We first demonstrate Property I by showing that the radial-trace transformations represented by equation (4), and analogously equation (7), are equivalent to a chain of two transformations. The first one is the transformation of the HOCIGs (or VOCIGs) to GOCIGs by a dip-dependent stretching of the offset axis; that is:

(16) |

(17) |

The second is the transformation of HOCIGs to the angle domain according to the relation

(18) |

The transformation of HOCIGs to GOCIGs by
equations (16) and (17)
follows directly from
equations (9) and (10).
Because the transformation is a dip-dependent stretching of the offset axis,
it shifts energy in the (*z*,*x*) plane.
Appendix A demonstrates that the amount of shift
in the (*z*,*x*) plane exactly
corrects for the image-point shift characterized by
equations (11)
and (12).

Appendix B demonstrates the geometrical property
that for energy dipping at an angle in the
the (*z*,*x*) plane, the wavenumber *k*_{n} along
the normal to the dip is linked to the wavenumbers along
(*z*,*x*) by the following relationships:

(19) |

We just demonstrated that the transformation to ADCIG
is independent from which type of offset-domain CIGs
we started from (HOCIG, VOCIG, or GOCIG).
Consequently, the imaging point must be common
to all kinds of ADCIGs.
Furthermore, the image point must
lie along each of the normals to the offset directions
passing through the respective image points.
In particular,
it must lie along the normal to the apparent geological dip,
and at the crossing point
of the the vertical line
passing through *I*_{xh} and the horizontal line
passing through *I*_{zh}.

Given these constraints, the validity of Property II [equations (13) and (14)] can be easily verified by trigonometry, assuming that the image-point shifts are given by the expressions in equations (9) and (10). However, we will now demonstrate Property II in an alternative way; that is, by analyzing a GOCIG computed from an event with no apparent geological dip (). This analysis provides intuitive understanding of the relation between offset-domain and angle domain CIGs when the migration velocity is incorrect. Furthermore, the analysis of a GOCIG with flat dip is representative of all the GOCIGs, as a rotation of Figure suggests.

Figure
shows the geometry of a GOCIG with flat apparent dip.
In this particular case,
the imaging condition for ADCIGs has a direct ``physical''
explanation.
The source and receiver rays can be associated
with the corresponding planar wavefronts propagating
in the same direction (and thus tilted by an angle with
respect to the horizontal).
The crosscorrelation of the plane waves creates
the angle-domain image point where the plane waves intersect.
is shifted vertically by with respect to
the offset-domain imaging point *I _{0}*.
In this case, there is also a direct connection between
the computation of ADCIGs in the image space
and the computation of ADCIGs in the data space
by plane-wave decomposition
of the full prestack wavefield obtained by
recursive survey sinking Prucha et al. (1999).

The interpretation of ADCIGs in the ``physical'' space
(Figure )
can also be easily connected to the effects
of applying slant stacks in the image space
(Figure ).
Migration of a prestack flat event
with too low a migration velocity
generates an incompletely focused hyperbola in the image space,
as sketched in Figure .
According to equation (4),
the tangent to the hyperbola at offset
has the slope
.This tangent intersects the vertical axis at a point
shifted by
from *I _{0}*.

In the more general case of dipping reflectors (i.e. with ), when ,the shift along the vertical is .This result is consistent with the geometric construction represented in Figure .

cig-flat-v1
Geometry of a GOCIG with flat apparent dip.
In this case,
the source and receiver rays can be associated
with the corresponding planar wavefronts propagating
in the same direction.
The crosscorrelation of the plane waves creates
the angle-domain image point where the plane waves intersect.
Figure 4 |

cig-image-v1
Graphical analysis of the application of
slant stacks to a GOCIG
when an event with flat apparent dip is migrated with
a low velocity.
The event is an incompletely focused
hyperbola in the image space.
The tangent of this hyperbola at crosses the vertical axis at .Figure 5 |

7/8/2003