High dimensional seismic data interpolation with weighted matching pursuit based on compressed sensing

Abstract

The spatial aliasing of seismic data is usually serious because of the sub-sampling rate of the acquisition system. It induces amplitude artifacts or blurs the migration result when the spatial aliasing is not removed before migration. The compressed sensing (CS) method has been proven to be an effective tool to restore a sub-sampled signal which is compressible in another domain. Since the wave-fronts of seismic data are sparse and linear in a local spatiotemporal window, they can be significantly compressed by linear Radon transform or Fourier transform. Therefore, seismic data interpolation can be considered as a CS problem. The approximate solution of a CS problem using L0-norm can be achieved by matching pursuit (MP) algorithm. MP becomes intractable due to the high computing cost induced by the increasing dimension of the problem. In order to tackle this issue, a variant of MP - weighted matching pursuit (WMP) - is presented in this paper. Since there is little spatial aliasing in the data of low frequency and the events are supposed to be linear, the linear Radon spectrogram of the interpolated data of low frequency can be used to predict the energy distribution of data of high frequency in a frequency-wavenumber (FK) domain. The predicted energy distribution is then utilized to form the weighted factor of WMP. With this factor, WMP possesses the ability to distinguish the linear events from the spatial aliasing in the FK domain. WMP is also proven to be an efficient algorithm. Since projection onto convex sets (POCS) is another common sparsity-based method, we use Fourier POCS and WMP to realize high-dimension interpolation in numerical examples. The numerical examples show that the interpolation result of WMP significantly improves the quality of seismic data, and the quality of the migration result is also improved by the interpolation.