Time migration may be thought of as hyperbola summation or semi-circle
superposition Schneider (1978).
In order to illustrate the kinematics of time migration when there
are lateral velocity variations, we first
superimpose time migration semi-circles along the diffraction curve of
Figure 2.
This geometrical construction is shown in
Figure 3, where
the radius of each circle is determined by the
RMS velocity that would be derived from a well drilled straight
down to the diffractor from the surface
at the diffractor location. This value is 8.846 kft/sec and is what we call
as the well RMS velocity. This is the migration velocity used throughout
this paper.
The points where the semi-circles add constructively
form the time-migrated image of the point diffractor. Figure 4
is an enlarged view showing
the semi-circle superposition near the apex of the *true-diffraction
curve*.
The envelope of semi-circles is displaced to the right of the true diffractor
position at 8 kft
and forms a distinctive cusp shape that opens to the right.
This shape is what we call the ``plume.'' The plume shows that the
various dip components of a point diffractor end up at different positions
on the time-migrated image. Thus the plume clearly shows that mis-focusing
occurs when time migration is applied to a medium with lateral velocity
variations.

Figure 3

Figure 4

The plume shape can be qualitatively
explained by examining the kinematics of time migration. For the left side
of the *true-diffraction curve* the migration velocity is too fast, so that
limb of the curve is overmigrated. For the right side of the
*true-diffraction curve* the migration velocity is too slow, so that limb
of the curve is undermigrated.

11/17/1997