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This paper reviews and introduces characterizations of the covariance function on all spheres that is isotropic and continuous, and characterizations of the covariance matrix function on all spheres whose entries are isotropic and continuous. These characterizations are used to derive some covariance (matrix) structures on all spheres, with certain polynomials obtained, besides some rational, (negative) power, and logarithmic models. [ABSTRACT FROM AUTHOR]

This paper is concerned with vector random fields on spheres with second-order increments, which are intrinsically stationary and mean square continuous and have isotropic variogram matrix functions. A characterization of the continuous and isotropic variogram matrix function on a sphere is derived, in terms of an infinite sum of the products of positive definite matrices and ultraspherical polynomials. It is valid for Gaussian or elliptically contoured vector random fields, but may not be valid for other non-Gaussian vector random fields on spheres such as a χ, log-Gaussian, or skew-Gaussian vector random field. Some parametric variogram matrix models are derived on spheres via different constructional approaches. A simulation study is conducted to illustrate the implementation of the proposed model in estimation and cokriging, whose performance is compared with that using the linear model of coregionalization. [ABSTRACT FROM AUTHOR]

This paper presents the characterization of the covariance matrix function of a Gaussian or second-order elliptically contoured vector random field on the sphere which is stationary, isotropic, and mean square continuous. This characterization involves an infinite sum of the products of positive definite matrices and Gegenbauer's polynomials, and may not be available for other non-Gaussian vector random fields on spheres such as a χ or log-Gaussian vector random field. We also offer two simple but efficient constructing approaches, and derive some parametric covariance matrix structures on spheres. [ABSTRACT FROM AUTHOR]