Articles | Volume 18, issue 2
Nonlin. Processes Geophys., 18, 243–250, 2011
https://doi.org/10.5194/npg-18-243-2011
Nonlin. Processes Geophys., 18, 243–250, 2011
https://doi.org/10.5194/npg-18-243-2011

Research article 28 Mar 2011

Research article | 28 Mar 2011

On the Kalman Filter error covariance collapse into the unstable subspace

A. Trevisan1 and L. Palatella2 A. Trevisan and L. Palatella
  • 1Istituto di Scienze dell'Atmosfera e del Clima del CNR, via Gobetti 101, 40129 Bologna, Italy
  • 2Istituto di Scienze dell'Atmosfera e del Clima del CNR, UdR di Lecce, via Lecce-Monteroni, km 1,200, 73100 Lecce, Italy

Abstract. When the Extended Kalman Filter is applied to a chaotic system, the rank of the error covariance matrices, after a sufficiently large number of iterations, reduces to N+ + N0 where N+ and N0 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 assimilation to the space spanned by the Lyapunov vectors with non-negative Lyapunov exponents. Theoretical arguments and numerical verification are provided to show that the asymptotic state and covariance estimates of the full EKF and of its reduced form, with assimilation in the unstable and neutral subspace (EKF-AUS) are the same. The consequences of these findings on applications of Kalman type Filters to chaotic models are discussed.