FUDLInter: Frequency-Space-Dependent Unsupervised Deep Learning Framework for 3-D and 5-D Seismic Data Interpolation

Deep learning (DL) has emerged as a focal point in addressing various challenges within the field of exploration seismology, prominently featuring applications in seismic data interpolation. Existing neural networks utilized in exploration seismology predominantly employ real-valued nonlinear transforms on time-space seismic data. Nevertheless, the seismic signal contains significant information in its phase, whereas the real-valued transforms meet with challenges to take into account the entire phase information of nonstationary seismic data. To surmount this challenge, we propose a novel framework termed frequency-space-dependent unsupervised DL interpolation (FUDLInter). The primary objective of FUDLInter is to interpolate high-dimensional seismic data within the frequency-space domain, thereby optimizing the exploitation of intricate information derived from the fast Fourier transform representation of seismic signals. In this framework, we meticulously explore and harness the capability of a complex-valued deep convolutional neural network employing the U-Net architecture, designated as CVU-Net. This network is designed to autonomously recover each frequency component of both 3-D and 5-D seismic data. We leverage the Bernoulli sampling technique and the nonmissing elements in the subsampled data to construct a data misfit model. The efficacy of the proposed method is evaluated using both high-dimensional synthetic data and field data examples. The interpolation results from the proposed FUDLInter method outperform those achieved by alternative methods, i.e., the projection onto convex sets with an adaptive threshold schedule (APOCS), damped rank-reduction (DRR), and DenseNet methods.