  In this work we marry multi-index Monte Carlo with ensemble Kalman filtering
(EnKF) to produce the multi-index EnKF method (MIEnKF). The MIEnKF method is
based on independent samples of four-coupled EnKF estimators on a multi-index
hierarchy of resolution levels, and it may be viewed as an extension of the
multilevel EnKF (MLEnKF) method developed by the same authors in 2020.
Multi-index here refers to a two-index method, consisting of a hierarchy of
EnKF estimators that are coupled in two degrees of freedom: time discretization
and ensemble size. Under certain assumptions, the MIEnKF method is proven to be
more tractable than EnKF and MLEnKF, and this is also verified in numerical
examples.
