Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution

TL;DR

This paper proves that in linear time-varying systems, the Kalman filter's error covariance matrices are asymptotically rank-deficient, confined to the unstable-neutral subspace, which supports subspace-based data assimilation methods.

Contribution

It provides a theoretical proof linking the rank of error covariance matrices to Lyapunov exponents in time-varying systems, justifying unstable-neutral subspace assimilation.

Findings

Error covariance matrices have rank bounded by the number of non-negative Lyapunov exponents.

Support of covariance matrices is confined to unstable-neutral Lyapunov vectors.

The results extend to autonomous systems as a special case.

Abstract

We prove that for linear, discrete, time-varying, deterministic system (perfect model) with noisy outputs, the Riccati transformation in the Kalman filter asymptotically bounds the rank of the forecast and the analysis error covariance matrices to be less than or equal to the number of non-negative Lyapunov exponents of the system. Further, the support of these error covariance matrices is shown to be confined to the space spanned by the unstable-neutral backward Lyapunov vectors, providing the theoretical justification for the methodology of the algorithms that perform assimilation only in the unstable-neutral subspace. The equivalent property of the autonomous system is investigated as a special case.