Reviews of the 2D seislet transform

As proposed by Fomel and Liu (2010), the seislet transform can be constructed using the lifting scheme (Sweldens, 1995). The procedure of the lifting scheme is as follows:

1. Arrange the input data as a sequence of records. In the seislet transform, the input data are common shot gather or seismic image.
2. Divide the arranged records into even and odd components $\mathbf{e}$ and $\mathbf{o}$. For the 2D seislet transform, each trace is a component and the seismic data are split into two parts according to trace indices.
3. Find the residual $\mathbf{r}$ 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.
4. Using the difference from the previous step to get a coarse approximation $\mathbf{c}$ of the data by updating the even component

   $$\mathbf{c} = \mathbf{e} + \mathbf{U[r]},$$ (2)

   where $\mathbf{U}$ is an update operator.
5. The coarse approximation $\mathbf{c}$ obtained from last step becomes the new data, and previous steps are applied to the new data at the next scale level.

The coarse approximation $\mathbf{c}$ from the final scale and residual difference $\mathbf{r}$ from all scales form the result of the seislet transform.

The prediction and update operators are defined, for example, by modifying operators in the construction of CDF 5/3 biorthogonal wavelets (Cohen et al., 1992) as follows:

$$\mathbf{P[e]}_k = \left(\mathbf{S}_k^{(+)}[\mathbf{e}_{k-1}]+ \mathbf{S}_k^{(-)}[\mathbf{e}_{k}] \right) / 2$$ (3)

and

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

where $\mathbf{S}_k^{(+)}$ and $\mathbf{S}_k^{(-)}$ can accomplish the prediction of a trace from its left and right neighbors, respectively. The predictions need to be applied at different scales. For 2D seislet transform, different scales mean different distances between traces. More accurate higher-order formulations are possible. Following the inverse lifting scheme from coarse scale to fine scale, the inverse seislet transform can be constructed.