Linearized Seismic AVO Inversion: A Variational Bayesian Learning Method

AVO (amplitude-varying-with-offset) is a seismic prestack inversion technique for estimating elastic parameters of the subsurface. The seismic AVO inversion problem can be formulated as a linearized or nonlinear inverse problem. It is a multidimensional and highly ill-posed inverse problem. It is affected by strong noise and measurement uncertainty. Therefore, in seismic AVO analysis, the goal is not only to find the model that best-fits the data, but also to characterize the uncertainty of the analysis. Uncertainty characterization of seismic analysis makes the geophysical interpretation more reliable. The Bayesian formalism generally constitutes a powerful approach for solving many seismic inverse problems. It allows combining available prior knowledge with the information contained in the seismic data. The solution of a Bayesian inverse problem is given by the joint posterior distribution of model parameters. Stochastic approximations of the true posterior are exemplified by the MCMC method and more recently by the particle filtering (PF) method. For most seismic inverse problems, stochastic approximation techniques are computationally unappealing. The maximum a posteriori (MAP) is the simplest Bayesian method for estimating model parameters. It provides the explanation that maximizes the posterior distribution. The maximum likelihood (ML) is a popular non-Bayesian method for estimating model parameters. It provides the explanation that maximizes the likelihood of the data. It assumes that the model parameters are deterministic and omits the pertinent prior knowledge on them. In practice, both MAP and ML point estimators have severe problems with over-fitting the data and model order estimation. In this paper, the recently-developed variational Bayesian learning is proposed as a new method for solving the linearized seismic AVO inversion problem. Variational Bayesian learning allows the true joint posterior distribution of model parameters to be approximated by a simpler approximating ensemble for which the required inferences are tractable. The main advantage in resorting to variational Bayesian learning is its robustness to the over-fitting problem, which is major in practice. This probabilistic machine learning method allows finding an optimum balance between the representation power and the model complexity of the data.