A circumstance in which the degree of noncommutativity of two differential operators has a simple physical meaning and an obviously significant geophysical application is the so-called monochromatic 15 wave-extrapolation equation in inhomogeneous media. Adding and subtracting a constant term to the unretarded equation gives us

(4) | ||

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 variation of velocity occurring in the lens term
is known only to a limited accuracy.
The application could be to determine *v*(*x*) by the extrapolation procedure.

For long lateral spatial wavelengths the terms commute.
Then diffraction may proceed in ignorance
of the lateral variation in *v*.
At shorter wavelengths the diffraction
and lensing effects must be interspersed.
So the real issue is not merely
computational convenience but the interplay
between data accuracy and the possible range
for velocity in the underlying model.

10/31/1997