Seismic low-frequency extrapolation in a latent space based on self-supervised learning

Anqi Jia, Jian Sun, Kristopher A. Innanen

Low-frequency seismic data are crucial for robust full-waveform inversion but are often unavailable in field acquisitions. This report presents a Koopman-Autoencoder framework for low-frequency extrapolation. By leveraging Koopman operator theory, we transform the nonlinear problem of frequency extrapolation into a linear dynamics learning task within a latent space. A self-supervised deep autoencoder discovers Koopman eigenfunctions that embed seismic data into a low-dimensional manifold where frequency evolution follows linear dynamics. We introduce and compare synchronous and step-wise training strategies to balance reconstruction fidelity with dynamical linearity. Extensive experiments across time-space, frequency-space, and source-receiver domains demonstrate our method's capability to accurately extrapolate low frequencies from band-limited data. Both proposed approaches significantly outperform conventional end-to-end methods in low-frequency extrapolation accuracy. In particular, the synchronous training strategy achieves superior performance in modeling multi-step frequency dynamics, demonstrating strong generalization to unseen frequencies.