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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.

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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 Goals

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Inverse Regional Ocean Modeling System (IROMS) Chua and Bennett (2001) Inverse Ocean Modeling System (IOMs) Moore et al. (2003) NL-ROMS, TL-ROMS, REP-ROMS, AD-ROMS To implement a representer-based generalized inverse method to solve weak constraint data assimilation into a non-linear model a 4D-variational data assimilation system for high-resolution basin- wide and coastal oceanic flows

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def: NL-ROMS: REP-ROMS: also referred to as Picard Iterations Approximation of NONLINEAR DYNAMICS (STEP 1)

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REP-ROMS: def: Representer Tangent Linear Model

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

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

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TL-ROMS: AD-ROMS: REP-ROMS: Small Errors 1)model missing dynamics 2)boundary conditions errors 3)Initial conditions errors (STEP 2)

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

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TL-ROMS: AD-ROMS: REP-ROMS: Integral Solutions …..

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TL-ROMS: AD-ROMS: REP-ROMS: Tangent Linear Propagator Integral Solutions …..

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TL-ROMS: AD-ROMS: REP-ROMS: Tangent Linear Propagator Integral Solutions

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TL-ROMS: AD-ROMS: REP-ROMS: Integral Solutions Adjoint Propagator

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TL-ROMS: AD-ROMS: REP-ROMS: Integral Solutions Adjoint Propagator Tangent Linear Propagator

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TL-ROMS: How is the tangent linear model useful for assimilation?

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

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

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

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

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

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1) Initial model-data misfit 2) Corrections to Model State 3) Find that minimizes ASSIMILATION (2)Modeling the Corrections Corrections to initial conditions Corrections to model dynamics and boundary conditions

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1) Initial model-data misfit 2) Correction to Model Initial Guess 3) Find that minimizes ASSIMILATION (2)Modeling the Corrections Corrections to initial conditions Corrections to model dynamics and boundary conditions Assume we seek to correct only the initial conditions STRONG CONSTRAINT

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

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ASSIMILATION (2)Modeling the Corrections ASSIMILATION (3)Cost Function 1) Initial model-data misfit 2) Correction to Model Initial Guess 3) Find that minimizes 2) corrections should not exceed our assumptions about the errors in model initial condition. 1) corrections should reduce misfit within observational error

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ASSIMILATION (3)Cost Function

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

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

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ASSIMILATION (3)Cost Function Minimize J

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4DVAR inversion Hessian Matrix def:

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4DVAR inversion IOM representer-based inversion Hessian Matrix def:

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

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

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

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IOM representer-based inversion Stabilized Representer Matrix Representer Coefficients Representer Matrix Assume a special assimilation case: Observations = Full model state at time T Diagonal Covariance with unit variance

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IOM representer-based inversion Stabilized Representer Matrix Representer Coefficients Representer Matrix Assume a special assimilation case: Observations = Full model state at time T Diagonal Covariance with unit variance

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IOM representer-based inversion Stabilized Representer Matrix Representer Coefficients Representer Matrix

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

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Representer Matrix Assume you want to compute the model spatial covariance at time T STEP 1) AD-ROMS

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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

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Representer Matrix Assume you want to compute the model spatial covariance at time T STEP 2) run TL-ROMS forced with STEP 3) multiply by and take expected value STEP 1) AD-ROMS

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Representer Matrix Assume you want to compute the model spatial covariance at time T STEP 2) run TL-ROMS forced with STEP 3) multiply by and take expected value STEP 1) AD-ROMS

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Representer Matrix model to model covariance

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Representer Matrix model to model covariance model to model covariance most general form

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Representer Matrix model to model covariance

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

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Representer Matrix model to model covariance data to data covariance if we sample at observation locations through if we introduce a non-diagonal initial condition covariance preconditioned data to data covariance

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… back to the system to invert ….

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

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

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

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

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Stabilized Representer Matrix Representer Coefficients WEAK CONSTRAINT How to solve for corrections ? Method of solution in IROMS

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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 STEP 3) Compute model-data misfit STEP 4) Solve for Representer Coeficients STEP 5) Compute corrections STEP 6) Update model state using REP-ROMS outer loop

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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 STEP 3) Compute model-data misfit STEP 4) Solve for Representer Coeficients STEP 5) Compute corrections STEP 6) Update model state using REP-ROMS outer loop inner loop

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How to evaluate the action of the stabilized Representer Matrix

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Baroclinic Jet Example -

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… end

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