Calibrated spatial moving average simulations

The spatial moving average (SMA) is a very natural type of spatial process that involves integrals or sums of independent and identically distributed random variables. Consequently, the mean and covariance function of the SMAs can be written down immediately in terms of their integrand or summand. Moreover, simulation from them is straightforward, and it does not require any large-matrix inversions. Although the SMAs generate a large class of spatial covariance functions, can we find easy-to-use SMAs, calibrated to be 'like' some of the usual covariance functions used in geostatistics? For example, is there an SMA that is straightforward to simulate from, whose covariance function is like the spherical covariance function? This article will derive such an SMA.