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Warping provides a mapping between different migration velocities
that is kinematically equivalent to velocity continuation for
plane-wave events.
Fomel (1997) showed this directly from the zero-offset
velocity continuation equation, but it is apparent intuitively if you
consider the effect map-migration Claerbout (1993) has on a planar dipping
events.
In this context, warping bears the same relationship to residual
migration as `map-migration' bears to conventional
zero-offset migration.
Map-migration and warping are both point-to-point operators; whereas
conventional zero-offset migration and residual migration are based on
a convolutional model. Warping, therefore, can be thought of as
`residual map-migration'.
The relationship between warp-function and velocity change can be
derived from kinematic map-migration equations.
The following three equations describe migration
of a zero-offset planar event at dipping with slowness, , with
velocity *v*:

| |
(6) |

| (7) |

| (8) |

Differentiating with respect to *v*, and eliminating the zero-offset
variables leads to the equations that describe
residual map-migration along Fomel's velocity rays, providing a link
between the warp-function and the residual velocity correction.
| |
(9) |

| (10) |

Using an algorithm based on map-migration may seem questionable when
we are considering an amplitude-sensitive issue such as reservoir
monitoring. However, for this application the shifts we apply are so
small (a few sample points), that such an approach is valid.

** Next:** Separating kinematics and dynamics
** Up:** Warping
** Previous:** Warping
Stanford Exploration Project

7/5/1998