Weak and Strong Constraint 4D variational data assimilation: Methods and Applications

Weak and Strong Constraint 4D variational data assimilation:

Methods 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

Bennet A. and B. ChuaOregon State University

Short review of 4DVAR theory (an alternative derivation of the representer method and

comparison between different 4DVAR approaches)

Overview of Current applications

Things we struggle with

STRONG Constraint WEAK Constraint (A) (B)

…we want to find the corrections e

Best Model Estimate (consistent with observations)

Initial Guess

ASSIMILATION Goal

STRONG Constraint WEAK Constraint (A) (B)

…we want to find the corrections e

ASSIMILATION Goal

(ˆ ( )) ()tt t= +u u sBest Model Estimate

Initial Guess Corrections

ASSIMILATION Goal



( ) ( )

( ) 000

ì ¶ïï = +ï ¶íïï = +ïî

N tt

u eu

u Fu

ASSIMILATION Goal



( )( ) ( )

( ) ( ) 0 00 0

ì ¶ +ïï = + +ï ¶íïï + = +ïî

N tt

su

eu

s

s

u F

u

ASSIMILATION Goal



( ) ( ) ( )

( ) 00

ì ¶ ¶ ¶ïï ++ = + +ï ¶ ¶ ¶íïï =ïî

N tt t

Ou N

us

us

s F

e

ASSIMILATION Goal

( )( )ˆ ) (- =tt tu suBest Model Estimate


Tangent Linear Dynamics

)

( )

( ) ( ( )

00

ì ¶ ¶ïï =ï ¶ ¶íïï =ïî

¶+ + + +

¶N t

tO

tN

su

s

u Fs

e

u

ASSIMILATION Goal

(( ,( )

(

) )

( )) 0

0 0

0

=

=

ttt

t

t e

es

Rs

Tangent Linear Propagator

( )( )ˆ ) (- =tt tu suBest Model Estimate


Integral Solution

ASSIMILATION Goal

( ) ( , )ˆ ) (( )00=- t tt tt u R euBest Model Estimate


ASSIMILATION Goal



( ') ˆ ( ) '0

= +òt

tH t dtd u

The Observations

ASSIMILATION Goal



ˆ( ') ( ) '0

= +òt

H t t dtd u

ˆ ( ') ( , ) ' ( )00

0 = +òt

H t t t t td ed R

Data misfit from initial guess

ASSIMILATION Goal

( ') ( , ) ' ( )ˆ00

0= +ò

t

H t t dt tt ed R


( ') ( , ') '0

0

T

tt t t dt=òG H R

def:

G is a mapping matrix of dimensions

observations X model space

ASSIMILATION Goal

ˆ0 = +d Ge


( ') ( , ') '0

0

T

tt t t dt=òG H R

def:



ˆ ˆ[ ] 10 0 0 0 0

1- -é ù é ù= - - +ê ú ê úë û ë û

TTJ d dC PGe e e e eG

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[ ] 10 0 0 0 0

1- -é ù é ù= - - +ê ú ê úë û ë û

TTJ d dC PGe e e e eG

Quadratic Linear Cost Function for residuals[ ]0J e



( ) ˆ

ˆ

1 10

1 0T T - - -+ - =G G G CeP dC

H14444444244444443

Minimize Linear Cost Function[ ]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

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

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

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 the ROMS inverse machinery:

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 inverse machinery

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

…careful

Data Assimilation is NOT a black box

…careful


Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems fitting vs. assimilating data)

…careful



…careful

Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics)



…careful

Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)




…careful



Assimilation of SSTa

Nt

True

True Initial Condition

Nt

True


Which model has correct dynamics?

Nt Model 1 Model 2

Assimilation of SSTa

Nt

True

True Initial Condition Wrong Model Good Model

Model 1 Model 2

Time Evolution of solutions after assimilation

Wrong Model

Good Model

DAY 0


Wrong Model

Good Model

DAY 1


Wrong Model

Good Model

DAY 2


Wrong Model

Good Model

DAY 3


Wrong Model

Good Model

DAY 4

Model 1 Model 2True


What if we apply more background constraints?

Model 1 Model 2

Assimilation of data at time Nt

True


True Gaussian Covariance Gaussian Covariance

Explained Variance 24% Explained Variance 83%


Nt


Weak Constraint

Strong Constraint

True Gaussian Covariance Gaussian Covariance



Nt


Weak Constraint

Strong Constraint

RMS difference from TRUE

Observations

Days

RM

S

Less constraint

More constraint



…careful



AHV=0AHT=0

AHV=4550AHT=1000

AHV=4550AHT=0

INSTABILITY of Linearized model SST[C]

Initial Condition

Day=5

Day=5Day=5

INSTABILITY of the linearized model (TLM)

TLMAHV=4550AHT=4550

TLMAHV=4550AHT=1000

TLMAHV=0AHT=0

Non Linear Model Initial

Guess

Misfit DAY=5



…careful



..need research to properly setup a coastal assimilation/forecasting system

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)

Download:

ROMS componentshttp://myroms.orgArango H.

IOM componentshttp://iom.asu.eduMuccino, J. et al.Chua and Bennet (2002)

Inverse Ocean Modeling Portal

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


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


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

Data point

Assimilation toolItalianconstraint

New challenges for young coastal oceanographers data assimilators

New challenges for young oceanographers

= +ny Ex

Model-Data Misfit(vector)

Parameters (vector)

Model (matrix)

Error (vector)

= +*

e.g. Correction to Initial condition

Correction to Boundary or ForcingBiological or Mixing parameters

more

Reconstructing the dispersion of a pollutant

X km

Y k

m

[conc]

TIME = 100

Where are the sources? You only know the solution at time=100

Assume you have a quasi perfect model, where you know diffusion K, velocity u and v

( ) ( )

( )ˆ1

T

T T

J-

ìï = - -ïïíï =ïïî

y Ex y Ex

x E E E y(1) Least Square Solution

2 2

2 2

T T T T Tu v K

t x y x y¶ ¶ ¶ ¶ ¶

+ + = +¶ ¶ ¶ ¶ ¶

Where x (the model parameters) are the unkown, y is the values of the tracers at time=100 (which you know) and E is the linear mapping of the initial condition x into y. Matrix E needs to be computed numerically.

True Solution

Reconstruction

Initial time

Initial time lsq. estimate

Final time

Final time lsq. estimate

Assume you guess the wrong model.

Say you think there is only diffusion

( ) ( )

( )ˆ1

T

T T

J-


y Ex y Ex


2 2

2 2

T T TK

t x y¶ ¶ ¶

= +¶ ¶ ¶

True Solution

Reconstruction

Initial time


Final time


Solution looks good at final time, but initial conditions are completely wrong and the values too high

Limit the size of the model parameters!(which means that the initial condition cannotexceed a certain size)

2 2

2 2

T T TK

t x y¶ ¶ ¶

= +¶ ¶ ¶

( ) ( )

( )ˆ1

T T

T T

J-

ìï = - - +ïïíï = +ïïî

y Ex y Ex x Sx

x E E S E y

(3) Weighted and TaperedLeast Square Solution

True Solution

Reconstruction

Initial time


Final time


Solution looks ok, the initial condition is still unable to isolate the source, given that you have a really bad model not including advection. However the initial condition is reasonable with in the diffusion limit, and the size of the initial condition is also within range.

Say you guess the right model

however velocities are not quite right

( )

( ) '

', '

( )'2 2

2 2

T T T Tu v

u v

Tu v K

t x y x y¶ ¶ ¶ ¶ ¶

+ + + + = +¶ ¶ ¶ ¶ ¶

is the error in velocity

( ) ( )

( )ˆ1

T

T T

J-


y Ex y Ex


Let us try again the strait least square estimate

True Solution

Reconstruction

Initial time


Final time


Solution looks great, but again the initial condition totally wrong both in the spatial structure and size.So in this case a small error in our model and too much focus on just fitting the data make the lsq solutionuseless in terms of isolating the source.

Again limit the size of the model parameters!

( ) ( )

( )ˆ1

T T

T T

J-

ìï = - - +ïïíï = +ïïî

y Ex y Ex x Sx

x E E S E y

(3) Weighted and TaperedLeast Square Solution

True Solution

Reconstruction

Initial time


Final time


Solution looks good, the initial condition is able to isolate the sources, the size of the initial condition is within the initial values.

If you do not have the correct model, it is alwaysa good idea to constrain your model parameters,

you will fit the data less but will have a smoother inversion.

What have we learned?

Weak and Strong Constraint 4D variational data assimilationfor coastal/regional applications

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

( )

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


What if we apply more smoothing?

Model 1 Model 2

Assimilation of data at time Nt

True


COLD PHASEensemble average

WARM PHASEensemble average

April Upwelling Site

Pt. Conception Pt. Conception


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%

Documents

Weak and Strong Constraint 4D variational data assimilation: Methods and Applications