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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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slide1

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.

slide2

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
slide3

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

slide4

NL-ROMS:

def:

REP-ROMS:

Approximation of NONLINEAR DYNAMICS

(STEP 1)

also referred to as Picard Iterations

slide5

def:

Representer Tangent Linear Model

REP-ROMS:

slide6

def:

Representer Tangent Linear Model

REP-ROMS:

Perturbation Tangent Linear Model

TL-ROMS:

Adjoint Model

AD-ROMS:

slide7

REP-ROMS:

TL-ROMS:

AD-ROMS:

slide8

REP-ROMS:

(STEP 2)

TL-ROMS:

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

AD-ROMS:

slide9

REP-ROMS:

TL-ROMS:

AD-ROMS:

slide10

Integral Solutions

REP-ROMS:

TL-ROMS:

…..

AD-ROMS:

slide11

Integral Solutions

REP-ROMS:

TL-ROMS:

…..

AD-ROMS:

Tangent Linear Propagator

slide12

Integral Solutions

REP-ROMS:

TL-ROMS:

AD-ROMS:

Tangent Linear Propagator

slide13

Integral Solutions

REP-ROMS:

TL-ROMS:

AD-ROMS:

Adjoint Propagator

slide14

Integral Solutions

REP-ROMS:

Tangent Linear Propagator

TL-ROMS:

AD-ROMS:

Adjoint Propagator

slide16

ASSIMILATION (1)

Problem Statement

1) Set of observations

2) Model trajectory

3) Find that minimizes

Sampling functional

TL-ROMS:

slide17

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:

slide18

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

slide19

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Model Tangent Linear trajectory

3) Find that minimizes

TL-ROMS:

slide20

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

slide21

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

slide22

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

slide23

ASSIMILATION (2)

Modeling the Corrections

1) Initial model-data misfit

2) Correction to Model Initial Guess

3) Find that minimizes

ASSIMILATION (3)

Cost Function

slide24

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.

slide25

ASSIMILATION (3)

Cost Function

slide26

is a mapping matrix of dimensions

observations X model space

def:

ASSIMILATION (3)

Cost Function

slide27

is a mapping matrix of dimensions

observations X model space

def:

ASSIMILATION (3)

Cost Function

slide28

Minimize J

ASSIMILATION (3)

Cost Function

slide29

def:

4DVAR inversion

Hessian Matrix

slide30

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

slide31

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

slide32

def:

What is the physical meaning of

the Representer Matrix?

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

slide33

Representer Matrix

TL-ROMS

AD-ROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

slide34

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

slide35

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

slide36

Representer Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

slide37

Representer Matrix

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

slide38

Representer Matrix

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

STEP 1) AD-ROMS

slide39

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

slide40

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

slide41

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

slide42

Representer Matrix

model to model covariance

slide43

Representer Matrix

model to model covariance

model to model covariance most general form

slide44

Representer Matrix

model to model covariance

slide45

Representer Matrix

model to model covariance

if we sample at observation locations through

data to data covariance

slide46

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

slide48

STRONG CONSTRAINT

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

slide49

WEAK CONSTRAINT

def:

4DVAR inversion

Hessian Matrix

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

Representer Matrix

slide50

WEAK CONSTRAINT

How to solve for corrections ?

 Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

slide51

WEAK CONSTRAINT

How to solve for corrections ?

 Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

slide52

WEAK CONSTRAINT

How to solve for corrections ?

 Method of solution in IROMS

IOM representer-based inversion

Representer Coefficients

Stabilized Representer Matrix

slide53

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

slide54

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

slide55

How to evaluate the action of

the stabilized Representer Matrix

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