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We propose a regularization method for ensemble Kalman filtering (EnKF) withelliptic observation operators. Commonly used EnKF regularization methodssuppress state correlations at long distances. For observations described byelliptic 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 spuriouslong-range correlations. This is particularly true for the PPE, in whichdistant vortex elements couple nonlinearly to induce pressure. Instead, theseinverse problems have a low effective dimension: low-dimensional projections ofthe 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 ofthe Jacobian of the observation operator. The identified eigenvectorsgeneralize the source and target modes of the multipole expansion,independently of the underlying spatial distribution of the problem. Givenrapid spectral decay, inference can be performed in the low-dimensionalsubspace spanned by the dominant eigenvectors. This low-rank EnKF is assessedon dynamical systems with Poisson observation operators, where we seek toestimate the positions and strengths of point singularities over time frompotential or pressure observations. We also comment on the broaderapplicability of this approach to elliptic inverse problems outside the contextof filtering.

A low-rank ensemble Kalman filter for elliptic observations