Recovery of seismic wavefields by an &lt;i&gt;l&lt;sub&gt;q&lt;/sub&gt;&lt;/i&gt;-norm constrained regularization method

Fengmin Xu, ,Yanfei Wang,,University Of Chinese Academy Of Sciences, Beijing 100049, China

doi:10.3934/ipi.2018048

Abstract

Reconstruction of the seismic wavefield from sub-sampled data is an important problem in seismic image processing, this is partly due to limitations of the observations which usually yield incomplete data. In essence, this is an ill-posed inverse problem. To solve the ill-posed problem, different kinds of regularization technique can be applied. In this paper, we consider a novel regularization model, called the \begin{document}$l_2$\end{document} - \begin{document}$l_{q}$\end{document} minimization model, to recover the original geophysical data from the sub-sampled data. Based on the lower bound of the local minimizers of the \begin{document}$l_2$\end{document} - \begin{document}$l_{q}$\end{document} minimization model, a fast convergent iterative algorithm is developed to solve the minimization problem. Numerical results on random signals, synthetic and field seismic data demonstrate that the proposed approach is very robust in solving the ill-posed restoration problem and can greatly improve the quality of wavefield recovery.

Highlights

In seismology, due to limitations of the observations, the observed data is usually incomplete, e.g., some traces are lost, or random sampling to save cost of seismic acquisition
This paper focuses on data restoration problem
Major issues affecting the effect of restoration are: sparse transforms, sampling, sparse modeling and solving methods

Summary

Introduction

Due to limitations of the observations, the observed data is usually incomplete, e.g., some traces are lost, or random sampling (insufficient sampling) to save cost of seismic acquisition. A key obstacle is how to invert the model using only incomplete, sub-sampled data [14, 18, 25]. Recovery of the original wavefield from incomplete observed data is generally an ill-posed problem. To successfully recover a signal without error, according to Nyquist-Shannon sampling theorem, the signal acquisition systems require that the sampling rate is twice the maximum frequency. This sampling theorem is hard to satisfy in practice.

Fengmin Xu and Yanfei Wang

Inverse Problems and Imaging

Trace number

Data with missing traces