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Residual moveout in ADCIGs

The inconsistencies between the migrated images at different aperture angles are the primary source of information for velocity updating during Migration Velocity Analysis (MVA). Figure [*] demonstrated how the reflector mispositioning caused by velocity errors can be exactly predicted by a kinematic migration that assumes the image point to lie on the normal to the apparent geological dip. However, this exact prediction is based on the knowledge of the true velocity model. Of course, this condition is not realistic when we are actually trying to estimate the true velocity model by MVA. In these cases, we first measure the inconsistencies between the migrated images at different aperture angles, and then we ``invert'' these measures into perturbations of the velocity model.

An effective and robust method for measuring inconsistencies between images is to compute semblance scans as a function of one ``residual moveout'' (RMO) parameter, and then pick the maxima of the semblance scan. This procedure is most effective when the residual moveout function used for computing the semblance scans closely approximates the true moveouts in the images. In this section, we use the kinematic properties that we derived and illustrated in the previous sections to derive two alternative RMO functions for scanning ADCIGs computed from wavefield-continuation migration.

As discussed above, the exact relationships derived in Appendix C cannot be used, because the true velocity function is not known. Thus we cannot realistically estimate the changes in ray-propagation directions caused by velocity perturbations. However, we can linearize the relations and estimate the reflector movement by assuming that the raypaths are stationary. This assumption is consistent with the typical use of measured RMO functions by MVA procedures. For example, in a tomographic MVA procedure the velocity is updated by applying a tomographic scheme that ``backprojects'' the image inconsistencies along unperturbed raypaths. Furthermore, the consequences of the errors introduced by neglecting ray bending are significantly reduced by the fact that RMO functions describe the movements of the reflectors relative to the reflector position imaged at normal incidence ($\gamma=0$), not the absolute movements of the reflectors with respect to the true (unknown) reflector position.

Appendix D derives two expressions for the RMO shift along the normal to the reflector $({\bf \Delta n}_{\rm RMO})$, under the assumptions of stationary raypaths and constant scaling of the slowness function by a factor $\rho$.The first expression is [equation (61)]:  
{\bf \Delta n}_{\rm RMO}=
 ...gamma}{\left(\cos^2\alpha - \sin^2\gamma\right)}
z_0\; {\bf n},\end{displaymath} (21)
where z0 is the depth at normal incidence.

The second RMO function is directly derived from the first by assuming flat reflectors $(\alpha=0)$[equation (62)]:  
{\bf \Delta n}_{\rm RMO}=
 z_0\;{\bf n}.\end{displaymath} (22)
As expected, in both expressions the RMO shift is null at normal incidence $(\gamma =0)$,and when the migration slowness is equal to the true slowness $(\rho=1)$.

According to the first expression [equation (21)], the RMO shift increases as a function of the apparent geological dip $\left\vert\alpha\right\vert$.The intuitive explanation for this behavior is that the rays become longer as the apparent geological dip increases, and consequently the effects of the slowness scaling increase. The first expression is more accurate than the second one when the spatial extent of the velocity perturbations is large compared to the raypath length, and consequently the velocity perturbations are uniformly felt along the entire raypaths. Its use might be advantageous at the beginning of the MVA process when slowness errors are typically large scale. However, it has the disadvantage of depending on the reflector dip $\alpha$,and thus its application is somewhat more complex.

The second expression is simpler and is not as dependent on the assumption of large-scale velocity perturbations as the first one. Its use might be advantageous for estimating small-scale velocity anomalies at a later stage of the MVA process, when the gross features of the slowness function have been already determined.

To test the accuracy of the two RMO functions we will use the migration results of a synthetic data set acquired over a spherical reflector. This data set was described in the previous section. Figure [*] illustrates the accuracy of the two RMO functions when predicting the actual RMO in the migrated images obtained with a constant slowness function with $\rho=1.04$.The four panels show the ADCIGs corresponding to different apparent reflector dip: a) $\alpha$ = 0; b) $\alpha$ = 30; c) $\alpha$ = 45; d) $\alpha$ = 60. Notice that the vertical axes change across the panels; in each panel the vertical axis is oriented along the direction normal to the respective apparent geological dip. The solid lines superimposed onto the images are computed using equation (21), whereas the dashed lines are computed using equation (22). As in Figure [*], the images extend beyond the termination of the analytical lines because of the finite-frequency nature of the truncation artifacts.

The migrated images displayed in Figure [*] were computed by setting both the true and the migration slowness function to be constant. Therefore, this case favors the first RMO function [equation (21)] because it nearly meets the conditions under which equation (21) was derived in Appendix D. Consequently, the solid lines overlap the migration results for all dip angles. This figure demonstrates that, when the slowness perturbation is sufficiently small (4 % in this case), the assumption of stationary raypaths causes only small errors in the predicted RMO.

On the contrary, the dashed lines predicted by the second RMO function [equation (22)] are an acceptable approximation of the actual RMO function only for small dip angles (up to 30 degrees). For large dip angles, a value of $\rho$ substantially higher than the correct one would be necessary to fit the actual RMO function with equation (22). If this effect of the reflector dip is not properly taken into account, the false indications provided by the inappropriate use of equation (22) can prevent the MVA process from converging.

Figure 14
ADCIGs for four different apparent reflector dips: a) $\alpha$ = 0; b) $\alpha$ = 30; c) $\alpha$ = 45; d) $\alpha$ = 60 with $\rho=1.04$.Superimposed onto the images are the RMO functions computed using equation (21) (solid lines), and using equation (22) (dashed lines). Notice that the vertical axes change across the panels; in each panel the vertical axis is oriented along the direction normal to the respective apparent geological dip.

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We analyzed the kinematic properties of ADCIGs in presence of velocity errors. We proved that in the angle domain the image point lies along the normal to the apparent reflector dip. This geometric property of ADCIGs makes them immune to the image-point dispersal and thus attractive for MVA.

We derived a quantitative relationship between image-point movements and traveltime perturbations caused by velocity errors, and verified its validity with a synthetic-data example. This relationship should be at the basis of velocity-updating methods that exploit the velocity information contained in ADCIGs.

Our analysis leads to the definition of two RMO functions that can be used to measure inconsistencies between migrated images at different aperture angles. The RMO functions describe the relative movements of the imaged reflectors only approximately, because they are derived assuming stationary raypaths. However, a synthetic example shows that, when the velocity perturbation is sufficiently small, one of the proposed RMO functions is accurate for a wide range of reflector dips and aperture angles.

The insights gained from our kinematic analysis explain the strong artifacts that affect conventional ADCIG in presence of steeply dipping reflectors. They also suggest a procedure for overcoming the problem: the computation of vertical-offset CIGs (VOCIGs) followed by the combination of VOCIGs with conventional HOCIGs. We propose a simple and robust scheme for combining HOCIGs and VOCIGs. A North Sea data example clearly illustrates both the need for and the advantages of our method for computing ADCIGs in presence of a vertical salt edge.

We thank Guojian Shan for helping in the development of the program that we used to migrate both the synthetic and the real data sets. We also thank Henri Calandra and TotalFinaElf for making the North Sea data set available to the Stanford Exploration Project (SEP).

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