The semi-circle superposition method allows us to visualize how the plume is
built up from constructive interference. It does not, however, offer a
convenient way to numerically compute the actual shape of the plume.
A more effective way to compute the time-migration response to
a point diffractor
is by summing along
hyperbolas Schneider (1978).
The analytical technique for summing along hyperbolas is called the ``common-
tangent'' construction.
It relies on the fact that the main contribution to the sum
comes from the
place where a time-migration hyperbola is tangent to the
*true-diffraction curve* of
Figure 2. The event at the tangency point is positioned
by time migration to the apex of the time-migration hyperbola.

The time migration hyperbola is given by:

(1) |

(2) |

(3) |

This
procedure is then carried out for each point along the *true-diffraction
curve* to yield the complete response of time migration to a point
diffractor.
Details of the algebra of
this computation are given in the Appendix and
the results are shown in Figure 5. In generating this
figure we have again used the vertical RMS well velocity that was
used in Figure 3.
Note that
the plume envelope in
Figure 4 matches the curve in Figure 5 exactly.
Thus the two ways of geometrically constructing
the time-migration response to lateral velocity variation are consistent with
each other.

Figure 5

11/17/1997