Kirchhoff migration

Remarks

Example: zero-offset data ($s = r$) over a constant speed $c$, with one point scatterer at $(x_0, z_0)$ and receivers at $(x, 0)$. Then $\tau(s, x) + \tau(x, r) = (2/c)\sqrt{(x - x_1)^2 + z_1^2}$ for an image point $(x_1, z_1)$. For $(x_1, z_1) = (x_0, z_0)$ this curve coincides with the event in the data, so all terms of the sum add in phase and $I$ is large. For other image points the curve crosses the event at most at a few traces, so $I$ is small. Non-example: Stolt migration maps the data in the Fourier domain by a change of variables in frequency and wavenumber. It does not sum along travel-time curves, so it is not Kirchhoff migration. In its basic form Kirchhoff migration uses one travel time per pair of points, so it handles poorly media in which waves reach a point along several paths.