OC - seislet : seislet transform construction with differential offset continuation a

Many of the geophysical data analysis problems, such as signal-noise separation and data regularization, are conveniently formulated in a transform domain, where the signal appears sparse. Classic transforms such as the Fourier transform or the digital wavelet transform, fail occasionally in processing complex seismic wavefields, because of the nonstationarity of seismic data in both time and space dimensions. We present a sparse multiscale transform domain specifically tailored to seismic reflection data. The new wavelet-like transform – the OC-seislet transform – uses a differential offset-continuation (OC) operator that predicts prestack reflection data in offset, midpoint, and time coordinates. It provides high compression of reflection events. In the transform domain, reflection events get concentrated at small scales. Its compression properties indicate the potential of OC-seislets for applications such as seismic data regularization or noise attenuation. Results of applying the method to both synthetic and field data examples demonstrate that the OC-seislet transform can reconstruct missing seismic data and eliminate random noise even in structurally complex areas. INTRODUCTION Digital wavelet transforms are excellent tools for multiscale data analysis. The wavelet transform is more powerful when compared with the classic Fourier transform, because it is better fitted for representing non-stationary signals. Wavelets provide a sparse representation of piecewise regular signals, which may include transients and singularities (Mallat, 2009). In recent years, many wavelet-like transforms that explore directional characteristics of images (Starck et al., 2000; Do and Vetterli, 2005; Pennec and Mallat, 2005; Velisavljevic, 2005) were proposed. The curvelet transform in particular has found important applications in seismic imaging and data analysis (Douma and de Hoop, 2007; Chauris and Nguyen, 2008; Herrmann et al., 2008). Fomel (2006) and Fomel and Liu (2010) investigated the possibility of designing a wavelet-like transform tailored specifically to seismic data and introduced it under the name of the seislet transform. Based on the digital wavelet transform (DWT), the seislet transform follows patterns of seismic events (such as local slopes in 2-D and frequencies in 1-D) when analyzing those events at different scales. The seislet