Bayesian Wavelet Regression for Spatial Estimation

We consider the problem of estimating the properties of an oil reservoir, like porosity and sand thickness, in an exploration scenario where only a few wells have been drilled. We use gamma ray records measured directly from the wells as well as seismic traces recorded around the wells. To model the association between the soil properties and the signals, we fit a linear regression model. Additionally we account for the spatial correla- tion structure of the observations using a correlation function that depends on the distance between two points. We transform the predictor variable using discrete wavelets and then perform a Bayesian variable selection us- ing a Metropolis search. We obtain predictions of the properties over the whole reservoir providing a probabilistic quantification of their uncertainties, thanks to the Bayesian nature of our method. The cross-validated results show that a very high accuracy can be achieved even with a very small number of wavelet coefficients.