  We introduce a new multilevel ensemble Kalman filter method (MLEnKF) which
consists of a hierarchy of independent samples of ensemble Kalman filters
(EnKF). This new MLEnKF method is fundamentally different from the preexisting
method introduced by Hoel, Law and Tempone in 2016, and it is suitable for
extensions towards multi-index Monte Carlo based filtering methods. Robust
theoretical analysis and supporting numerical examples show that under
appropriate regularity assumptions, the MLEnKF method has better complexity
than plain vanilla EnKF in the large-ensemble and fine-resolution limits, for
weak approximations of quantities of interest. The method is developed for
discrete-time filtering problems with finite-dimensional state space and linear
observations polluted by additive Gaussian noise.
