A Valid Matérn Class of Cross-Covariance Functions for Multivariate Random Fields With Any Number of Components

We introduce a valid parametric family of cross-covariance functions for multivariate spatial random fields where each component has a covariance function from a well-celebrated Matérn class. Unlike previous attempts, our model indeed allows for various smoothnesses and rates of correlation decay for any number of vector components. We present the conditions on the parameter space that result in valid models with varying degrees of complexity. Practical implementations, including reparametrizations to reflect the conditions on the parameter space and an iterative algorithm to increase the computational efficiency, are discussed. We perform various Monte Carlo simulation experiments to explore the performances of our approach in terms of estimation and cokriging. The application of the proposed multivariate Matérn model is illustrated on two meteorological datasets: Temperature/Pressure over the Pacific Northwest