  We design and analyse the performance of a multilevel ensemble Kalman filter
method (MLEnKF) for filtering settings where the underlying state-space model
is an infinite-dimensional spatio-temporal process. We consider underlying
models that needs to be simulated by numerical methods, with discretization in
both space and time. The multilevel Monte Carlo (MLMC) sampling strategy,
achieving variance reduction through pairwise coupling of ensemble particles on
neighboring resolutions, is used in the sample-moment step of MLEnKF to produce
an efficient hierarchical filtering method for spatio-temporal models. Under
sufficient regularity, MLEnKF is proven to be more efficient for weak
approximations than EnKF, asymptotically in the large-ensemble and
fine-numerical-resolution limit. Numerical examples support our theoretical
findings.
