  Spatial processes with nonstationary and anisotropic covariance structure are
often used when modelling, analysing and predicting complex environmental
phenomena. Such processes may often be expressed as ones that have stationary
and isotropic covariance structure on a warped spatial domain. However, the
warping function is generally difficult to fit and not constrained to be
injective, often resulting in `space-folding.' Here, we propose modelling an
injective warping function through a composition of multiple elemental
injective functions in a deep-learning framework. We consider two cases; first,
when these functions are known up to some weights that need to be estimated,
and, second, when the weights in each layer are random. Inspired by recent
methodological and technological advances in deep learning and deep Gaussian
processes, we employ approximate Bayesian methods to make inference with these
models using graphics processing units. Through simulation studies in one and
two dimensions we show that the deep compositional spatial models are quick to
fit, and are able to provide better predictions and uncertainty quantification
than other deep stochastic models of similar complexity. We also show their
remarkable capacity to model nonstationary, anisotropic spatial data using
radiances from the MODIS instrument aboard the Aqua satellite.
