Sensitivity of the estimation of $(\gamma,\phi)$ to velocity errors

We start with a simple 2-D qualitative analysis of the effects of velocity errors on both kinds of ADCIGs that we presented; that is, ADCIGs computed before imaging and ADCIGs computed after imaging. This first step illustrates the sensitivity of the estimation of the angles $(\gamma,\phi)$to velocity errors. Figure 11 shows the effects of velocity errors on ADCIGs computed before imaging. It shows the same panels as Figure 1 [i.e. wavefield at zero-offset (a), wavefield at fixed midpoint (b), and ADCIG (c)], but when the migration velocity is 10% higher than the correct one. At the correct depth of the flat reflector (z=700 meters), the energy for both reflectors has started defocusing and it forms time-reversed hyperbolas. Most of the energy for the flat reflector has not reached the zero-time line yet, causing the flat reflector to be imaged deeper than the correct depth. In the ADCIG both events frown downward, and thus they indicate too high of a migration velocity. The residual moveout caused by velocity errors in ADCIGs is thus qualitatively similar to the moveout observed in conventional surface-offset CIGs computed by Kirchhoff migration. If the velocity function is too low the reflections will smile upward; if the velocity function is too high the reflections will frown downward.

 

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Figure 11 Illustration of the sensitivity to velocity errors of the ADCIGs computed before imaging. The panels are the same as in Figure 1; that is, they display orthogonal slices of the prestack wavefield after downward continuation to the depth of 700 meters, but the velocity used for downward continuation was 10% higher than the correct one. Panel a) displays the zero-offset section, panel b) displays the common-midpoint gather at 1,410 meters, and panel c) shows the complete (i.e. for all depths) ADCIG at 1,410 meters.

 

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Figure 12 Illustration of the sensitivity of ADCIGs computed after imaging to velocity errors. The panels are the same as in Figure 4, but after migration with a velocity 10% higher than the correct one. Panel a) displays the ODCIG and panel b) the ADCIG$\left(\gamma\right)$.Notice that the range of $\gamma$ increases from Figure 4b to Figure 12b.

Figure 12 shows the effects of velocity errors on ADCIGs computed after imaging. It shows the same panels as Figure 4 [i.e. ODCIG (a), and ADCIG (b)], but when the migration velocity is 10% higher than the correct one. The energy is defocused in the ODCIGs and the events have a hyperbolic moveout (Figure 12a); the apexes of the hyperbolas are deeper than the reflectors' true depths. Slant stacks transform the hyperbolas into frowns (Figure 12b) that are similar to the frowns shown in Figure 11c. However, there is a subtle but important difference between the two cases. The p_xh range does not change between the ADCIG shown in Figure 1c and the one shown in Figure 11c, whereas the $\gamma$ range increases when the velocity is too high (Figure 12b). This increase is due to the increase in the apparent vertical wavelength in the image, which correspondingly causes a decrease of k_z in equation (16). Or, from a different viewpoint, if we were to use equation (9) to map p_xh into $\gamma$,the mapping would be affected by both the increase in $v\left(z,x\right)$ and the increase in apparent geological dip $\alpha_x$ - and the corresponding decrease in $\cos\alpha_x$.This simple example illustrates the fact that the estimates of $\gamma$ and $\phi$are similarly sensitive to the accuracy of the local velocity $v\left(z,x,y \right)$,regardless of whether ADCIGs are computed before or after imaging.