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The characteristics of the model dynamics are critical in the performance of(ensemble) Kalman filters. In particular, as emphasized in the seminal work ofAnna Trevisan and co-authors, the error covariance matrix is asymptoticallysupported by the unstable-neutral subspace only, i.e., it is spanned by thebackward Lyapunov vectors with non-negative exponents. This behavior is at thecore of algorithms known as Assimilation in the Unstable Subspace, although aformal proof was still missing.  This paper provides the analytical proof of the convergence of the Kalmanfilter covariance matrix onto the unstable-neutral subspace when the dynamicsand the observation operator are linear and when the dynamical model iserror-free, for any, possibly rank-deficient, initial error covariance matrix.The rate of convergence is provided as well. The derivation is based on anexpression that explicitly relates the error covariances at an arbitrary timeto the initial ones. It is also shown that if the unstable and neutraldirections of the model are sufficiently observed and if the column space ofthe initial covariance matrix has a non-zero projection onto all of the forwardLyapunov vectors associated with the unstable and neutral directions of thedynamics, the covariance matrix of the Kalman filter collapses onto anasymptotic sequence which is independent of the initial covariances. Numericalresults are also shown to illustrate and support the theoretical findings.

Degenerate Kalman filter error covariances and their convergence onto the unstable subspace