The wavefront synthesis algorithm explained in the previous chapter implicitly assumes that a data set is collected with source and receiver spacings short enough to avoid spatial aliasing and with a large receiver cable long enough to avoid artifacts caused by a truncated aperture. In practice, however, the data are usually collected with a finite source spacing and a finite receiver length, as well as some missing near offset traces. The former can introduce spatial aliasing, while the latter causes the problems of end-effects.
If we are only interested in structural imaging, weak or broken images are not a significant problem. Such image artifacts will disappear after the stacking of several images generated by different plane-wave synthesis. If we are interested in angle-dependent reflectivity, however, each image from a different plane-wave needs to be analyzed separately. Therefore, the missing traces should be interpolated to produce a complete image without artifacts in it.
Fowler 1985 proposes a method that finds zero-offset traces by fitting a polynomial to the amplitudes as a function of offset after NMO correction. However, this method requires accurate stacking velocity before interpolation. In this chapter, I introduce an interpolation scheme for missing traces in near and far offset, which does not require velocity estimation for each events. First I review the forward model of a shot experiment and wave stack that allows the wavefront synthesis. I then present the detailed algorithm of the interpolation scheme, along with some examples of synthetic and Marmousi data sets.