Download - The Inverse Regional Ocean Modeling System:
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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.
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• A brief overview of the Inverse Regional Ocean Modeling System
• How do we assimilate data using the ROMS set of models
• Examples, (a) zonal baroclinic jet and (b) mesoscale eddies in the Southern California Current
Goals
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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
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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)
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( ) (( ) ( ) )B BN t tt
¶= + - ++
¶uuA F eu
uREP-ROMS: BB
B
N¶º
¶
= -
u
Au
s u u
def:
Representer Tangent Linear Model
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( )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:
Representer Tangent Linear Model
Perturbation Tangent Linear Model
Adjoint Model
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( )tt
¶=
¶+
sAs e
T
t¶
=¶
Aλ
λ
TL-ROMS:
AD-ROMS:
( ) (( ) ( ) )B BN t tt
¶= + - ++
¶uuA F eu
uREP-ROMS:
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( )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)
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( )tt
¶= +
¶s
As eTL-ROMS:
AD-ROMS:
REP-ROMS:
T
t¶
=¶
Aλ
λ
( )( ) ( ) ( )B BN t tt
¶= + - + +
¶u
u A u u F e
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( )tt
¶= +
¶s
As eTL-ROMS:
AD-ROMS:
REP-ROMS:
T
t¶
=¶
Aλ
λ
( )( ) ( ) ( )B BN t tt
¶= + - + +
¶u
u A u u F e
Integral Solutions
( ) ( , ) ( ) ( (', ) ') '0
0 0
Nt
Nt
NN t tt t t ttt d= +ò R es R s …..
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( )tt
¶= +
¶s
As eTL-ROMS:
AD-ROMS:
REP-ROMS:
T
t¶
=¶
Aλ
λ
( )( ) ( ) ( )B BN t tt
¶= + - + +
¶u
u A u u F e
Tangent Linear Propagator
Integral Solutions
( ) ( ) (( , ')( ' ') , )0
0 0
Nt
NN Nt
t t dtt t tt t= +òR R es s …..
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( )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
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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
t t dtt t tt t= +òR R es s
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
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TL-ROMS:
AD-ROMS:
REP-ROMS:
( ) ( ) (( , ')( ' ') , )0
0 0
Nt
NN Nt
t t dtt t tt t= +òR R es s
Integral Solutions
( ) ( )( , ) ( ', () ) '0
0 00
NT TN
t
Nt
t t tt t t t dt= +òR R fλ λ
[ ]( , ) ( ', )( ) ( ( ')) ( ) ( ') '0
00
N
N N
t
N Bt
t t N t dtt t t t t= + + +òR eu u FR u
Adjoint Propagator
Tangent Linear Propagator
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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?
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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
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TL-ROMS:
( , ) ( ) ( ', ) '( () ')0
0 0
Nt
N NNt
t t t t t t dtt = +òR s R es
(ˆ ( )) ()tt t= +u u sBest Model Estimate
Initial Guess Corrections
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
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TL-ROMS:
ˆ®d1) Initial model-data misfit
2) Model Tangent Linear trajectory
3) Find that minimizes
( )t® s
( , ) ( ) ( ', ) '( () ')0
0 0
Nt
N NNt
t t t t t t dtt = +òR s R es
( )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
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TL-ROMS:
( , ) ( ) ( ') )'( ( , ) '0
0 0
Nt
N NNt
t tt dtt ttt= +ò R es R s
ASSIMILATION (2) Modeling the Corrections
(( )() )0 000 t ttt = - =e es
( )t® s
( )ts (ˆ ( )) '' '0
T
ttt dt® - ò H sd
ˆ®d1) Initial model-data misfit
2) Model Tangent Linear trajectory
3) Find that minimizes
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TL-ROMS:
ASSIMILATION (2) Modeling the Corrections
( )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
ˆ®d
(( )() )0 000 t ttt = - =e es
1) Initial model-data misfit
2) Model Tangent Linear trajectory
3) Find that minimizes
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ˆ®d1) Initial model-data misfit
2) Corrections to Model State
3) Find that minimizes
( )t®e'
ˆ ( ') ( ' ', ( '' ' ' ')')0 0
T t
t ttt t t dt dt® - ò òd H R e
ASSIMILATION (2) Modeling the Corrections
( )0te
( )te
Corrections to initial conditions
Corrections to model dynamics and boundary conditions
( )te
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ˆ®d1) Initial model-data misfit
2) Correction to Model Initial Guess
3) Find that minimizes
( )0t®e
ˆ ( ' () , ') ')(0
0 0
T
tt t t t dt® - ò H R ed
ASSIMILATION (2) Modeling the Corrections
( )0te
( )te
Corrections to initial conditions
Corrections to model dynamics and boundary conditions
( )0te
Assume we seek to correct only the initial conditions
STRONG CONSTRAINT
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ASSIMILATION (2) Modeling the Corrections
ASSIMILATION (3) Cost Function
ˆ®d1) Initial model-data misfit
2) Correction to Model Initial Guess
3) Find that minimizes
( )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
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ASSIMILATION (2) Modeling the Corrections
ASSIMILATION (3) Cost Function
ˆ®d1) Initial model-data misfit
2) Correction to Model Initial Guess
3) Find that minimizes
( )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
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
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ASSIMILATION (3) Cost Function
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
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ASSIMILATION (3) Cost Function
( ') ( , ') ' ( ') ( , '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
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ASSIMILATION (3) Cost Function
ˆ ˆ[ ] ( ') ( , ') ' ( ') ( , ') '0 0
10 0 0 0 0
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 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
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ASSIMILATION (3) Cost Function
ˆ ˆ[ ] ( ') ( , ') ' ( ') ( , ') '0 0
10 0 0 0 0
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 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
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( ) 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:
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( ) 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:
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( ) 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
Stabilized Representer Matrix
Representer Coefficients
µ TºR GPG
Representer Matrix
( ') ( , ') '0
0
T
tt t t dt=òG H R
def:
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Representer Matrix
( ') ( , ') '0
0
T
tt t t dt=òG H R
def:What is the physical meaning ofthe Representer Matrix?
( )( ) ˆ
ˆ
1
0
n
T T
-+ =C PG e dGPG
P β14444442444444314444244443
IOM representer-based inversion
Stabilized Representer Matrix
Representer Coefficients
µ TºR GPG
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IOM representer-based inversion
Stabilized Representer Matrix
Representer Coefficients
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
( )( ) ˆ
ˆ
1
0
n
T T
-+ =C PG e dGPG
P β14444442444444314444244443
TL-ROMS AD-ROMS
µR
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IOM representer-based inversion
Stabilized Representer Matrix
Representer Coefficients
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
( )( ) ˆ
ˆ
1
0
n
T T
-+ =C PG e dGPG
P β14444442444444314444244443
Assume a special assimilation case:
( ) ( ) t t T= -
=
I
P
H
I
Observations = Full model state at time T
Diagonal Covariance with unit variance
µR
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IOM representer-based inversion
Stabilized Representer Matrix
Representer Coefficients
µ ( ' ) ( , ') ' ( ', ) ( ' ) '0 0
0 0
T TT
t tt T t t dt t t t T dt º - -ò òI IRR RI
Representer Matrix
( )( ) ˆ
ˆ
1
0
n
T T
-+ =C PG e dGPG
P β14444442444444314444244443
Assume a special assimilation case:
Observations = Full model state at time T
Diagonal Covariance with unit variance
µR
( ) ( ) t t T= -
=
I
P
H
I
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IOM representer-based inversion
Stabilized Representer Matrix
Representer Coefficients
Representer Matrix
( )( ) ˆ
ˆ
1
0
n
T T
-+ =C PG e dGPG
P β14444442444444314444244443
µR
µ ( , ) ( , )0 0Tt T T tº R RR
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR
)( () T TT ssAssume you want to compute the model spatial covariance at time T
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR
)( () T TT ssAssume you want to compute the model spatial covariance at time T
(( , )( ) )0 0T T tt T=R sλ
STEP 1) AD-ROMS
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR
)( () T TT ssAssume you want to compute the model spatial covariance at time T
( , ) (( ) )0 0T T tt T=R sλ
STEP 1) AD-ROMS
( )
(( , ) () , ( ))0 0
0
T
t
Tt T T tT = sRs Rλ
144444424444443
STEP 2) run TL-ROMS forced with ( )0tλ
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR
)( () T TT ssAssume you want to compute the model spatial covariance at time T
( )
(( , ) () , ( ))0 0
0
T
t
Tt T T tT = sRs Rλ
144444424444443
STEP 2) run TL-ROMS forced with ( )0tλ
STEP 3) multiply by and take expected value
( , ) ( , )) ( ) ( )) (( 0 0T TTT Tt tT T T T=
I
s sR R ss1444442444443
( )T Ts
(( , )( ) )0 0T T tt T=R sλ
STEP 1) AD-ROMS
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR
)( () T TT ssAssume you want to compute the model spatial covariance at time T
( )
( ) ( , ) ( , ) ( )0 0
0
T
t
T t T T t T=s R R sλ
144444424444443
STEP 2) run TL-ROMS forced with ( )0tλ
STEP 3) multiply by and take expected value( )T Ts
( ) ( , ) ( )0 0Tt T t T=R sλ
STEP 1) AD-ROMS
µ( , ) ( ,) ( ))( 0 0T Tt T T tT T = ºR Rss R
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR ( )( ) T TT= ss
model to model covariance
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR ( )( ) T TT= ss
µ ( , ') ( '', )( ', '')0
TT
tt t t t tt t dº ò R RR ( ') ( '')Tt t= ss
model to model covariance
model to model covariance most general form
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR ( )( ) T TT= ss
model to model covariance
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR ( )( ) T TT= ss
model to model covariance
µ ( ') ( , ') ' ( ', ) ( ') '0 0
0 0
T TT T
t tt t t dt t t t dtº ò òI R HR H R
data to data covariance
ˆˆ T= dd
if we sample at observation locations through ( ') '0
( )T
tt dtò H
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Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR ( )( ) T TT= ss
model to model covariance
µ ( ') ( , ') ' ( ', ) ( ') '0 0
0 0
T TT T
t tt t t dt t t t dtº ò òI R HR H R
µ ( ') ( , ') ' ( ', ) ( ') '0 0
0 0
T TT T
t tt t t dt t t t dtº ò òP R HR H R
data to data covariance
ˆˆ T= dd
if we sample at observation locations through ( ') '0
( )T
tt dtò H
if we introduce a non-diagonal initial condition covariance
preconditioned data to data covariance
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… back to the system to invert ….
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( ) 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
Stabilized Representer Matrix
Representer Coefficients
µ TºR GPG
Representer Matrix
( ') ( , ') '0
0
T
tt t t dt=òG H R
def:STRONG CONSTRAINT
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4DVAR inversion
IOM representer-based inversion
Hessian Matrix
Stabilized Representer Matrix
Representer Coefficients
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
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IOM representer-based inversion
Stabilized Representer Matrix
Representer Coefficients
ˆ
ˆ( ') ( ', '') ( '') ' '' ( ', '') ( '') ) ' ''( '0 0 0 0
1
T T T T
T T
t t t t
n
t t t t dt dt t t t dt dtt
-é ù é ù+ =ê ú ê úë û ë ûò ò ò ò
P
G C G C C deG
β14444444444444444444244444444444444444443 1444444444444444442444444444444444443
WEAK CONSTRAINT
How to solve for corrections ? Method of solution in IROMS
( )te
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Stabilized Representer Matrix
Representer Coefficients
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
IOM representer-based inversion
''( , '') ( ', '') ( '( ) ') ' '
0
T
T T
t
tn
tt t t t dt tt dt = òò C R He β
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IOM representer-based inversion
Stabilized Representer Matrix
Representer Coefficients
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
tt t t t dt tt dt = òò C R He β
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Method of solution in IROMS
''( , '') ( ', '') ( '( ) ') ' '
0
T
T T
t
tn
tt t t t dt tt dt = òò C R He β
STEP 1) Produce background state using nonlinear model starting from initial guess.
ˆ( ) [ , ( ), , ]0 0B t t t t=u Nu 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
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Method of solution in IROMS
''( , '') ( ', '') ( '( ) ') ' '
0
T
T T
t
tn
tt t t t dt tt dt = òò C R He β
STEP 1) Produce background state using nonlinear model starting from initial guess.
ˆ( ) [ , ( ), , ]0 0B t t t t=u Nu 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
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µˆ
''
ˆ
( '')
( ')
( )
ˆ ˆ( ) ( ', ) ( ', '') ( , '') ( ) '' '0 0
Adjoint Solution
Convolution of 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
Sampling of Tangent Linear solution
T
t
t
t
dtò
f
τ
1444444444444444444444444424444444444444444444444444434144444444444444444444444444444424444444444444444444444444444 3
+Cεψ
444444
How to evaluate the action of the stabilized Representer Matrix µP
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Baroclinic Jet Example -
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… end
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µˆ
''
ˆ
( '')
( ')
( )
ˆ ˆ( ) ( ', ) ( ', '') ( , '') ( ) '' '0 0
Adjoint Solution
Convolution of 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
Sampling of Tangent Linear solution
T
t
t
t
dtò
f
τ
1444444444444444444444444424444444444444444444444444434144444444444444444444444444444424444444444444444444444444444 3
+Cεψ
444444
µˆ
ˆ
( ')
( ) ( ')
ˆ ˆ ˆ( ) ( ', ) ( ', '') ( '') '' '0 0 0
Convolution of Adjoint Solution
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
Sampling of Tangent Linear solution
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
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µ
ˆ( )
ˆ ˆ ˆ( ) ( )0
Sampling of Tangent Linear solution
T
t
t
t t dt= +òP H Cε
τ
ψ τ ψ144444424444443