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 assimilation system 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: REP-ROMS: TL-ROMS: AD-ROMS: Representer Tangent Linear Model Perturbation Tangent Linear Model TL-ROMS: Adjoint Model AD-ROMS:

REP-ROMS: TL-ROMS: AD-ROMS:

REP-ROMS: TL-ROMS: AD-ROMS: Small Errors model missing dynamics (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 TL-ROMS: 1) Set of observations 2) Model trajectory 3) Find that minimizes Sampling functional TL-ROMS:

ASSIMILATION (1) Problem Statement TL-ROMS: 1) Set of observations 2) Model trajectory 3) Find that minimizes Sampling functional TL-ROMS: Best Model Estimate Initial Guess Corrections

Modeling the Corrections ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Model Tangent Linear trajectory 3) Find that minimizes TL-ROMS: Best Model Estimate Initial Guess Corrections

Modeling the Corrections ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Model Tangent Linear trajectory 3) Find that minimizes TL-ROMS:

Modeling the Corrections 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

Modeling the Corrections 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

STRONG CONSTRAINT 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

Modeling the Corrections ASSIMILATION (2) Modeling the Corrections 1) Initial model-data misfit 2) Correction to Model Initial Guess 3) Find that minimizes ASSIMILATION (3) Cost Function

Modeling the Corrections 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

observations X model space is a mapping matrix of dimensions observations X model space def: ASSIMILATION (3) Cost Function

observations X model space 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

IOM representer-based inversion def: 4DVAR inversion Hessian Matrix IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

What is the physical meaning of the Representer Matrix? def: IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

IOM representer-based inversion Representer Matrix TL-ROMS AD-ROMS IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

IOM representer-based inversion 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

IOM representer-based inversion 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

IOM representer-based inversion 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 and take 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 and take 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

model to model covariance Representer Matrix model to model covariance if we sample at observation locations through data to data covariance

model to model 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 preconditioned data to data covariance

… back to the system to invert ….

IOM representer-based inversion STRONG CONSTRAINT def: 4DVAR inversion Hessian Matrix IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

IOM representer-based inversion WEAK CONSTRAINT def: 4DVAR inversion Hessian Matrix IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix

How to solve for corrections ?  Method of solution in IROMS WEAK CONSTRAINT How to solve for corrections ?  Method of solution in IROMS IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix

How to solve for corrections ?  Method of solution in IROMS WEAK CONSTRAINT How to solve for corrections ?  Method of solution in IROMS IOM representer-based inversion Representer Coefficients Stabilized Representer Matrix

How to solve for corrections ?  Method of solution in IROMS 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 -

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