Strict positive definiteness under axial symmetry on the sphere

Pier Giovanni Bissiri, Ana Paula Peron, Emilio Porcu

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


Axial symmetry for covariance functions defined over spheres has been a very popular assumption for climate, atmospheric, and environmental modeling. For Gaussian random fields defined over spheres embedded in a three-dimensional Euclidean space, maximum likelihood estimation techiques as well kriging interpolation rely on the inverse of the covariance matrix. For any collection of points where data are observed, the covariance matrix is determined through the realizations of the covariance function associated with the underlying Gaussian random field. If the covariance function is not strictly positive definite, then the associated covariance matrix might be singular. We provide conditions for strict positive definiteness of any axially symmetric covariance function. Furthermore, we find conditions for reducibility of an axially symmetric covariance function into a geodesically isotropic covariance. Finally, we provide conditions that legitimate Fourier inversion in the series expansion associated with an axially symmetric covariance function.

Original languageBritish English
Pages (from-to)723-732
Number of pages10
JournalStochastic Environmental Research and Risk Assessment
Issue number5
StatePublished - 1 May 2020


  • Axial symmetry
  • Covariance function
  • Fourier inversion
  • Reducibility


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