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

TL;DR

This paper investigates the limitations of the ensemble Kalman filter (EnKF) in assimilating observations, revealing that the degrees of freedom for signal (DFS) is bounded by ensemble size, affecting analysis confidence and requiring localization and inflation strategies.

Contribution

It introduces the concept of DFS in EnKF, demonstrates its bounds and effects, and extends the theory to localization schemes, providing insights into practical data assimilation challenges.

Findings

DFS is bounded by ensemble size, leading to underestimation with more observations.

Localization schemes help mitigate DFS underestimation.

DFS diagnostics offer explanations for observed EnKF behaviors.

Abstract

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.