Methods and Algorithms for Scientific Computing

Covariance Models and Simulation Algorithm for Stationary Vector Random Fields on Spheres Crossed with Euclidean Spaces


This paper focuses on vector random fields defined on $\mathbb{S}^d\times \mathbb{R}^k$, $d \geq 2$ and $k \geq 1$, with covariance functions that depend on the geodesic distance in $\mathbb{S}^d$ and on the separation vector in $\mathbb{R}^k$. First, we propose parametric families of nonseparable covariance functions with closed-form expressions and explicit spectral representations. Then, we derive an algorithm for fast simulation of such random fields, which combines spectral simulation methods in $\mathbb{S}^d$ and $\mathbb{R}^k$ previously introduced in the literature and relies on importance sampling techniques. We provide computer codes and practical guidelines and describe the advantages of our proposal in comparison to other methods.


  1. nonseparable sphere-time covariance functions
  2. Gaussian random fields
  3. spectral measure
  4. spectral simulation

MSC codes

  1. 62H11
  2. 86A32

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Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Covariance Models and Simulation Algorithm for Stationary Vector Random Fields on Spheres Crossed with Euclidean Spaces

Authors: Xavier Emery, Alfredo Alegría, Daisy Arroyo


Type: Compressed Code Files

Contents: ZIP file contains seven computer codes in GNU Octave's language and a file with details on these codes:

  • simu_besselcos.m
  • simu_multistable.m
  • experiments.m
  • Gegenbauer.m
  • stablernd.m
  • zetarnd.m
  • display_S2.m


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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A3114 - A3134
ISSN (online): 1095-7197


Submitted: 8 October 2020
Accepted: 16 June 2021
Published online: 13 September 2021


  1. nonseparable sphere-time covariance functions
  2. Gaussian random fields
  3. spectral measure
  4. spectral simulation

MSC codes

  1. 62H11
  2. 86A32



Funding Information

ANID : PIA AFB180004, 1210050, PAI79160084

Funding Information

Fondo Nacional de Desarrollo Científico y Tecnológico : 11170529

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