Bayesian spatial inversion and conjugate selection Gaussian prior models

Abstract

We study conjugate prior models in Bayesian spatial inversion. The spatial Kriging model may be phrased in a conjugate Bayesian inversion setting with a Gaussian prior model and a Gauss-linear likelihood function, resulting in a Gaussian posterior model. Spatial variables with unimodal, symmetric spatial histograms can be represented by this Kriging model. We generalize this Gaussian prior model by a selection mechanism, and this selection Gaussian prior model may represent multimodal, skewed, and/or peaked spatial variables. Also this selection Gaussian prior model is conjugate with respect to Gauss-linear likelihood functions. Hence the posterior model is selection Gaussian and analytically tractable. Efficient algorithms for simulation of and prediction in the selection Gaussian posterior model are defined. Model parameter inference in a maximum likelihood setting, which is simplified by the conjugate property, is also discussed. Moreover, we demonstrate that any conjugate prior model can be generalized by selection and still remain conjugate with respect to the actual likelihood function. Lastly, a seismic inversion case study is presented, and improvements of 20--40% in prediction mean-square-error, relative to traditional Gaussian inversion, are found.