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

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Spatial modelling with R-INLA: A review (2018, methodology): “…that are valid on R 3 are valid on the sphere if chordal distances are used, and some families of covariance functions are valid when great circle distances are used, but validity of a covariance function using great circle distance does not follow from validity on R 2 or R 3 (Huang et al., 2011).”
Excursion probability of Gaussian random fields on sphere (2014, background): “For instance, Huang, Zhang and Robeson [16] have recently shown that the restriction of some commonly used stationary isotropic covariance functions on R may not be a valid covariance functions on the sphere (when the Euclidean metric is replaced by the spherical metric).”
A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space (2013, methodology): “Although the validity of usual parametric variogram models is not guaranteed in non-Euclidean spaces (Curriero, 2006), both the spherical and the exponential models are valid in the spherical geometry (Huang et al., 2011) and thus can be used in this case.”