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 *t _{0}* yields sea-floor multiple reflections
at times .In the exploding-reflector conceptual model,
a sea-floor one-way travel-time depth
of

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 , it seems natural that these
boundary conditions should
be knowledge of *P* and at .But 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.

10/31/1997