Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities

Baehr, Christophe

ESAIM: Modélisation mathématique et analyse numérique, Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 921-945.

To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear filtering for the mobile signal is therefore those of an acquisition process contaminated by random errors. This will provide a Feynman-Kac distribution flow for the conditional laws and an N particle approximation with a $𝒪$ $\left(\frac{1}{\sqrt{N}}\right)$ asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.

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