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

Appendix: The lifting scheme for DWT

The lifting scheme (Sweldens, 1995) provides a convenient approach for defining wavelet transforms by breaking them down into the following steps:

1. Divide data into even and odd components, $\mathbf{e}$ and $\mathbf{o}$.
2. Find a residual difference, $\mathbf{r}$, between the odd component and its prediction from the even component:
   

   $$\mathbf{r} = \mathbf{o} - \mathbf{P[e]}\;,$$ (20)

   
   
   where $\mathbf{P}$ is a prediction operator.
3. Find a coarse approximation, $\mathbf{c}$, of the data by updating the even component:
   

   $$\mathbf{c} = \mathbf{e} + \mathbf{U[r]}\;,$$ (21)

   
   
   where $\mathbf{U}$ is an update operator.
4. The coarse approximation, $\mathbf{c}$, becomes the new data, and the sequence of steps is repeated at the next scale.

The Cohen-Daubechies-Feauveau (CDF) 5/3 biorthogonal wavelets (Cohen et al., 1992) are constructed by making the prediction operator a linear interpolation between two neighboring samples,

$$\mathbf{P[e]}_k = \left(\mathbf{e}_{k-1} + \mathbf{e}_{k}\right)/2\;,$$ (22)

and by constructing the update operator to preserve the running average of the signal (Sweldens and Schröder, 1996), as follows:

$$\mathbf{U[r]}_k = \left(\mathbf{r}_{k-1} + \mathbf{r}_{k}\right)/4\;.$$ (23)

Furthermore, one can create a high-order CDF 9/7 biorthogonal wavelet transform by using CDF 5/3 biorthogonal wavelets twice with different lifting operator coefficients (Lian et al., 2001). The transform is easily inverted according to reversing the steps above:

1. Start with the coarsest scale data representation $\mathbf{c}$ and the coarsest scale residual $\mathbf{r}$.
2. Reconstruct the even component $\mathbf{e}$ by reversing the operation in equation A-2, as follows:
   

   $$\mathbf{e} = \mathbf{c} - \mathbf{U[r]}\;,$$ (24)

   
   

3. Reconstruct the odd component $\mathbf{o}$ by reversing the operation in equation A-1, as follows:
   

   $$\mathbf{o} = \mathbf{r} + \mathbf{P[e]}\;,$$ (25)

   
   

4. Combine the odd and even components to generate the data at the previous scale level and repeat the sequence of steps.

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