Simultaneous Estimation of Reflectivity and Geologic Texture: Least-Squares Migration with a Hierarchical Bayesian Model

Abstract

In many geophysical inverse problems, smoothness assumptions on the underlying geology are utilized to mitigate the effects of poor resolution and noise in the data and to improve the quality of the inferred model parameters. Within a Bayesian inference framework, a priori assumptions about the probabilistic structure of the model parameters impose such a smoothness constraint or regularization. We consider the particular problem of inverting seismic data for the subsurface reflectivity of a 2-D medium, where we assume a known velocity field. In particular, we consider a hierarchical Bayesian generalization of the Kirchhoff-based least-squares migration (LSM) problem. We present here a novel methodology for estimation of both the optimal image and regularization parameters in a least-squares migration setting. To do so we utilize a Bayesian statistical framework that treats both the regularization parameters and image parameters as random variables to be inferred from the data. Hence rather than fixing the regularization parameters prior to inverting for the image, we allow the data to dictate where to regularize. In order to construct our prior model of the subsurface and regularization parameters, we define an undirected graphical model (or Markov random field) where vertices represent reflectivity values, and edges between vertices model the degree of correlation (or lack thereof) between the vertices. Estimating optimal values for the vertex parameters gives us an image of the subsurface reflectivity, while estimating optimal edge strengths gives us information about the local “texture” of the image, which, in turn, may tell us something about the underlying geology. Subsequently incorporating this information in the final model produces more clearly visible discontinuities in the final image. The inference framework is verified on a 2-D synthetic dataset, where the hierarchical Bayesian imaging results significantly outperform standard LSM images.