The most prevalent spatial data setting is, arguably, that of so-called geostatistical data, data that arise as random variables observed at fixed spatial locations. Collection of such data in space and in time has grown enormously in the past two decades. With it has grown a substantial array of methods to analyze such data. Here, we attempt a review of a fully model-based perspective for such data analysis, the approach of hierarchical modeling fitted within a Bayesian framework. The benefit, as with hierarchical Bayesian modeling in general, is full and exact inference, with proper assessment of uncertainty. Geostatistical modeling includes univariate and multivariate data collection at sites, continuous and categorical data at sites, static and dynamic data at sites, and datasets over very large numbers of sites and long periods of time. Within the hierarchical modeling framework, we offer a review of the current state of the art in these settings.
Keywords: Gaussian processes; Markov chain Monte Carlo; big spatial data; data assimilation; data fusion; integrated nested Laplace approximation; multivariate spatial processes; spatiotemporal processes.