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Hayder Salman Department of Mathematics UNC-Chapel Hill

Towards Nonlinear Filtering in Lagrangian Data Assimilation. Hayder Salman Department of Mathematics UNC-Chapel Hill. Collaborators: Chris Jones, Kayo Ide. Sponsored by . The augmented approach for Lagrangian Data Assimilation (LaDA) :.

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Hayder Salman Department of Mathematics UNC-Chapel Hill

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  1. Towards Nonlinear Filtering in Lagrangian Data Assimilation Hayder Salman Department of Mathematics UNC-Chapel Hill Collaborators: Chris Jones, Kayo Ide Sponsored by

  2. The augmented approach for Lagrangian Data Assimilation (LaDA): • Traditionally, velocity field reconstructed from drifter observations • reconstructed velocity field assimilated into the model • problematic since drifter positions and velocity related nonlinearly • nonlinear observation operator • Introducing the augmented state vector • results in a linear observation operator • drifter positions assimilated directly into the model

  3. Kalman Filter: • At analysis we update the state vector with • This produces the correct (Bayesian) solution provided • observation operator is linear • likelihood is Gaussian • prior is Gaussian • Need to rethink last point - generally not true for nonlinear models

  4. Nonlinear Attributes of Lagrangian Data: • Important observation • very simple simple Eulerian velocity fields can give rise to Lagrangian chaos • Augmented system is strongly nonlinear in observation space (i.e. with respect to Lagrangian drifter trajectories. Lagrangian coherent structures in double gyre ocean model Ottino ARFM (1990)

  5. Filter Performance Near Saddle: • The nonlinearity associated with chaotic advection is problematic in LaDA • filter divergence observed near a Lagrangian saddle • Augmented system is strongly nonlinear with respect to the space of the drifters - we need a Lagrangian specific component for our LaDA filter

  6. Nonlinear Filtering for LaDA: • A full solution to the nonlinear problem requires • computing the transitional PDF from the Fokker-Planck equation (*) • computing the posterior PDF from the prior and likelihood using Baye’s rule • We would like to identify a specific structure in (*) to improve the approximation of the PDF in observation space • Under certain assumptions, the PDF associated with the evolution of a drifter is related to an advection-diffusion equation of a passive tracer !!! • We are exploiting the simplification associated with this property to formulate a more general method for LaDA.

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