  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.
