  Multivariate spatial fields are of interest in many applications, including
climate model emulation. Not only can the marginal spatial fields be subject to
nonstationarity, but the dependence structure among the marginal fields and
between the fields might also differ substantially. Extending a recently
proposed Bayesian approach to describe the distribution of a nonstationary
univariate spatial field using a triangular transport map, we cast the
inference problem for a multivariate spatial field for a small number of
replicates into a series of independent Gaussian process (GP) regression tasks
with Gaussian errors. Due to the potential nonlinearity in the conditional
means, the joint distribution modeled can be non-Gaussian. The resulting
nonparametric Bayesian methodology scales well to high-dimensional spatial
fields. It is especially useful when only a few training samples are available,
because it employs regularization priors and quantifies uncertainty. Inference
is conducted in an empirical Bayes setting by a highly scalable stochastic
gradient approach. The implementation benefits from mini-batching and could be
accelerated with parallel computing. We illustrate the extended transport-map
model by studying hydrological variables from non-Gaussian climate-model
output.
