Covariance Matrix Functions of Isotropic Vector Random Fields

An isotropic scalar or vector random field is a second-order random field in Rd$\mathbb {R}^d$ (d ⩾ 2), whose covariance function or direct/cross covariance functions are isotropic. While isotropic scalar random fields have been well developed and widely used in various sciences and industries, the theory of isotropic vector random fields needs to be investigated for applications. The objective of this article is to study properties of covariance matrix functions associated with vector random fields in Rd$\mathbb {R}^d$ which are stationary, isotropic, and mean square continuous, and derives the characterizations of the covariance matrix function of the Gaussian or second-order elliptically contoured vector random field in Rd$\mathbb {R}^d$. In particular, integral or spectral representations for isotropic and continuous covariance matrix functions are derived.