Geometric anisotropy arises when the variogram of a spatial random field varies with direction. We propose a Bayesian inference for geometrically anisotropic random fields on regular lattice. The class of random fields we focus on arises from fractional Laplacian differencing on the lattice. Furthermore, with diminishing lattice spacing, these fields approximate certain continuum anisotropic Matern class of models. We demonstrate our methodology by analyzing data on ocean chlorophyll concentrations obtained from MODIS-Aqua project of NASA.