Seismic Low-Frequency Extrapolation via Physics-Aware Conditional Diffusion Models

Low-frequency (LF) seismic data are crucial for stable full-waveform inversion (FWI), yet remain difficult to acquire due to source bandwidth limitations and noise. We propose a novel physics-aware conditional diffusion framework to extrapolate LF wavefields from observed high-frequency (HF) data. Our method models LF extrapolation as a conditional generation task, using a neural network trained via denoising diffusion to approximate the gradient of the log-probability of LF data given the HF observations, also known as the conditional score function. During sampling, we enforce physical consistency by introducing a soft constraint based on the plane-wave partial differential equation, leveraging the observation that HF and LF wavefields share similar local slopes. This guides the generative process toward kinematically plausible extrapolations. We demonstrate on synthetic datasets that our method improves amplitude coherence and continuity in complex waveform settings. Figure 1 shows a scheme with an overview of the proposed methodology. The approach also supports uncertainty quantification through posterior sampling: each diffusion trajectory yields a different plausible LF reconstruction. This capability is particularly valuable in FWI, where LF data strongly influence the recovery of long-wavelength velocity models and proper initialization helps mitigate local minima. By analyzing variability across LF realizations, we gain insights into inversion uncertainty. Overall, the method combines data-driven generation with physics-based guidance to enable realistic, interpretable LF reconstructions with direct relevance for seismic inversion workflows.