    OC-seislet: seislet transform construction with differential offset continuation  Next: OC-seislet structure Up: Theoretical basis Previous: Theoretical basis

## The lifting scheme for the seislet transform

The lifting scheme (Sweldens, 1995) provides a convenient approach for designing digital wavelet transforms. The general recipe is as follows:

1. Organize the input data as a sequence of records. For OC-seislet transform of 2-D seismic reflection data, the input is in the frequency'-midpoint wavenumber'-`offset' domain after the log-stretched NMO correction (Bolondi et al., 1982), and the transform direction is offset.
2. Divide the data records (along the offset axis in the case of the OC-seislet transform) into even and odd components and . This step works at one scale level.
3. Find the residual difference between the odd component and its prediction from the even component: (1)

where is a prediction operator. For example, one can obtain Cohen-Daubechies-Feauveau (CDF) 5/3 biorthogonal wavelets (Cohen et al., 1992) by defining the prediction operator as a linear interpolation between two neighboring samples, (2)

where is an index number at the current scale level.
4. Find an approximation of the data by updating the even component: (3)

where is an update operator. Constructing the update operator for CDF 5/3 biorthogonal wavelets aims at preserving the running average of the signal (Sweldens and Schröder, 1996): (4)

5. The coarse approximation becomes the new data, and the sequence of steps is repeated on the new data to calculate the transform coefficients at a coarser scale level.

Next, we define new prediction and update operators using offset-continuation operators.    OC-seislet: seislet transform construction with differential offset continuation  Next: OC-seislet structure Up: Theoretical basis Previous: Theoretical basis

2013-07-26