  We propose a regularization method for ensemble Kalman filtering (EnKF) with
elliptic observation operators. Commonly used EnKF regularization methods
suppress state correlations at long distances. For observations described by
elliptic partial differential equations, such as the pressure Poisson equation
(PPE) in incompressible fluid flows, distance localization cannot be applied,
as we cannot disentangle slowly decaying physical interactions from spurious
long-range correlations. This is particularly true for the PPE, in which
distant vortex elements couple nonlinearly to induce pressure. Instead, these
inverse problems have a low effective dimension: low-dimensional projections of
the observations strongly inform a low-dimensional subspace of the state space.
We derive a low-rank factorization of the Kalman gain based on the spectrum of
the Jacobian of the observation operator. The identified eigenvectors
generalize the source and target modes of the multipole expansion,
independently of the underlying spatial distribution of the problem. Given
rapid spectral decay, inference can be performed in the low-dimensional
subspace spanned by the dominant eigenvectors. This low-rank EnKF is assessed
on dynamical systems with Poisson observation operators, where we seek to
estimate the positions and strengths of point singularities over time from
potential or pressure observations. We also comment on the broader
applicability of this approach to elliptic inverse problems outside the context
of filtering.
