Bayesian trans-gaussian kriging with log-log transformed skew data

Abstract

Besides the assumption of stationarity it is conventional among geostatistical practitioners to make the assumption of Gaussianity when applying the linear kriging methodology. The standard procedure in geostatistics is to estimate an empirical variogram or covariance function, fit a theoretical variogram or covariance model to the empirical estimate by means of least squares, apply linear kriging to the data by plugging in the estimated positive semi-definite theoretical covariance model, report uncertainties of the predictions by means of the plug-in kriging variance and Gaussian-based confidence intervals. The above approach has certain deficiencies: The empirical variogram or covariance estimates at the different lags are highly correlated and, therefore, can be very misleading. Once the empirical variogram is badly estimated, the same is also true for the fitted theoretical model. Plugging the estimated theoretical covariance model in the kriging predictor and neglecting the uncertainty of the covariance estimate has the following consequences: the plug-in kriging predictor is neither a linear nor best predictor. the kriging variance that is based on a plug-in estimate is not the true variance of this highly non-linear plug-in kriging predictor and actually underestimates the true unknown variance [5]. In hardly any application the assumption of Gaussianity is fulfilled. Most environmental processes like ore grades, soil and air measurements show highly right-skewed marginal distributions. All these negative facts were the motivation for us to look for more advanced kriging methodologies that relax the Gaussian assumption and the disadvantage of not taking into account the uncertainty of the covariance function. One of the first papers that addressed the above deficiencies and influenced our work was De Oliveira [15]. He developed a Bayesian trans-Gaussian kriging where uncertainties could be specified on the trend, the covariance function and the parameter of the transformation function. The transformation he used to make a skew random field Gaussian was the Box-Cox transformation. Motivated by the conjugacy of the normal-inverse-gamma family to the normal sampling distribution he exactly used such kind of prior for the sill, the range and the trend parameters of his model. His approach distinguishes from our approach by the fact that his prior is informative. Berger et al. [3] were the first to investigate also non-informative reference priors for correlated stochastic processes. The disadvantage of their approach is that the assumed random field must be Gaussian. Some time later the paper [16] appeared where also a non-informative prior for the transformed Gaussian model was proposed, but with fixed range parameter while the sill and the nugget are variable. Our approach seems to be the first completely non-informative approach to trans-Gaussian Bayesian kriging. We avoid the specification of a prior distribution by means of a parametric bootstrap, where the sampling distribution of maximum likelihood estimates is taken as the posterior for unknown parameters. © 2009 Springer Berlin Heidelberg.