OC-seislet: seislet transform construction with differential offset continuation

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):

$$\mathbf{P[e]}_k = \left(\mathbf{S}_k^{(+)}[\mathbf{e}_{k-1}] + \mathbf{S}_k^{(-)}[\mathbf{e}_{k}]\right)/2$$ (5)

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

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

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

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

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

where the subscripts $1$ and $2$ represent the first and the second stage. $\alpha$ , $\beta$ , $\gamma$ , and $\delta$ are defined numerically as follows:

$$\alpha = -1.586134342,$$

$$\beta = -0.052980118,$$

$$\gamma = 0.882911076,$$

$$\delta = 0.443506852.$$

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:

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

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

OC-seislet: seislet transform construction with differential offset continuation