Signal and noise separation in prestack seismic data using velocity-dependent seislet transform |

To define seislet transform, we follow the general recipe of the lifting scheme for the discrete wavelet transform, as described by Sweldens and Schröder (1996). The construction is reviewed in Appendix A. Designing pattern-based prediction operator and update operator for seismic data is key in the seislet framework. In the seislet transform, the basic data components can be different, e.g., traces or common-offset gathers, and the prediction and update operators shift components according to different patterns.

The prediction and update operators for a simple seislet transform are
defined by modifying the biorthogonal wavelet construction in
equations from Appendix A as follows:

where is even components of data at the th transform scale, is residual difference between the odd component of data and its prediction from the even component at the th transform scale, and and are operators that predict a component from its left and right neighbors correspondingly by shifting them according to their patterns. The details are explained in Appendix A.

To get the relationship between prediction operator
and slope pattern , the plane-wave destruction
operation (Fomel, 2002) can be defined in a linear operator notation as

where stands for the identity operator, is local slope pattern, and is an operator for prediction of trace from trace according to the slope pattern . A trace is predicted by shifting it according to the local seismic event slopes. Prediction of a trace from a distant neighbor can be accomplished by simple recursion, i.e., predicting trace from trace is simply

If is a reference trace, then the prediction of trace is .

The predictions need to operate at different scales, which, in this case, mean different separation distances between the data elements, e.g., traces in PWD-seislet transform. Equations 1 and 2, in combination with the forward and inverse lifting schemes, provide a complete definition of the seislet framework. For different kinds of slope-based seislets, one needs to define the corresponding slope pattern .

Signal and noise separation in prestack seismic data using velocity-dependent seislet transform |

2015-10-24