A fast butterfly algorithm for generalized Radon transforms

Field data

We now consider a 2D field seismic gather shown in Figure 15. Its Fourier transform is shown in Figure 16. Due to the comparatively wide frequency bandwidth, $N$ cannot be chosen too small (here the range of $\Phi=f\sqrt{\tau^2+p^2h^2}$ is about 306). The input sampling sizes are $N_t=1500$ , $N_h=240$ , while the output sizes are chosen as $N_{\tau }=1500$ , $N_p=800$ . Although this small dataset is not very suitable for showcasing the fast algorithm, our method runs in 6.62 s for $N=128$ , $q_{k_1}=q_{x_1}=7$ , $q_{k_2}=q_{x_2}=5$ (Figure 17), still outperforming the velocity scan which takes about 10 s (Figure 18). Note that the simplest interpolation is used in the velocity scan, any other higher order interpolation should take longer computation time.

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Figure 15. 2D field CMP gather. $N_t=1500$ , $N_h=240$ . $\Delta t=0.004$ s, $\Delta h=0.0125$ km.

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Figure 16. The Fourier transform (absolute value) on time axis of the field data in Figure 15.

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Figure 17. $N_{\tau }=1500$ , $N_p=800$ . Output of the fast butterfly algorithm applied to the field data in Figure 15. $N=128$ , $q_{k_1}=q_{x_1}=7$ , $q_{k_2}=q_{x_2}=5$ . CPU time: 6.62 s.

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Figure 18. $N_{\tau }=1500$ , $N_p=800$ . Output of the velocity scan applied to the field data in Figure 15. CPU time: 9.91 s.

A fast butterfly algorithm for generalized Radon transforms