(4) | ||
Results To test the methodology I decided to start with a structurally simple 2-D line from a land dataset from Columbia provided by Ecopetrol. Figure 1 shows the estimated velocity for the data. Note how it is generally v(z) with some deviation, especially in the lower portion of the image. Figure 2 shows the result of performing split-step phase shift migration and Figure 3 shows the resulting angle gathers Sava (2000). Note how the image is generally well focused and the gathers with some slight variation below three kilometers at x=3.5. Figure 4 shows the moveout of the gathers in Figure 3. Note the traditional `W' pattern associated with the velocity anomaly can be seen in cross-section at depth.
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Figure 1 Initial velocity model. |
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Figure 2 Initial migration using the velocity shown in Figure 1. |
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Figure 3 Every 10th migrated gather using the velocity shown in Figure 1. |
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Figure 4 Moveout of the gathers shown in Figure 3. |
To start we need to solve the problem without accounting for model variance. If we solve for using fitting goals (4) our updated velocity is shown in Figure 5. The change of the velocity is generally minor, with an increase in the high velocity structure at x=3.5, z=3.2. The resulting image and migration gathers are shown in Figures 6 and 7. The resulting image is slightly better focused below the anomaly and the migration gathers are, as expected, a little flatter.
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Figure 5 New velocity obtained by inverting for using fitting goals (4). |
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Figure 6 New image obtained by inverting for using fitting goals (4) using the velocity shown in Figure 5. |
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Figure 7 New gathers obtained by inverting for using fitting goals (4) using the velocity shown in Figure 5. |
If we apply equation (3) using the when estimating our improved velocity model we can find the right amount of noise to add to our fitting goals. We can now resolve for accounting for the model variability. Figure 8 shows four such realizations. Note that they have the same general structure as seen in Figure 5 but within additional texture that is accounted for by covariance description. If we migrate with these new velocity models we get the images and migrated gathers shown in Figures 9 and 10. In printed form these images appear identical, or close to identical. If watched as a movie, amplitude differences can be observed.