  In many environmental applications involving spatially-referenced data,
limitations on the number and locations of observations motivate the need for
practical and efficient models for spatial interpolation, or kriging. A key
component of models for continuously-indexed spatial data is the covariance
function, which is traditionally assumed to belong to a parametric class of
stationary models. However, stationarity is rarely a realistic assumption.
Alternative methods which more appropriately model the nonstationarity present
in environmental processes often involve high-dimensional parameter spaces,
which lead to difficulties in model fitting and interpretability. To overcome
this issue, we build on the growing literature of covariate-driven
nonstationary spatial modeling. Using process convolution techniques, we
propose a Bayesian model for continuously-indexed spatial data based on a
flexible parametric covariance regression structure for a convolution-kernel
covariance matrix. The resulting model is a parsimonious representation of the
kernel process, and we explore properties of the implied model, including a
description of the resulting nonstationary covariance function and the
interpretational benefits in the kernel parameters. Furthermore, we demonstrate
that our model provides a practical compromise between stationary and highly
parameterized nonstationary spatial covariance functions that do not perform
well in practice. We illustrate our approach through an analysis of annual
precipitation data.
