Sparsity in compressive sensing

Reducing the cost of seismic exploration is a long-standing problem. Compressive sensing (CS) provides a new routine for recovering signals when the seismic acquisition is under the criteria of Nyquist theory. The basic assumptions of CS are: (1) the signal is sparse under a certain transform, and (2) the acquisition or measurement is random in a certain way. The ratio of compressive samplings to Nyquist samplings is determined by the sparsity of the signal under random measurements. Sparsity has been introduced to seismic exploration from three sources: (1) borrowing from image processing, (2) training from seismic data, and (3) special design for seismic data. We review the sparse transforms used in seismic data acquisition and processing under the framework of CS and then discuss a new way of designing sparse transforms.