    OC-seislet: seislet transform construction with differential offset continuation  Next: Simple example Up: Theoretical basis Previous: The lifting scheme for

## 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):   (5)   (6)

where and 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: (7) (8) (9) (10)

where the subscripts and represent the first and the second stage. , , , and are defined numerically as follows:            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: (11) (12)

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

2013-07-26