The Inverse Regional Ocean Modeling System: Development and Application to Data Assimilation of Coastal Mesoscale Eddies. Di Lorenzo, E., Moore, A., H

The Inverse Regional Ocean Modeling System:

Development and Application to Data Assimilation of Coastal Mesoscale Eddies.

Di Lorenzo, E., Moore, A., H. Arango, B. Chua, B. D. Cornuelle , A. J. Miller and Bennett A.

•Overview of the Inverse Regional Ocean Modeling System

• Implementation - How do we assimilate data using the ROMS set of models

• Examples, (a) Coastal upwelling (b) mesoscale eddies in the Southern California Current

Goals

Inverse Regional Ocean Modeling System (IROMS)

Chua and Bennett (2001)

Inverse Ocean Modeling System (IOMs)

Moore et al. (2003)

NL-ROMS, TL-ROMS, REP-ROMS, AD-ROMS

To implement a representer-based generalized inverse method to solve weak constraint data assimilation into a non-linear model

a 4D-variational data assimilation system for high-resolution basin-wide and coastal oceanic flows

N ttt

¶= +

¶+eu F

u( ) ( ) ( )

( ) (( ) ( ) )B BN t tt

¶= + - ++

¶uuA F eu

uB

B

N¶º

¶u

Au

def:NL-ROMS:

REP-ROMS:

also referred to as Picard Iterations

( )( ) ( )n

B BnN t

t¶

= + - +¶

A uuu Fu

n ®u u

Approximation of NONLINEAR DYNAMICS

do =1n ® ¥

enddo

1B

n-ºu u

(STEP 1)

( ) (( ) ( ) )B BN t tt

¶= + - ++

¶uuA F eu

uREP-ROMS: BB

B

N¶º

¶

= -

u

Au

s u u

def:

( )tt

¶=

¶+

sAs e

T

t¶

=¶

Aλ

λ

TL-ROMS:

AD-ROMS:

( ) (( ) ( ) )B BN t tt

¶= + - ++

¶uuA F eu

uREP-ROMS: BB

B

N¶º

¶

= -

u

Au

s u u

def:

( )tt

¶= +

¶s

As e

T

t¶

=¶

Aλ

λ

TL-ROMS:

AD-ROMS:

( ) (( ) ( ) )B BN t tt

¶= + - + +

¶uuA F eu

uREP-ROMS:

Small Errors 1) model missing

dynamics

2) boundary conditions errors

3) Initial conditions errors

(STEP 2)

( )tt

¶= +

¶s

As eTL-ROMS:

AD-ROMS:

REP-ROMS:

T

t¶

=¶

Aλ

λ

( )( ) ( ) ( )B BN t tt

¶= + - + +

¶u

u A u u F e

( )tt

¶= +

¶s

As eTL-ROMS:

AD-ROMS:

REP-ROMS:

T

t¶

=¶

Aλ

λ

( )( ) ( ) ( )B BN t tt

¶= + - + +

¶u

u A u u F e

( ) ( ) (( , ')( ' ') , )0

0 0

Nt

NN Nt

t t dtt t tt t= +òR R es s

Tangent Linear Propagator

Integral Solutions

TL-ROMS:

AD-ROMS:

REP-ROMS:

T

t¶

=¶

Aλ

λ

( )( ) ( ) ( )B BN t tt

¶= + - + +

¶u

u A u u F e

( ) ( ) (( , ')( ' ') , )0

0 0

Nt

NN Nt


Integral Solutions

( ) ( )( , ) ( ', () ) '0

0 00

NT TN

t

Nt

t t tt t t t dt= +òR R fλ λ

( )tt

¶= +

¶s

As e

Adjoint Propagator

TL-ROMS:

AD-ROMS:

REP-ROMS:

( ) ( ) (( , ')( ' ') , )0

0 0

Nt

NN Nt


Integral Solutions

( ) ( )( , ) ( ', () ) '0

0 00

NT TN

t

Nt


[ ]( , ) ( ', )( ) ( ( ')) ( ) ( ') '0

00

N

N N

t

N Bt

t t N t dtt t t t t= + + +òR eu u FR u

Adjoint Propagator


TL-ROMS:

AD-ROMS:

REP-ROMS:

( ) ( ) (( , ')( ' ') , )0

0 0

Nt

NN Nt


Integral Solutions

( ) ( )( , ) ( ', () ) '0

0 00

NT TN

t

Nt


[ ]( , ) ( ', )( ) ( ( ')) ( ) ( ') '0

00

N

N N

t

N Bt

t t N t dtt t t t t= + + +òR eu u FR u

Adjoint Propagator


R

TR

TL-ROMS:

( ) ( , ) ( ) ( ', ) ( ') '0

0 0

Nt

N N Nt

t t t t t t t dt= +òs R s R e

How is the tangent linear model useful for assimilation?

TL-ROMS:

( , ) ( ) ( ', ) '( () ')0

0 0

Nt

N NNt

t t t t t t dtt = +òR s R es

ASSIMILATION (1) Problem Statement

® d1) Set of observations

2) Model trajectory

3) Find that minimizes

( )t® u

Sampling functional

ˆ ( )tu ˆ ( )( ') '0

T

ttt dt® - ò ud H

TL-ROMS:

( , ) ( ) ( ', ) '( () ')0

0 0

Nt

N NNt


(ˆ ( )) ()tt t= +u u sBest Model Estimate

Initial Guess Corrections

ASSIMILATION (1) Problem Statement

® d1) Set of observations

2) Model trajectory


( )t® u

Sampling functional

ˆ ( )tu ˆ ( )( ') '0

T

ttt dt® - ò ud H

TL-ROMS:

ˆ ( ') ( ) '0

T

tt t dt® º - òd d H u1) Initial model-data misfit

2) Model Tangent Linear trajectory


( )t® s

( , ) ( ) ( ', ) '( () ')0

0 0

Nt

N NNt


( )ts (ˆ ( )) '' '0

T

ttt dt® - ò H sd

ASSIMILATION (2) Modeling the Corrections

(ˆ ( )) ()tt t= +u u sBest Model Estimate

Initial Guess Corrections

TL-ROMS:

( , ) ( ) ( ') )'( ( , ) '0

0 0

Nt

N NNt

t tt dtt ttt= +ò R es R s


(( )() )0 000 t ttt = - =e es

( )t® s

( )ts (ˆ ( )) '' '0

T

ttt dt® - ò H sd

1) Initial model-data misfit



ˆ ( ') ( ) '0

T

tt t dt® º - òd d H u

TL-ROMS:


( )0te

( )te

Corrections to initial conditions

Corrections to model dynamics and boundary conditions

( , ) ( ) ( ') )'( ( , ) '0

0 0

Nt

N NNt

t tt dtt ttt= +ò R es R s

( )t® s

( )ts (ˆ ( )) '' '0

T

ttt dt® - ò H sd

(( )() )0 000 t ttt = - =e es

1) Initial model-data misfit



ˆ ( ') ( ) '0

T

tt t dt® º - òd d H u

ˆ® d1) Initial model-data misfit

2) Corrections to Model State


( )t® e'

( )

( 'ˆ ( ') ( ' ', ') ' '') '0 0

T t

t t

t

t t t dt dtt® - ò òs

d eH R1444444442444444443


( )0te

( )te



( )te


2) Correction to Model Initial Guess


( )0t® e

ˆ ( ' () , ') ')(0

0 0

T

tt t t t dt® - ò H R ed


( )0te

( )te



( )0te

Assume we seek to correct only the initial conditions

STRONG CONSTRAINT


ASSIMILATION (3) Cost Function




( )0t® e

ˆ ( ' () , ') ')(0

0 0

T

tt t t t dt® - ò H R ed( )0te

ˆ ˆ[ ] ( ') ( , ') ' ( ') ( , ') '0 0

0 0 00 01

TT T

t tJ t t t dt t t t dt-é ù é ù

= - -ê ú ê úê ú ê úë û ë ûò òe e eH R RCd d Hε

10 0T -+ Pe e






( )0t® e

ˆ ( ' () , ') ')(0

0 0

T

tt t t t dt® - ò H R ed( )0te

2) corrections should not exceed our assumptions about the errors in model initial condition.

1) corrections should reduce misfit within observational error

ˆ ˆ[ ] ( ') ( , ') ' ( ') ( , ') '0 0

0 0 00 01

TT T



10 0T -+ Pe e


ˆ ˆ[ ] ( ') ( , ') ' ( ') ( , ') '0 0

0 0 00 01

TT T



10 0T -+ Pe e


( ') ( , ') ' ( ') ( , 'ˆ[ ] 'ˆ )0 0

0 0 001

0

T T

t t

T

t t t dt t t t dtJ -é ù é ù= - -ê ú ê ú

ê ú ê úë û ë ûò òe e eH R RCd Hdε

10 0T -+ Pe e

( ') ( , ') '0

0

T

tt t t dt=òG H R

def:

G is a mapping matrix of dimensions

observations X model space


ˆ ˆ[ ] ( ') ( , ') ' ( ') ( , ') '0 0

10 0 0 0 0

TT T


= - -ê ú ê úê ú ê úë û ë ûò òe d H R e C d H R eε

10 0T -+e P e

( ') ( , ') '0

0

T

tt t t dt=òG H R

def:

G is a mapping matrix of dimensions

observations X model space

ˆ ˆ[ ] 1 10 0 0 0 0

TTJ - -é ù é ù= - - +ê ú ê úë û ë ûd dC Pe eG eGe eε


ˆ ˆ[ ] ( ') ( , ') ' ( ') ( , ') '0 0

10 0 0 0 0

TT T


= - -ê ú ê úê ú ê úë û ë ûò òe d H R e C d H R eε

10 0T -+e P e

ˆ ˆ[ ] 1 10 0 0 0 0

TTJ - -é ù é ù= - - +ê ú ê úë û ë ûd dC Pe eG eGe eε

( ) ˆ

ˆ

1 10

1 0T T - - -+ - =G G G CeP dC

H14444444244444443

Minimize J

( ) ˆ

ˆ

1 10

1 0T T - - -+ - =G G G CeP dC

H14444444244444443

4DVAR inversion

Hessian Matrix

( ') ( , ') '0

0

T

tt t t dt=òG H R

def:

( ) ˆ

ˆ

1 10

1 0T T - - -+ - =G G G CeP dC

H14444444244444443

( )( ) ˆ

ˆ0

1

n

T T

-+ =dGP CG eGP

P β14444442444444314444244443

4DVAR inversion

IOM representer-based inversion

Hessian Matrix

( ') ( , ') '0

0

T

tt t t dt=òG H R

def:

( ) ˆ

ˆ

1 10

1 0T T - - -+ - =G G G CeP dC

H14444444244444443

( )( ) ˆ

ˆ0

1

n

T T

-+ =dGP CG eGP

P β14444442444444314444244443

4DVAR inversion


Hessian Matrix

Stabilized Representer Matrix

Representer Coefficients

µ TºR GPG

Representer Matrix

( ') ( , ') '0

0

T

tt t t dt=òG H R

def:

( ) ˆ

ˆ

1 1 10 0T T

- - -+ - =G C G P e G C d

H14444444244444443

4DVAR inversion


Hessian Matrix



µ TºR GPG

Representer Matrix

( ') ( , ') '0

0

T

tt t t dt=òG H R

def:

Data to Data Covariance

( )( ) ˆ

ˆ

1

0

n

T T

-+ =C PG e dGPG

P β14444442444444314444244443

Representer Matrix

( ') ( , ') '0

0

T

tt t t dt=òG H R

def:

How to understand the physical meaning of the Representer Matrix?

( )( ) ˆ

ˆ

1

0

n

T T

-+ =C PG e dGPG

P β14444442444444314444244443




µ TºR GPG

µ TºR GPG

Representer Matrix

µ ( ') ( , ') ' ( ', ) ( ') '0 0

0 0

T TT T

t tt t t dt t t t dtº ò òP R HR H R

Representer Matrix

TL-ROMS AD-ROMS

µ ( ') ( , ') ' ( ', ) ( ') '0 0

0 0

T TT T

t tt t t dt t t t dtº ò òP R HR H R

Representer Matrix

Assume a special assimilation case:( ) ( )

0 0

T

t t T= -

=

I

ss

H

P Observations = Full model state at time T

Diagonal Covariance with unit variance

µ ( ' ) ( , ') ' ( ', ) ( ' ) '0 0

0 00 0

T TT

t

T

tt T t t dt t t t T dt º - -ò òI ss IRR R

Representer Matrix

Assume a special assimilation case:

Observations = Full model state at time T

Diagonal Covariance with unit variance

( ) ( )

0 0

T

t t T= -

=

I

ss

H

P

Representer Matrix

µ ( , ) ( , )0 00 0T Tt T T tº ss RR R

Representer Matrix

Assume you want to compute the model spatial covariance at time T

µ ( , ) ( , )0 00 0T Tt T T tº ss RR R

( ) ( )TT Ts s

Representer Matrix

( ) ( )TT Ts sAssume you want to compute the model spatial covariance at time T

,( () ) 00TT t=s R s

( )( )( , ) ( , )

(

( )

, )

)

,

(

( )

0 0

0 0

0 0

0 0

T T

T

t T t TT

t T

T

T t

=

=

sR R

RsR s

s s s

µ ( , ) ( , )0 00 0T Tt T T tº ss RR R

Representer Matrix

model to model covariance

( ) ( )TT T= s sµ ( , ) ( , )0 00 0T Tt T T tº ss RR R

Temperature Temperature Covariancefor grid point n

Representer Matrix



Temperature Temperature Covariancefor grid point n

Temperature Velocity Covariancefor grid point n

Representer Matrix

model to model covariance most general form

µ ( , ') ( '','' )( ', ) 0 00 0T Tt t tt t tº s RsR R ( ') ( '')Tt t= s s



Representer Matrix

model to model covariance most general form

µ ( , ') ( '','' )( ', ) 0 00 0T Tt t tt t tº s RsR R ( ') ( '')Tt t= s s



µ ( ') ( , ') ' ( ', ) ( ') '0 0

0 0

T TT T

t tt t t dt t t t dtº ò òP R HR H R ˆ ˆT= dd

if we sample at observation locations through ( ') '0

( )T

tt dtò H

data to data covariance

… back to the system to invert ….

( ) ˆ

ˆ

1 10

1 0T T - - -+ - =G G G CeP dC

H14444444244444443

( )( ) ˆ

ˆ0

1

n

T T

-+ =dGP CG eGP

P β14444442444444314444244443

4DVAR inversion


Hessian Matrix



µ TºR GPG

Representer Matrix

( ') ( , ') '0

0

T

tt t t dt=òG H R

def:STRONG CONSTRAINT

4DVAR inversion


Hessian Matrix



µ (( ') (', '') ' '''')0 0

TT T

t tt dt ttt t d

é ùº +ê úë ûò òR GCG C

Representer Matrix

ˆ

( ') ( '') ˆ' ''( ' (, '') ( ' ' '(, '' ))) '' ''0 0 0 0

1

T TT T T T

t t t t

n

t t t dt t tdt dt dt t tt

-é ù é ù+ =ê ú ê úë û ë ûò ò ò ò

P

G dCG GC C e

β14444444444444444444244444444444444444443 1444444444444444442444444444444444443

( ')( ) ( ( ˆ', ')') ( )

ˆ ( , ')0

1 1 1 0T TT

tt ttt t dt t

t t

- - -é ù+ - =ê úë ûò eC CG dG CG

H144444444444424444444444443 ( ) ( ') ( , ') '

T

tt t t t dt=òG H R

def:WEAK CONSTRAINT



ˆ

( ') ( ', '') ( '') ˆ' '( ', ' (' ')' '' ' ( '' ))0 00 0

1

TT T

t

T TT

t t t

n

dt dtt t t t dt dt tt t t

-é ù+ê ú é ù =êë úû ë ûò òò ò

P

G e dG C CG C

β144444444444444444442444444444444444444 144444444444444444244444444444444444343

WEAK CONSTRAINT

How to solve for corrections ? Method of solution in IROMS

( )te


''( , '') ( ', '') ( '( ) ') ' '

0

T

T T

t

tn

tt t t t dt tt dt = òò C R He β




ˆ

( ') ( '') ˆ' ''( ' (, '') ( ' ' '(, '' ))) '' ''0 0 0 0

1

T TT T T T

t t t t

n

t t t dt t tdt dt dt t tt

-é ù é ù+ =ê ú ê úë û ë ûò ò ò ò

P

G dCG GC C e

β14444444444444444444244444444444444444443 1444444444444444442444444444444444443

WEAK CONSTRAINT

How to solve for corrections ? Method of solution in IROMS

( )te

nβP̂ x ˆ = d

''( , '') ( ', '') ( '( ) ') ' '

0

T

T T

t

tn


Method of solution in IROMS

''( , '') ( ', '') ( '( ) ') ' '

0

T

T T

t

tn


STEP 1) Produce background state using nonlinear model starting from initial guess.

ˆ( ) [ , ( ), , ]0 0B t t t t=u N u F

[ ]ˆ( ) ( , ) ( ', ) ( ) ( ') '0

00 0

t

F Bt

t t t t t N t dt= + +òu R u R u F

ˆ ˆ( ') ( ) '0

Tn

tt t dt= - òd d H u

ˆˆ n =dPβ

[ ]ˆ ˆ( ) ( , ) ( ( ')', ) ( ) ( ') ' ( ', ) '0 0

0 0

t tn

Bt t

t t t t t N t dt t dtt t= + + +ò òu R u R u F eR

STEP 2) Run REP-ROMS linearized around background state to generate first estimate of model trajectory

STEP 3) Compute model-data misfit

do 0n N= ®ˆ ( ) ( )0 0

Ft t=u u

STEP 4) Solve for Representer Coeficients

STEP 5) Compute corrections

STEP 6) Update model state using REP-ROMS

outer loop

Method of solution in IROMS

''( , '') ( ', '') ( '( ) ') ' '

0

T

T T

t

tn


STEP 1) Produce background state using nonlinear model starting from initial guess.

ˆ( ) [ , ( ), , ]0 0B t t t t=u N u F

[ ]ˆ( ) ( , ) ( ', ) ( ) ( ') '0

00 0

t

F Bt

t t t t t N t dt= + +òu R u R u F

ˆ ˆ( ') ( ) '0

Tn

tt t dt= - òd d H u

ˆˆ n =dPβ

[ ]ˆ ˆ( ) ( , ) ( ( ')', ) ( ) ( ') ' ( ', ) '0 0

0 0

t tn

Bt t

t t t t t N t dt t dtt t= + + +ò òu R u R u F eR

STEP 2) Run REP-ROMS linearized around background state to generate first estimate of model trajectory

STEP 3) Compute model-data misfit

do 0n N= ®ˆ ( ) ( )0 0

Ft t=u u

STEP 4) Solve for Representer Coeficients

STEP 5) Compute corrections

STEP 6) Update model state using REP-ROMS

outer loop

inner loop

µˆ

''

ˆ

( '')

( ')

( )

ˆ ˆ( ) ( ', ) ( ', '') ( , '') ( ) '' '0 0

Adjoint Solution

Convolution of Adjoint Solution

Tang

t T TT T

t t t

t

t

t

t t t t t t t t dt dt dt= ò ò ò

f

P H R C R H

λ

τ

β β) ) )

1444444444442444444444443144444444444444444424444444444444444443

ˆ

( ')

( )

ˆ0

ent Linear model forced with

Sampling of Tangent Linear solution

T

t

t

t

dtò

f

τ

1444444444444444444444444244444444444444444444444444344144444444444444444444444444444424444444444444444444444444444443

+Cεβ

4444

How to evaluate the action of the stabilized Representer Matrix µP

µ''

ˆ

ˆ

( '')

( ')

( )

ˆ ˆ( ) ( ', ) ( ', '' ( , '')) ''( ')0 0

Adjoint Solution


Tang

t TT T

t

T

t t

t

t

t

t t t t dtt t t t dt dt= ò ò ò

f

P H R HC R

λ

τ

ββ1444444444442444444444443

144444444444444444424444444444444444443

) ) )

ˆ

( ')

( )

ˆ0



T

t

t

t

dtò

f

τ

1444444444444444444444444244444444444444444444444444344144444444444444444444444444444424444444444444444444444444444443

+Cεβ

4444


µˆ

''

ˆ

( '')

( '

( )

)

( 'ˆ ˆ( ) ( ', ) , '') ( , '') ( ) ' ''00

Adjoint Solution

Convolution of Adjoint Soluti

T

on

ang

T TT

t

t

T

t t

t

t

t

t t t t t dt dtt t t dt= ò òò

f

R C R HP H

τ

λ

ββ) ) )

1444444444442444444444443144444444444444444424444444444444444443

ˆ

( ')

( )

ˆ0



T

t

t

t

dtò

f

τ

1444444444444444444444444244444444444444444444444444344144444444444444444444444444444424444444444444444444444444444443

+Cεβ

4444


µˆ

''

ˆ

( ''

(

)

)

( ')

ˆ( ', ) ( ', '') ( , '') ( ) '' 'ˆ( )0 0

Adjoint Solution


T

on

ang

t T TT T

t t t

t

t

t

t t t t t t t dt dt tt d= ò ò ò

f

R C R HP H

τ

λ

ββ) ) )

1444444444442444444444443144444444444444444424444444444444444443

ˆ

( ')

( )

ˆ0



T

t

t

t

dtò

f

τ

1444444444444444444444444244444444444444444444444444344144444444444444444444444444444424444444444444444444444444444443

+Cεβ

4444


µˆ

''

ˆ

( ''

(

)

)

( ')

ˆ( ', ) ( ', '') ( , '') ( ) '' 'ˆ( )0 0

Adjoint Solution


T

on

ang

t T TT T

t t t

t

t

t

t t t t t t t dt dt tt d= ò ò ò

f

R C R HP H

τ

λ

ββ) ) )

1444444444442444444444443144444444444444444424444444444444444443

ˆ

( ')

( )

ˆ0



T

t

t

t

dtò

f

τ

1444444444444444444444444244444444444444444444444444344144444444444444444444444444444424444444444444444444444444444443

+Cεβ

4444


ROMS Coastal Upwelling

Model Topography

10km res, Baroclinic, Periodic NS, CCS Topographic Slope with Canyons and Seamounts, full options!

ShelfOpen Ocean

200 km

Seasonal Wind Stress

ROMS Coastal Upwelling

Final Time

Initial Condition

0t

Nt

Temperature Sea Surface Heigth

10km res, Baroclinic, Periodic NS, CCS Topographic Slope with Canyons and Seamounts, full options!

= 1 JAN ‘06

= 4 JAN ‘06

The ASSIMILATION Setup

Final Time

Initial Condition

0t

Nt

Problem: Assimilate a passive tracer, which was perfectly measured.

Passive Tracer

( )0obsP t

Model Dynamics are Correct

Nt True

1 10 0

T TJ - -= +n C n e P eε

( )

22

0

2

0

H H

T TT K T

t T

Kt z

T

¶ ¶+ Ñ = Ñ +

¶ ¶=

u

( )0 te Corrections only to initial state

True Diagonal CovarianceNt

Model Dynamics are Correct1 1

0 0T TJ - -= +n e Pn C eε

( )0obsP t

True Diagonal Covariance Gaussian CovarianceNt0t

How about the initial condition?

Model Dynamics are Correct1 1

0 0T TJ - -= +n e Pn C eε

Explained Variance 92% Explained Variance 89%

TrueNt

True Initial Condition

Model Dynamics are Correct

Diagonal Covariance

Explained Variance 75%


Gaussian Covariance



Strong Constraint

What if we allow for error in the model dynamics?

TrueNt


( ) te Time Dependent corrections to model dynamics and boundary conditions

( )

22

0

2

0

H H

T TT K T

t

K

Tt z

T

¶ ¶+ Ñ = Ñ +

¶ ¶=

u

True Gaussian Covariance



Nt


Weak Constraint

True Gaussian Covariance Gaussian Covariance



Nt


Weak Constraint

Strong Constraint

Data Misfit(comparison)

Weak Constraint

Strong Constraint

What if we really have substantial model errors, or bias?

( )

22

2

0 0

H H

T TT K T K

t z

T t T

¶ ¶+ Ñ = Ñ +

¶ ¶

=

u

Assimilation of data at time Nt

True


Nt

Model 1 Model 2


True


Which is the model with the correct dynamics?

Nt

Model 1 Model 2True

True Initial Condition Wrong Model Good Model

Time Evolution of solutions after assimilation

Wrong Model

Good Model

DAY 0


Wrong Model

Good Model

DAY 1


Wrong Model

Good Model

DAY 2


Wrong Model

Good Model

DAY 3


Wrong Model

Good Model

DAY 4

Model 1 Model 2True

True Initial Condition Wrong Model Good Model

What if we apply more smoothing?

Model 1 Model 2


True


( )0obsT t

( )obs NT t

Assimilating SST

Prior (First Guess)

( )0obsT t

( )obs NT t

Assimilating SST Wrong assumption in1 1

0 0T TJ --= +n n e PC eε

Prior (First Guess) After Assimilation

( )0obsT t

( )obs NT t

Assimilating SSTNo smoothing

Prior (First Guess) After Assimilation

( )0obsT t

( )obs NT t

Assimilating SSTGaussian Covariance

SmoothingPrior (First Guess) After Assimilation

RMS difference from TRUE

Observations

Days

RM

S

Diagonal CASE

Gaussian

Assimilation of Mesoscale Eddies in the Southern California Bight

TRUE

1st GUESS

IOM solution

Free Surface Surface NS Velocity SST

Assimilated data:TS 0-500m Free surface Currents 0-150m

Normalized Misfit

Datum

Assimilated data:TS 0-500m Free surface Currents 0-150m

(a)

TS

V U

Assimilation of Mesoscale Eddies in the Southern California Bight

Happy data assimilation and forecasting!

Di Lorenzo, E., Moore, A., H. Arango, Chua, B. D. Cornuelle, A. J. Miller and Bennett A. (2005) The Inverse Regional Ocean Modeling System (IROMS): development and application to data assimilation of coastal mesoscale eddies. Ocean Modeling (in preparation)

… end

( )0obsP t

( )obs NP t True Assimilation EXP 1 Assimilation EXP 2


( , )0 0t tCGaussian( , )0 0t tCDiagonal

( )0obsP t

( )obs NP t True

( , )0 0CGaussian( , '')t tCDIagonal

Data Misfit(comparison)

Weak Constraint

Strong Constraint

Model Dynamics are InCorrect

( )0obsP t



Wrong ( )0obsT t ( , '')t tCGaussian

( )0obsP t



Wrong ( )0obsT t ( , '')t tCGaussian

µˆ

''

ˆ

( '')

( ')

( )

ˆ ˆ( ) ( ', ) ( ', '') ( , '') ( ) '' '0 0

Adjoint Solution


Ta

t T TT T

t t t

t

t

t

t t t t t t t t dt dt dt= ò ò ò

f

P H R C R H

λ

τ

ψ ψ) ) )

144444444444244444444444314444444444444444442444444444444444444434

ˆ

( ')

( )

ˆ0

ngent Linear model forced with


T

t

t

t

dtò

f

τ

1444444444444444444444444424444444444444444444444444434144444444444444444444444444444424444444444444444444444444444 3

+Cεψ

444444

µˆ

ˆ

( ')

( ) ( ')

ˆ ˆ ˆ( ) ( ', ) ( ', '') ( '') '' '0 0 0


Tangent Linear model forced with

Sa

T t T

t t t

t

t

t

t t t t t t dt dt dt=ò ò òf

f

P H R C

τ

ψ λ144444444424444444443

14444444444444444442444444444444444444434

ˆ( ) mpling of Tangent Linear solution t

+Cε

τ

ψ

14444444444444444444444442444444444444444444444444434

µˆ

ˆ

ˆ

( ) ( ')

( )

ˆ ˆ ˆ( ) ( ', ) ( ') '0 0

Tangent Linear model forced with


T t

t t

t

t

t

t t t t dt dt= +ò òf

P H R f Cε

τ

τ

ψ ψ1444444442444444443

144444444444444444444244444444444444444444434

µ

ˆ( )

ˆ ˆ ˆ( ) ( )0


T

t

t

t t dt= +òP H Cε

τ

ψ τ ψ144444424444443

True Gaussian Covariance Gaussian Covariance



Nt


Model Dynamics are Incorrect

( )0obsT t

( )obs NT t

( )0obsSSH t ( , ) ( )0obsU V t

( )obs NSSH t ( , ) ( )obs NU V t

Coastal Upwelling Baroclinic -Passive Sources

Documents

The Inverse Regional Ocean Modeling System: Development and Application to Data Assimilation of Coastal Mesoscale Eddies. Di Lorenzo, E., Moore, A., H