Filtering a nonlinear slow-fast system with strong fast forcing. (English) Zbl 1202.62128

Gershgorin, Boris; Majda, Andrew

Commun. Math. Sci. 8, No. 1, 67-92 (2010).

The authors study the problem of the filtering of atmospheric signals with multiple time scales. They start by introducing the nonlinear multiple time test model with fast forcing (a model introduced first by the same authors in 2008) and give more interesting motivations and concrete application for this model. They study the exact solutions and exactly solvable statistics giving a way to find an analytic solution, extending some results given by R. Kubo [J. Math. Phys. 4, 174–183 (1963; Zbl 0135.45102)] and A.J. Majda et al [Proc. Natl. Acad. Sci. USA 96, No.26, 14687–14691 (1999; Zbl 0966.86003)]. They also present some more numerical examples. We have to remark, that the authors pointed out the non-Gaussianity of the statistics of the fast mode.
In the next part, they introduced two adequate filters: the exact Nonlinear Extended Kalman Filter (NEKF) and the linear Kalman Filter (KF) with model errors for the slow-fast system with fast forcing (both filters are based on the classical Kalman filter. They compare the performance of both filters under various filtering parameters.
An important result of this study presents their discovering that for the observation time and observation variance larger than certain values and all three types of observations, the linear KF with model errors performs surprisingly better than the exact NEKF on the slowly varying amplitude of the fast mode.

Reviewer: Romeo Negrea (Timisoara)

MSC:

62M20	Inference from stochastic processes and prediction
86A10	Meteorology and atmospheric physics
93E11	Filtering in stochastic control theory

Keywords:

nonlinear model; extended Kalman filter

Citations:

Zbl 0135.45102; Zbl 0966.86003