using spectral approximations without a large extra computational burden.

Low-rank approximations, in which the covariance matrix is approximated by a low-rank matrix plus a diagonal matrix, can greatly reduce the burden of memory and computation relative to the exact likelihood \[@cressiejohannesson [@eidsvik]\]. However, for the kinds of applications we have in mind, in which the diagonal component of the covariance matrix does not dominate the small-scale variation of the process, these low-rank approximations tend to work poorly and are not a viable option \[@steinkorea\].

Covariance tapering replaces the covariance matrix of interest by a sparse covariance matrix with similar local behavior \[@furrer\]. There is theoretical support for this approach \[@kaufman [@wangloh]\], but the tapered covariance matrix must be very sparse to help a great deal with calculating the log determinant of the covariance matrix, in which case @steintaper finds that composite likelihood approaches will often be preferable. There is scope for combining covariance tapering with the approach presented here in that sparse matrices lead to efficient matrix–vector multiplication, which is also essential for our implementation of computing (\[ascore\]) based on iterative methods to do the matrix solves. @sang show that covariance tapering and low-rank approximations can also sometimes be profitably 