Topographic correlation, power-law covariance functions, and diffusion

Evidence from agricultural uniformity trials strongly indicates that the covariance function of yield in the plane, Γ(s), decays ultimately as s −1 , the inverse of distance (see §2 and the figure). A diffusion model (1) is examined in a varying number of spatial dimensions: in three dimensions it proves to generate an s −1 covariance law, (6). In §4 modifications of the model are examined which have the effect of removing singularities of Γ(s) at s = 0. In §5 I consider the three-dimensional model in a slab of finite or semi-infinite thickness, with varying boundary conditions, and with drift perpendicular to the face of the slab. In this way I find the various covariance functions (25), (26), (28), (29), (30), (39) and (40).