the inverse regional ocean modeling system: development and application to data assimilation of...
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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.
•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
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:
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
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 ( ') ( ) '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
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
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
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
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:
( ) 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
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
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
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
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
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
… 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
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 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
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
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
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
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
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
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
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%
Explained Variance 92%
Gaussian Covariance
Explained Variance 83%
Explained Variance 89%
Strong Constraint
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
True Gaussian Covariance
Explained Variance 24%
Explained Variance 99%
Nt
True Initial Condition
Weak Constraint
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
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
True Initial Condition
Nt
Model 1 Model 2
Assimilation of data at time Nt
True
True Initial Condition
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
Time Evolution of solutions after assimilation
Wrong Model
Good Model
DAY 1
Time Evolution of solutions after assimilation
Wrong Model
Good Model
DAY 2
Time Evolution of solutions after assimilation
Wrong Model
Good Model
DAY 3
Time Evolution of solutions after assimilation
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
Assimilation of data at time Nt
True
True Initial Condition
( )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
Assimilation of data at time Nt
( , )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
( )obs NP t True Assimilation EXP 1 Assimilation EXP 2
Assimilation of data at time Nt
Wrong ( )0obsT t ( , '')t tCGaussian
( )0obsP t
( )obs NP t True Assimilation EXP 1 Assimilation EXP 2
Assimilation of data at time Nt
Wrong ( )0obsT t ( , '')t tCGaussian
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
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
( )0obsT t
( )obs NT t
( )0obsSSH t ( , ) ( )0obsU V t
( )obs NSSH t ( , ) ( )obs NU V t
Coastal Upwelling Baroclinic -Passive Sources
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