  This paper investigates covariance operator estimation via thresholding. For
Gaussian random fields with approximately sparse covariance operators, we
establish non-asymptotic bounds on the estimation error in terms of the
sparsity level of the covariance and the expected supremum of the field. We
prove that thresholded estimators enjoy an exponential improvement in sample
complexity compared with the standard sample covariance estimator if the field
has a small correlation lengthscale. As an application of the theory, we study
thresholded estimation of covariance operators within ensemble Kalman filters.
