


 Iterative deblending of simultaneoussource seismic data using seisletdomain shaping regularization  

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Appendix
A
Review of seislet transform
Fomel (2006) and Fomel and Liu (2010) proposed a digital waveletlike transform, which is defined with the help of the waveletlifting scheme (Sweldens, 1995) combined with local planewave destruction. The waveletlifting utilizes predictability of even traces from odd traces of 2D seismic data and finds a difference
between them, which can be expressed as:

(19) 
where
is the prediction operator.
A coarse approximation
of the data can be achieved by updating the even component:

(20) 
where
is the updating operator.
The digital wavelet transform can be inverted by reversing the liftingscheme operations as follows:

(21) 

(22) 
The foward transform starts with the finest scale (the original sampling) and goes to the coarsest scale. The inverse transfrom starts with the coarsest scale and goes back to the finest scale. At the start of forward transform,
and
corresponds to the even and odd traces of the data domain. At the start of the inverse transform,
and
will have just one trace of the coarsest scale of the seislet domain.
The above prediction and update operators can be defined, for example, as follows:

(23) 
and

(24) 
where
and
are operators that predict a trace from its left and right neighbors, correspondingly, by shifting seismic events according to their local slopes.
This scheme is analogous to CDF biorthogonal wavelets (Cohen et al., 1992). The predictions need to operate at different scales, which means different separation distances between traces. Taken through different scales, equations A1A6 provide a simple definition for the 2D seislet transform. More accurate versions are based on other schemes for the digital wavelet transform (Liu et al., 2009a).



 Iterative deblending of simultaneoussource seismic data using seisletdomain shaping regularization  

Next: About this document ...
Up: Chen et al.: Deblending
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