It is found that only non-decaying eigenvectors are required to represent both the analysis and forecast error covariance matrices. Consequently, for the autonomous systems considered, precise ensemble based representations of the analysis and forecast error covariance matrices become possible when the number of ensemble perturbations becomes equal to the number of non-decaying eigenvectors of the dynamics propagator.
The manner in which ensemble perturbations should be recycled in order to precisely represent the error covariances of the system is given by the computationally inexpensive Ensemble Transform Kalman Filter (ET KF). ET KF recycling may be viewed as a sophisticated form of the breeding method. The computationally more expensive system simulation recycling approach could deliver qualitatively accurate estimates of the error covariances although the observation error covariance forcing would be systematically overestimated. The simplest form of the breeding method would yield highly rank deficient and qualitatively inaccurate error covariances.
Apart from allowing one to assess the value of various ensemble recycling techniques in an idealized context, we also note that the solutions provide some insight into how dynamics and observations conspire to constrain and structure error correlations. It also provides a computationally inexpensive way of assessing the error reducing value of various feasible configurations of the global observational network.