Schneider [1978] states the analytic representation for the Huygens secondary wavelet

(19) |

Equation (19) states the *two* -dimensional Huygens wavelet,
not the *three* -dimensional wavelet
(which differs in some minor aspects).
Although waves from point sources are mainly spherical,
the focusing of bent layers is mainly a two-dimensional focusing,
i.e.,
bent layers are more like cylinders than spheres.

You might wonder why anyone would prefer approximations, given the exact inverse transform (19). The difficulty of graphing (19) shows up in practice as a difficulty in convolving it with data. That is why early Kirchhoff migrations were generally recognizable by precursor noise above a flat sea floor. Chapters , , and , are largely devoted to extensions of (19) that are valid with variable velocity and that are better representations on a data mesh.

In the Fourier domain, the Huygens secondary source function
is simple and smooth.
It is a straightforward matter to evaluate the function on a rectangular mesh
and inverse transform with `ft1axis()` and `ft2axis()` .
Figure 8
shows the result on a 256 64 point mesh.

Figure 8

(In practice the mesh would be about 1024 256 or more, but the coarser mesh used here provides a plot of suitable detail). Because of the difficulty in plotting functions that resemble an impulsive doublet, a second plot of the time integral, (with gentle band limiting) is displayed in the lower part of Figure 8.

10/31/1997