Ensemble Kalman filters with self exclusion

We explore and analyze the properties of ensemble Kalman filters (EnKFs) using “self exclusion,” in which the gain matrix in a given ensemble member’s update employs covariances estimated from all other members, but not the given member. This approach and variants, which seek to represent the uncertainty in the gain, dates almost to the beginning of EnKF research and can clearly be beneficial in practice (Houtekamer and Mitchell 1998; Buehner 2021), yet has only been thoroughly studied for a scalar state. In higher dimensions, we find that self exclusion always results in larger analysis variance than the standard EnKF, though relative to the actual analysis error variance it may be biased either too large (for very low-dimensional states) or too small (in higher dimensions). Importantly, the additional analysis variance from self exclusion is “correct” in that it is concentrated in degrees of freedom whose analysis accuracy suffers most from sampling error in the gain.