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For completeness, I conclude this section with two simple examples of
forward interpolation in seismic data processing.
Figure shows a 3D impulse response of Stolt migration
Stolt (1978), computed by using 2point linear
interpolation and 8point Bspline interpolation. As noted by
Ronen (1982) and Harlan (1982),
inaccurate interpolation may lead to spurious artifact events in
Stoltmigrated image. Indeed, we see several artifacts for the image
with linear interpolation (the left plots in Figure .)
The artifacts are removed by a more accurate interpolation method (the
right plots in Figure .)
stolt
Figure 25 Stolt migration impulse response. Left: using
linear interpolation. Right: using seventhorder Bspline
interpolation. Migration artifacts are removed by a more accurate
forward interpolation method.
Another simple example is radial trace transform
Ottolini (1982) Figure shows a land shot
gather contaminated by nearly radial groundroll. As discussed by
Claerbout (1983), Henley (1999), and
Brown and Claerbout (2000), one can effectively eliminate
groundroll noise by applying radial trace transform, followed by
highpass 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 groundroll noise and the transform error after
going back to the original domain. Comparing the results of using
linear and thirdorder Bspline interpolation, we see once again that
the transform artifacts are removed with a more accurate interpolation
scheme.
radialdat
Figure 26 Groundrollcontaminated 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 thirdorder
Bspline interpolation.
Next: Inverse Interpolation and Data
Up: Forward Interpolation
Previous: Beyond Bsplines
Stanford Exploration Project
9/5/2000