  We introduce methods for estimating the spectral density of a random field on
a $d$-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 $O(n\log n)$ time and $O(n)$ memory. However, these so-called
periodic imputations are motivated mainly by their ability to produce accurace
spectral density estimates. In addition, we introduce a parametric filtering
method that is designed to reduce periodogram smoothing bias. The paper
contains theoretical results studying 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.
