[Paper Review] Wave-equation-based inversion with amortized variational Bayesian inference

Ali Siahkoohi, Rafael Orozco+2 more|arXiv (Cornell University)|Mar 29, 2022

Seismic Imaging and Inversion Techniques4 citations

TL;DR

This paper proposes a physics-informed seismic imaging method that uses a pretrained conditional normalizing flow (NF) to learn a data-driven prior from low- and high-fidelity seismic image pairs. By reparameterizing the unknown model via the NF's invertible mapping and optimizing a physics-guided data misfit with a Gaussian prior on the latent space, the approach produces high-quality, artifact-minimized images even under noisy or out-of-distribution conditions.

ABSTRACT

Solving inverse problems involving measurement noise and modeling errors requires regularization in order to avoid data overfit. Geophysical inverse problems, in which the Earth's highly heterogeneous structure is unknown, present a challenge in encoding prior knowledge through analytical expressions. Our main contribution is a generative-model-based regularization approach, robust to out-of-distribution data, which exploits the prior knowledge embedded in existing data and model pairs. Utilizing an amortized variational inference objective, a conditional normalizing flow (NF) is pretrained on pairs of low- and high-fidelity migrated images in order to achieve a low-fidelity approximation to the seismic imaging posterior distribution for previously unseen data. The NF is used after pretraining to reparameterize the unknown seismic image in an inversion scheme involving physics-guided data misfit and a Gaussian prior on the NF latent variable. Solving this optimization problem with respect to the latent variable enables us to leverage the benefits of data-driven conditional priors whilst being informed by physics and data. The numerical experiments demonstrate that the proposed inversion scheme produces seismic images with limited artifacts when dealing with noisy and out-of-distribution data.

Motivation & Objective

• Address the challenge of incorporating realistic prior knowledge in geophysical inverse problems where analytical priors may introduce bias.
• Overcome limitations of traditional regularization in ill-posed inverse problems involving noisy data and modeling errors.
• Leverage existing high-fidelity seismic data to learn a conditional prior that generalizes to unseen, out-of-distribution data.
• Integrate data-driven priors with physics-based inversion to maintain interpretability while improving image quality.
• Develop a scalable, amortized inference framework that avoids solving variational inference afresh for each new data instance.

Proposed method

• Train a conditional normalizing flow (NF) on paired low- and high-fidelity seismic images using an amortized variational inference objective.
• Use a block-triangular NF architecture to model the conditional distribution $ p(\mathbf{x} \mid \mathbf{y}) $, where $ \mathbf{y} $ is low-fidelity data and $ \mathbf{x} $ is the high-fidelity image.
• Minimize the Kullback-Leibler divergence between the NF output and a standard normal latent distribution via a differentiable objective involving log-likelihood and Jacobian determinant regularization.
• Reparameterize the unknown seismic model $ \delta\mathbf{m} $ as $ \delta\mathbf{m} = T_{\mathrm{w}_2}^{-1}(T_{\mathrm{w}_1}(\delta\mathbf{m}_{\text{RTM}}), \mathbf{z}) $, where $ \mathbf{z} $ is a latent variable.
• Solve the inversion by optimizing the maximum a posteriori (MAP) estimate over the latent variable $ \mathbf{z} $, minimizing a physics-based data misfit and a Gaussian prior on $ \mathbf{z} $.
• Initialize optimization at $ \mathbf{z}_0 = \mathbf{0} $, leveraging the prior knowledge embedded in the NF to improve convergence and reduce artifacts.

Experimental results

Research questions

• RQ1 Can a data-driven conditional prior learned from seismic image pairs improve inversion robustness under noise and distribution shift?
• RQ2 How does combining a conditional normalizing flow prior with physics-based data misfit enhance image quality compared to standard reverse-time migration?
• RQ3 To what extent does amortized variational inference with NFs enable efficient, generalizable inference for unseen seismic data?
• RQ4 Can the invertibility of NFs mitigate bias in out-of-distribution scenarios where high-fidelity training data is scarce?
• RQ5 What is the impact of using a Gaussian prior on the latent space in conjunction with a learned conditional prior on the final inversion outcome?

Key findings

• The proposed method successfully reconstructs seismic reflectors with limited artifacts in a deep, noisy section of the Parihaka dataset, which was not part of the training distribution.
• The initial guess, derived from the NF and RTM image, shows improved amplitudes and structure compared to standard reverse-time migration, indicating better initialization.
• The final MAP estimate, obtained after 5 passes over shot records (equivalent to ~5 RTMs), recovers most subsurface reflectors with high fidelity despite low signal-to-noise ratio.
• The use of a Gaussian prior on the latent variable $ \mathbf{z} $, combined with the NF reparameterization, regularizes the inversion and prevents overfitting to noisy data.
• The method generalizes well to out-of-distribution data, as demonstrated on deeper, more heterogeneous sections of the Parihaka dataset not seen during training.
• The framework is computationally efficient due to amortized inference, avoiding per-instance variational inference and enabling fast deployment on new data.