The Assimilation in the Unstable Subspace (AUS) view on the (ensemble) Kalman filter tells that, in the absence of model error, the error covariance matrix is asymptotically projected on the unstable subspace, where all the uncertainty is confined. In the geosciences, this could drastically reduce the number of effective degrees of freedom. This property was conjectured by Anna Trevisan and co-authors and that we proved in a series of papers published in 2017 (Gurumoorthy et al., Bocquet et al., Bocquet and Carrassi).

Colin Grudzien, Alberto Carrassi and I have examined how the AUS view on the Kalman filter is modified when one adds additive model error to the data assimilation system. The error covariance matrix's projection onto the stable subspace does not vanish anymore, but it can be shown that it is bounded due to dynamical dissipation. Hence, the AUS picture has still some relevance in presence of additive model error. Unfortunately, the picture would be spoiled by transient instability which could make the projection on the stable subspace momentarily very large.

Our results have been published in the SIAM/ASA Journal on Uncertainty Quantification. It is entitled Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error.