Figure 7
illustrates the geometry of the offset-domain CIGs
for a single event recorded at the surface for
the source location *S*
and receiver location *R*.
The crucial assumption of our geometric construction
is that the traveltime
along the source ray summed with the traveltime
along the receiver ray is the same for all the
offset directions
and equal
to the recording time of the event
.

In this sketch,
the migration velocity is assumed
to be lower than the true velocity,
and thus the reflections are imaged too shallow
and above the point where the source
ray crosses the receiver ray (*SR*).
The line passing through *SR*, and
bisecting the angle formed by the source and receiver ray,
is oriented at an angle with respect
to the vertical direction.
The angle is the apparent geological dip
of the event after imaging.
It would correspond to the true geological dip
if the migration velocity were correct.
Half of the angle formed between
the source and receiver ray
is the aperture angle .

When HOCIGs are computed,
the end point of the source ray (*S*_{xh}) and
the end point of the receiver ray (*R*_{xh}) are at the same depth.
The imaging point *I*_{xh} is in the middle between
*S*_{xh} and *R*_{xh}
and the imaging offset is *x*_{h}=*R*_{xh}-*S*_{xh}.
Similarly,
when VOCIGs are computed,
the end point of the source ray (*S*_{zh}) and
the end point of the receiver ray (*R*_{zh})
are at the same horizontal location.
The imaging point *I*_{zh} is in the middle between
*S*_{zh} and *R*_{zh}
and the imaging offset is *z*_{h}=*R*_{zh}-*S*_{zh}.
When the offset direction is oriented along
the apparent geological dip (what we called the optimally focusing offset direction),
the end point of the source ray is *S _{0}* and
the end point of the receiver ray is

The offsets along the different directions are linked by the following simple relationship, which can be readily derived by trigonometry applied to Figure 7; that is,

(1) | ||

(2) |

Also the shift of the imaging points
*I*_{xh} and *I*_{zh}
can be easily expressed in terms of the offset *h _{0}*
and the angles and as:

(3) | ||

(4) |

The fact that all three imaging points are aligned along the apparent geological dip allows our transformation to remove the image-point dispersal, and it is crucial to the effectiveness of DDOCIGs. In other words, to transform one set of CIGs into another set we just need to transform the offset axis; the image is then automatically shifted along the apparent geological dip by the right amount. Appendix A demonstrates this fact.

The proposed CIG transformation is a simple
dip-dependent non-uniform stretching of the
the offset-axis according to the relationships
in equations (1) and (2).
The transformation is easily implemented in the wavenumber (*k*_{z},*k*_{x}) domain,
by taking advantage of the well
known relationship .

After both the HOCIGs and the VOCIGs are transformed,
they can be merged together.
A simple scheme to merge them is a weighted average,
where the weights *w*_{xh} and *w*_{zh} are set to

(5) | ||

(6) |

Figure 8

11/11/2002