Alberto Carrassi and myself have published a new paper on the exploration of the model dynamics on an advanced ensemble data assimilation method, namely the Iterative ensemble Kalman smoother (IEnKS), a nonlinear 4D ensemble-variational method.
The performance of (ensemble) Kalman filters used for data assimilation in the geosciences critically depends on the dynamical properties of the evolution model. A key aspect is that the error covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., it is spanned by the backward Lyapunov vectors with non-negative exponents. This paper first generalizes those results to the case of the Kalman smoother in a linear, Gaussian and perfect model scenario. We also provide square-root formulae for the filter and smoother that make the connection with ensemble formulations of the Kalman filter and smoother.
We then discuss how this neat picture is modified when the dynamics are nonlinear and chaotic. A numerical investigation is carried out to study the approximate confinement of the anomalies for both the EnKF and the IEnKS, in a perfect model scenario. The confinement is characterized using geometrical angles that determine the relative position of the anomalies with respect to the unstable-neutral subspace. The alignment of the anomalies and of the unstable-neutral subspace is more pronounced when observation precision or frequency, as well as the data assimilation window length for the IEnKS, are increased. These results also suggest that the IEnKS and the deterministic EnKF realize in practice (albeit implicitly) the paradigm behind the approach of Anna Trevisan and co-authors known as the assimilation in the unstable subspace.