weak and strong constraint 4dvar in the r egional o cean m odeling s ystem ( roms ): development and...
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Weak and Strong Constraint 4DVAR in the Regional Ocean Modeling System (ROMS):
Development and Applications
Di Lorenzo, E.Georgia Institute of Technology
Arango, H.Rutgers University
Moore, A. and B. PowellUC Santa Cruz
Cornuelle, B and A.J. MillerScripps Institution of Oceanography
Short review of development and theory (an alternative derivation of the representer method)
Current applications(some pending issues)
Inverse Regional Ocean Modeling System (ROMS)
Chua and Bennett (2001)
Inverse Ocean Modeling System (IOMs)
Moore et al. (2004)
NL-ROMS, TL-ROMS, REP-ROMS, AD-ROMS
To implement a representer-based generalized inverse method to solve weak constraint data assimilation problems
a representer-based 4D-variational data assimilation system for high-resolution basin-wide and coastal oceanic flows
Di Lorenzo et al. (2007)
OCEAN INIT IALIZE
FINALIZE
RUN
S4DVAR_OCEAN
IS4DVAR_OCEAN
W4DVAR_OCEAN
ENSEMBLE_OCEAN
NL_OCEAN
TL_OCEAN
AD_OCEAN
PROPAGATOR
KERNELNLM, TLM, RPM, ADM
physicsbiogeochemicalsedimentsea ice
Optimal pertubations
ADM eigenmodes
TLM eigenmodes
Forcing singular vectors
Stochastic optimals
Pseudospectra
ADSEN_OCEAN
SANITY CHECK S
PERT_OCEAN
PICARD_OCEAN
GRAD_OCEAN
TLCHECK _OCEAN
RP_OCEAN
ESMF
AIR_OCEAN
MASTER
ean M ode
earch C o m
Non Linear Model
Tangent Linear Model
Representer Model
Adjoint Model
Sensitivity Analysis
Data Assimilation
1) Incremental 4DVAR Strong Constrain
2) Indirect Representer Weak and Strong Constrain
3) PSAS
Ensemble Ocean Prediction
Stability Analysis Modules
ROMS Block Diagram NEW Developments
Arango et al. 2003Moore et al. 2004Di Lorenzo et al. 2007
Download:
ROMS componentshttp://myroms.orgArango H.
IOM componentshttp://iom.asu.eduMuccino, J. et al.
Inverse Ocean Modeling Portal
STRONG Constraint WEAK Constraint (A) (B)
…we want to find the corrections e
Best Model Estimate (consistent with observations)
Initial Guess
ASSIMILATION Goal
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
Quadratic Linear Cost Function for residuals[ ]0J e
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
Quadratic Linear Cost Function for residuals[ ]0J e
ASSIMILATION 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 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 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 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
[ ]0
0
0¶
=¶J ee
( ) 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
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 10
1 0T T - - -+ - =G G G CeP dC
H14444444244444443
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:
An example of Representer Functions for the Upwelling System
Computed using the TL-ROMS and AD-ROMS
An example of Representer Functions for the Upwelling System
Computed using the TL-ROMS and AD-ROMS
Applications of inverse ROMS:
Baroclinic coastal upwelling: synthetic model experiment to test the development
CalCOFI Reanalysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. Di Lorenzo, Miller, Cornuelle and Moisan
Intra-Americas Seas Real-Time DAPowell, Moore, Arango, Di Lorenzo, Milliff et al.
Coastal Baroclinic Upwelling System Model Setupand Sampling Array
section
1) The representer system is able to initialize the forecast extracting dynamical information from the observations.
2) Forecast skill beats persistence
Applications of inverse ROMS:
Baroclinic coastal upwelling: synthetic model experiment to test the development
10 day assimilationwindow
10 day forecast
SKILL of assimilation solution in Coastal UpwellingComparison with independent observations
SKILL
DAYS
Climatology
Weak
Strong
Persistence
Assimilation Forecast
Di Lorenzo et al. 2007; Ocean Modeling
Day=0
Day=2
Day=6
Day=10
Day=0
Day=2
Day=6
Day=10
Assimilation solutions
Day=14
Day=18
Day=22
Day=26
Day=14
Day=18
Day=22
Day=26
Forecast
Day=14
Day=18
Day=22
Day=26
Day=14
Day=18
Day=22
Day=26
April 3, 2007
Intra-Americas Seas Real-Time DAPowell, Moore, Arango, Di Lorenzo, Milliff et al. www.myroms.org/ias
CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. Di Lorenzo, Miller, Cornuelle and Moisan
Things we struggle with …
Tangent Linear Dynamics can be very unstable in realistic settings.
Background and Model Error COVARIANCE functions are Gaussian and implemented through the use of the diffusion operator.
Fitting data vs. improving the dynamical trajectory of the model.
Assimilation of surface Salinity
Nt
True
True Initial Condition
Nt
True
True Initial Condition
Which model has correct dynamics?
Nt
Assimilation of surface Salinity
Nt
Model 1 Model 2
True
True Initial Condition Wrong Model Good Model
Model 1 Model 2
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
RMS difference from TRUE
Observations
Days
RM
S
Less constraint
More constraint
Applications of inverse ROMS (cont.)
Improve model seasonal statistics using surface and open boundary conditions as the only controls.
Predictability of mesoscale flows in the CCS: explore dynamics that control the timescales of predictability.
Mosca et al. – (Georgia Tech)
inverse machinery of ROMS can be applied to regional ocean climate studies …
inverse machinery of ROMS can be applied to regional ocean climate studies …
EXAMPLE:Decadal changes in the CCS upwelling cells Chhak and Di Lorenzo, 2007; GRL
SSTa Composites
1
2
3
4
Observed PDO indexModel PDO index
Warm PhaseCold Phase
Chhak and Di Lorenzo, 2007; GRL
-50
-100
-150
-250
-200
-350
-300
-450
-400
-500
-140W-130W
-120W30N
40N
50N
-50
-100
-150
-250
-200
-350
-300
-450
-400
-500
-140W-130W
-120W30N
40N
50N
COLD PHASEensemble average
WARM PHASEensemble average
April Upwelling Site
Pt. Conception
Chhak and Di Lorenzo, 2007; GRL
Pt. Conception
dep
th [
m]
Tracking Changes of CCS Upwelling Source Waters during the PDOusing adjoint passive tracers enembles
Con
cen
trati
on
An
om
aly
Model PDO PDO lowpassedSurface0-50 meters(-) 50-100 meters(-) 150-250 meters
year
Changes in depth of Upwelling Cell (Central California)and PDO Index Timeseries
Chhak and Di Lorenzo, 2007; GRL
Ad
join
t Tra
cer
Arango, H., A. M. Moore, E. Di Lorenzo, B. D. Cornuelle, A. J. Miller, and D. J. Neilson, 2003: The ROMS tangent linear and adjoint models: A comprehensive ocean prediction and analysis system. IMCS, Rutgers Tech. Reports.
Moore, A. M., H. G. Arango, E. Di Lorenzo, B. D. Cornuelle, A. J. Miller, and D. J. Neilson, 2004: A comprehensive ocean prediction and analysis system based on the tangent linear and adjoint of a regional ocean model. Ocean Modeling, 7, 227-258.
Di Lorenzo, E., Moore, A., H. Arango, Chua, B. D. Cornuelle, A. J. Miller, B. Powell and Bennett A., 2007: Weak and strong constraint data assimilation in the inverse Regional Ocean Modeling System (ROMS): development and application for a baroclinic coastal upwelling system. Ocean Modeling, doi:10.1016/j.ocemod.2006.08.002.
References
( )
2
2
0 0
¶ ¶=- Ñ +
¶ ¶
=
×P TP K
t z
P t P
u
Adjoint passive tracers ensembles( )P t
uphysical circulation independent of ( )P t
Australia
Asia
USA
Canada
Pacific Model Grid SSHa
(Feb. 1998)
Regional Ocean Modeling System (ROMS)
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
COLD PHASEensemble average
WARM PHASEensemble average
April Upwelling Site
Pt. Conception Pt. Conception
Chhak and Di Lorenzo, 2007; GRL
What if we really have substantial model errors?
( )
2
2
0 0
¶ ¶+ Ñ =
¶ ¶
=
×P TP K
t z
P t P
u
Current application of inverse ROMS in the California Current System (CCS):
1)CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. NASA - Di Lorenzo, Miller, Cornuelle and Moisan
2)Predictability of mesoscale flow in the CCS: explore dynamics that control the timescales of predictability. Mosca and Di Lorenzo
3)Improve model seasonal statistics using surface and open boundary conditions as the only controls.
Comparison of SKILL score of IOM assimilation solutions with independent observations
HIRES: High resolution sampling array
COARSE: Spatially and temporally aliased sampling array
RP-ROMS with CLIMATOLOGY as BASIC STATE
RP-ROMS with TRUE as BASIC STATE
RP-ROMS WEAK constraint solution
Instability of the Representer Tangent Linear Model (RP-ROMS)
SKILL SCORE
TRUE Mesoscale Structure
SSH[m]
SST[C]
ASSIMILATION SetupCalifornia Current
Sampling:(from CalCOFI program)5 day cruise 80 km stations spacing
Observations:T,S CTD cast 0-500mCurrents 0-150mSSH
Model Configuration:Open boundary cond.nested in CCS grid
20 km horiz. Resolution20 vertical layersForcing NCEP fluxesClimatology initial cond.
SSH [m]
WEAK day=5
STRONG day=5
TRUE day=5
ASSIMILATION Results
1st GUESS day=5
WEAK day=5
STRONG day=5
ASSIMILATION Results
ERRORor
RESIDUALS
SSH [m]
1st GUESS day=5
WEAK day=0
STRONG day=0
TRUE day=0
Reconstructed Initial Conditions
1st GUESS day=0
Normalized Observation-Model Misfit
Assimilated data:TS 0-500m Free surface Currents 0-150m
TS
VU
observation number
Error Variance ReductionSTRONG Case = 92%WEAK Case = 98%