Regularization of the ensemble Kalman filter using a non-stationary spatial convolution model

Applications of the ensemble Kalman filter to high-dimensional problems are feasible only with small ensembles. This necessitates a kind of regularization of the analysis (observation update) problem. We propose a regularization technique based on a new non-stationary, non-parametric spatial model on the sphere. The model termed the Locally Stationary Convolution Model is a constrained version of the general Gaussian process convolution model. The constraints on the location-dependent convolution kernel include local isotropy, positive definiteness as a function of distance, and smoothness as a function of location. The model allows for a rigorous definition of the local spectrum, which is required to be a smooth function of spatial wavenumber. We propose and test an ensemble filter in which prior covariances are postulated to obey the Locally Stationary Convolution Model. The model is estimated online in a two-stage procedure. First, ensemble perturbations are bandpass filtered in several wavenumber bands to extract aggregated local spatial spectra. Second, a neural network recovers the local spectra from sample variances of the filtered fields. In simulation experiments, the new filter was capable of outperforming several existing techniques. With small to moderate ensemble sizes, the improvement was substantial.