Acoustic Waveform Inversion with Image-to-Image Schr\"odinger Bridges

Abstract

Recent developments in application of deep learning models to acoustic Full Waveform Inversion (FWI) are marked by the use of diffusion models as prior distributions for Bayesian-like inference procedures. The advantage of these methods is the ability to generate high-resolution samples, which are otherwise unattainable with classical inversion methods or other deep learning-based solutions. However, the iterative and stochastic nature of sampling from diffusion models along with heuristic nature of output control remain limiting factors for their applicability. For instance, an optimal way to include the approximate velocity model into diffusion-based inversion scheme remains unclear, even though it is considered an essential part of FWI pipeline. We address the issue by employing a Schr\"odinger Bridge that interpolates between the distributions of ground truth and smoothed velocity models. To facilitate the learning of nonlinear drifts that transfer samples between distributions we extend the concept of Image-to-Image Schr\"odinger Bridge ($\text{I}^2\text{SB}$) to conditional sampling, resulting in a conditional Image-to-Image Schr\"odinger Bridge (c$\text{I}^2\text{SB}$) framework. To validate our method, we assess its effectiveness in reconstructing the reference velocity model from its smoothed approximation, coupled with the observed seismic signal of fixed shape. Our experiments demonstrate that the proposed solution outperforms our reimplementation of conditional diffusion model suggested in earlier works, while requiring only a few neural function evaluations (NFEs) to achieve sample fidelity superior to that attained with supervised learning-based approach. The supplementary code implementing the algorithms described in this paper can be found in the repository https://github.com/stankevich-mipt/seismic_inversion_via_I2SB.