To the interpreting
*geologist,*
lateral velocity variation produces a strange
distortion in the seismic section.
And the distortion is worse than it looks.
The
*geophysicist*
is faced with the challenge of trying to deal with
lateral velocity variation
in a quantitative manner.
First, how can reliable estimates
of the amount of lateral velocity variation be arrived at?
Then, do we dare use these estimates for reprocessing data?

Our studies of
*dip*
and
*offset*
have resulted in straightforward procedures to handle them,
even when they are simultaneously present.
Unfortunately, increasing lateral velocity variation
leads to increasing confusion--confusion we must try to overcome.
Strong lateral velocity variation overlies
the largest oil field in North America, Prudhoe Bay.
Luckily, however, we have many idealized examples that are easy to understand.
Any ``ultimate'' theory would have to
explain these examples as limiting cases.

Let us review.
The double-square-root equation presumably works if
the square roots are expanded and if we accept
the usual limitation of accuracy with angle.
Our problem with the DSR is that it merely tells us
how to migrate and stack
*
once the velocity is known.
*
Kjartansson's method of determining
the distribution of (some function of) *v*(*x*,*z*) assumes
straight rays, no dip, and a single, planar reflector.
On the other hand, stacking along with prestack partial migration
allows any scattering geometry but
enables determination of *v*(*z*) only under the presumption
that there is no lateral variation of velocity.
Clearly, there are many gaps.
We begin with comprehensible, special cases but
ultimately sink into a sea of confusion.

- Replacement velocity: freezing the water
- Lateral shift of the hyperbola top
- Phantom diffractor
- Wavefront healing
- Fault-plane reflection
- Misuse of v(x) for depth migration
- First-order effects, the lens term
- The migrated time section: an industry kludge

10/31/1997