Seislet transform and seislet frame

2-D seislet transform

If we view seismic data as collections of traces, we can predict one trace from the other by following local slopes of seismic events. Such a prediction is a key operation in the method of plane-wave destruction (Fomel, 2002). In fact, it is the minimization of prediction error that provides a criterion for estimating local slopes (Claerbout, 1992). For completeness, we include a review of plane-wave destruction in the appendix.

The prediction and update operators for a simple seislet transform are defined by modifying the biorthogonal wavelet construction in equations 4-5 as follows:

$$\mathbf{P[e]}_k = \left(\mathbf{S}_k^{(+)}[\mathbf{e}_{k-1}] + \mathbf{S}_k^{(-)}[\mathbf{e}_{k}]\right)/2$$ (12)

$$\mathbf{U[r]}_k = \left(\mathbf{S}_k^{(+)}[\mathbf{r}_{k-1}] + \mathbf{S}_k^{(-)}[\mathbf{r}_{k}]\right)/4\;,$$ (13)

where $\mathbf{S}_k^{(+)}$ and $\mathbf{S}_k^{(-)}$ are operators that predict a trace from its left and right neighbors correspondingly by shifting seismic events according to their local slopes. The predictions need to operate at different scales, which, in this case, mean different separation distances between the traces. Equations 12-13, in combination with the forward and inverse lifting schemes 1-2 and 6-7, provide a complete definition of the 2-D seislet transform.

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Figure 2. Synthetic seismic image (a) and local slopes estimated by plane-wave destruction (b).

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Figure 3. Wavelet transform (a) and seislet transform (b) of the synthetic image from Figure 2.

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Figure 4. Transform coefficients sorted from large to small, normalized, and plotted on a decibel scale. Solid line: seislet transform. Dashed line: wavelet transform.

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Figure 5. Randomly selected representative basis functions for wavelet transform (a) and seislet transform (b).

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Figure 6. Synthetic image reconstruction using only 1% of significant coefficients (a) by inverse wavelet transform (b) by inverse seislet transform. Compare with Figure 2a.

Figure 2a shows a synthetic seismic image from Claerbout (2008). After estimating local slopes from the image by plane-wave destruction (Figure 2b), we applied the 2-D seislet transform described above. The transform is shown in Figure 3b and should be compared with the corresponding wavelet transform in Figure 3a. Apart from the fault and unconformity regions, where the image is not predictable by continuous local slopes, the 2-D seislet transform coefficients are small, which enables an effective compression. In contrast, the wavelet transform has small residual coefficients at fine scales but develops large coefficients at coarser scales. Figure 4 shows a comparison between the decay of coefficients (sorted from large to small) between the wavelet transform and the seislet transform. A significantly faster decay of the seislet coefficients is evident.

Effectively, the wavelet transform in this case is equivalent to the 2-D seislet transform with the erroneous zero slope. Figure 5 shows example basis functions for the wavelet and 2-D seislet transform used in this example. If using only a small number of the most significant coefficients, the wavelet transform fails to reconstruct the most important features of the original image, while the 2-D seislet transform achieves an excellent reconstruction (Figure 6). We use the method of soft thresholding (Donoho, 1995) for selecting the most significant coefficients.

Seislet transform and seislet frame