Comparison of the projection onto convex sets and iterative hard thresholding methods for seismic data interpolation and denoising

Abstract

Because of the environment limitations, irregularity appears in the observed seismic data. In addition, the observed seismic data contains random noise from the acquisition equipment and surrounding environment, which affects the performances of multi-channel techniques, such as surface related multiple elimination (SRME) and amplitude variation with offset (AVO) analysis. The projection onto convex sets (POCS) method, known as an efficient interpolation method, is suitable for high signal-to-noise ratio (SNR) situations; however, the existing random noise may affect its final performance. In our previously published paper, the POCS formula was deduced in the view of iterative hard thresholding (IHT) method using a projection operator. In this paper, more physical illustrations about its detailed deduction are provided to show the differences between IHT and POCS in noise-free and noisy situations with easy understanding for readers. Then, performances of the POCS and IHT methods are compared in both noise-free and noisy situations, in terms of seismograms, frequency wavenumber (FK) spectra and single traces. For noise-free data, both the POCS and IHT methods can achieve good interpolation results. For noisy data, the POCS method is unsuitable because of the observed noisy data insertion, while the IHT performance is satisfactory because it uses a thresholding operator to eliminate random noise. Numerical examples on noise-free datasets demonstrate the validities of the POCS and IHT methods for interpolation. Tests on noisy data contaminated with additive white Gaussian noise prove the ability and superiority of the IHT method with anti-noise property compared with the POCS method.