We randomly generate 10,000 models from uniform distributions defined with the following parameter ranges:
(21) | ||
fig2
Figure 2. A comparison of error histograms from 10 000 randomly generated models for the results from 1-D computations (Blue) and those from the proposed method (Orange) according to equation 17 in the single layer case with both vertical and horizontal velocity variations. We can observe relatively smaller errors in a larger proportion of the models from the latter, which corroborate our theoretical findings. |
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For an additional example, we consider a similar setup with only horizontal velocity gradient (), which leads to the 1-D traveltime derivative simply being . The result of this experiment is shown in Figure 3 with the mean, the standard deviation, and the maximum error from the proposed approach equal to 0.24, 0.48 and 4.28 per cent in comparison with 1.41, 2.45 and 17.58 per cent from the usual 1-D computation. An analogous conclusion to the previous case can be drawn.
fig3
Figure 3. A comparison of error histograms from 10 000 randomly generated models for the results from 1-D computations (Blue) and those from the proposed method (Orange) according to equation 17 in the single layer case with only horizontal velocity variation. We can observe relatively smaller errors in a larger proportion of the models from the latter, which corroborate our theoretical findings. |
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