Theory of Diffraction-stack Migration

A prestack migration image $m({\bf {x}})$ for a common shot gather (CSG) can be computed with the diffraction-stack migration formula in the frequency domain

$$m({\bf {x}}) = \sum_{\omega} \sum_r \alpha(\omega,{\bf {x}},{\bf {r}},{\bf {s}})... ...f {x}})^*]}^{migration~kernel} ~\overbrace{ D({\bf {r}}\vert{\bf {s}})}^{CSG},$$ (1)

where $D({\bf {r}}\vert{\bf {s}})$ is the CSG in the frequency domain for a shot at ${\bf {s}}$ and receiver at ${\bf {r}}$ , $G({\bf {x}}\vert{\bf {s}})$ is the background Green's function which is a solution to the acoustic Helmholtz equation for a source at ${\bf {s}}$ and observed at ${\bf {x}}$ , and $m({\bf {x}})$ is the migration image at the trial image point ${\bf {x}}$ . Here, the $G({\bf {x}}\vert{\bf {x}}')$ ( ${\bf {x}}'$ may be at ${\bf {s}}$ or ${\bf {r}}$ ) and $D({\bf {r}}\vert{\bf {s}})$ spectra depend on angular frequency $\omega$ , but its notation is silent. The preconditioning function $\alpha(\omega,{\bf {x}},{\bf {r}},{\bf {s}})$ is defined by the user, and can compensate for the bandlimited source wavelet, obliquity factor, acquisition footprint, and geometrical spreading.

If $G({\bf {x}}\vert{\bf {x}}')$ is the asymptotic Green's function computed by ray tracing and only accounts for single scattering events, then equation  is the general formula for diffraction-stack migration, which is also known as Kirchhoff migration (KM). The KM point scatterer response of $\sum_{\omega} \sum_r \alpha(\omega,{\bf {x}},{\bf {r}},{\bf {s}})~G({\bf {x}}\vert{\bf {s}})^* G({\bf {r}}\vert{\bf {x}})^* D({\bf {r}}\vert{\bf {s}})$ is computed by specifying a trial image point at ${\bf {x}}$ and summing the energy in the CSG along the hyperbola-like red curve in Figure . This summed energy value is placed at ${\bf {x}}$ and the result is the migration image $m({\bf {x}})$ , and only accounts for primary reflections in the data.

This migration operation can also be interpreted as a dot-product of the kernel fingerprint $G({\bf {x}}\vert{\bf {s}})G({\bf {r}}\vert{\bf {x}})$ with the CSG fingerprint $D({\bf {r}}\vert{\bf {s}})$ (Schuster, 2002), and the result is $m({\bf {x}})$ . If the trial image point is at an actual scatterer, then the fingerprints of the CSG and migration kernel will be a good match and the dot-product will return a large value. If the trial image point is far from any reflector, then the CSG and kernel fingerprints will be mismatched and the dot-product will yield a low magnitude. This description defines the dot-product interpretation of migration, and I will now refer to $G({\bf {x}}\vert{\bf {s}})G({\bf {r}}\vert{\bf {x}})$ and $D({\bf {r}}\vert{\bf {s}})$ as kernel and data fingerprints, respectively.

Figure: Simple diffraction-stack migration operator (red hyperbola) superimposed on the data, which only accounts first arrival scattering information.