Forward interpolation |

For completeness, I conclude this section with two simple examples of forward interpolation in seismic data processing. Figure 25 shows a 3-D impulse response of Stolt migration (Stolt, 1978), computed by using 2-point linear interpolation and 8-point B-spline interpolation. As noted by Ronen (1982) and Harlan (1982), inaccurate interpolation may lead to spurious artifact events in Stolt-migrated images. Indeed, we see several artifacts in the image with linear interpolation (the left plots in Figure 25). The artifacts are removed if we use a more accurate interpolation method (the right plots in Figure 25).

stolt
Stolt-migration impulse response. Left: using
linear interpolation. Right: using seventh-order B-spline
interpolation. Migration artifacts are removed by a more accurate
forward interpolation method.
Figure 25. |
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Another simple example is the radial trace transform (Ottolini, 1982). Figure 26 shows a land shot gather contaminated by nearly radial ground-roll. As discussed by Claerbout (1983), Henley (1999,2000), and Brown and Claerbout (2000b,a), one can effectively eliminate ground-roll noise by applying a radial trace transform followed by high-pass filtering and the inverse radial transform. Figure 27 shows the result of the forward radial transform of the shot gather in Figure 26 in the radial band of the ground-roll noise and the transform error after we go back to the original domain. Comparing the results of using linear and third-order B-spline interpolation, we see once again that the transform artifacts are removed with a more accurate interpolation scheme.

radialdat
Ground-roll-contaminated shot
gather used in a radial transform test
Figure 26. | |
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radial
Radial trace transform results. Top: radial
trace domain. Bottom: residual error after the inverse transform.
The error should be zero in a radial band from 0 to 0.65 km/s radial
velocity. Left: using linear interpolation. Right: using third-order
B-spline interpolation.
Figure 27. |
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Forward interpolation |

2014-02-21