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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. Goals. A brief overview of the Inverse Regional Ocean Modeling System

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The inverse regional ocean modeling system

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.


The inverse regional ocean modeling system

Goals

  • A brief overview of the Inverse Regional Ocean Modeling System

  • How do we assimilate data using the ROMS set of models

  • Examples, (a) zonal baroclinic jet and (b) mesoscale eddies in the Southern California Current


The inverse regional ocean modeling system

Inverse Ocean Modeling System (IOMs)

Chua and Bennett (2001)

To implement a representer-based generalized inverse method to solve weak constraint data assimilation into a non-linear model

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

Moore et al. (2003)

Inverse Regional Ocean Modeling System (IROMS)

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


The inverse regional ocean modeling system

NL-ROMS:

def:

REP-ROMS:

Approximation of NONLINEAR DYNAMICS

(STEP 1)

also referred to as Picard Iterations


The inverse regional ocean modeling system

def:

Representer Tangent Linear Model

REP-ROMS:


The inverse regional ocean modeling system

def:

Representer Tangent Linear Model

REP-ROMS:

Perturbation Tangent Linear Model

TL-ROMS:

Adjoint Model

AD-ROMS:


The inverse regional ocean modeling system

REP-ROMS:

TL-ROMS:

AD-ROMS:


The inverse regional ocean modeling system

REP-ROMS:

(STEP 2)

TL-ROMS:

  • Small Errors

  • model missing dynamics

  • boundary conditions errors

  • Initial conditions errors

AD-ROMS:


The inverse regional ocean modeling system

REP-ROMS:

TL-ROMS:

AD-ROMS:


The inverse regional ocean modeling system

Integral Solutions

REP-ROMS:

TL-ROMS:

…..

AD-ROMS:


The inverse regional ocean modeling system

Integral Solutions

REP-ROMS:

TL-ROMS:

…..

AD-ROMS:

Tangent Linear Propagator


The inverse regional ocean modeling system

Integral Solutions

REP-ROMS:

TL-ROMS:

AD-ROMS:

Tangent Linear Propagator


The inverse regional ocean modeling system

Integral Solutions

REP-ROMS:

TL-ROMS:

AD-ROMS:

Adjoint Propagator


The inverse regional ocean modeling system

Integral Solutions

REP-ROMS:

Tangent Linear Propagator

TL-ROMS:

AD-ROMS:

Adjoint Propagator


The inverse regional ocean modeling system

How is the tangent linear model useful for assimilation?

TL-ROMS:


The inverse regional ocean modeling system

ASSIMILATION (1)

Problem Statement

1) Set of observations

2) Model trajectory

3) Find that minimizes

Sampling functional

TL-ROMS:


The inverse regional ocean modeling system

Best Model

Estimate

Corrections

Initial Guess

ASSIMILATION (1)

Problem Statement

1) Set of observations

2) Model trajectory

3) Find that minimizes

Sampling functional

TL-ROMS:


The inverse regional ocean modeling system

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Model Tangent Linear trajectory

3) Find that minimizes

TL-ROMS:

Best Model

Estimate

Corrections

Initial Guess


The inverse regional ocean modeling system

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Model Tangent Linear trajectory

3) Find that minimizes

TL-ROMS:


The inverse regional ocean modeling system

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Model Tangent Linear trajectory

3) Find that minimizes

TL-ROMS:

 Corrections to initial conditions

 Corrections to model dynamics and boundary conditions


The inverse regional ocean modeling system

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Corrections to Model State

3) Find that minimizes

 Corrections to initial conditions

 Corrections to model dynamics and boundary conditions


The inverse regional ocean modeling system

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Correction to Model Initial Guess

3) Find that minimizes

Assume we seek to correct only the initial conditions

STRONG CONSTRAINT

 Corrections to initial conditions

 Corrections to model dynamics and boundary conditions


The inverse regional ocean modeling system

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Correction to Model Initial Guess

3) Find that minimizes

ASSIMILATION (3)

Cost Function


The inverse regional ocean modeling system

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Correction to Model Initial Guess

3) Find that minimizes

ASSIMILATION (3)

Cost Function

1) corrections should reduce misfit within observational error

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


The inverse regional ocean modeling system

ASSIMILATION (3)

Cost Function


The inverse regional ocean modeling system

is a mapping matrix of dimensions

observations X model space

def:

ASSIMILATION (3)

Cost Function


The inverse regional ocean modeling system

is a mapping matrix of dimensions

observations X model space

def:

ASSIMILATION (3)

Cost Function


The inverse regional ocean modeling system

Minimize J

ASSIMILATION (3)

Cost Function


The inverse regional ocean modeling system

def:

4DVAR inversion

Hessian Matrix


The inverse regional ocean modeling system

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion


The inverse regional ocean modeling system

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix


The inverse regional ocean modeling system

def:

What is the physical meaning of

the Representer Matrix?

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix


The inverse regional ocean modeling system

Representer Matrix

TL-ROMS

AD-ROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix


The inverse regional ocean modeling system

Representer Matrix

Assume a special assimilation case:

 Observations = Full model state at time T

 Diagonal Covariance with unit variance

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix


The inverse regional ocean modeling system

Representer Matrix

Assume a special assimilation case:

 Observations = Full model state at time T

 Diagonal Covariance with unit variance

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix


The inverse regional ocean modeling system

Representer Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix


The inverse regional ocean modeling system

Representer Matrix

Assume you want to compute the model spatial covariance at time T


The inverse regional ocean modeling system

Representer Matrix

Assume you want to compute the model spatial covariance at time T

STEP 1) AD-ROMS


The inverse regional ocean modeling system

Representer Matrix

Assume you want to compute the model spatial covariance at time T

STEP 1) AD-ROMS

STEP 2) run TL-ROMS forced with


The inverse regional ocean modeling system

Representer Matrix

Assume you want to compute the model spatial covariance at time T

STEP 1) AD-ROMS

STEP 2) run TL-ROMS forced with

STEP 3) multiply by andtake expected value


The inverse regional ocean modeling system

Representer Matrix

Assume you want to compute the model spatial covariance at time T

STEP 1) AD-ROMS

STEP 2) run TL-ROMS forced with

STEP 3) multiply by andtake expected value


The inverse regional ocean modeling system

Representer Matrix

model to model covariance


The inverse regional ocean modeling system

Representer Matrix

model to model covariance

model to model covariance most general form


The inverse regional ocean modeling system

Representer Matrix

model to model covariance


The inverse regional ocean modeling system

Representer Matrix

model to model covariance

if we sample at observation locations through

data to data covariance


The inverse regional ocean modeling system

Representer Matrix

model to model covariance

if we sample at observation locations through

data to data covariance

if we introduce a non-diagonal initial condition covariance

preconditioneddata to data covariance


The inverse regional ocean modeling system

… back to the system to invert ….


The inverse regional ocean modeling system

STRONG CONSTRAINT

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix


The inverse regional ocean modeling system

WEAK CONSTRAINT

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix


The inverse regional ocean modeling system

WEAK CONSTRAINT

How to solve for corrections ?

 Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix


The inverse regional ocean modeling system

WEAK CONSTRAINT

How to solve for corrections ?

 Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix


The inverse regional ocean modeling system

WEAK CONSTRAINT

How to solve for corrections ?

 Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix


The inverse regional ocean modeling system

Method of solution in IROMS

STEP 1) Produce background state using nonlinear model starting from initial guess.

STEP 2) Run REP-ROMS linearized around background state to generate first estimate of model trajectory

outer loop

STEP 3) Compute model-data misfit

STEP 4) Solve for Representer Coeficients

STEP 5) Compute corrections

STEP 6) Update model state using REP-ROMS


The inverse regional ocean modeling system

Method of solution in IROMS

STEP 1) Produce background state using nonlinear model starting from initial guess.

STEP 2) Run REP-ROMS linearized around background state to generate first estimate of model trajectory

outer loop

STEP 3) Compute model-data misfit

inner loop

STEP 4) Solve for Representer Coeficients

STEP 5) Compute corrections

STEP 6) Update model state using REP-ROMS


The inverse regional ocean modeling system

How to evaluate the action of

the stabilized Representer Matrix


The inverse regional ocean modeling system

Baroclinic Jet Example -


The inverse regional ocean modeling system

… end


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