Summary: When modelling continuous regional data, it is common to model the regional variables themselves. For example, in geographic modelling, it is usual to specify the inverse covariance matrix in a simple form. However, in many situations it may be plausible to model an underlying continuous-space process. In order to postulate reasonable models, the behaviour of the derived regional process needs to be known. For exact Gaussian maximum likelihood estimation the covariance structure of the derived process may be needed for a large number of parameter values of the underlying process, and so may be computationally infeasible. This paper considers approximate relationships between the original and derived covariance structures for irregular regions. The derived covariance structures are compared with some geographic models in common use, and alternative models are postulated.