WaveDiffusion: Joint Latent Diffusion for Physically Consistent Seismic and Velocity Generation

TL;DR

The paper tackles physically consistent generation of seismic and velocity data by reframing FWI as a generative task with a shared latent space. It introduces WaveDiffusion, a two-stage framework: Stage 1 builds a vector-quantized encoder–decoder that jointly reconstructs seismic signals and velocity maps, and Stage 2 applies a latent diffusion model to refine latent codes toward solutions that approximately satisfy the acoustic wave equation $\frac{\partial^2 s(\mathbf{x}, t)}{\partial t^2} = v^2(\mathbf{x}) \nabla^2 s(\mathbf{x}, t) + S(\mathbf{x_s}, t)$. The authors demonstrate that diffusion progressively moves arbitrary latent points toward PDE-consistent outputs, enabling the generation of high-fidelity, diverse seismic–velocity pairs on OpenFWI benchmarks. They also show that these generated pairs can augment training for supervised FWI models, improving performance in low-data regimes and when fine-tuning on challenging samples. While no formal PDE guarantees are provided, the empirical results highlight diffusion as an effective mechanism to bias latent generation toward physics-consistent regions, bridging generative modeling with physics-based problem-solving."

Abstract

Full Waveform Inversion (FWI) is a critical technique in subsurface imaging, aiming to reconstruct high-resolution subsurface properties from surface measurements. Acoustic FWI involves two physical modalities, seismic waveforms and velocity maps, which are governed by the acoustic wave equation. Prior works primarily focus on the inverse problem, modeling the relationship between seismic and velocity as an image-to-image translation task. In this work, we study their relationship from a generative perspective. Our aim is to explore and characterize the latent space structure, and identify latent vectors that generate seismic-velocity pairs consistent with the governing partial differential equation (PDE). Specifically, we model seismic and velocity data jointly from a shared latent space via a diffusion process. In experiments, we find that diffusion progressively refines arbitrary latent vectors into ones that yield approximately physics-consistent seismic-velocity pairs, even without explicit physics constraints. This provides empirical evidence of PDE-consistency in latent diffusion, where sampling is biased toward PDE-valid solutions. In latent space, satisfying the acoustic wave equation can be approximated through sampling and gradient descent. We formalize this physics-consistent latent modeling task and quantify it through extensive experiments. On large-scale OpenFWI benchmarks, our approach produces high-fidelity, diverse, and physically consistent seismic-velocity pairs, demonstrating the potential of a data-driven latent diffusion for physically consistent generation in a complex scientific domain.

Figures (15)

• Figure 1: Overview of WaveDiffusion. WaveDiffusion generates paired samples that satisfy the governing PDE via a joint latent diffusion framework. The diffusion guides the sampling process toward physically valid solutions through denoising, progressively mapping non-solution points (gray squares in valleys) to valid solutions (colored stars at peaks).
• Figure 2: Comparison of inversion performance. Mean Absolute Error (MAE) across various OpenFWI datasets for encoder-decoder-based models and diffusion refinements. Our models perform competitively against BigFWI-B, with diffusion providing slight refinements. Yellow stars indicate datasets where our model outperforms the BigFWI-B baseline.
• Figure 3: Generated samples Comparison.
• Figure 4: Visualization of deviation from PDE during diffusion. Seismic data of a CVB example at different timesteps during the forward (left half) and backward diffusion processes (right half). Rows 1-3 show generated seismic channels $\hat{s}$ by the joint diffusion model. Row 4 shows the generated velocity map $\hat{v}$. Rows 5-7 show the ground truth seismic data $s_{\hat{v}}$ calculated for the generated $\hat{v}$. Row 8 shows the deviation, visualized as the channel-stacked difference between $s_{\hat{v}}$ and $\hat{s}$. Noise increases the deviation during the forward diffusion, and the reverse process reduces discrepancies.
• Figure 5: Deviation from the governing PDE. The L2 distance is calculated between generated seismic data $\hat{s}$ and ground truth $s_{\hat{v}}$ of the generated velocity map $\hat{v}$ using a finite difference solver.