  Fractional Gaussian fields provide a rich class of spatial models and have a
long history of applications in multiple branches of science. However,
estimation and inference for fractional Gaussian fields present significant
challenges. This book chapter investigates the use of the fractional Laplacian
differencing on regular lattices to approximate to continuum fractional
Gaussian fields. Emphasis is given on model based geostatistics and likelihood
based computations. For a certain range of the fractional parameter, we
demonstrate that there is considerable agreement between the continuum models
and their lattice approximations. For that range, the parameter estimates and
inferences about the continuum fractional Gaussian fields can be derived from
the lattice approximations. Interestingly, regular lattice approximations
facilitate fast matrix-free computations and enable anisotropic
representations. We illustrate the usefulness of lattice approximations via
simulation studies and by analyzing sea surface temperature on the Indian
Ocean.
