(9) | ||

(10) |

Mathematicians look at the problem this way:
Consider any fixed wave propagation angle
so is a constant.
Now let frequency (and hence *k*_{x}) tend together to infinity.
The terms in *BCP* and *CBP* grow in proportion to the second power
of frequency,
whereas those in (*BC*-*CB*)*P* grow as lower powers.
There is however, a catch.
The material properties have a ``wavelength'' too.
We can think of (*dv*/*dx*)/*v* as a spatial wavenumber
for the material just as *k*_{x} is a spatial wavenumber for the wave.
If the material contains a step function change in its properties,
that is an infinite spatial frequency (*dv*/*dx*)/*v* for the material.
Then the (*BC*-*CB*)*P* terms dominate near the place where
one material changes to another.
If we drop the (*BC*-*CB*)*P* terms,
we'll get the transmission coefficient incorrect,
although everything would be quite fine everywhere else
except at the boundary.

A question is,
to what degree do the terms commute?
The problem is just that of
focusing a slide projector.
Adjusting the focus knob amounts to
repositioning the thin-lens term
in comparison to the free-space diffraction term.
There is a small range of knob positions over
which no one can notice any difference,
and a larger range over which
the people in the back row are not disturbed by misfocus.
Much geophysical data processing amounts to downward extrapolation of data.
The
**lateral velocity variation**

occurring in the **lens term**
is known only to a limited accuracy
and we often wish to determine *v*(*x*)
by the extrapolation procedure.

In practice it seems best to
forget the (*BC*-*CB*)*P* terms because we hardly ever know
the material properties very well anyway.
Then we split,
doing the shift and the thin-lens part analytically
while doing the diffraction part by a numerical method.

12/26/2000