Seislet transform

The construction of seislet transform Fomel and Liu (2010) follows the basics of the second-generation wavelet transform. The forward transform starts with the finest scale (the original sampling) and goes to the coarsest scale. The inverse transform starts with the coarsest scale and goes back to the finest scale Chen et al. (2014b). The forward and inverse seislet transforms can be expressed as:

$$\mathbf{r}=\mathbf{o}-\mathbf{P\left[e\right]},$$ (3)

$$\mathbf{c}=\mathbf{e}+\mathbf{U\left[r\right]},$$ (4)

$$\mathbf{e}=\mathbf{c}-\mathbf{U\left[r\right]},$$ (5)

$$\mathbf{o}=\mathbf{r}+\mathbf{P\left[e\right]},$$ (6)

where $\mathbf{P}$ is the prediction operator, $\mathbf{U}$ is the updating operator. $\mathbf{r}$ denotes the difference between true odd trace and predicted odd trace (from even trace), $\mathbf{c}$ denotes a coarse approximation of the data. At the start of the forward transform, e and o correspond 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 seislet transform differs from the wavelet transform in that the prediction and updating operators utilize the local slope of seismic profiles to predict and update the even and odd traces. The above prediction and update operators can be defined as follows:

$$\mathbf{P}\left[\mathbf{e}\right]_k=\left(\mathbf{P}^{(+)}_k\left[\mathbf{e}_{k-1}\right]+\mathbf{P}^{(-)}_k\left[\mathbf{e}_k\right]\right)/2,$$ (7)

$$\mathbf{U}\left[\mathbf{r}\right]_k=\left(\mathbf{P}^{(+)}_k\left[\mathbf{r}_{k-1}\right]+\mathbf{P}^{(-)}_k\left[\mathbf{r}_k\right]\right)/4,$$ (8)

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. The local slope can be calculated using a robust algorithm as introduced in Fomel (2002).