  The analysis of area-level aggregated summary data is common in many
disciplines including epidemiology and the social sciences. Typically, Markov
random field spatial models have been employed to acknowledge spatial
dependence and allow data-driven smoothing. In this paper, we exploit recent
theoretical and computational advances in continuous spatial modeling to carry
out the reconstruction of an underlying continuous spatial surface. In
particular, we focus on models based on stochastic partial differential
equations (SPDEs). We also consider the interesting case in which the aggregate
data are supplemented with point data. We carry out Bayesian inference, and in
the language of generalized linear mixed models, if the link is linear, an
efficient implementation of the model is available via integrated nested
Laplace approximations. For nonlinear links, we present two approaches: a fully
Bayesian implementation using a Hamiltonian Monte Carlo algorithm, and an
empirical Bayes implementation, that is much faster, and is based on Laplace
approximations. We examine the properties of the approach using simulation, and
then estimate an underlying continuous risk surface for the classic Scottish
lip cancer data.
