Seismic applications of forward interpolation

For completeness, I conclude this section with two simple examples of forward interpolation in seismic data processing. Figure  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 ). The artifacts are removed if we use a more accurate interpolation method (the right plots in Figure ).

 

stolt
Figure 27 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  shows a land shot gather contaminated by nearly radial ground-roll. As discussed by Claerbout (1983), Henley (1999, 2000), and Brown and Claerbout (2000a,b), one can effectively eliminate ground-roll noise by applying a radial trace transform followed by high-pass filtering and the inverse radial transform. Figure  shows the result of the forward radial transform of the shot gather in Figure  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 28 Ground-roll-contaminated shot gather used in a radial transform test

 

radial
Figure 29 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.