On the Kalman Filter error covariance collapse into the unstable subspace

When the Extended Kalman Filter is applied to a chaotic system, the rank of the error covariance matri- ces, after a sufficiently large number of iterations, reduces to N + + N 0 where N + and N 0 are the number of positive and null Lyapunov exponents. This is due to the collapse into the unstable and neutral tangent subspace of the solution of the full Extended Kalman Filter. Therefore the solution is the same as the solution obtained by confining the assimila- tion to the space spanned by the Lyapunov vectors with non- negative Lyapunov exponents. Theoretical arguments and numerical verification are provided to show that the asymp- totic state and covariance estimates of the full EKF and of its reduced form, with assimilation in the unstable and neu- tral subspace (EKF-AUS) are the same. The consequences of these findings on applications of Kalman type Filters to chaotic models are discussed.