Generative AI and Bayesian methods for seismic imaging and uncertainty estimation

An inverse problem involves deducing the causes from observed effects within a system, often requiring the solution of partial differential equations that describe the underlying physics. In geophysical seismic imaging, the objective is to reconstruct subsurface structures, such as velocity and density fields, by analyzing seismic waveforms recorded at the Earth’s surface. This process involves solving the non-linear wave equation to model seismic wave propagation through the Earth. Full Waveform Inversion (FWI) is a deterministic technique that employs gradient-based methods. Despite its potential, FWI faces challenges such as non-uniqueness, local minima, and computational complexity, highlighting the critical need for advanced methods to address these issues and quantify uncertainties in subsurface property estimation.

Bayesian inference provides a robust framework for solving inverse problems and estimating uncertainties by applying Bayes' theorem. This approach derives a posterior probability density function for model parameters based on observed data. In this study, we present an innovative method that parametrizes unknowns using Generative Adversarial Networks (GANs), enabling the creation of realistic subsurface representations by learning the prior distribution in a latent space. Once trained, the GAN remains fixed, serving as a generative prior for Bayesian posterior sampling.

We compare and evaluate four posterior sampling methods, i.e. the Metropolis-adjusted Langevin Algorithm (MALA), variational Bayesian inference using normalizing flows (NF), inference neural networks (INN), and Stein Variational Gradient Descent (SVGD). The performance of these methods is assessed in terms of computational efficiency and accuracy in capturing the posterior distribution. By integrating deep generative priors with advanced Bayesian sampling techniques, we demonstrate significant improvements in handling the high dimensionality and non-linearity inherent in geophysical inverse problems. This work contributes to the development of advanced methods for seismic imaging and uncertainty quantification, aligning with the need for robust, data-driven approaches in the field of geophysics.