Bayesian Inference in Gaussian Model-based Geostatistics

In a geostatistical analysis, spatial interpolation or smoothing of the observed values is often carried out by a procedure known as kriging. In its basic form, kriging involves the construction of a linear predictor for an unobserved value of the process, and the form of this linear predictor is chosen with reference to the covariance structure of the data as estimated by a data-analytic tool known as the variogram. Often, no explicit underlying stochastic model is declared. We adopt a model-based approach to this class of problems, by which we mean that we start with an explicit stochastic model and derive associated methods of parameter estimation, interpolation and smoothing by the application of general statistical principles. In particular, we use Bayesian methods of inference so as to make proper allowance for the uncertainty associated with estimating the unknown values of model parameters. To illustrate the model-based approach we analyse data on precipitation levels in Paraná State, Brazil.