What limits the number of observations that can be effectively assimilated by EnKF?

The ability of ensemble Kalman filter (EnKF) algorithms to extract information from observations is analyzed with the aid of the concept of the degrees of freedom for signal (DFS). A simple mathematical argument shows that DFS for EnKF is bounded from above by the ensemble size, which entails that assimilating much more observations than the ensemble size automatically leads to DFS underestimation. Since DFS is a trace of the posterior error covariance mapped onto the normalized observation space, underestimated DFS implies overconfidence (underdispersion) in the analysis spread, which, in a cycled context, requires covariance inflation to be applied. The theory is then extended to cases where covariance localization schemes (either B-localization or R-localization) are applied to show how they alleviate the DFS underestimation issue. These findings from mathematical argument are demonstrated with a simple one-dimensional covariance model. Finally, the DFS concept is used to form speculative arguments about how to interpret several puzzling features of LETKF previously reported in the literature such as why using less observations can lead to better performance, when optimal localization scales tend to occur, and why covariance inflation methods based on relaxation to prior information approach are particularly successful when observations are inhomogeneously distributed. A presumably first application of DFS diagnostics to a quasi-operational global EnKF system is presented in Appendix.