Multivariate Spartan spatial random field models

• A novel family of vector spatial random fields with local interactions is introduced.
• Permissibility conditions for the model parameters are determined.
• Explicit forms of the matrix covariance function are derived in one, two, and three dimensions.
• Simulations of bivariate random processes and two-dimensional random fields are presented.
• The new vector random field formalism provides opportunities for computationally efficient interpolation and simulation methods.

This paper introduces a family of stationary multivariate spatial random fields with D scalar components that extend the scalar model of Gibbs random fields with local interactions (i.e., Spartan spatial random fields). We derive permissibility conditions for Spartan multivariate spatial random fields with a specific structure of local interactions. We also present explicit expressions for the respective matrix covariance functions obtained at the limit of infinite spectral cutoff in one, two and three spatial dimensions. Finally, we illustrate the proposed covariance models by means of simulated bivariate time series and two-dimensional random fields.