Bayesian Dix inversion

Abstract

We have developed a Bayesian method for Dix inversion and illustrated it with examples from the North Sea. The method is a constrained Dix inversion in which the uncertainty of the estimated interval velocities is an integral part of the solution. The method combines available geologic prior knowledge with the information in the picked rms velocities so that the prior model stabilizes and constrains the inversion. The definition of layers or intervals is flexible. One possibility is the classical layering whereby the top and bottom of the intervals are defined directly from the times of the picked rms velocities, but the intervals can be defined arbitrarily and independent of the picked velocities. Even a pseudocontinuous velocity profile with dense time sampling can be predicted with corresponding uncertainty. The Dix inversion is conventionally formulated as a 1D inversion problem in the locations with picked rms velocities. Under the assumption that Dix inversion is adequate, we have defined a method for spatial prediction at any position in a 3D model. This involves spatial smoothing and interpolation, and provides a method for building a velocity cube with the corresponding uncertainty on any grid. Explicit analytic expressions were found for both the optimal solution and the uncertainty. In general, the uncertainty of the estimated interval velocities increases with depth and with decreasing layer thickness. The uncertainty of the spatial prediction decreases with increasing spatial correlation. Because the solution is of an explicit analytic form, the Bayesian Dix inversion is computationally fast and does not require stochastic simulation.