Intermediate Wavenumber Resolution with Interbed Multiples

Figure 4.9 illustrates that transmission tomography (smearing residuals along rabbit ears) fills in the low wavenumber part of the spectrum, while reflection migration (smearing residuals along ellipse) fills in the high wavenumbers. Now I show that interbed multiples can fill in some of the intermediate wavenumbers denoted by the blue dots.

Figure 4.12a depicts the interbed multiple rays for a thin-bed model with a diffractor at the lower interface. Each order of the multiple will be associated with a different mirror ray, where the depth of the mirror scatterer deepens with the order of the multiple. Therefore the raypath lengthens with order of multiple, and the wavepath thickens as well. I conclude that the mirror wavepath that intersects the thin bed^4.6 thickens progressively with the order of the multiple, and so should fill in some of the ``intermediate wavenumber'' gap (Jannane et al., 1989) illustrated by the blue dots in Figure 4.9.

The above analysis can be quantified as in the previous sections by analyzing the model resolution function. In this case, the forward modeling kernel $G({\bf {g}}\vert{\bf {x}})G({\bf {x}}\vert{\bf {s}})$ is replaced by one that generates an internal multiple rather than a direct wave or primary reflection^4.7. The phase term in the Green's function will be replaced by a summation of times corresponding to each leg of the raypaths seen in Figure 4.12a. The migration kernel is also modified by terms that will image the internal multiple to one of its bounce points in the thin layer.