Forward interpolation

Seismic applications of 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
Figure 25. 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.

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 Figure 26. Ground-roll-contaminated shot gather used in a radial transform test

radial
Figure 27. 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.

Forward interpolation