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OC-seislet structure

We define the OC-seislet transform by specifying prediction and update operators with the help of the offset-continuation operator. Prediction and update operators for the OC-seislet transform are specified by modifying the biorthogonal wavelet construction in equations 2 and 4 as follows (Fomel and Liu, 2010; Fomel, 2006):

$\displaystyle \mathbf{P[e]}_k$ $\displaystyle =$ $\displaystyle \left(\mathbf{S}_k^{(+)}[\mathbf{e}_{k-1}] +
\mathbf{S}_k^{(-)}[\mathbf{e}_{k}]\right)/2$ (5)
$\displaystyle \mathbf{U[r]}_k$ $\displaystyle =$ $\displaystyle \left(\mathbf{S}_k^{(+)}[\mathbf{r}_{k-1}] +
\mathbf{S}_k^{(-)}[\mathbf{r}_{k}]\right)/4\;,$ (6)

where $ \mathbf{S}_k^{(+)}$ and $ \mathbf{S}_k^{(-)}$ are operators that predict the data record (a common-offset section) by differential offset continuation from its left and right neighboring common-offset sections with different offsets. Offset continuation operators provide the physical connection between data records. The theory of offset continuation is reviewed in Appendix A.

One can also employ a higher-order transform, for example, by using the template of the CDF 9/7 biorthogonal wavelet transform, which is used in JPEG-2000 compression (Lian et al., 2001). There is only one stage (one prediction and one update) for the CDF 5/3 wavelet transform, but there are two cascaded stages and one scaling operation for CDF 9/7 wavelet transform. Prediction and update operators for a high-order OC-seislet transform are defined as follows:

$\displaystyle \mathbf{P}_1[\mathbf{e}]_k=(\mathbf{S}_k^{(+)}[\mathbf{e}_{k-1}]+ \mathbf{S}_k^{(-)}[\mathbf{e}_{k}]) \cdot {\alpha}\;,$ (7)

$\displaystyle \mathbf{U}_1[\mathbf{r}]_k=(\mathbf{S}_k^{(+)}[\mathbf{r}_{k-1}]+ \mathbf{S}_k^{(-)}[\mathbf{r}_{k}]) \cdot {\beta}\;,$ (8)

$\displaystyle \mathbf{P}_2[\mathbf{e}]_k=(\mathbf{S}_k^{(+)}[\mathbf{e}_{k-1}]+ \mathbf{S}_k^{(-)}[\mathbf{e}_{k}]) \cdot {\gamma}\;,$ (9)

$\displaystyle \mathbf{U}_2[\mathbf{r}]_k=(\mathbf{S}_k^{(+)}[\mathbf{r}_{k-1}]+ \mathbf{S}_k^{(-)}[\mathbf{r}_{k}]) \cdot {\delta}\;,$ (10)

where the subscripts $ 1$ and $ 2$ represent the first and the second stage. $ \alpha$ , $ \beta$ , $ \gamma$ , and $ \delta$ are defined numerically as follows:
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle -1.586134342,$  
$\displaystyle \beta$ $\displaystyle =$ $\displaystyle -0.052980118,$  
$\displaystyle \gamma$ $\displaystyle =$ $\displaystyle 0.882911076,$  
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle 0.443506852.$  

One can combine equations 1, 3, 7, and 8 to finish the first stage, and repeatedly process the result by using equations 1, 3, 9, and 10. The scale normalization factors correspond to the CDF 9/7 biorthogonal wavelet transform (Daubechies and Sweldens, 1998). Scaling and coefficients are as follows:

$\displaystyle \mathbf{e}=\mathbf{e} \cdot K\;,$ (11)

$\displaystyle \mathbf{o}=\mathbf{o} \cdot (1/K)\;,$ (12)

where $ K$ = 1.230174105.

We used the high-order version of OC-seislet transform to process the synthetic and field data examples used in this paper.


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Next: Simple example Up: Theoretical basis Previous: The lifting scheme for

2013-07-26