Mathematical Geosciences

, Volume 47, Issue 6, pp 699–717 | Cite as

Isotropic Covariance Matrix Functions On All Spheres



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.


Absolutely monotone function Covariance Cross covariance  Direct covariance Elliptically contoured random field Gaussian random field  Positive definite matrix 



This work was supported in part by U.S. Department of Energy under Grant DE-SC0005359. The author wishes to thank the reviewers for the valuable comments and suggestions which helped to improve the presentation of this paper tremendously.


  1. Bapat RB, Raghavan TES (1997) Nonnegative matrices and applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  2. Bingham NH (1973) Positive definite functions on spheres. Proc Camb Phil Soc 73:145–156CrossRefGoogle Scholar
  3. Cheng D, Xiao Y (2014) Excursion probability of Gaussian random fields on sphere. arXiv:1401.5498v1
  4. Christensen JPR, Ressel P (1978) Functions operating on positive definite matrices and a theorem of Schoenberg. Trans Am Math Soc 243:89–95CrossRefGoogle Scholar
  5. Cressie N, Johannesson G (2008) Fixed rank kriging for very large spatial data sets. J R Stat Soc Ser B 70:209–226CrossRefGoogle Scholar
  6. Dimitrakopoulos RD, Mustapha H, Gloaguen E (2010) High-order statistics of spatial random fields: exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena. Math Geosci 42:65–99CrossRefGoogle Scholar
  7. Du J, Ma C (2011) Spherically invariant vector random fields in space and time. IEEE Trans Signal Proc 59:5921–5929CrossRefGoogle Scholar
  8. Du J, Ma C, Li Y (2013) Isotropic variogram matrix functions on spheres. Math Geosci 45:341–357CrossRefGoogle Scholar
  9. Feller W (1971) An introduction to probability theory and its applications, vol II, 2nd edn. Wiley, New YorkGoogle Scholar
  10. Gangolli R (1967a) Abstract harmonic analysis and Lévy’s Brownian motion of several parameters. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol II, Pt. 1. University of California Press, Berkeley, pp 13–30Google Scholar
  11. Gangolli R (1967b) Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann Inst H Poincaré B 3:121–226Google Scholar
  12. Gaspari G, Cohn SE (1999) Construction of correlations in two and three dimensions. Q J R Meteorol Soc 125:723–757CrossRefGoogle Scholar
  13. Gaspari G, Cohn SE, Guo J, Pawson S (2006) Construction and application of covariance functions with variable length-fields. Q J R Meteorol Soc 132:1815–1838CrossRefGoogle Scholar
  14. Gneiting T (2013) Strictly and non-strictly positive definite functions on spheres. Bernoulli 19:1327–1349CrossRefGoogle Scholar
  15. Gradshteyn IS, Ryzhik IM (2007) Tables of integrals, series, and products, 7th edn. Academic Press, AmsterdamGoogle Scholar
  16. Hannan EJ (1970) Multiple time series. Wiley, New YorkCrossRefGoogle Scholar
  17. Huang C, Zhang H, Robeson SM (2011) On the validity of commonly used covariance and variogram functions on the sphere. Math Geosci 43:721–733CrossRefGoogle Scholar
  18. Johns RH (1963a) Stochastic processes on a sphere. Ann Math Stat 34:213–218CrossRefGoogle Scholar
  19. Johns RH (1963b) Stochastic processes on a sphere as applied to meteorological 500-millibar forecasts. In: Proceedings of symposium on time series analysis. Wiley, New York, pp 119–124Google Scholar
  20. Jun M, Stein M (2007) An approach to producing space-time covariance functions on sphere. Technometrics 49:468–479CrossRefGoogle Scholar
  21. Leonenko N, Sakhno L (2012) On spectral representation of tensor random fields on the sphere. Stoch Anal Appl 31:167–182Google Scholar
  22. Ma C (2011a) Vector random fields with second-order moments or second-order increments. Stoch Anal Appl 29:197–215CrossRefGoogle Scholar
  23. Ma C (2011b) Covariance matrix functions of vector \(\chi ^2\) random fields in space and time. IEEE Trans Commun 59:2254–2561CrossRefGoogle Scholar
  24. Ma C (2012) Stationary and isotropic vector random fields on spheres. Math Geosci 44:765–778CrossRefGoogle Scholar
  25. Ma C (2013) K-distributed vector random fields in space and time. Stat Probab Lett 83:1143–1150CrossRefGoogle Scholar
  26. Ma C (2014) Isotropic covariance matrix polynomials on spheres (manuscript)Google Scholar
  27. Marinucci D, Peccati G (2011) Random fields on the sphere: representation, limit theorems and cosmological applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  28. Matheron G (1989) The internal consistency of models in geostatistics. In: Armstrong M (ed) Geostatistics, vol 1. Kluwer Academic Publishers, Netherlands, pp 21–38Google Scholar
  29. Minozzo M, Ferracuti L (2012) On the existence of some skew-normal stationary processes. Chilean J Stat 3:159–172Google Scholar
  30. McLeod MG (1986) Stochastic processes on a sphere. Phys. Earth Plan. Interiors 43:283–299CrossRefGoogle Scholar
  31. Roy R (1973) Estimation of the covariance function of a homogeneous process on the sphere. Ann Stat 1:780–785CrossRefGoogle Scholar
  32. Roy R (1976) Spectral analysis for a random process on the sphere. Ann Inst Stat Math 28:91–97CrossRefGoogle Scholar
  33. Schoenberg I (1938) Metric spaces and completely monotone functions. Ann Math 39:811–841CrossRefGoogle Scholar
  34. Schoenberg I (1942) Positive definite functions on spheres. Duke Math J 9:96–108CrossRefGoogle Scholar
  35. Weaver A, Courtier P (2001) Correlation modelling on the sphere using a generalized diffusion equation. Q J R Meteorol Soc 127:1815–1846CrossRefGoogle Scholar
  36. Widder DV (1946) The Laplace transform. Princeton University Press, PrincetonGoogle Scholar
  37. Yadrenko AM (1983) Spectral theory of random fields. Optimization Software, New YorkGoogle Scholar
  38. Yaglom AM (1987) Correlation theory of stationary and related random functions, vol I. Springer, New YorkGoogle Scholar

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© International Association for Mathematical Geosciences 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and PhysicsWichita State UniversityWichitaUSA
  2. 2.School of Mathematics and StatisticsHubei Engineering UniversityXiaoganChina

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