Meet the parabolic wave equation

At the time the parabolic equation was introduced to petroleum prospecting (1969), it was well known that ``wave theory doesn't work.'' At that time, petroleum prospectors analyzed seismic data with rays. The wave equation was not relevant to practical work. Wave equations were for university theoreticians. (Actually, wave theory did work for the surface waves of massive earthquakes, scales 1000 times greater than in exploration). Even for university workers, finite-difference solutions to the wave equation didn't work out very well. Computers being what they were, solutions looked more like ``vibrations of a drum head'' than like ``seismic waves in the earth.'' The parabolic wave equation was originally introduced to speed finite-difference wave modeling. The following introduction to the parabolic wave equation is via my original transformation method.

The difficulty prior to 1969 came from an inappropriate assumption central to all then-existing seismic wave theory, namely, the horizontal layering assumption. Ray tracing was the only way to escape this assumption, but ray tracing seemed to ignore waveform modeling. In petroleum exploration almost all wave theory further limited itself to vertical incidence. The road to success lay in expanding ambitions from vertical incidence to include a small angular bandwidth around vertical incidence. This was achieved by abandoning much known, but cumbersome, seismic theory.

A vertically downgoing plane wave is represented mathematically by the equation  

$$P(t,x,z) \ \eq \ P_0 \ \ e^{{-} \,i \omega \,(t \,-\, z/v) }$$ (2)

In this expression, P₀ is absolutely constant. A small departure from vertical incidence can be modeled by replacing the constant P₀ with something, say, Q(x,z), which is not strictly constant but varies slowly.  

$$P(t,x,z) \ \eq \ Q(x,z) \ \ e^{{-} \, i \, \omega \, (t \, - \, z/v) }$$ (3)

Inserting (3) into the scalar wave equation $P_{xx} +\,P_{zz} \,=\, P_{tt} / v^2$ yields

$$\begin{eqnarray} {\partial^2 \ \over \partial x^2} \ \ Q \ \ +\ \ \left( {i \om... ...} \ \ \ +\ \ {\partial^2 Q \over \partial z^2} \ \ \ \ &=&\ \ \ 0\end{eqnarray}$$ (4)

The wave equation has been reexpressed in terms of Q(x,z). So far no approximations have been made. To require the wavefield to be near to a plane wave, Q(x,z) must be near to a constant. The appropriate means (which caused some controversy when it was first introduced) is to drop the highest depth derivative of Q, namely, Q_zz. This leaves us with the parabolic wave equation  

$${\partial Q \over \partial z} \ \eq {v \over \,-\,2\, i \omega} \ {\partial^2 Q \over \partial x^2}$$ (5)

When I first introduced equation (5) for use in seismology, I thought its most important property was this: For a wavefield close to a vertically propagating plane wave, the second x-derivative is small, hence the z-derivative is small. Thus, the finite-difference method should allow a very large $\Delta z$ and thus be able to treat models more like the earth, and less like a drumhead. I soon realized that the parabolic wave equation is also just what is needed for seismic imaging because you can insert it in an equation like (1). (Curiously, equation (5) also happens to be the Schroedinger equation of quantum mechanics.)

I called equation (5) the $15^\circ$ equation. After using it for about a year I discovered a way to improve on it by estimating the dropped $\partial_{zz}$ term. Differentiate equation (5) with respect to z and substitute the result back into equation (4) getting  

$${\partial^2 Q \over \partial x^2} \ \ +\ \ {2\, i \omega \o... ...i \omega} \ {\partial^3 Q \over \partial z \partial x^2} \eq 0$$ (6)

I named equation (6) the $45^\circ$ migration equation. It is first order in $\partial_z$,so it requires only a single surface boundary condition, however, downward continuation will require something more complicated than equation (1).

The above approach, the transformation approach, was and is very useful. But people were confused by the dropping and estimating of the $\partial_{zz}$derivative, and a philosophically more pleasing approach was invented by Francis Muir, a way of getting equations to extrapolate waves at wider angles by fitting the dispersion relation of a semicircle by polynomial ratios.