Monte-Carlo Bayesian look-ahead inversion of walkaway vertical seismic profiles

We describe a method to invert a walkaway vertical seismic profile (VSP) and predict elastic properties (P-wave velocity, S-wave velocity and density) in a layered model looking ahead of the deepest receiver. Starting from Bayes's rule, we define a posterior distribution of layered models that combines prior information (on the overall variability of and correlations among the elastic properties observed in well logs) with information provided by the VSP data. This posterior distribution of layered models is sampled by a Monte-Carlo method. The sampled layered models agree with prior information and fit the VSP data, and their overall variability defines the uncertainty in the predicted elastic properties. We apply this technique first to a zero-offset VSP data set, and show that uncertainty in the long-wavelength P-wave velocity structure results in a sizable uncertainty in the predicted elastic properties. We then use walkaway VSP data, which contain information on the long-wavelength P-wave velocity (in the reflection moveout) and on S-wave velocity and density contrasts (in the change of reflectivity with offset). The uncertainty of the look-ahead prediction is considerably decreased compared with the zero-offset VSP, and the predicted elastic properties are in good agreement with well-log measurements.