Degenerate Kalman Filter Error Covariances and Their Convergence onto the Unstable Subspace
Résumé
The characteristics of the model dynamics are critical in the performance of (ensemble) Kalman
filters. In particular, as emphasized in the seminal work of Anna Trevisan and coauthors, the
error covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., it
is spanned by the backward Lyapunov vectors with nonnegative exponents. This behavior is at the
core of algorithms known as assimilation in the unstable subspace, although a formal proof was still
missing. This paper provides the analytical proof of the convergence of the Kalman filter covariance
matrix onto the unstable-neutral subspace when the dynamics and the observation operator are linear
and when the dynamical model is error free, for any, possibly rank-deficient, initial error covariance
matrix. The rate of convergence is provided as well. The derivation is based on an expression that
explicitly relates the error covariances at an arbitrary time to the initial ones. It is also shown that
if the unstable and neutral directions of the model are sufficiently observed and if the column space
of the initial covariance matrix has a nonzero projection onto all of the forward Lyapunov vectors
associated with the unstable and neutral directions of the dynamics, the covariance matrix of the
Kalman filter collapses onto an asymptotic sequence which is independent of the initial covariances.
Numerical results are also shown to illustrate and support the theoretical findings.