On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere

Covariance and variogram functions have been extensively studied in Euclidean space. In this article, we investigate the validity of commonly used covariance and variogram functions on the sphere. In particular, we show that the spherical and exponential models, as well as power variograms with 0 < α ≤ 1, are valid on the sphere. However, two Radon transforms of the exponential model, Cauchy model, the hole-effect model and power variograms with 1 < α ≤ 2 are not valid on the sphere. A table that summarizes the validity of commonly used covariance and variogram functions on the sphere is provided.