Kirchhoff migration

**Seismic Data Analysis**
Series	Investigations in Geophysics
Author	Öz Yilmaz
DOI	http://dx.doi.org/10.1190/1.9781560801580
ISBN	ISBN 978-1-56080-094-1
Store	SEG Online Store

Dictionary entry for Kirchhoff migration (edit)
<translate> </translate> <translate> Migration (q.v.) by integrating along diffraction curves, in effect integrating with the Kirchhoff equation (q.v.) and placing the results at the crests of the diffraction curves. See Sheriff and Geldart (1995, 327–329). </translate>

Claerbout ^[1] uses the harbor example in Figure 4.1-9 to describe the physical principles of migration. Assume that a storm barrier exists at some distance z₃ from the beach and that there is a gap in the barrier. Imagine a calm afternoon breeze that comes from the ocean as a plane incident wave. The wavefront is parallel to the storm barrier. As we walk along the beach line, we see a wavefront different from a plane wave. The gap on the storm barrier has acted as a secondary source and generated the semicircular wavefront that is propagating toward the beach.

If we did not know about the storm barrier and the gap, we might want to lay out a receiver cable along the beach to record the approaching waves. This experiment is illustrated in Figure 4.1-10 with the recorded time section. Physicists call the gap on the barrier a point aperture. It is somewhat similar to a point source, since both generate circular wavefronts. However, the amplitudes on the wavefront that propagate outward from a point source are isotropic, while those from a point aperture are angle-dependent. The point aperture on the barrier acts as a Huygens’ secondary source.

From the beach experiment, we find that Huygens’ secondary source responds to a plane incident wave and generates a semicircular wavefront in the x − z plane. The response in the x − t plane is the diffraction hyperbola shown in Figure 4.1-11.

Imagine that the subsurface consists of points along each reflecting horizon that behave much as the gap on the storm barrier. From Figure 4.1-12, these points act as Huygens’ secondary sources and produce hyperbolic traveltime trajectories. Moreover, as the sources (the points on the reflecting interface) get closer to each other, superposition of the hyperbolas produces the response of the actual reflecting interface (Figure 4.1-13). In terms of the harbor example, this is like assuming that the barrier is wiped out by a storm so that the primary incident plane wave reaches the beach without modification. The diffraction hyperbolas, which are caused by sharp discontinuities at both ends of the reflector in Figure 4.1-13, remain. These hyperbolas are equivalent to diffractions seen at fault boundaries on stacked sections.

In summary, we find that reflectors in the subsurface can be visualized as being made up of many points that act as Huygens’ secondary sources. We also find that the zero-offset section consists of a superposition of the many hyperbolic traveltime responses. Moreover, when there are discontinuities (faults) along the reflector, diffraction hyperbolas often stand out.

Figure 4.1-9 The gap in the barrier acts as Huygens’ secondary source, causing the circular wavefronts that approach the beach line. Adapted from Claerbout ^[1].
Figure 4.1-10 Waves recorded along the beach generated by Huygens’ secondary source (the gap in the barrier in Figure 4.1-9) have a hyperbolic traveltime trajectory.
Figure 4.1-11 A point that represents a Huygens’ secondary source (a) produces a diffraction hyperbola on the zero-offset time section (b). The vertical axis in this section is two-way time, while the vertical axis in the time section in Figure 4.1-10 is one-way time.
Figure 4.1-12 Superposition of the zero-offset responses (b) of a discrete number of Huygens’ secondary sources as in (a).
Figure 4.1-13 Superposition of the zero-offset responses (b) of a continuum of Huygens’ secondary sources as in (a).