Splitting the parabolic equation

In discussing and solving the parabolic wave equation it is convenient to rearrange it to recognize the role of an averaged stratified medium of velocity of $\bar v(z)$and departures from it.

$$\begin{eqnarray} {\partial P \over \partial z } & = & i\,\omega\ \left( {1 \ove... ...hin\ lens} \quad\quad\quad \ +\quad {\rm diffraction} \nonumber\end{eqnarray}$$ (9)

The shift term in (9) commutes with the thin-lens term, that is, ABP=BAP. the shift term also commutes with the diffraction term because ACP=CAP. But the thin-lens term and the diffraction term do not commute with one another because $(BC-CB)P\ne 0$, because

$$0\quad\ne\quad (BC-CB)P \eq v(x,z)\ \left[ \left( -2 {d^2\... ...x,z)\over dx} {\partial\over\partial x} \right] \ P \quad\quad$$ (10)

Mathematicians look at the problem this way: Consider any fixed wave propagation angle so $vk_x/\omega$ is a constant. Now let frequency $\omega$ (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.