  Multivariate geostatistics is based on modelling all covariances between all
possible combinations of two or more variables at any sets of locations in a
continuously indexed domain. Multivariate spatial covariance models need to be
built with care, since any covariance matrix that is derived from such a model
must be nonnegative-definite. In this article, we develop a conditional
approach for spatial-model construction whose validity conditions are easy to
check. We start with bivariate spatial covariance models and go on to
demonstrate the approach's connection to multivariate models defined by
networks of spatial variables. In some circumstances, such as modelling
respiratory illness conditional on air pollution, the direction of conditional
dependence is clear. When it is not, the two directional models can be
compared. More generally, the graph structure of the network reduces the number
of possible models to compare. Model selection then amounts to finding possible
causative links in the network. We demonstrate our conditional approach on
surface temperature and pressure data, where the role of the two variables is
seen to be asymmetric.
