Application to lateral velocity variation

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$^\circ$ wave-extrapolation equation in inhomogeneous media. Adding and subtracting a constant term to the unretarded $15^\circ$ equation gives us

$$\begin{eqnarray} {\partial U \over \partial z}\ \ \ &=& \ \ \ \left[ {i \omega \... ...\ retardation\ +\ thin\ lens\ +\ diffraction) \ {\it U}} \nonumber\end{eqnarray}$$ (4)

Inspection of (4) shows that the retardation term commutes with the thin-lens term and with the free-space diffraction term. But the thin-lens term and the diffraction term do not commute with one another. In practice it seems best to split, doing the thin-lens part analytically and the diffraction part by the Crank-Nicolson method. Then stability is assured because the stability of each separate problem is known. Also, the accuracy of the analytic solution is an attractive feature. Now the question is, to what degree do these two 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 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.