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To understand how velocity errors influence the results of shot-profile
migration, we consider the simple case of a reflector of dipping angle
, located at (*x*_{t},*z*_{t}) of a homogeneous medium of velocity *v*_{t}.
As shown in Figure 2, a shot is at surface location *x*_{s}
and a receiver is on the surface with coordinate *x*=*x*_{r}.
We define to be the angle of the incident
ray to the vertical and to be the angle of the reflected ray to the
vertical. Suppose we migrate shot records with velocity *v*_{m}, and we want
to determine the location of the image of the reflector
on the migrated sections.
For a fixed shot location, the travel time of
a reflection event is a function of the location of observation.
Now we apply Goldin's three-step method.
The first step is to find the travel time of the
reflection event observed by a receiver at *x*=*x*_{r} on the earth's surface.

| |
(1) |

where
The next step is to find by solving Eiconal equation

| |
(2) |

with equation (1) as the boundary condition.
In a case of homogeneous media, the problem can be solved analytically.
| |
(3) |

It can be shown that (Goldin, 1982)
| |
(4) |

where
and is a parameter to be determined.
The last step is imaging. Let the travel time of the reflection event observed
at point (*x*,*z*) be equal to the travel time from the shot to this point.

| |
(5) |

Combining equations (1), (3) and (5), we
have relation
| |
(6) |

To determine the in equation (4), we replace *x* and
*z* in equation (6) with equation (4).
In Appendix A, I show that
| |
(7) |

and
| |
(8) |

where
Thus, for given location (*x*_{t},*z*_{t}) of a reflector
and the relationship between and
, we can compute the location (*x*,*z*) of the migrated image.
We consider, as an example, a plane reflector of dipping angle ,

| |
(9) |

that is illuminated by a shot at *x*=*x*_{s}.
The relation between and follows Snell's law,
| |
(10) |

where is the open angle between the incident (or reflected) ray and
the normal of the reflector.
Substituting this relation and equation (9) into
equations (7) and (8) offers us formulas
to compute the coordinates of the migrated image.
In Figure 3 are plotted the migrated images
for two different values.
When , the migrated image is stretched out and curves upwards;
when , the migrated image shrinks and curves downwards.
In Figure 3 is also drawn a set
of migration ellipses. As expected, the migrated images are the
envelopes of those migration ellipses.
**envbed
**

Figure 3 When the reflected energy from a plane reflector
in a homogeneous medium
is migrated with a velocity
of (a) -10% error; (b) +10% error, the image of the reflector,
defined by the envelope of the migration ellipses, is distorted from
the actual image of the reflector.
The dipping angle of the reflector is .

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Stanford Exploration Project

1/13/1998