Assume a wideband source, so
,
and
high frequencies so that the asymptotic Green's function
can be used:

where is the time for energy to propagate from the source at to the interrogation point at , and is the geometrical spreading term. Here we assume that the forward Fourier transform kernel is while the kernel for the inverse Fourier transform is .

Assuming a single
source and an homogeneous model with a buried scatterer (see Figure 3.3), and
substituting equation 3.8 into equation 3.6 yields

where is the transform of , and the terms associated with geometrical spreading and have been neglected in the last expression. The double time derivative sharpens up the image, which corrects for the image smoothing due to the summation over the geophone coordinates (as well as the smoothing from a possible summation over source coordinates if more than one shot gather is used).

Equation 3.9 says that at each image point is computed by summing the data residuals along a hyperbola-like curve, where the dashed hyperbola shown in Figure 3.3 is denoted by . Here denotes the propagation time from the source to the buried scatterer, and denotes the propagation time from the scatterer to the geophone at the surface. Equation 3.9 approximates the prestack Kirchhoff migration equation for a single source gather.

As an example, the top picture in
Figure 3.4 depicts the synthetic shot gather for a point scatterer
in a homogeneous medium, and the bottom picture depicts the result of applying
an equation similar to that of equation 3.9.
Note that the point scatterer in the bottom picture is well imaged.