Application of randomized sampling schemes to curvelet‐based sparsity‐promoting seismic data recovery

Reconstruction of seismic data is routinely used to improve the quality and resolution of seismic data from incomplete acquired seismic recordings. Curvelet‐based Recovery by Sparsity‐promoting Inversion, adapted from the recently‐developed theory of compressive sensing, is one such kind of reconstruction, especially good for recovery of undersampled seismic data. Like traditional Fourier‐based methods, it performs best when used in conjunction with randomized subsampling, which converts aliases from the usual regular periodic subsampling into easy‐to‐eliminate noise. By virtue of its ability to control gap size, along with the random and irregular nature of its sampling pattern, jittered (sub)sampling is one proven method that has been used successfully for the determination of geophone positions along a seismic line. In this paper, we extend jittered sampling to two‐dimensional acquisition design, a more difficult problem, with both underlying Cartesian and hexagonal grids. We also study what we term separable and non‐separable two‐dimensional jittered samplings. We find hexagonal jittered sampling performs better than Cartesian jittered sampling, while fully non‐separable jittered sampling performs better than separable jittered sampling. Two other 2D randomized sampling methods, Poisson Disk sampling and Farthest Point sampling, both known to possess blue‐noise spectra, are also shown to perform well.