(t), d

NLROMS, za xa(t)

Choose δz

Run TLROMS

R-1

Run ADROMS

ConjugateGradient

Algorithm

NLROMS, zb

NLROMS, za

Primal 4D-Var Algorithm (I4D-Var)

Outer-loop

Inner-loop

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Contours of J

Conjugate Gradient (CG) Methods

An Example: ROMS CCS

30km, 10 km & 3 km grids, 30- 42 levelsVeneziani et al (2009)Broquet et al (2009)Moore et al (2010)

COAMPSforcing

ECCO openboundaryconditions

fb(t), Bf

bb(t), Bb

xb(0), Bx

Previous assimilationcycle

Observations (y)

CalCOFI &GLOBEC

SST &SSH

Argo

TOPP Elephant SealsIngleby andHuddleston (2007)

Data from Dan Costa

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4D-Var Configuration

• Case studies for a representative case3-10 March, 2003.

• 1 outer-loop, 100 inner-loops• 7 day assimilation window• Prior D: x Lh=50 km, Lv=30m, σ from clim

f Lτ=300km, LQ=100km, σ from COAMPS b Lh=100 km, Lv=30m, σ from clim

• Super observations formed• Obs error R (diagonal):

SSH 2 cmSST 0.4 Chydrographic 0.1 C, 0.01psu

Primal, strongDual, strongDual, weakJmin

4D-Var Performance

3-10 March, 2003(10km, 42 levels)

Summary• Strong constraint incremental 4D-Var, primal

formulation: define IS4DVARDrivers/is4dvar_ocean.h

• Matrix-less iterations to identify cost functionminimum using TLROMS and ADROMS

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References• Broquet, G., C.A. Edwards, A.M. Moore, B.S. Powell, M. Veneziani

and J.D. Doyle, 2009: Application of 4D-variational data assimilation to the California Current System. Dyn. Atmos. Oceans, 48, 69-91.

• Courtier, P., J.-N. Thépaut and A. Hollingsworth, 1994: A strategy for operational implemenation of 4D-Var using an incremental approach. Q. J. R. Meteorol. Soc., 120, 1367-1388.

• Ingleby, B. and M. Huddleston, 2007: Quality control of ocean temperature and salinity profiles - historical and real-time data. J. Mar. Systems, 65, 158-175.

• Moore, A.M., H.G. Arango, G. Broquet, B.S. Powell, J. Zavala-Garayand A.T. Weaver, 2010: The Regional Ocean Modeling System (ROMS) 4-dimensional data assimilation systems: Part I. Ocean Modelling, Submitted.

• Veneziani, M., C.A. Edwards, J.D. Doyle and D. Foley, 2009: A central California coastal ocean modeling study: 1. Forward model and the influence of realistic versus climatological forcing. J. Geophys. Res., 114, C04015, doi:10.1029/2008JC004774.

• Wikle, C.K. and L.M. Berliner, 2007: A Baysian tutorial for data assimilation. Physica D, 230, 1-16.