Subspace Diffusion Posterior Sampling for Travel-Time Tomography

TL;DR

This work proposes a posterior sampling process for PDE-based inverse problems by solving the associated adjoint-state equation in a plug-and-play fashion and presents a subspace-based dimension reduction technique, enabling solving PDE-based inverse problems from coarse to refined grids, for conditional sampling acceleration.

Abstract

Diffusion models have been widely studied as effective generative tools for solving inverse problems. The main ideas focus on performing the reverse sampling process conditioned on noisy measurements, using well-established numerical solvers for gradient updates. Although diffusion-based sampling methods can produce high-quality reconstructions, challenges persist in nonlinear PDE-based inverse problems and sampling speed. In this work, we explore solving PDE-based travel-time tomography based on subspace diffusion generative models. Our main contributions are twofold: First, we propose a posterior sampling process for PDE-based inverse problems by solving the associated adjoint-state equation. Second, we resorted to the subspace-based dimension reduction technique for conditional sampling acceleration, enabling solving the PDE-based inverse problems from coarse to refined grids. Our numerical experiments showed satisfactory advancements in improving the travel-time imaging quality and reducing the sampling time for reconstruction.

Figures (9)

• Figure 1: The Subspace-DPS framework consists of two fundamental components: the local adjoint-state solver for PDE-based gradient update and the multi-scale decomposition based on downsampling. An extra denoising step is performed for the reconstruction provided by the conditional sampling.
• Figure 2: (a) The Marmousi dataset was collected by randomly extracting 128x128-sized sub-images from geological structure simulation images. (b) The KIT4 dataset was generated by randomly placing non-overlapping geometric shapes within a 128x128 area and assigning different speed field values to distinct regions.
• Figure 3: There are three geometries for placing the source-receiver pairs: (a)(b) For the Marmousi dataset, they are arranged in horizontal and vertical patterns; (c) For the KIT4 dataset, they are arranged in a circular pattern. As illustrated, the red pentagrams represent the locations of the sources, and the white triangles represent the locations of the receivers.
• Figure 4: $D_F(\mathbf{U}_{k \mid k-1}; t)$ plots for two datasets are computed with respect to the trained full-space score model ${\boldsymbol s}_0({\boldsymbol x}_t, t)$, where the horizontal axis represents the range of the sampling time, while the vertical axis indicates the values of the Orthogonal Fisher divergence. For a given divergence threshold $10^{-3}$, the optimal downsampling times $t_k$ for any subspace sequence are identified as the times when the corresponding divergences reach that threshold.
• Figure 5: The ray paths travel from each transmitter to the receivers, following the route that minimizes the travel time. (a) In the horizontal geometry, the ray paths are prone to traverse the high-speed fault region to reach the receivers. (b) In the vertical geometry, the ray paths encounter different faults with different speed values. (c) In the surrounding pattern, the rays tend to bypass the subregions with lower speed values and instead pass through higher-speed areas.

Theorems & Definitions (8)

• theorem 1
• theorem 2
• corollary 1
• theorem 3
• theorem 3
• theorem 3
• corollary 1
• theorem 3