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Kinematic properties of ADCIGs

The transformation to the angle domain, as defined by equations (3-4) for HOCIGs and equations (6-7) for VOCIGs, acts on each offset-domain CIG independently. Therefore, when the reflected energy does not focus at zero offset, the transformation to the angle domain shifts the image point along the direction orthogonal to the offset. The horizontal-offset image point (Ixh) shifts vertically, and the vertical-offset image point (Izh) shifts horizontally. We will demonstrate the two following important properties of this normal shift:
 I)
The normal shift corrects for the effects of the offset direction on the location of the image point; that is, the transformation to the angle domain shifts the image points from different locations in the offset domain (Ixh, Izh and I0) to the same location in the angle domain (${I}_{\gamma}$).
II)
The image location in the angle domain (${I}_{\gamma}$)lies on the normal to the apparent geological dip passing through the crossing point of the source and receiver rays ($\bar{I}$). ${I}_{\gamma}$ is located at the crossing point of the lines passing through S0 and R0 and orthogonal to the source ray and receiver ray, respectively. The shift along the normal to the reflector, caused by the transformation to angle domain, is thus equal to:  
 \begin{displaymath}
{{\bf \Delta n}_{\gamma}} = \left({I}_{\gamma}-I_0\right) =
...
 ...{\tan{\gamma}}{\bf n}=
\tan^2\gamma \;
{{\bf \Delta n}_{h_0}}
,\end{displaymath} (13)
where ${\bf \Delta n}_{h_0}=\left(\widetilde{h_0}/\tan{\gamma}\right){\bf n}$ is the normal shift in the geological-dip domain. The total normal shift caused by incomplete focusing at zero offset is thus equal to:  
 \begin{displaymath}
{\bf \Delta n}_{\rm tot} = 
\left({I}_{\gamma}-\bar{I}\right...
 ...\tan^2\gamma\right)=
\frac{{\bf \Delta n}_{h_0}}{\cos^2\gamma}.\end{displaymath} (14)

Figure [*] illustrates Properties I and II. These properties are far from obvious and their demonstration constitutes one of the main results of this paper. They also have several important consequences; the three results most relevant to migration velocity analysis are:

1.
ADCIGs obtained from HOCIGs and VOCIGs can be constructively averaged, in contrast to the original HOCIGs and VOCIGs. We will exploit this property to introduce a robust algorithm for creating a single set of ADCIGs that is insensitive to geological dips, and thus is ready to be analyzed for velocity information.
2.
The reflector-point dispersal that negatively affects offset-domain CIGs is corrected in the ADCIGs, at least at first order. If we assume the raypaths to be stationary, for a given reflecting segment the image points for all aperture angles $\gamma$ share the same apparent dip, and thus they are all aligned along the normal to the apparent reflector dip.
3.
From equation (14), invoking Fermat's principle and applying simple trigonometry, we can also easily derive a relationship between the total normal shift ${\bf \Delta n}_{\rm tot}$ and the total traveltime perturbation caused by velocity errors as follows:  
 \begin{displaymath}
{\bf \Delta n}_{\rm tot}=
-\frac{\Delta t}{2 S\cos\gamma} {\bf n},\end{displaymath} (15)
where S is the background slowness around the image point and $\Delta t$ is defined as the difference between the perturbed traveltime and the background traveltime. We will exploit this relationship to introduce a simple and accurate expression for measuring residual moveouts from ADCIGs.

next up previous print clean
Next: Demonstration of kinematic properties Up: Kinematic properties of Common Previous: Kinematic properties of Common
Stanford Exploration Project
7/8/2003