In an operations environment, 3-D is much harder to cope with than 2-D.
Therefore, it may be expedient to suppose that 3-D migration
can be achieved merely by application of 2-D migration twice, once
in the *x*-direction and once in the *y*-direction.
The previous section would lead you to believe that such an expedient process
would result in a significant degradation of accuracy.
In fact, the situation is
*much*
better than might be supposed.
It has been shown by Jakubowicz and Levin [1983]
that, wonder of wonders, for a constant-velocity medium,
the expedient process is exact.

The explanation is this:
migration consists of more than downward continuation.
It also involves imaging, that is, the selection of data at *t*=0.
In principle, downward continuation is first completed,
for both the *x* and the *y* directions.
After that, the imaging condition is applied.
In the expedient process there are four steps:
downward continuation in *x*, imaging,
downward continuation in *y*, and finally
a second imaging.
Why it is that the expedient procedure gives the correct result
seems something of a puzzle, but the validity of the result
is easy to demonstrate.

First note that substitution of (7) into (8) gives (9) where

(7) | ||

(8) | ||

(9) |

The Jakubowicz justification is somewhat more mathematical, but may be paraphrased as follows. First note that substitution of (10) into (11) gives (12) where

(10) | ||

(11) | ||

(12) |

The validity of the Jakubowicz result goes somewhat beyond its proof. Our two-dimensional geophysicist may be migrating other offsets besides zero offset. (In chapter nonzero-offset data is migrated). If a good job is done, all the reflected energy moves up to the apex of the zero-offset hyperbola. Then the cross-plane migration can handle it if it can handle zero offset. So offset is not a problem. But can a good job be done of bringing all the energy up to the apex of the zero-offset hyperbola?

Difficulty arises when the velocity of the earth is depth-dependent,
as it usually is.
Then the Jakubowicz proof fails, and so does the expedient 3-D method.
With a 2-D survey you have the problem that the sideswipe planes
require a different migration velocity than the vertical plane.
Rays propagating to the side take longer to reach the high-velocity
media deep in the earth.
So sideswipes usually require a lower migration velocity.
If you really want to do three-dimensional migration with *v*(*z*),
you should forget about separation and do it the hard way.
Since we know how to transpose
the hard way really isn't much harder.

10/31/1997