  The sample covariance matrix of a random vector is a good estimate of the
true covariance matrix if the sample size is much larger than the length of the
vector. In high-dimensional problems, this condition is never met. As a result,
in high dimensions the Ensemble Kalman Filter's (EnKF) ensemble does not
contain enough information to specify the prior covariance matrix accurately.
This necessitates the need for regularization of the analysis (observation
update) problem. We propose a regularization technique based on a new spatial
model. The model is a constrained version of the general Gaussian process
convolution model. The constraints include local stationarity and smoothness of
local spectra. We regularize EnKF by postulating that its prior covariances
obey the spatial model. Placing a hyperprior distribution on the model
parameters and using the likelihood of the prior ensemble data allows for an
optimized use of both the ensemble and the hyperprior. The respective estimator
is shown to be consistent. Its neural Bayes implementation proved to be both
accurate and computationally efficient. In simulation experiments, the new
technique led to substantially better EnKF performance than several existing
techniques.
