Seismic migration

Remarks

Example: take zero-offset data over a medium with constant speed $c$. A point scatterer at $(x_0, z_0)$ produces an event at two-way time $$t(x) = \frac{2}{c} \sqrt{(x - x_0)^2 + z_0^2}.$$ This is a branch of a hyperbola in the $(x, t)$ plane. Migration sums the data along such curves, so the energy of this event focuses at $(x_0, z_0)$. Non-example: the unmigrated section, which plots each reflection at depth $c\, t/2$ directly below the recording point, is not migrated; it places the scatterer energy on the whole hyperbola. The image $F^{*} d$ has correct positions but distorted amplitudes, because $F^{*}$ is not the inverse of $F$; least-squares migration solves $\min_R \norm{F R - d}^2$ instead. Common methods are Kirchhoff migration, which sums along travel-time curves, and wave-equation methods such as reverse time migration.