Geometric Analysis of Conditional Bias-Informed Kalman Filters

Abstract

This paper presents a comparative geometric analysis of the conditional bias (CB)-informed Kalman filter (KF) with the Kalman filter (KF) in the Euclidean space. The CB-informed KFs considered include the CB-penalized KF (CBPKF) and its ensemble extension, the CB-penalized Ensemble KF (CBEnKF). The geometric illustration for the CBPKF is given for the bi-state model, composed of an observable state and an unobservable state. The CBPKF co-minimizes the error variance and the variance of the Type-II error. As such, CBPKF-updated state error vectors are larger than the KF-updated, the latter of which is based on minimizing the error variance only. Different error vectors in the Euclidean space imply different eigenvectors and covariance ellipses in the state space. To characterize the differences in geometric attributes between the two filters, numerical experiments were carried out using the Lorenz 63 model. The results show that the CBEnKF yields more accurate confidence regions for encompassing the truth, smaller errors in the ensemble mean, and larger norms for Kalman gain and error covariance matrices than the EnKF, particularly when assimilating highly uncertain observations.