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

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 |

2013-07-26