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

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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. Powell

UC Santa Cruz

Cornuelle, B and A.J. Miller

Scripps Institution of Oceanography

Bennet A. and B. Chua

Oregon 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

ASSIMILATION Goal

Initial Guess

Best Model Estimate (consistent with observations)

(A)

(B)

WEAK Constraint

STRONG Constraint

…we want to find the correctionse

Best Model

Estimate

Corrections

Initial Guess

ASSIMILATION Goal

(A)

(B)

WEAK Constraint

STRONG Constraint

…we want to find the correctionse

Best Model

Estimate

Corrections

Initial Guess

ASSIMILATION Goal

Best Model

Estimate

Corrections

Initial Guess

ASSIMILATION Goal

Best Model

Estimate

Corrections

Initial Guess

ASSIMILATION Goal

Best Model

Estimate

Corrections

Initial Guess

ASSIMILATION Goal

Tangent Linear Dynamics

ASSIMILATION Goal

Best Model

Estimate

Corrections

Initial Guess

Integral Solution

Tangent Linear Propagator

ASSIMILATION Goal

Best Model

Estimate

Corrections

Initial Guess

ASSIMILATION Goal

Best Model

Estimate

Corrections

Initial Guess

The Observations

ASSIMILATION Goal

Best Model

Estimate

Corrections

Initial Guess

Data misfit from initial guess

ASSIMILATION Goal

is a mapping matrix of dimensions

observations X model space

def:

Data misfit from initial guess

ASSIMILATION Goal

is a mapping matrix of dimensions

observations X model space

def:

Data misfit from initial guess

Quadratic Linear Cost Function for residuals

is a mapping matrix of dimensions

observations X model space

Quadratic Linear Cost Function for residuals

2) corrections should not exceed our assumptions about the errors in model initial condition.

1) corrections should reduce misfit within observational error

is a mapping matrix of dimensions

observations X model space

Minimize Linear Cost Function

def:

4DVAR inversion

Hessian Matrix

def:

4DVAR inversion

Hessian Matrix

Representer-based inversion

def:

4DVAR inversion

Hessian Matrix

Representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

def:

4DVAR inversion

Hessian Matrix

Representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

An example of Representer Functionsfor the Upwelling System

Computed using

the TL-ROMS and AD-ROMS

An example of Representer Functionsfor 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 Setup

and Sampling Array

section

- Applications of inverse ROMS:
- Baroclinic coastal upwelling: synthetic model experiment to test inverse machinery

1) The representer system is able to initialize the forecast extracting dynamical information from the observations.

2) Forecast skill beats persistence

10 day assimilation

window

10 day forecast

SKILL of assimilation solution in Coastal UpwellingComparison with independent observations

Weak

SKILL

Strong

Climatology

Persistence

Assimilation Forecast

DAYS

Di Lorenzo et al. 2007; Ocean Modeling

Day=0

Day=2

Day=6

Day=10

Assimilation solutions

Day=0

Day=2

Day=6

Day=10

Day=14

Day=18

Day=22

Day=26

Day=14

Day=18

Day=22

Day=26

Forecast

Day=14

Day=14

Day=18

Day=18

Day=22

Day=22

Day=26

Day=26

Intra-Americas Seas Real-Time DAPowell, Moore, Arango, Di Lorenzo, Milliff et al.

www.myroms.org/ias

April 3, 2007

CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. Di Lorenzo, Miller, Cornuelle and Moisan

…careful

…careful

Data Assimilation is NOT a black box

…careful

Data Assimilation is NOT a black box

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

…careful

Data Assimilation is NOT a black box

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

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

…careful

Data Assimilation is NOT a black box

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

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

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

…careful

Data Assimilation is NOT a black box

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

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

Assimilation of SSTa

True Initial Condition

True

Assimilation of SSTa

True Initial Condition

Which model has correct dynamics?

Model 1

Model 2

True

Wrong Model

Good Model

True Initial Condition

Model 1

Model 2

True

Time Evolution of solutions after assimilation

Wrong Model

DAY 0

Good Model

Time Evolution of solutions after assimilation

Wrong Model

DAY 1

Good Model

Time Evolution of solutions after assimilation

Wrong Model

DAY 2

Good Model

Time Evolution of solutions after assimilation

Wrong Model

DAY 3

Good Model

Time Evolution of solutions after assimilation

Wrong Model

DAY 4

Good Model

What if we apply more background constraints?

Wrong Model

Good Model

True Initial Condition

True

Model 1

Model 2

Assimilation of data at time

True Initial Condition

True

Model 1

Model 2

Strong

Constraint

Weak

Constraint

True Initial Condition

Explained Variance 24%

Explained Variance 83%

Gaussian Covariance

Gaussian Covariance

True

Explained Variance 99%

Explained Variance 89%

Strong

Constraint

Weak

Constraint

True Initial Condition

Explained Variance 24%

Explained Variance 83%

Gaussian Covariance

Gaussian Covariance

True

Explained Variance 99%

Explained Variance 89%

RMS difference from TRUE

Less constraint

RMS

More constraint

Days

Observations

…careful

Data Assimilation is NOT a black box

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

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

INSTABILITY of Linearized model

SST

[C]

AHV=0

AHT=0

Initial Condition

Day=5

AHV=4550

AHT=0

AHV=4550

AHT=1000

Day=5

Day=5

INSTABILITY of the linearized model (TLM)

Misfit DAY=5

Non Linear

Model Initial

Guess

TLMAHV=4550

AHT=4550

TLMAHV=0

AHT=0

TLMAHV=4550

AHT=1000

…careful

Data Assimilation is NOT a black box

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

- ..need research to properly setup a coastal assimilation/forecasting system
- Improve model seasonal statisticsusing 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 Ocean Modeling Portal

Download:

ROMS components

http://myroms.org

Arango H.

IOM componentshttp://iom.asu.edu

Muccino, J. et al.

Chua and Bennet (2002)

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

Observed PDO index

Model PDO index

SSTa Composites

Cold Phase

Warm Phase

1

2

3

4

Chhak and Di Lorenzo, 2007; GRL

-50

-50

-100

-100

-150

-150

-200

-200

-250

-250

-300

-300

-350

-350

-400

-400

-450

-450

50N

50N

-500

-500

40N

40N

-140W

-140W

-130W

30N

-130W

30N

-120W

-120W

Tracking Changes of CCS Upwelling Source Waters during the PDOusing adjoint passive tracers enembles

WARM PHASEensemble average

COLD PHASEensemble average

April Upwelling Site

Pt. Conception

Pt. Conception

depth [m]

Chhak and Di Lorenzo, 2007; GRL

Changes in depth of Upwelling Cell (Central California)

and PDO Index Timeseries

Concentration Anomaly

Adjoint Tracer

year

Model PDO

PDO lowpassed

Surface

0-50 meters

(-) 50-100 meters

(-) 150-250 meters

Chhak and Di Lorenzo, 2007; GRL

References

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.

New challenges for young coastal oceanographers data assimilators

Italianconstraint

Assimilation tool

Data point

New challenges for young oceanographers

Model-Data Misfit

(vector)

Model (matrix)

Parameters (vector)

Error (vector)

e.g. Correction to Initial condition

Correction to Boundary or ForcingBiological or Mixing parameters

more

Reconstructing the dispersion of a pollutant

TIME = 100

[conc]

Y km

X km

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

(1) Least Square Solution

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

Initial time

Final time

Initial time lsq. estimate

Final time lsq. estimate

Reconstruction

Assume you guess the wrong model.

Say you think there is only diffusion

(1) Least Square Solution

True Solution

Initial time

Final time

Initial time lsq. estimate

Final time lsq. estimate

Reconstruction

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 cannot

exceed a certain size)

(3) Weighted and TaperedLeast Square Solution

True Solution

Initial time

Final time

Initial time lsq. estimate

Final time lsq. estimate

Reconstruction

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

is the error in velocity

Let us try again the strait least square estimate

(1) Least Square Solution

True Solution

Initial time

Final time

Initial time lsq. estimate

Final time lsq. estimate

Reconstruction

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 solution

useless in terms of isolating the source.

Again limit the size of the model parameters!

(3) Weighted and TaperedLeast Square Solution

True Solution

Initial time

Final time

Initial time lsq. estimate

Final time lsq. estimate

Reconstruction

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

What have we learned?

If you do not have the correct model, it is always

a good idea to constrain your model parameters,

you will fit the data less but will

have a smoother inversion.

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

Inverse Ocean Modeling System (IOMs)

Chua and Bennett (2001)

To implement a representer-based generalized inverse method to solve weak constraint data assimilation problems

NL-ROMS, TL-ROMS, REP-ROMS, AD-ROMS

Moore et al. (2004)

Inverse Regional Ocean Modeling System (ROMS)

Di Lorenzo et al. (2007)

a representer-based 4D-variational data assimilationsystem for high-resolution basin-wide and coastal oceanic flows

ROMS Block Diagram NEW Developments

Stability Analysis Modules

Non Linear Model

Tangent Linear Model

Representer Model

Adjoint Model

Sensitivity Analysis

Data Assimilation

1) Incremental 4DVARStrong Constrain

2) Indirect Representer Weak and Strong Constrain

3) PSAS

Arango et al. 2003Moore et al. 2004Di Lorenzo et al. 2007

Ensemble Ocean Prediction

Adjoint passive tracers ensembles

physical circulation independent of

Regional Ocean Modeling System (ROMS)

Pacific Model Grid

SSHa

(Feb. 1998)

Canada

Asia

USA

Australia

What if we apply more smoothing?

Wrong Model

Good Model

True Initial Condition

True

Model 1

Model 2

Assimilation of data at time

True Initial Condition

True

Model 1

Model 2

April Upwelling Site

WARM PHASEensemble average

COLD PHASEensemble average

Pt. Conception

Pt. Conception

Chhak and Di Lorenzo, 2007; GRL

What if we really have substantial model errors?

- Current application of inverse ROMS in the California Current System (CCS):
- CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. NASA - Di Lorenzo, Miller, Cornuelle and Moisan
- Predictability of mesoscale flow in the CCS: explore dynamics that control the timescales of predictability. Mosca and Di Lorenzo
- Improve model seasonal statisticsusing 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

Instability of the Representer Tangent Linear Model (RP-ROMS)

SKILL SCORE

RP-ROMS WEAK constraint solution

RP-ROMS with TRUE as BASIC STATE

RP-ROMS with CLIMATOLOGY as BASIC STATE

ASSIMILATION SetupCalifornia Current

Sampling:

(from CalCOFI program)

5 day cruise

80 km stations spacing

Observations:

T,S CTD cast 0-500m

Currents 0-150m

SSH

Model Configuration:

Open boundary cond.nested in CCS grid

20 km horiz. Resolution20 vertical layersForcing NCEP fluxesClimatology initial cond.

TRUE Mesoscale Structure

SSH

[m]

SST

[C]

SSH [m]

ASSIMILATION Results

STRONG

day=5

TRUE

day=5

WEAK

day=5

1st GUESS

day=5

ASSIMILATION Results

SSH [m]

STRONG

day=5

ERROR

or

RESIDUALS

WEAK

day=5

1st GUESS

day=5

Reconstructed Initial Conditions

STRONG

day=0

TRUE

day=0

1st GUESS

day=0

WEAK

day=0

Normalized Observation-Model Misfit

T

S

U

V

observation number

Assimilated data:

TS 0-500m

Free surface

Currents 0-150m

Error Variance Reduction

STRONG Case = 92%WEAK Case = 98%