The parabolic wave-equation operator
can be split into two parts,
a complicated part called the
*diffraction*
or
*migration*
part, and an easy part called the
*lens*
part.
The **lens equation** applies a time shift that is a function of *x*.
The **lens equation** acquires its name because it acts just like
a thin optical lens when a light beam enters on-axis (vertically).
Corrections for nonvertical incidence
are buried somehow in the diffraction part.
The **lens equation** has an analytical solution,
namely, .It is better to use this analytical solution than to use a finite-difference
solution because there are no approximations in it to go bad.
The only reason the **lens equation** is mentioned at all
in a chapter on finite differencing
is that the companion diffraction equation
must be marched forward along with the **lens equation**,
so the analytic solutions are marched along in small steps.

12/26/2000