OC-seislet: seislet transform construction with differential offset continuation

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 $\mathbf{e}$ and $\mathbf{o}$ . This step works at one scale level.
3. Find the residual difference $\mathbf{r}$ between the odd component and its prediction from the even component:

   $$\mathbf{{r}} = \mathbf{{o}} - \mathbf{P[{e}]}\;,$$ (1)

   
   
   where $\mathbf{P}$ 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,

   $$\mathbf{P[e]}_k = \left(\mathbf{e}_{k-1} + \mathbf{e}_{k}\right)/2\;,$$ (2)

   
   
   where $k$ is an index number at the current scale level.
4. Find an approximation $\mathbf{c}$ of the data by updating the even component:

   $$\mathbf{{c}} = \mathbf{{e}} + \mathbf{U[{r}]}\;,$$ (3)

   
   
   where $\mathbf{U}$ 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):

   $$\mathbf{U[r]}_k = \left(\mathbf{r}_{k-1} + \mathbf{r}_{k}\right)/4\;.$$ (4)

   
   

5. The coarse approximation $\mathbf{{c}}$ 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