Seislet transform and seislet frame

Discussion

Data size	1-D FFT	1-D DWT	1-D seislet	2-D seislet
$1024 \times 1024$	0.06	0.03	0.17	1.03
$512 \times 512$	0.02	0.01	0.04	0.22

Table 1. CPU times (in seconds) for different transforms.

How efficient are the proposed algorithms? The CPU times, in our implementation, are shown in Table 1. They confirm that, while the seislet transform and frame can be more expensive than FFT or DWT, they are still comfortably efficient in practice. In applications of the 2-D seislet transform, the main cost may not be in the transform itself but in iterative estimation of the slope fields. In practical large-scale applications, it is advantageous to break the input data in parts and process them in parallel.

How effective are the seislet transform and frame in compressing seismic data? In the case of the 2-D seislet transform that requires a slope field, it appears that one would need to store this field in addition to the compressed data. However, since we force the estimated slopes to be smooth, the slope field can be easily compressed with one of the classic compression algorithms. Consider the example in Figure 2. Suppose that we apply lossy compression and require 99% of the energy to be preserved. The seislet transform compression ratio in this case is less that 1% while the corresponding wavelet transform ratio is 26%. Applied to the smooth slope field from Figure 2b, the wavelet transform compresses it to about 0.1%. This example shows that compressing seismic data with the seislet transform and the corresponding slope field with the wavelet transform can be significantly more effective that trying to compress seismic data with the wavelet transform.

Seislet transform and seislet frame