  There is a growing interest in developing covariance functions for processes
on the surface of a sphere due to wide availability of data on the globe.
Utilizing the one-to-one mapping between the Euclidean distance and the great
circle distance, isotropic and positive definite functions in a Euclidean space
can be used as covariance functions on the surface of a sphere. This approach,
however, may result in physically unrealistic distortion on the sphere
especially for large distances. We consider several classes of parametric
covariance functions on the surface of a sphere, defined with either the great
circle distance or the Euclidean distance, and investigate their impact upon
spatial prediction. We fit several isotropic covariance models to simulated
data as well as real data from NCEP/NCAR reanalysis on the sphere. We
demonstrate that covariance functions originally defined with the Euclidean
distance may not be adequate for some global data.
