Due to some physical limitations, such as the presence of obstacles in the field, feathering on the sea, and economical consideration, such as reducing recorded data intentionally in order to save acquisition cost, and other different reasons, seismic records are typically irregularly sampled, or regularly but under-sampled. Seismic data interpolation is such a crucial part in the whole seismic data processing workflow that deals with this problem. It can compensate for the acquisition limitation and help prepare a dense and regularly sampled seismic dataset for subsequent seismic imaging and inversion. Meanwhile, seismic data reconstruction is also an important procedure to improve the amplitude quality and to remove sampling artifacts, which is of significant importance for subsequent processing workflows including high-resolution processing, amplitude preservation migration and amplitude-versus-offsets (AVO) analysis.

During the past decades, there have been several classical methods for seismic data reconstruction. A number of fixed basis sparsity-promoting transforms have been proposed for restoring seismic data, e.g. the Fourier transform Chen et al. (2014a); Naghizadeh (2012); Sacchi et al. (1998); Zhong et al. (2015), the Radon transform Trad et al. (2002); Wang et al. (2010); Yu et al. (2007), the curvelet transform Shahidi et al. (2013); Herrmann and Hennenfent (2008); Liu et al. (2015a) and the seislet transform Gan et al. (2015,2016a); Fomel and Liu (2010). Spatial predication filters are capable of interpolating aliased data by utilizing non-aliased low frequency data Parsani (1999). Wave equation methods always require prior distribution of underground parameters, which are computational expensive Ronen (1987). Rank reduction methods are based on the fact that missing traces and random noise can increase the rank of matrix, such as multichannel singular spectrum analysis (MSSA) Oropeza and Sacchi (2011); Gao et al. (2013).

Interpolation of irregularly sampled data has been well solved by using a number of methods belonging to the emerging research field: compressive sensing Donoho (2006). The interpolation problem is transformed into a denoising problem in some sparse transform domains. The commonly used approach for dealiased seismic interpolation is the prediction based approach Spitz (1991), which uses the low-frequency components to design a prediction error filter for interpolating high-frequency missing components (Spitz's method).

Gan et al. (2015) proposes a sparse transform based interpolation method based on the seislet transform. The under-sampled regularly sampled data will be sparse in the seislet domain provided that an accurate local slope field can be obtained. To get such accurate slope map, a lot of iterations should be carried out with low-frequency constraints. In this letter, we propose a new algorithm that does not require the iterative slope estimation. The slope map can be easily estimated through velocity-slope transformation Liu and Liu (2013). In the new method, we can use a fairly small number of projection onto convex sets (POCS) iterations to get a very good interpolation result. Both synthetic and field data examples are used to show the performance. Since the velocity-slope transformation is valid only for prestack seismic data, where data structure is relatively simple and the slope is generally positive, the proposed method can only be used in prestack seismic data reconstruction.