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 results have been published in Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

In the sequel paper, entitled Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error published Nonlinear Processes in Geophysics, we look into the Kalman filter equations in presence of additive model error and project them onto a Lyapunov filtration. We show that there is an upwelling of the errors injected by model error into the stable subspace into the unstable subspace. We discuss how this could be counteracted by inflation. This mechanism is illustrated on numerical examples.