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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

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):

  • 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


Kalman Filter (LaDA):

  • 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


Nonlinear Attributes of Lagrangian Data (LaDA):

  • 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)


Filter Performance Near Saddle: (LaDA)

  • 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


Nonlinear Filtering for LaDA: (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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