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Synthetic data -- irregular sampling

Going back to the five steps of the butterfly algorithm, it is clear that the input data $ g(\mathbf {k})$ is only involved at the very first step. Besides, for every $ (A,B)$ the operation connecting $ g(\mathbf {k})$ and $ \delta_t^{AB}$ amounts to a matrix-vector multiplication (see equation 23), which does not at all require the input data to be uniformly distributed (the same argument applies to the output data $ u(\mathbf {x})$ ). Therefore, our algorithm can be easily extended to handle the following problem:

$\displaystyle (Rd)(\tau,p)=\iint d(\sqrt{\tau^2+p^2(h_1^2+h_2^2)},h_1,h_2)\,dh_1\,dh_2,$ (28)

where $ d(t,h_1,h_2)$ is a 3D function. All we need is to introduce a new variable for the absolute offset $ h=\sqrt{h_1^2+h_2^2}$ , and reorder the values $ d(t,h_1,h_2)$ according to $ h$ . Figure 12 shows such synthetic data sampled on $ N_t=1000$ , $ N_{h_1}=N_{h_2}=128$ . The output is obtained on $ N_{\tau }=1000$ , $ N_p=128$ . The fast algorithm (Figure 13) runs in only 1.67 s for $ N=64$ , $ q_{k_1}=q_{k_2}=q_{x_1}=q_{x_2}=5$ (here the range of $ \Phi=f\sqrt{\tau^2+p^2(h_1^2+h_2^2)}$ is about 162), while the velocity scan (Figure 14) takes more than 125 s.

data-4
data-4
Figure 12.
3D synthetic CMP gather. $ N_t=1000$ , $ N_{h_1}=N_{h_2}=128$ . $ \Delta t=0.004$ s, $ \Delta h_1=\Delta h_2=0.08$ km.
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fmod-4
fmod-4
Figure 13.
$ N_{\tau }=1000$ , $ N_p=128$ . Output of the fast butterfly algorithm applied to the synthetic data in Figure 12. $ N=64$ , $ q_{k_1}=q_{k_2}=q_{x_1}=q_{x_2}=5$ . CPU time: 1.67 s. Purple curve overlaid is the true slowness.
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dimod-4
dimod-4
Figure 14.
$ N_{\tau }=1000$ , $ N_p=128$ . Output of the velocity scan applied to the synthetic data in Figure 12. CPU time: 125.54 s. Purple curve overlaid is the true slowness.
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Next: Field data Up: Numerical examples Previous: Synthetic data

2013-07-26