Why the wave equation is rarely used for migration

Life would be simpler if migration could be done with the scalar wave equation instead of the paraxial equation. Indeed, migration can be done with the scalar wave equation, and there are some potential advantages (Hemon [1978], Kosloff and Baysal [1983]). But more than 99% of current industrial migration is done with the paraxial equation.

The main problem with the scalar wave equation is that it will generate unwanted internal multiple reflections. The exploding-reflector concept cannot deal with multiple reflections. Primary reflections can be modeled with only upcoming waves, but multiple reflections involve both up and downgoing paths. The multiple reflections observed in real life are completely different from those predicted by the exploding-reflector concept. For the sea-floor multiple reflection, a sea-floor two-way travel-time depth of t₀ yields sea-floor multiple reflections at times $2t_0 , \ 3t_0 , \ 4t_0 , ...$.In the exploding-reflector conceptual model, a sea-floor one-way travel-time depth of t₀ yields sea-floor multiple reflections at times $3t_0 , \ 5t_0 , \ 7t_0 , ...$.In building a telescope, microscope, or camera, the designer takes care to suppress backward reflected light because it creates background noise on the image. Likewise, in building a migration program we do not want to have energy moving around that does not contribute to the focused image. The scalar wave equation with space-variable coefficients will generate such energy. This unwanted energy is especially troublesome if it is coherent and migrates to a time when primaries are weak. It is annoying, as the reflection of a bright window seen on a television screen is annoying. So if you were trying to migrate with the scalar wave equation, you would make the velocity as smooth as possible.

Another difficulty of imaging with the scalar wave equation arises with evanescent waves. These are the waves that are exponentially growing or decaying with depth. Nature extrapolates waves forward in time, but we are extrapolating them in depth. Growing exponentials can have tiny sources, even numerical round-off, and because they grow rapidly, some means must be found to suppress them.

A third difficulty of imaging with the scalar wave equation derives from initial conditions. The scalar wave equation has a second depth z-derivative. This means that two boundary conditions are required on the z-axis. Since data is recorded at $z\,=\,0$, it seems natural that these boundary conditions should be knowledge of P and ${\partial P}/{\partial z}$ at $\,z=\,0$.But ${\partial P}/{\partial z}$ isn't recorded.

Luckily, in building an imaging device that operates wholly within a computer, we have ideal materials to work with, i.e., reflectionless lenses. Instead of the scalar wave equation of the real world we have the paraxial wave equation.