Accurate Interpolation of Ambient Noise Empirical Green's Functions by Denoising Diffusion Probabilistic Model and Implicit Neural Representation

Abstract

Empirical Green's functions (EGFs) extracted from seismic ambient noise have been widely used to image Earth's interior structures, and the resolution of EGF-based tomography depends on the spatial density of seismic stations. However, due to cost and logistical constraints, it is often difficult to deploy dense seismic networks suitable for high-resolution tomography. While reliable interpolation of EGFs at unsampled locations could enhance tomographic resolution, the task remains inherently challenging and underexplored due to the dispersive nature of EGFs. In this study, we introduce DIER (diffusion-assisted implicit EGF representation), a self-supervised learning framework that integrates implicit neural representation with denoising diffusion probabilistic models to achieve high-fidelity EGF interpolation. In DIER, the diffusion process is conditioned on station coordinates to guide the transformation from random noise into EGF waveforms, which allows flexible reconstruction of five-dimensional EGF fields without labeled data or synthetic waveforms. We demonstrate the effectiveness of DIER through continent-scale EGF interpolation across the United States. The results show that DIER significantly outperforms the conventional radial basis function-based interpolation approach by generating EGFs with markedly improved phase alignment and dispersion characteristics. Surface wave tomography using the phase velocities derived from the interpolated EGFs also closely matches a reference model constructed from data acquired by a much denser seismic network. Our findings suggest that DIER provides a promising and cost-effective approach toward high-resolution ambient noise tomography in regions with sparse station coverage.