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Kinematic properties of Common Image Gathers

In this section we analyze the kinematic properties of CIGs, with particular emphasis on the case when velocity errors prevent the image from focusing at zero offset, causing the reflected energy to be imaged over a range of offsets. We will start by analyzing the kinematics of offset-domain CIGs.

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 ($\bar{I}$). The line passing through $\bar{I}$, and bisecting the angle formed by the source and receiver ray, is oriented at an angle $\alpha$ with respect to the vertical direction. The angle $\alpha$ 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 $\gamma$.

When HOCIGs are computed, the end point of the source ray (Sxh) and the end point of the receiver ray (Rxh) are at the same depth. The imaging point Ixh is midway between Sxh and Rxh, and the imaging half offset is xh=Rxh-Ixh. Similarly, when VOCIGs are computed, the end point of the source ray (Szh) and the end point of the receiver ray (Rzh) are at the same horizontal location. The imaging point Izh is midway between Szh and Rzh, and the imaging half offset is zh=Rzh-Izh. When the offset direction is oriented along the apparent geological dip $\alpha$(what we called the geological-dip offset direction), the end point of the source ray is S0 and the end point of the receiver ray is R0. The imaging point I0 is midway between S0 and R0, and the imaging half offset is ${\bf h}_0=R_0-I_0$.Notice that the geological-dip half offset ${\bf h}_0$is a vector, because it can be oriented arbitrarily with respect to the coordinate axes.

Figure [*] shows that both Ixh and Izh lie on the line passing through $S_0, I_0~\rm{and}~R_0$.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:  
\left\vert S_{x_h}-S_0\right\vert=\left\vert R_{x_h}-R_0\rig...
 ...\vert S_{z_h}-S_0\right\vert=\left\vert R_{z_h}-R_0\right\vert,\end{displaymath} (8)
which in turn are at the basis of the collinearity of I0, Ixh and Izh.

The offsets along the different directions are linked by the following simple relationship, which can be readily derived by trigonometry applied to Figure [*]:
x_h& = &\frac{\widetilde{h_0}}{\cos\alpha},
\\ z_h& = &-\frac{\widetilde{h_0}}{\sin\alpha},\end{eqnarray} (9)

where $\widetilde{h_0}={\bf n}\times{\bf h}_0$.Notice that the definition of $\widetilde{h_0}$is such that its sign depends on whether I0 is before or beyond $\bar{I}$, and that for flat events ($\alpha=0$) we have $\widetilde{h_0}=x_h$.

Although Ixh and Izh are both collinear with I0, they are shifted with respect to each other and with respect to I0. The shifts of the imaging points Ixh and Izh with respect to I0 can be easily expressed in terms of the offset ${\bf h}_0$and the angles $\alpha$ and $\gamma$ as follows:
{\bf \Delta I}_{x_h}& = \left(I_{x_h}-I_0\right) & ={\bf h}_0\t...
 ...I_{z_h}-I_0\right) & =-{\bf h}_0\frac{\tan{\gamma}}{\tan{\alpha}}.\end{eqnarray} (11)
The shift between Ixh and Izh prevents us from constructively averaging HOCIGs with VOCIGs to create a single set of offset-domain CIGs.

Notice the dependence of ${\bf \Delta I}_{x_h}$ and ${\bf \Delta I}_{z_h}$ on the aperture angle $\gamma$.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.

Figure 3
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 (Ixh, Izh,I0) to the same angle-domain image point ${I}_{\gamma}$.


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Next: Kinematic properties of ADCIGs Up: Biondi and Symes: ADCIGs Previous: Computation of Common Image
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