Introduction

Most of the time, seismic data processing need a regular and dense dataset input, which is of extreme importance for obtaining a high-resolution result. However, during the data acquisition process, many different reasons may result in the missing traces, including economic reasons, ground surface limitations, and regulatory reasons. Seismic data reconstruction is such a pre-condition procedure that can be used to remove sampling artifacts, filling the gaps, and to improve amplitude analysis, which is indispensable for the subsequent processing steps including high-resolution processing, wave-equation migration, multiple suppression, amplitude-versus-offset (AVO) or amplitude-versus-azimuth (AVAZ) analysis, and time-lapse studies (Abma and Kabir, 2005; Chen et al., 2014a; Trad et al., 2002; Zhong et al., 2015; Wang et al., 2010; Abma and Kabir, 2006; Naghizadeh and Sacchi, 2010; Liu and Sacchi, 2004).

In recent years, because of the popularity of compressive sensing (CS) based applications (Candès et al., 2006b), there exists a new paradigm for seismic data acquisition that can potentially reduce the survey time and increase the data resolution (Herrmann, 2010). Compressive sensing (CS) is a relatively new paradigm in signal processing that has recently received a lot of attention. The theory indicates that the signal which is sparse under some basis may still be recovered even though the number of measurements is deemed insufficient by Shannon's criterion. The principle of CS involves solving a least-square minimization problem with a $L_1$ norm penalty term of the reconstructed model, which requires compromising a least-square data-misfit constraint and a sparsity constraint over the reconstructed model. The iterative shrinkage thresholding (IST) and the projection onto convex sets (POCS) are two common approaches used to solve the minimization problem in the exploration geophysics field.

Inspired from the fast iterative shrinkage-thresholding algorithm (FISTA) introduced in Beck and Teboulle (2009), we propose a similar faster version of POCS (FPOCS). Sparsity of seismic data has been explored utilizing different transforms, such as Fourier transform, curvelet (Candès et al., 2006a) and synchrosqueezed wavelet transform (Chen et al., 2014c). We compare the sparseness of different well-known sparse transforms by displaying the transform domain and drawing the transform domain coefficients decaying curves. The comparison shows that the seislet transform is obviously sparser than other alternative sparse transforms. Thus, we use the seislet transform (Chen et al., 2014b; Fomel and Liu, 2010) as the sparsity promoting transform in the compressive sensing data recovery framework in order to explore its related behaviors. Both synthetic and field data examples show that the proposed seislet based FPOCS can obtain better and faster data recovery than the $f$-$k$ transform based POCS method.

The contributions of the paper can be divided into three aspects. (1) We extend the acceleration strategy used previously in the IST approach to POCS approach, and compare the performance difference of IST and POCS (and related FISTA and FPOCS) in seismic data with different noise level and pointed out that the selection of IST or POCS depends on the noise level of seismic data. (2) We compare the transform domain sparsity of different well-known sparse transforms in terms of the plotted sparse coefficients and coefficients decaying diagrams, and find out that the seislet transform has a much sparser transform domain structure than Fourier transform, wavelet transform, and the curvelet transform. (3) The seislet-based CS approach for seismic data reconstruction is initially investigated and the performance of seislet-based approach and $f$-$k$ based approach are compared in terms of the reconstruction signal-to-noise ratio (SNR), local similarity comparison, and visual observation.