h_{g}^{2} = h - ^{2}b,
^{2} |
(10) |

(11) |

(12) |

Figure 2

Each fixed sample in (*m*,*h*,*t*) space is mapped to
a curve in (*m*+*b*,*h*_{g},*t*) space. Here, the curve lies in a radial
plane, within the data volume (*m*,*h*,*t*), defined by . It is
possible to show Gardner (1993) that the mapping can be accomplished by (*f*,*k*) migration
of the radial-plane data with the "velocity" .Thus, the computational requirements for velocity-independent DMO are . Although
it is not within the scope of this article, Gardner 1993 also provides a
log-stretch-(*f*,*k*) version of this DMO when *h* (common-offset section) is fixed.

After DMO, the new output volume consists of zero-offset but
unmigrated data. Arrival times are specified by (11)
and as a result are completely hyperbolic functions of *h*_{g}. The new data set can
be NMO corrected, and stacked, to produce a zero-offset section.
Any form of standard time migration may then be applied to
produce a proper subsurface image.

Note also that if each new equivalent offset is migrated after NMO but prior to stack, then each point on the diffraction curve of Figure will result in one of the circles indicated in Figure . After stack, the envelope of the circles, as also indicated in Figure , will be the equal-traveltime ellipse of (2). Since the radii of each of these circles is given by the velocity-independent equation (7), one can postulate that constant-velocity-prestack imaging requires no initial velocity information.

Figure 3

10/10/1997