Off-the-grid vertical seismic profile data regularization by a compressive sensing method

Different from the surface survey, the vertical seismic profile (VSP) survey deploys sources on the surface and geophones in a well. VSP provides higher resolution information of subsurface structures. The faults that cannot be imaged with surface seismic data may be detected with VSP data, and detailed analysis of fracture zones can be achieved with multicomponent VSP. However, one of the main problems is that the sources seldom are acquired on a regular grid in realistic VSP surveys. The irregular samplings cause serious artifacts in migration or imaging, such that data regularization must be implemented first. We have developed a compressive sensing (CS)-based method to regularize nonstationary VSP data. Our method directly operates on irregularly gridded data sets, which is a key contribution compared to the existing CS-based reconstruction methods that work on regular grids. The CS framework consists of a sparsity constraint and a penalty term. We have used the curvelet transform for sparsity constraint of nonstationary events in the regularization term and the nonequispaced Fourier transform to regularize the VSP data in a penalty term. An alternative directional method of multipliers is used for solving the optimization problem. Our method is tested on synthetic, field 2D and 3D VSP data sets. Our method obtains improved reconstructions on continuities of the events and produces fewer artifacts compared to the well-known antileaking Fourier transform method. © 2020 Society of Exploration Geophysicists.