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Review of seislets

The seislet transform was introduced by Fomel (2006) and extended by Fomel and Liu (2010) and Liu and Fomel (2010). The seislet construction is based on the discrete wavelet transform (DWT) combined with seismic data patterns, such as local slopes or frequencies. Fomel (2002) developed a local plane-wave destruction (PWD) operation to predict local plane-wave events, where an all-pass digital filter is used to approximate the time shift between two neighboring traces. The inverse operation, plane-wave construction (Fomel and Guitton, 2006; Fomel, 2010), predicts a seismic trace from its neighbors by following locally varying slopes of seismic events and has been used for designing a PWD-seislet transform, which is a particular kind of the seislet transforms based on slope patterns. Liu and Liu (2013) proposed a velocity-dependent (VD) slope as a pattern in VD-seislet transform, where the normal moveout (NMO) equation serves as a bridge between local slopes and scanned NMO velocities.

To define seislet transform, we follow the general recipe of the lifting scheme for the discrete wavelet transform, as described by Sweldens and Schröder (1996). The construction is reviewed in Appendix A. Designing pattern-based prediction operator $\mathbf{P}$ and update operator $\mathbf{U}$ for seismic data is key in the seislet framework. In the seislet transform, the basic data components can be different, e.g., traces or common-offset gathers, and the prediction and update operators shift components according to different patterns.

The prediction and update operators for a simple seislet transform are defined by modifying the biorthogonal wavelet construction in equations from Appendix A as follows:

$\displaystyle \mathbf{P[e]}_k$ $\textstyle =$ $\displaystyle \left(\mathbf{R}_k^{(+)}[\mathbf{e}_{k-1}] +
\mathbf{R}_k^{(-)}[\mathbf{e}_{k}]\right)/2$ (1)
$\displaystyle \mathbf{U[r]}_k$ $\textstyle =$ $\displaystyle \left(\mathbf{R}_k^{(+)}[\mathbf{r}_{k-1}] +
\mathbf{R}_k^{(-)}[\mathbf{r}_{k}]\right)/4\;,$ (2)

where $\mathbf{e}_k$ is even components of data at the $k$th transform scale, $\mathbf{r}_k$ is residual difference between the odd component of data $\mathbf{o}$ and its prediction from the even component at the $k$th transform scale, and $\mathbf{R}_k^{(+)}$ and $\mathbf{R}_k^{(-)}$ are operators that predict a component from its left and right neighbors correspondingly by shifting them according to their patterns. The details are explained in Appendix A.

To get the relationship between prediction operator $\mathbf{R}_k$ and slope pattern $\sigma$, the plane-wave destruction operation (Fomel, 2002) can be defined in a linear operator notation as

\mathbf{d} = \mathbf{D(\sigma)\,s}\;,
\end{displaymath} (3)

where seismic section $\mathbf{s} = \left[\mathbf{s}_1 \;
\mathbf{s}_2 \; \ldots \; \mathbf{s}_N\right]^T$ is a collection of traces, and $\mathbf{d}$ is the destruction residual. The general structure of $\mathbf{D}$ is defined as follows (Fomel and Guitton, 2006; Fomel, 2010)
\mathbf{D(\sigma)} =
..._{N-1,N}(\sigma_{N-1}) & \mathbf{I} \\
\end{displaymath} (4)

where $\mathbf{I}$ stands for the identity operator, $\sigma_i$ is local slope pattern, and $\mathbf{R}_{i,j}(\sigma_i)$ is an operator for prediction of trace $j$ from trace $i$ according to the slope pattern $\sigma_i$. A trace is predicted by shifting it according to the local seismic event slopes. Prediction of a trace from a distant neighbor can be accomplished by simple recursion, i.e., predicting trace $k$ from trace $1$ is simply
\mathbf{R}_{1,k} = \mathbf{R}_{k-1,k}\,
\end{displaymath} (5)

If $\mathbf{s}_r$ is a reference trace, then the prediction of trace $\mathbf{s}_k$ is $\mathbf{R}_{r,k} \mathbf{s}_r$.

The predictions need to operate at different scales, which, in this case, mean different separation distances between the data elements, e.g., traces in PWD-seislet transform. Equations 1 and 2, in combination with the forward and inverse lifting schemes, provide a complete definition of the seislet framework. For different kinds of slope-based seislets, one needs to define the corresponding slope pattern $\sigma$.

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Next: VD-slope pattern for primary Up: Theory Previous: Theory