Isotropic Variogram Matrix Functions on Spheres

Juan Du, Chunsheng Ma, Yang Li

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


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 χ2, 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.

Original languageEnglish (US)
Pages (from-to)341-357
Number of pages17
JournalMathematical Geosciences
Issue number3
StatePublished - Apr 2013

Bibliographical note

Funding Information:
Acknowledgements The authors wish to thank the Editor-in-Chief, an associate editor, two reviewers and Dr. James Stapleton for their valuable comments and suggestions which helped to improve the presentation of this paper. Ma’s work is supported in part by US Department of Energy under Grant DESC0005359.


  • Absolutely monotone function
  • Cross variogram
  • Direct variogram
  • Elliptically contoured random field
  • Gaussian random field
  • Gegenbauer's polynomials
  • Positive definite matrix


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