  Multivariate spatial-statistical models are often used when modeling
environmental and socio-demographic processes. The most commonly used models
for multivariate spatial covariances assume both stationarity and symmetry for
the cross-covariances, but these assumptions are rarely tenable in practice. In
this article we introduce a new and highly flexible class of nonstationary and
asymmetric multivariate spatial covariance models that are constructed by
modeling the simpler and more familiar stationary and symmetric multivariate
covariances on a warped domain. Inspired by recent developments in the
univariate case, we propose modeling the warping function as a composition of a
number of simple injective warping functions in a deep-learning framework.
Importantly, covariance-model validity is guaranteed by construction. We
establish the types of warpings that allow for cross-covariance symmetry and
asymmetry, and we use likelihood-based methods for inference that are
computationally efficient. The utility of this new class of models is shown
through two data illustrations: a simulation study on nonstationary data and an
application on ocean temperatures at two different depths.
