Lecture 3: Dual 4D-Var · 2010. 6. 30. · Innovation vector × H Observation ... finite-amp...

Lecture 3: Dual 4D-Var

Outline

• 4D-Var recap• Dual 4D-Var (4D-PSAS & R4D-Var)• The ROMS 4D-PSAS & R4D-Var algorithms• Weak constraint 4D-Var

,y R

Data Assimilation: Recap

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

Notation & Nomenclature: Recap

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

TS

x ζuv

(0)( )( )( )

ttt

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

xf

zbη

1

N

y

y

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

y ( )( )H= − bd y x

Statevector

Controlvector

Observationvector

Innovationvector

×

HObservation

operator

Prior

( (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

= − bd y Hx

Prior

Incremental Formulation: Recap

(Courtier et al., 1994)

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢

•••••••

⎥⎢ ⎥⎢•

⎥⎢ ⎥⎢ ⎥⎣

•

⎦•

••

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢

•••••••

⎥⎢ ⎥⎢•

⎥⎢ ⎥⎢ ⎥⎣

•

⎦•

zPrimalSpace

yObservation

vector

zDual

Space

Primal vs Dual Formulation: RecapVector of

increments

The Solution: Recap

= +a bz z Kd

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

Analysis:

Gain (dual form):

Gain (primal form):

( )T T− − − −= +1 1 1 1K D G R G G R

Two Spaces: Recap

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

Gain (primal):

( )T T− − − −= +1 1 1 1K D G R G G R

obs obsN N×

model modelN N×obs modelN N

Two Spaces: Recap

G maps from model spaceto observation space

GT maps from observation spaceto model space

Primal Formulation: Recap

= +a bz z KdAnalysis:Goal of 4D-Var is to identify:

11 T 1 T 1( )δ−− − −= = +z Kd D G R G G R d

Solve the equivalent linear system:1 T 1 T 1( )δ− − −+ =D G R G z G R d

by minimizing:1 T 1 T 1 11 1( )

2 2T T TJ δ δ δ− − − −= + − +z D G R G z z G R d d R d

( ) ( )1 11 12 2

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

Dual Formulation


1T T( )δ−

= = +z Kd DG GDG R dSolve the equivalent linear system:

by minimizing:

T1( ) ( )2

T TI = + −w w GDG R w w d

T T( ) ; δ+ = =GDG R w d z DG w

then compute: Tδ =z DG w

There is no physicalsignificance attached to w

Contours of J

Conjugate Gradient (CG) Methods

Matrix-less Operations

TGDG δThere are no matrix multiplications!

Zonal shear flow


There are no matrix multiplications!

Adjoint Model

TGDG δ

Zonal shear flow



Covariance

TGDG δ

Zonal shear flow



Tangent LinearModel

TGDG δ

Zonal shear flow

TGDG δ

A covarianceZonal shear flow





Tangent LinearModel

TGDG δ

Zonal shear flow

Physical-space Statistical Analysis System(PSAS) – Da Silva et al. (1995)

Dual 4D-PSAS Algorithm

(define W4DPSAS, w4dpsas_ocean.h)

TG w

T( )I∂ ∂ = + −w GDG R w d

Xb(t), d

Choose w

Run ADROMS

D

Run TLROMS

ConjugateGradient

Algorithm

w

NLROMS, zb

NLROMS, za Xa(t)

TDG w

obs space

ADROMS forced by w

TGDG w

Run ADROMS

D

TaG w

δ az

Dual 4D-PSAS Algorithm

(define W4DPSAS, w4dpsas_ocean.h)

Choose w

Run ADROMS

D

Run TLROMS

ConjugateGradient

Algorithm

NLROMS, zb

NLROMS, za

Run ADROMS

D

Outer-loopInner-loop

The method of representers (R4D-Var)Bennett (2002)

The Dual of State-Space

ROMS state-vector increments:

( )( )

( ) ( )( )( )

tt

t ttt

δδ

δ δδδ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

TS

x ζuv

The set of all continuous, linear functionals of δx(t) iscalled the dual of δx

For example, ym=Gδz belongs to the dual of δx

Zonal shear flow

The Dual of State-SpaceConsider the assimilation window t=[0,T] for thezonal shear flow…

δ z

Zonal shear flow


×××

× Observations

Consider the assimilation window t=[0,T] for thezonal shear flow with 3 observations.

δ z

Tangent LinearModel

δG z

Zonal shear flow


×××

× Observations

Consider the assimilation window t=[0,T] for thezonal shear flow with 3 observations.

Tangent LinearModel

δG z

Zonal shear flow


×××

× Observations

Tangent LinearModel

δG z

Zonal shear flow


× Observations

Tangent LinearModel

δG z

Zonal shear flow


× Observations

ym=

Tangent LinearModel

δG z

Zonal shear flow


× Observations

ym=1my2my3my

Tangent LinearModel

δG z

Zonal shear flow


× Observations

d=1 1m oy y−2 2m oy y−3 3m oy y−

×××

Innovationvector


The innovation vector belongs to the dual of δx(t).

According to Riesz representation theorem:

Let1

2

(0)( )( )

(T)

tt

δδδ

δ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

xx

u x

x

for t=[0,T]

T T T1 1 1 2 2 2 3 3 3; ; ;m o m o m oy y y y y y− = − = − =ρ u ρ u ρ u

where ρi are referred to as “representer functions.”


Let1

2

(0)( )( )

(T)

i

i

i i

i

tt

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

rr

ρ r

r

for t=[0,T]

and ( )(t) (t)i= rR

Representer Functions

δ

Zonal shear flow

Observationlocation


TGDG δ

Zonal shear flow

Observationlocation

TGDG δ

A covariance

= A representer

Green’s Function

Zonal shear flow


Zonal shear flow

3

1(t)= (t) (t)= (t)+ (t)i i

iw

=

+ ∑a b bx x r x wR

The analysis increments can be written as theweighted sum of the representers

( )(t) (t)i= rR


Zonal shear flow

The analysis increments can be written as theweighted sum of the representers

3

1(t)= (t) (t)= (t)+ (t)i i

iw

=

+ ∑a b bx x r x wR( )(t) (t)i= rR


1(T)r

2 (T)r

3 (T)r


1T T( )δ−


by minimizing:

T1( ) ( )2


T T( ) ; (0)δ+ = = ≡GDG R w d z DG w wR

then compute:

The elements of w are theweighting coefs for the ri(t)

Indirect Representer Algorithm(Egbert et al, 1994)

T (0) ; (t)= (t)δ δ δ= ≡ =z DG w w x z wR M RTLROMS

Indirect R4D-Var algorithm for a

single outer-loop TG w

T( )I∂ ∂ = + −w GDG R w d

Xb(t)

Choose w

Run ADROMS

D

Run TLROMS

ConjugateGradient

Algorithm

w

NLROMS, zb

TLROMS, δza δXa(t)

TDG w

obs space

ADROMS forced by w

TGDG w

Run ADROMS

D

TaG w

δ az

Indirect R4D-Var Algorithm

(define W4DVAR, w4dvar_ocean.h)

TG w

T( )I∂ ∂ = + −w GDG R w d

Xb(t)

w

Xa(t)

TDG w

obs space

ADROMS forced by w

TGDG w

TaG w

δ az

Choose w

Run ADROMS

D

Run TLROMS

ConjugateGradient

Algorithm

RPROMS

RPROMS, za

Run ADROMS

D

NLROMS, zb

dRPROMS=finite-ampTLROMS

Choose w

Run ADROMS

D

Run TLROMS

ConjugateGradient

Algorithm

RPROMS

RPROMS, za

Run ADROMS

NLROMS, zb

D

Indirect R4D-Var Algorithm

(define W4DVAR, w4dvar_ocean.h)

Outer-loopInner-loop

Weak Constraint 4D-Var

1 1( ) ( , )( ( ), ( ), ( ))i i i i i it M t t t t t− −=x x f bNonlinear ROMS (NLROMS):

Nonlinear ROMS (NLROMS) with model error:

1 1( ) ( , )( ( ), ( ), ( ), ( ))i i i i i i it M t t t t t t− −=x x f b εModel error prior: 0Model error prior covariance:

4D-Var control vector:

(0)( )( )( )

ttt

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

xf

zbη Correction for modelerror

Q(no explicit timecorrelation in Q, but thereis some in practice)

Weak Constraint 4D-Var

1 1( ) ( , ) ( )i i i it t t tδ δ− −=x M uTangent linear ROMS (TLROMS):

( )( )

( )( )( )

i

ii

i

i

tt

ttt

δδ

δδδ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

xf

ubη 4D forcing for TLROMS

( )itδ =η 0Strong constraint:

Two Spaces

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

Gain (primal):

( )T T− − − −= +1 1 1 1K D G R G G R

obs obsN N×

model modelN N×obs modelN N

Two Spaces

( )model x times f b xN N N N N N= + + +Weak constraint:

( )model x times f bN N N N N= + +Strong constraint:

Weak constraint is only practical in dual formulationof 4D-Var since Nobs is unaffected:

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

obs obsN N×

Mechanics of Dual 4D-Var: Preconditioning


1T T( )δ−


by minimizing:

T1( ) ( )2


T T( ) ; δ+ = =GDG R w d z DG w

Preconditioning via the change of variable 1 2−=v R w

Mechanics of Dual 4D-Var: Lanczos vectors

Lanczos formulation of conjugate gradient algorithmin observation space is used (congrad.F).

1T T( )−

= +K DG GDG R

Dual formulation of gain matrix:

Dual formulation of practical gain matrix:

T 1 2 1 T 1 2k k k k

− − −=K DG R V T V R

Many practical diagnostic applications using thisformulation (Lectures 4 & 5).

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

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)

Issues, Things to do, & Coming Soon

• Slow convergence of dual 4D-Var compared to primalformulation:

- w has no physical significance, so need not be physically realizable

- minimum residual method may be the answer(El Akkraoui and Gauthier, 2010)

Tδ =z DG w

Summary• Strong and weak constraint 4D-Var, dual

formulation: define W4DPSASDrivers/w4dpsas_ocean.hdefine W4DVARDrivers/w4dvar_ocean.h

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

References• Bennett, A.F., 2002: Inverse Modeling of the Ocean and

Atmosphere. Cambridge University Press, 234pp.• Courtier, P., 1997: Dual formulation of four-dimensional variational

assimilation. Q. J. R. Meteorol. Soc.,123, 2449-2461.• Da Silva, A., J. Pfaendtner, J. Guo, M. Sienkiewicz and S. Cohn,

1995: Assessing the effects of data selection with DAO's physical-space statistical analysis system. Proceedings of the second international WMO symposium on assimilation of observations in meteorology and oceanography, Tokyo 13-17 March, 1995. WMO.TD 651, 273-278.

• 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 baroclinic coastal upwelling system. Ocean Modeling, 16, 160-187.

• Egbert, G.D., A.F. Bennett and M.C.G. Foreman, 1994: TOPEX/POSEIDON tides estimated using a global inverse method. J. Geophys. Res., 99, 24,821-24,852.

• El Akkraoui, A. and P. Gauthier, 2010: Convergence properties of the primal and dual forms of variational data assimilation. Q. J. R. Meteorol. Soc., 136, 107-115.

References• Moore, A.M., H.G. Arango, G. Broquet, B.S. Powell, J. Zavala-Garay

and A.T. Weaver, 2010: The Regional Ocean Modeling System (ROMS) 4-dimensional data assimilation systems: Part I. Ocean Modelling, Submitted.

Lecture 3: Dual 4D-Var · 2010. 6. 30. · Innovation vector × H Observation ... finite-amp...

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