  We develop a multi-level restricted Gaussian maximum likelihood method for
estimating the covariance function parameters and computing the best unbiased
predictor. Our approach produces a new set of multi-level contrasts where the
deterministic parameters of the model are filtered out thus enabling the
estimation of the covariance parameters to be decoupled from the deterministic
component. Moreover, the multi-level covariance matrix of the contrasts exhibit
fast decay that is dependent on the smoothness of the covariance function. Due
to the fast decay of the multi-level covariance matrix coefficients only a
small set is computed with a level dependent criterion. We demonstrate our
approach on problems of up to 512,000 observations with a Matern covariance
function and highly irregular placements of the observations. In addition,
these problems are numerically unstable and hard to solve with traditional
methods.
