OC-seislet: seislet transform construction with differential offset continuation

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 (Do and Vetterli, 2005; Pennec and Mallat, 2005; Velisavljevic, 2005; Starck et al., 2000) were proposed. The curvelet transform in particular has found important applications in seismic imaging and data analysis (Herrmann et al., 2008; Chauris and Nguyen, 2008; Douma and de Hoop, 2007). 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 transform's compression ability finds applications in common data processing tasks such as data regularization and noise attenuation. However, the problem of pattern detection limits its further applications. In 2-D, conflicting slopes at a single data point are difficult to detect reliably even using advanced methods (Fomel, 2002). It is also difficult to estimate local slopes in the presence of strong noise. A similar situation occurs in the 1-D case, in which it is difficult to exactly represent a known seismic signal using a limited set of frequencies.

Offset continuation is a process of seismic data transformation between different offsets (Fomel, 2003c; Bolondi et al., 1982; Deregowski and Rocca, 1981; Salvador and Savelli, 1982). Different types of dip moveout (DMO) operators (Hale, 1991) can be regarded as continuation to zero offset and derived as solutions to initial-value problems with the offset-continuation differential equation. In the shot-record domain, offset continuation transforms to shot continuation, which describes the process of transforming reflection seismic data along shot location (Bagaini and Spagnolini, 1996; Fomel, 2003b; Spagnolini and Opreni, 1996). The 3-D analog is known as azimuth moveout or AMO (Fomel, 2003a; Biondi et al., 1998). Bleistein and Jaramillo (2000) developed a general platform for Kirchhoff data mapping, which includes offset continuation as a special case.

In this paper, we propose to incorporate offset continuation as the prediction operator into the seislet transform. We design the transform in the log-stretch-frequency domain, where each frequency slice can be processed independently and in parallel. We expect the new seislet transform to perform better than the previously proposed seislet transform by plane-wave destruction, PWD-seislet transform (Fomel and Liu, 2010), in cases of moderate velocity variations and complex structures that generate conflicting dips in the data.

OC-seislet: seislet transform construction with differential offset continuation