Stochastic full waveform inversion with deep generative prior for uncertainty quantification

TL;DR

The paper tackles uncertainty quantification in seismic Full Waveform Inversion (FWI) by embedding a deep generative prior into a Bayesian framework. It couples the adjoint-state gradient for efficient PDE-based learning with four variational/inference approaches—explicit density via normalizing flows, implicit inference networks, SVGD, and conventional MCMC (MALA)—to approximate the posterior over a latent space given data $d_{obs}$. A GAN trained on a 3D Overthrust dataset provides the prior over velocity fields, enabling stochastic realizations conditioned on observations while accounting for inherent uncertainties. The results show that SVGD and MALA yield richer, multimodal posterior variability, whereas neural-network-based methods offer continuous posteriors with unlimited samples; collectively, the framework advances uncertainty-aware seismic inversion by fusing physics-based solving with deep generative priors.

Abstract

To obtain high-resolution images of subsurface structures from seismic data, seismic imaging techniques such as Full Waveform Inversion (FWI) serve as crucial tools. However, FWI involves solving a nonlinear and often non-unique inverse problem, presenting challenges such as local minima trapping and inadequate handling of inherent uncertainties. In addressing these challenges, we propose leveraging deep generative models as the prior distribution of geophysical parameters for stochastic Bayesian inversion. This approach integrates the adjoint state gradient for efficient back-propagation from the numerical solution of partial differential equations. Additionally, we introduce explicit and implicit variational Bayesian inference methods. The explicit method computes variational distribution density using a normalizing flow-based neural network, enabling computation of the Bayesian posterior of parameters. Conversely, the implicit method employs an inference network attached to a pretrained generative model to estimate density, incorporating an entropy estimator. Furthermore, we also experimented with the Stein Variational Gradient Descent (SVGD) method as another variational inference technique, using particles. We compare these variational Bayesian inference methods with conventional Markov chain Monte Carlo (McMC) sampling. Each method is able to quantify uncertainties and to generate seismic data-conditioned realizations of subsurface geophysical parameters. This framework provides insights into subsurface structures while accounting for inherent uncertainties.

Figures (13)

• Figure 1: Diagram illustrating Full Waveform Inversion with Parameterization, we parameterize the subsurface velocity field $\textbf{m} = G_\theta(\mathbf{z})$, and we solve acoustic wave equation to predict the synthetic seismic data $d_\text{pred}$. By evaluating the data residual with observed seismic data $d_\text{obs}$, we obtain the gradient using adjoint state method and subsequently update model parameters using optimization algorithms.
• Figure 2: Overview of the dataset and generated images in this work. a 3D Overthrust dataset with dimensions $94\times401\times401$ and b training images cropped from the dataset, with dimensions $40\times120$. c Training images and d generated fake images from GAN.
• Figure 3: Correlation between grids of the GAN-generated images. The cross indicates the grid point used to compute the correlation.
• Figure 4: Minimizing KL Divergence $\text{KL}\left[q_{\phi}(\mathbf{z}) \, \| \, p(\mathbf{z}|d_{\text{obs}})\right]$ using gradient descent with the TensorFlow Adam optimizer to approximate a Gaussian mixture in 1D space. This is an illustration explaining the process of minimizing KL Divergence in variational Bayesian inference. The left figure shows distributions at the first epoch, and the right figure shows the approximation result.
• Figure 5: Structure of the implicit inference method using inference network