A fast butterfly algorithm for generalized Radon transforms |
We start with a simple 2D example of square sampling. Figure 3 is a synthetic CMP gather sampled on . Figure 4 shows the absolute value of its Fourier transform on time axis. These band-limited data allow us to shorten the computational range for , which can be crucial as depends on this range. In model space, the sampling sizes are chosen as . Figure 5 is the output of the fast butterfly algorithm for , (here the range of is about 125). Figure 6 is the output of the velocity scan. The two methods yield nearly the same results. The fast algorithm runs in only s of CPU time, while the velocity scan takes about 37 s. In Figure 7, we plot the difference between the results of the fast algorithm and the direct evaluation of equation 3, where the relative error is . For reference, if we let and run the same test, the error decreases to and the running time is 3.63 s.
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Figure 3. 2D synthetic CMP gather. . s, km. |
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Figure 4. The Fourier transform (absolute value) on time axis of the synthetic data in Figure 3. |
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Figure 5. . Output of the fast butterfly algorithm applied to the synthetic data in Figure 3. , . CPU time: 1.75 s. Purple curve overlaid is the true slowness. |
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Figure 6. . Output of the velocity scan applied to the synthetic data in Figure 3. CPU time: 37.23 s. Purple curve overlaid is the true slowness. |
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Figure 7. Difference between the results of the fast algorithm and the direct evaluation of equation 3 plotted at the same scale as in Figure 5. |
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A fast butterfly algorithm for generalized Radon transforms |