  In this work we combine ideas from multi-index Monte Carlo and
  ensemble Kalman filtering (EnKF) to produce a highly efficient
  filtering method called multi-index EnKF (MIEnKF). MIEnKF 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, when strong coupling between solutions on
  neighboring numerical resolutions is attainable, the MIEnKF method
  is proven to be more tractable than EnKF and MLEnKF. Said efficiency
  gains are also verified numerically in a series of test problems.
