Multivariate spatial modeling of conditional dependence in microscale soil elemental composition data

The mobility and environmental impacts of toxic trace elements are regulated by their reactions with soils, which are complex heterogeneous mixtures of minerals and organic matter. We describe an experiment that maps the composition of elements on an individual soil sand grain using X-ray fluorescence microprobe analyses, after the grain is treated with arsenic solutions, resulting in multivariate spatial lattice maps of elemental abundance. To understand the behavior of arsenic in soils, it is important to disentangle the complex multivariate relationships among the elements in the sample. The abundance of most elements, including arsenic, correlates strongly with that of iron; but conditional on the amount of iron, some elements mitigate or potentiate the accumulation of arsenic. This problem motivates our work to define conditional correlation in spatial lattice models and give general conditions under which two components are conditionally uncorrelated given the rest. We describe how to enforce that two components are conditionally uncorrelated given a third in parametric models, which provides a basis for likelihood ratio tests for conditional correlation between arsenic and chromium given iron. We show how to apply our results to big datasets using the Whittle likelihood, and we demonstrate through simulation that tapering improves Whittle likelihood parameter estimates governing cross covariance.