Generalization of Bayesian inversion with application to PP reflection in a vertical transverse isotropy medium

Abstract

ABSTRACTSeismic inversion of amplitude variation with offset is an ill‐conditioned problem, in which small errors in the observed data result in large errors in the estimates, and therefore regularization functions are necessary. The Bayes' theorem regularizes ill‐posed problems using the statistical properties of the model parameters of interest; however, our knowledge of these statistical properties is poor. Meanwhile, the Bayesian framework is limited to particular forms of a priori probability distributions, due to poor knowledge of the statistical properties of individual probability distributions. Moreover, each particular a priori probability distribution requires a reformulation of the inversion under a Bayesian framework, which is not practically preferred. Here, we construct a Bayesian framework that enables the use of various types of a priori probability distributions without the need to reformulate the problem and to obtain well‐established statistical information on the model parameters of interest, specifically a lower bound on the variance of the a priori model, for example. Fisher information. Consequently, different probability distributions that best address the parameters of interest are first fitted using the maximum likelihood estimator . A lower bound on the variance of each a priori model is then estimated, which provides adequate statistical information between different model parameters; hence, it is ideally suited for the Bayesian framework. Thereafter, an iterative approach is proposed that utilizes the Hessians of the data and model spaces, and an adaptive learning rate to compute the optimal directions for model updates. The approach is applied to synthetic and real seismic data to estimate the elastic and seismic anisotropy parameters of shale formations. The regularization from the a priori probability density successfully stabilizes the updates of the variant‐sensitivity parameters by imposing the correlation information in the optimization process.