  We discuss properties of hierarchical Bayesian inversion through the ensemble
Kalman filter (EnKF). Our focus will be primarily on deriving continuous-time
limits for hierarchical inversion in the linear case. An important
characteristic of the EnKF for inverse problems is that the updated particles
are preserved by the linear span of the initial ensemble. By incorporating
certain hierarchical approaches we show that we can break away from the induced
subspace property. We further consider a number of variants of the EnKF such as
covariance inflation and localization, where we derive their continuous-time
limits. We verify these results with various numerical experiments through a
linear elliptic partial differential equation.
