the inverse regional ocean modeling system:
DESCRIPTION
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. Goals. Overview of the Inverse Regional Ocean Modeling System - PowerPoint PPT PresentationTRANSCRIPT
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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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•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
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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:
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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:
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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
( ) ( ) (( , ')( ' ') , )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:
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
R
TR
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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 ( ') ( ) '0
T
tt t dt® º - òd d H u1) 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
1) Initial model-data misfit
2) Model Tangent Linear trajectory
3) Find that minimizes
ˆ ( ') ( ) '0
T
tt t dt® º - òd d H u
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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
(( )() )0 000 t ttt = - =e es
1) Initial model-data misfit
2) Model Tangent Linear trajectory
3) Find that minimizes
ˆ ( ') ( ) '0
T
tt t dt® º - òd d H u
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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 t
t
t t t dt dtt® - ò òs
d eH R1444444442444444443
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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( ) ˆ
ˆ
1 1 10 0T T
- - -+ - =G C G P e G C d
H14444444244444443
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:
Data to Data Covariance
( )( ) ˆ
ˆ
1
0
n
T T
-+ =C PG e dGPG
P β14444442444444314444244443
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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
IOM representer-based inversion
Stabilized Representer Matrix
Representer Coefficients
µ TºR GPG
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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
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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
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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
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Representer Matrix
µ ( , ) ( , )0 00 0T Tt T T tº ss RR R
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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
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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
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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
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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
Temperature Velocity Covariancefor grid point n
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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
model to model covariance
( ) ( )TT T= s sµ ( , ) ( , )0 00 0T Tt T T tº ss RR R
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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
model to model covariance
( ) ( )TT T= s sµ ( , ) ( , )0 00 0T Tt T T tº ss RR R
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
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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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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 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
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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 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
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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
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µ''
ˆ
ˆ
( '')
( ')
( )
ˆ ˆ( ) ( ', ) ( ', '' ( , '')) ''( ')0 0
Adjoint Solution
Convolution of 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
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
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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
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
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µˆ
''
ˆ
( ''
(
)
)
( ')
ˆ( ', ) ( ', '') ( , '') ( ) '' 'ˆ( )0 0
Adjoint Solution
Convolution of Adjoint Soluti
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
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
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µˆ
''
ˆ
( ''
(
)
)
( ')
ˆ( ', ) ( ', '') ( , '') ( ) '' 'ˆ( )0 0
Adjoint Solution
Convolution of Adjoint Soluti
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
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
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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
![Page 55: The Inverse Regional Ocean Modeling System:](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815718550346895dc4b777/html5/thumbnails/55.jpg)
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
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The ASSIMILATION Setup
Final Time
Initial Condition
0t
Nt
Problem: Assimilate a passive tracer, which was perfectly measured.
Passive Tracer
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( )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
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True Diagonal CovarianceNt
Model Dynamics are Correct1 1
0 0T TJ - -= +n e Pn C eε
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( )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%
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TrueNt
True Initial Condition
Model Dynamics are Correct
Diagonal Covariance
Explained Variance 75%
Explained Variance 92%
Gaussian Covariance
Explained Variance 83%
Explained Variance 89%
Strong Constraint
![Page 61: The Inverse Regional Ocean Modeling System:](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815718550346895dc4b777/html5/thumbnails/61.jpg)
What if we allow for error in the model dynamics?
TrueNt
True Initial Condition
( ) 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
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True Gaussian Covariance
Explained Variance 24%
Explained Variance 99%
Nt
True Initial Condition
Weak Constraint
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True Gaussian Covariance Gaussian Covariance
Explained Variance 24% Explained Variance 83%
Explained Variance 99% Explained Variance 89%
Nt
True Initial Condition
Weak Constraint
Strong Constraint
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Data Misfit(comparison)
Weak Constraint
Strong Constraint
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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
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Assimilation of data at time Nt
True
True Initial Condition
Nt
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Model 1 Model 2
Assimilation of data at time Nt
True
True Initial Condition
Which is the model with the correct dynamics?
Nt
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Model 1 Model 2True
True Initial Condition Wrong Model Good Model
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Time Evolution of solutions after assimilation
Wrong Model
Good Model
DAY 0
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Time Evolution of solutions after assimilation
Wrong Model
Good Model
DAY 1
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Time Evolution of solutions after assimilation
Wrong Model
Good Model
DAY 2
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Time Evolution of solutions after assimilation
Wrong Model
Good Model
DAY 3
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Time Evolution of solutions after assimilation
Wrong Model
Good Model
DAY 4
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Model 1 Model 2True
True Initial Condition Wrong Model Good Model
What if we apply more smoothing?
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Model 1 Model 2
Assimilation of data at time Nt
True
True Initial Condition
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( )0obsT t
( )obs NT t
Assimilating SST
Prior (First Guess)
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( )0obsT t
( )obs NT t
Assimilating SST Wrong assumption in1 1
0 0T TJ --= +n n e PC eε
Prior (First Guess) After Assimilation
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( )0obsT t
( )obs NT t
Assimilating SSTNo smoothing
Prior (First Guess) After Assimilation
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( )0obsT t
( )obs NT t
Assimilating SSTGaussian Covariance
SmoothingPrior (First Guess) After Assimilation
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RMS difference from TRUE
Observations
Days
RM
S
Diagonal CASE
Gaussian
![Page 81: The Inverse Regional Ocean Modeling System:](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815718550346895dc4b777/html5/thumbnails/81.jpg)
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
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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
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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)
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… end
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( )0obsP t
( )obs NP t True Assimilation EXP 1 Assimilation EXP 2
Assimilation of data at time Nt
( , )0 0t tCGaussian( , )0 0t tCDiagonal
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( )0obsP t
( )obs NP t True
( , )0 0CGaussian( , '')t tCDIagonal
Data Misfit(comparison)
Weak Constraint
Strong Constraint
Model Dynamics are InCorrect
![Page 95: The Inverse Regional Ocean Modeling System:](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815718550346895dc4b777/html5/thumbnails/95.jpg)
( )0obsP t
( )obs NP t True Assimilation EXP 1 Assimilation EXP 2
Assimilation of data at time Nt
Wrong ( )0obsT t ( , '')t tCGaussian
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( )0obsP t
( )obs NP t True Assimilation EXP 1 Assimilation EXP 2
Assimilation of data at time Nt
Wrong ( )0obsT t ( , '')t tCGaussian
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µˆ
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( )
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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
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( ')
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ngent Linear model forced with
Sampling of Tangent Linear solution
T
t
t
t
dtò
f
τ
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+Cεψ
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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
( ) ( ')
( )
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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( )
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Sampling of Tangent Linear solution
T
t
t
t t dt= +òP H Cε
τ
ψ τ ψ144444424444443
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True Gaussian Covariance Gaussian Covariance
Explained Variance 24% Explained Variance 91%
Explained Variance 99% Explained Variance 99%
Nt
True Initial Condition
Model Dynamics are Incorrect
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( )0obsT t
( )obs NT t
( )0obsSSH t ( , ) ( )0obsU V t
( )obs NSSH t ( , ) ( )obs NU V t
Coastal Upwelling Baroclinic -Passive Sources