Introduction and Method

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

$$m({\bf {x}}) = \sum_{\omega} \sum_r \alpha(\omega,{\bf {x}},{\bf {r}},{\bf {s}})... ...f {x}})^*]}^{migration~kernel} ~\overbrace{ D({\bf {r}}\vert{\bf {s}})}^{CSG},$$ (9)

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 Kirchhoff migration (KM). The KM point scatterer response of $\sum_{\omega} \sum_r \alpha(\omega,{\bf {x}},{\bf {r}},{\bf {s}})~G({\bf {r}}\vert{\bf {x}})^*G({\bf {x}}\vert{\bf {s}})^*D({\bf {r}}\vert{\bf {s}})$ is computed by specifying a trial image point at ${\bf {x}}$ and summing the energy in the CSG along the hyperbola-like colored curves in Figure a. 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.