Another way to the parabolic wave equation

Here we review the historic ``transformation method'' of deriving the parabolic wave equation.

A vertically downgoing plane wave is represented mathematically by the equation  

$$P(t,x,z) \ \eq \ P_0 \ \ e^{{-} \,i \omega \,(t \,-\, z/v) }$$ (54)

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.  

$$P(t,x,z) \ \eq \ Q(x,z) \ \ e^{{-} \, i \, \omega \, (t \, - \, z/v) }$$ (55)

Inserting (55) into the scalar wave equation $P_{xx} +\,P_{zz} \,=\, P_{tt} / v^2$ yields

$$\begin{eqnarray} {\partial^2 \ \over \partial x^2} \ \ Q \ \ +\ \ \left( {i \om... ...} \ \ \ +\ \ {\partial^2 Q \over \partial z^2} \ \ \ \ &=&\ \ \ 0\end{eqnarray}$$ (56)

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  

$${\partial Q \over \partial z} \ \eq {v \over \,-\,2\, i \omega} \ {\partial^2 Q \over \partial x^2}$$ (57)

I called equation (57) 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 (57) with respect to z and substitute the result back into equation (56) getting  

$${\partial^2 Q \over \partial x^2} \ \ +\ \ {2\, i \omega \o... ...i \omega} \ {\partial^3 Q \over \partial z \partial x^2} \eq 0$$ (58)

I named equation (58) 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 (53).

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.