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Appendix A

Review of seislet transform

Fomel (2006) and Fomel and Liu (2010) proposed a digital wavelet-like transform, which is defined with the help of the wavelet-lifting scheme (Sweldens, 1995) combined with local plane-wave destruction. The wavelet-lifting utilizes predictability of even traces from odd traces of 2-D seismic data and finds a difference $ \mathbf{r}$ between them, which can be expressed as:

$\displaystyle \mathbf{r}=\mathbf{o}-\mathbf{P\left[e\right]},$ (19)

where $ \mathbf{P}$ is the prediction operator. A coarse approximation $ \mathbf{c}$ of the data can be achieved by updating the even component:

$\displaystyle \mathbf{c}=\mathbf{e}+\mathbf{U\left[r\right]},$ (20)

where $ \mathbf{U}$ is the updating operator.

The digital wavelet transform can be inverted by reversing the lifting-scheme operations as follows:

$\displaystyle \mathbf{e}=\mathbf{c}-\mathbf{U\left[r\right]},$ (21)

$\displaystyle \mathbf{o}=\mathbf{r}+\mathbf{P\left[e\right]}.$ (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, $ \mathbf{e}$ and $ \mathbf{o}$ corresponds to the even and odd traces of the data domain. At the start of the inverse transform, $ \mathbf{c}$ and $ \mathbf{r}$ 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:

$\displaystyle \mathbf{P}\left[\mathbf{e}\right]_k=\left(\mathbf{P}^{(+)}_k\left[\mathbf{e}_{k-1}\right]+\mathbf{P}^{(-)}_k\left[\mathbf{e}_k\right]\right)/2,$ (23)


$\displaystyle \mathbf{U}\left[\mathbf{r}\right]_k=\left(\mathbf{P}^{(+)}_k\left[\mathbf{r}_{k-1}\right]+\mathbf{P}^{(-)}_k\left[\mathbf{r}_k\right]\right)/4,$ (24)

where $ \mathbf{P}^{(+)}_k$ and $ \mathbf{P}^{(-)}_k$ 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 A-1-A-6 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).

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