A convergent alternating projections onto convex sets method for data reconstruction

Abstract

The alternating projections onto convex sets (POCS) method is one of the well-known data reconstruction methods in the seismic field. We first review the POCS theory in mathematics and then point out that many POCS methods reported in the literature have overlooked the convergence condition, which is a fundamental and important aspect in POCS. We further develop a convergent POCS (CP) method that uses an existing POCS method as the algorithm framework and adds a convergence condition before a convex set is updated, and we mathematically prove the convergence of this proposed CP method. Several examples are used to demonstrate the success of the CP method. The CP method is a reconstruction scheme robust to variations in the input data and the thresholding schedule.