weak and strong constraint 4dvar in the r egional o cean m odeling s ystem ( roms ): development and...

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%