MODEL-BASED GEOSTATISTICS

Conventional geostatistical methodology solves the problem of predicti ng the realized value of a linear functional of a Gaussian spatial sto chastic process S(x) based on observations Y-i = S(x(i))+Z(i) at sampl ing locations x(i), where the Z(i) are mutually independent, zero-mean Gaussian random variables. We describe two spatial applications for w hich Gaussian distributional assumptions are clearly inappropriate. Th e first concerns the assessment of residual contamination from nuclear weapons testing on a South Pacific island, in which the sampling meth od generates spatially indexed Poisson counts conditional on an unobse rved spatially varying intensity of radioactivity; we conclude that a conventional geostatistical analysis oversmooths the data and underest imates the spatial extremes of the intensity. The second application p rovides a description of spatial variation in the risk of campylobacte r infections relative to other enteric infections in part of north Lan cashire and south Cumbria. For this application, we treat the data as binomial counts at unit postcode locations, conditionally on an unobse rved relative risk surface which we estimate. The theoretical framewor k for our extension of geostatistical methods is that, conditionally o n the unobserved process S(x), observations at sample locations x(i) f orm a generalized linear model with the corresponding values of S(x(i) ) appearing as an offset term in the linear predictor. We use a Bayesi an inferential framework, implemented via the Markov chain Monte Carte method, to solve the prediction problem for non-linear functionals of S(x), making a proper allowance for the uncertainty in the estimation of any model parameters.