Spectral density estimation for random fields via periodic embeddings

We introduce methods for estimating the spectral density of a random field on a [Formula: see text]-dimensional lattice from incomplete gridded data. Data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function. The imputations are convenient computationally, in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in [Formula: see text] time and [Formula: see text] memory. However, these so-called periodic imputations are motivated mainly by their ability to produce accurate spectral density estimates. In addition, we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias. The paper contains theoretical results on properties of the imputed-data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios. We present an application to a gridded satellite surface temperature dataset with missing values.