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The Inverse Regional Ocean Modeling System:

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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
- 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

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

NL-ROMS:

def:

REP-ROMS:

Approximation of NONLINEAR DYNAMICS

(STEP 1)

also referred to as Picard Iterations

def:

Representer Tangent Linear Model

REP-ROMS:

def:

Representer Tangent Linear Model

REP-ROMS:

Perturbation Tangent Linear Model

TL-ROMS:

Adjoint Model

AD-ROMS:

REP-ROMS:

TL-ROMS:

AD-ROMS:

REP-ROMS:

(STEP 2)

TL-ROMS:

- Small Errors
- model missing dynamics
- boundary conditions errors
- Initial conditions errors

AD-ROMS:

REP-ROMS:

TL-ROMS:

AD-ROMS:

Integral Solutions

REP-ROMS:

TL-ROMS:

…..

AD-ROMS:

Integral Solutions

REP-ROMS:

TL-ROMS:

…..

AD-ROMS:

Tangent Linear Propagator

Integral Solutions

REP-ROMS:

TL-ROMS:

AD-ROMS:

Tangent Linear Propagator

Integral Solutions

REP-ROMS:

TL-ROMS:

AD-ROMS:

Adjoint Propagator

Integral Solutions

REP-ROMS:

Tangent Linear Propagator

TL-ROMS:

AD-ROMS:

Adjoint Propagator

How is the tangent linear model useful for assimilation?

TL-ROMS:

ASSIMILATION (1)

Problem Statement

1) Set of observations

2) Model trajectory

3) Find that minimizes

Sampling functional

TL-ROMS:

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:

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

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Model Tangent Linear trajectory

3) Find that minimizes

TL-ROMS:

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

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

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

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Correction to Model Initial Guess

3) Find that minimizes

ASSIMILATION (3)

Cost Function

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.

ASSIMILATION (3)

Cost Function

is a mapping matrix of dimensions

observations X model space

def:

ASSIMILATION (3)

Cost Function

is a mapping matrix of dimensions

observations X model space

def:

ASSIMILATION (3)

Cost Function

Minimize J

ASSIMILATION (3)

Cost Function

def:

4DVAR inversion

Hessian Matrix

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

def:

What is the physical meaning of

the Representer Matrix?

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

Representer Matrix

TL-ROMS

AD-ROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

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

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

Representer Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

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

Representer Matrix

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

STEP 1) AD-ROMS

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

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

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

Representer Matrix

model to model covariance

Representer Matrix

model to model covariance

model to model covariance most general form

Representer Matrix

model to model covariance

Representer Matrix

model to model covariance

if we sample at observation locations through

data to data covariance

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

… back to the system to invert ….

STRONG CONSTRAINT

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

WEAK CONSTRAINT

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

WEAK CONSTRAINT

How to solve for corrections ?

Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

WEAK CONSTRAINT

How to solve for corrections ?

Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

WEAK CONSTRAINT

How to solve for corrections ?

Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

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

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

How to evaluate the action of

the stabilized Representer Matrix

Baroclinic Jet Example -

… end