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Demonstration of kinematic properties of ADCIGs

Properties I and II can be demonstrated in several ways. In this paper, we will follow an indirect path that might seem circuitous but will allow us to gather further insights on the properties of ADCIGs.

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:  
 \begin{displaymath}
\widetilde{h_0}=x_h\cos\alpha,
\;\;\;\;\;\; 
{\rm or}
\;\;\;\;\;\;
\widetilde{h_0}=-z_h\sin\alpha;\end{displaymath} (16)
or in the wavenumber domain,  
 \begin{displaymath}
k_{h_0}=\frac{k_{x_h}}{\cos\alpha},
\;\;\; \;\;\; 
{\rm or}
\;\;\; \;\;\;
k_{h_0}=-\frac{k_{z_h}}{\sin\alpha};\end{displaymath} (17)
where kh0 is the wavenumber associated with $\widetilde{h_0}$,and kxh and kzh are the wavenumbers associated with xh and zh.

The second is the transformation of HOCIGs to the angle domain according to the relation  
 \begin{displaymath}
\tan \gamma = 
-\frac{k_{h_0}}{k_{n}},\end{displaymath} (18)
where kn is the wavenumber associated with the direction normal to the reflector. This direction is identified by the line passing through $\bar{I}$ and ${I}_{\gamma}$in Figures [*] and [*].

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 $\alpha$ in the the (z,x) plane, the wavenumber kn along the normal to the dip is linked to the wavenumbers along (z,x) by the following relationships:  
 \begin{displaymath}
k_{n}=
\frac{k_z}{\cos \alpha}=
\frac{k_{x}}{\sin \alpha}.\end{displaymath} (19)
Substituting equations (17) and (19) into equation (18), we obtain equations (4) and (7). The graphical interpretation of this analytical result is immediate. In Figure [*], the transformation to GOCIG [equations (17)] moves the imaging point Ixh (or Izh) to I0, and the transformation to the angle domain [equation (18)] moves I0 to ${I}_{\gamma}$.This sequence of two shifts is equivalent to the direct shift from Ixh (or Izh) to ${I}_{\gamma}$caused by the transformation to the angle domain applied to a HOCIG (or VOCIG).

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 ${I}_{\gamma}$ 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 Ixh and the horizontal line passing through Izh.

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 ($\alpha=0$). 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 $\gamma$ with respect to the horizontal). The crosscorrelation of the plane waves creates the angle-domain image point ${I}_{\gamma}$ where the plane waves intersect. ${I}_{\gamma}$ is shifted vertically by $\widetilde{h_0}\tan \gamma$ with respect to the offset-domain imaging point I0. 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 $\widetilde{h_0}=x_h$ has the slope $\partial z/\partial x_h= -\tan \gamma$.This tangent intersects the vertical axis at a point shifted by ${\bf \Delta n}_{\gamma }=\widetilde{h_0}\tan \gamma {\bf n}$ from I0.

In the more general case of dipping reflectors (i.e. with $\alpha\neq 0$ ), when $x_h=\widetilde{h_0}/\cos\alpha$,the shift along the vertical is $x_h\tan\gamma {\bf n}=\left(\widetilde{h_0}\tan\gamma/\cos\alpha\right){\bf n}$.This result is consistent with the geometric construction represented in Figure [*].

 
cig-flat-v1
Figure 4
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 ${I}_{\gamma}$ where the plane waves intersect.

cig-flat-v1
view

 
cig-image-v1
Figure 5
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 ${\bf h}_0$crosses the vertical axis at ${I}_{\gamma}$.

cig-image-v1
view


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Next: Robust computation of ADCIGs Up: Kinematic properties of Common Previous: Kinematic properties of ADCIGs
Stanford Exploration Project
7/8/2003