Soil pollution data collection typically studies multivariate measurements at sampling locations, e.g., lead, zinc, copper
or cadmium levels. With increased collection of such multivariate geostatistical spatial data, there arises the need for flexible
explanatory stochastic models. Here, we propose a general constructive approach for building suitable models based upon convolution
of covariance functions. We begin with a general theorem which asserts that, under weak conditions, cross convolution of covariance
functions provides a valid cross covariance function. We also obtain a result on dependence induced by such convolution. Since,
in general, convolution does not provide closed-form integration, we discuss efficient computation.
We then suggest introducing such specification through a Gaussian process to model multivariate spatial random effects within
a hierarchical model. We note that modeling spatial random effects in this way is parsimonious relative to say, the linear
model of coregionalization. Through a limited simulation, we informally demonstrate that performance for these two specifications
appears to be indistinguishable, encouraging the parsimonious choice. Finally, we use the convolved covariance model to analyze
a trivariate pollution dataset from California.
Keywords convolution - coregionalization - Fourier transforms - Gaussian spatial process - hierarchical model - Markov chain Monte Carlo - spectral density