Compressed sensing approach to 3D spatially irregular seismic data reconstruction in frequency-space domain

The reconstruction of spatially irregular data is a critical and challenging task in seismic exploration, essential for subsequent tasks like velocity analysis, reservoir inversion, and wave-equation migration imaging. Fortunately, compressed sensing (CS) overcomes the limitations of the Nyquist sampling theorem, providing cost-effective theoretical support for interpolating observed data, and has been widely proven effective in various fields. However, in seismic exploration, the current CS-based methods often struggle to consistently achieve satisfactory reconstruction. We theoretically demonstrate that the deficiency of 3D reconstruction based on CS for spatially irregular data arises from the inconsistency between the random sampling domain and the reconstruction domain. Sampling is randomly performed in the spatial domain, while reconstruction relies on the sparsity in the spatiotemporal domain, conflicting with the principles of CS and limits the accuracy and effectiveness of the reconstruction. To address this, we propose that CS-based reconstruction should be performed solely in the random sampling dimension, ensuring consistency between the two domains. For 3D spatially irregular seismic data, the random sampling dimension corresponds to the spatial domain, such as in time or frequency slices. For 2D CS-based reconstruction in the spatial domain of 3D spatially irregular data, we identify the limitations of the time-slices method, and numerical experiments demonstrate that the frequency-slices method outperforms traditional 3D spatiotemporal CS reconstruction with higher accuracy. Due to the complex characteristics on the frequency slices, a unitive transform cannot capture the sparsity of all components, future work will focus on developing more adaptive sparse transform methods.