A low-rank ensemble Kalman filter for elliptic observations

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