  We consider the problem of determining the optimal block (or subsample) size
for a spatial subsampling method for spatial processes observed on regular
grids. We derive expansions for the mean square error of the subsampling
variance estimator, which yields an expression for the theoretically optimal
block size. The optimal block size is shown to depend in an intricate way on
the geometry of the spatial sampling region as well as characteristics of the
underlying random field. Final expressions for the optimal block size make use
of some nontrivial estimates of lattice point counts in shifts of convex sets.
Optimal block sizes are computed for sampling regions of a number of commonly
encountered shapes. Numerical studies are performed to compare subsampling
methods as well as procedures for estimating the theoretically best block size.
