To analyze the
kinematic properties of HOCIGs and VOCIGs,
it is useful to observe
that they are
just particular cases
of offset-domain gathers.
In general, the offset can be oriented
along any arbitrary direction.
In particular,
the offset direction
aligned with the apparent geological
dip of the imaged event has unique properties.
We will refer to this offset as the *geological-dip offset*,
and the corresponding CIGs as
Geological Offset CIGs, or GOCIGs.

Figure illustrates the geometry of the different kinds of offset-domain CIGs for a single 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 (). The line passing through , 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. Half of the angle formed between the source and receiver ray is the apparent 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 midway between
*S*_{xh} and *R*_{xh},
and the imaging half offset is *x*_{h}=*R*_{xh}-*I*_{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 midway between
*S*_{zh} and *R*_{zh},
and the imaging half offset is *z*_{h}=*R*_{zh}-*I*_{zh}.
When the offset direction is oriented along
the apparent geological dip (what we called the geological-dip offset direction),
the end point of the source ray is *S _{0}* and
the end point of the receiver ray is

Figure
shows that both *I*_{xh} and *I*_{zh}
lie on the line passing through .This is an important property
of the offset-domain CIGs and is based on
a crucial constraint imposed on our geometric construction;
that is,
the traveltime
along the source ray summed with the traveltime
along the receiver ray is the same for all the
offset directions,
and is equal to the recording time of the event.
The independence of the total traveltimes from the offset directions
is a direct consequence of taking the zero lag of the crosscorrelation
in the imaging conditions of equation (2)
and (5).
This constraint,
together with the assumption of locally
constant velocity that we discussed above,
directly leads to the following equalities:

(8) |

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

(9) | ||

(10) |

where .Notice that the definition of is such that its sign depends on whether *I _{0}* is before or beyond
, and that for flat events ()
we have .

Although *I*_{xh} and *I*_{zh}
are both collinear with *I _{0}*,
they are shifted with respect to each
other and with respect to

(11) | ||

(12) |

Notice the dependence of and on the aperture
angle .This dependence
causes events with different aperture angles to be imaged
at different locations,
even if they originated at the same reflecting point in the subsurface.
This phenomenon is related to the well known
*reflector-point dispersal* in common midpoint gathers.
In this context,
this dispersal is a consequence of using a wrong imaging velocity,
and we will refer to it as *image-point dispersal*.
We will now discuss how the transformation
to ADCIGs overcomes the problems related to
the image-point shift and thus
removes, at least at first order, the image-point dispersal.

cig-image-dip-v2
Geometry of an angle-domain CIG for a single event
migrated with the wrong velocity.
The transformation to the angle domain
shifts all the offset-domain
image points (Figure 3 I_{xh}, I_{zh},I)
to the same angle-domain image point ._{0} |

7/8/2003