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.
• 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
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:
Representer Tangent Linear Model
( )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
( )tt
¶=
¶+
sAs e
T
t¶
=¶
Aλ
λ
TL-ROMS:
AD-ROMS:
( ) (( ) ( ) )B BN t tt
¶= + - ++
¶uuA F eu
uREP-ROMS:
( )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
Integral Solutions
( ) ( , ) ( ) ( (', ) ') '0
0 0
Nt
Nt
NN t tt t t ttt d= +ò R es R s …..
( )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 …..
( )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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
( ) 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
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:
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
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
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
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
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
Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR
)( () T TT ssAssume you want to compute the model spatial covariance at time T
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
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
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
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
Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR ( )( ) T TT= ss
model to model covariance
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
Representer Matrix
µ ( , ) ( , )0 0Tt T T tº R RR ( )( ) T TT= ss
model to model covariance
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
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
… 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
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
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
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
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
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
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
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
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
Baroclinic Jet Example -
… end
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
µ
ˆ( )
ˆ ˆ ˆ( ) ( )0
Sampling of Tangent Linear solution
T
t
t
t t dt= +òP H Cε
τ
ψ τ ψ144444424444443
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