artifacts caused by sparsely sampled wavefields

As demonstrated in Rickett and Sava (2002), ADCIGs suffer from poor sampling of the source or receiver wavefield. This effect is not very obvious from the derivation in Sava and Fomel (2003); it is, however, more evident from the perspective of shot-profile migration. Appendix A analytically demonstrates that in locally constant-velocity media, when the receiver wavefield is densely sampled, the circular wavefronts of the downward continued receiver wavefields can be well reconstructed; hence the multi-offset imaging condition in equation (1) produces a linear relationship between subsurface half offset h_x and depth z for a specific CMP location x as follows:  

$$z = - \tan \alpha x - \tan \gamma h_x,$$ (4)

where $\alpha$ is the geological dip and $\gamma$ is the reflection angle. If we take the derivative of z with respect to h_x, we get  

$$\tan \gamma = - \frac{\partial z}{\partial h_x} = - \frac{k_{h_x}}{k_z},$$ (5)

which is exactly the same as equation (2). In other words, for well-sampled receiver wavefields, one shot will generate SODCIGs with a straight line. The slope of that line in local-offset domain is the tangent of the reflection angle $\gamma$. Hence, radial-trace or slant-stack transform in the subsurface-offset domain will produce artifact-free angle gathers, and a straight line in SODCIGs would be strictly mapped to a single point in the ADCIGs.

Let us consider the extreme case for poor sampling, where we have only one receiver. As demonstrated in Appendix B, when we have only one shot and one receiver, the multi-offset imaging condition in equation (1) no longer produces a linear relationship between subsurface half offset h_x and depth z. Instead z and h_x are non-linearly related as follows:  

$$z^2 = -h_x^2 + (r-s)h_x+\frac{(r-s)^2}{\left[\tan(-\gamma -\alpha) + \tan(-\gamma+\alpha)\right]^2},$$ (6)

where s and r are the source location and receiver location respectively, and $\alpha$ is the geological dip. In such situations, the linear transformation from subsurface-offset domain to angle domain will definitely produce unwanted artifacts. Note that though here I limit my discussion to the effect in ADCIGs caused by poorly sampled receiver wavefields, reciprocity implies that the same should be true for poorly sampled source wavefields.

For example, Figure shows a synthetic shot gather for a flat reflector at z=2000 meters, with a constant velocity v=4000 m/s. The shot is located at x=-1000 meters. For an image point at x=0 meter, z=2000 meters, the reflection angle should be $\gamma = \arctan(1000/2000) = 26.6^\circ$. Figure (a) shows the migrated image obtained by using all the receivers shown in Figure . Figure (a) shows the SODCIG for surface location x=0 meter; it shows a perfectly straight line intersecting at (0,2000) in the subsurface-offset domain (h_x, z). Figure (a) shows the ADCIG obtained by slant-stacking the corresponding SODCIGs in Figure (a). This result also shows a point located at about $26.6^\circ$, which matches the theoretical result well. Figure (b), in contrast, shows the migration result obtained by using only one receiver located at x=1000 meters. The image is an ellipse, which is basically a prestack impulse response. The SODCIG at x=0 meter is shown in Figure (b), and as predicted by equation (6), it is a circle instead of a straight line in the subsurface-offset domain. The corresponding ADCIG is shown in Figure (b). Instead of a nicely focused point at $26.6^\circ$, we can identify a downward-frowning curve.

 

flat.shot Figure 1 Synthetic shot gather for a flat reflector where the shot is located at x=-1000 meters. The modeling is done using finite difference.

 

flat.img Figure 2 Migration results for only one shot. Panel (a) shows the migrated image for all the receivers shown in Figure , while (b) shows the migrated image for only one receiver located at x=1000 meters.

 

flat.sodcig Figure 3 SODCIGs at x=0 meter for different results. (a) SODCIG extracted from the result in Figure (a), where a straight line is obtained, since the receiver wavefield is well sampled. (b) SODCIG extracted from the result in Figure (b), where a circle instead of a straight line is obtained, since the wavefield is extremely poorly sampled.

 

flat.adcig Figure 4 ADCIGs at x=0 for different results. (a) ADCIG computed from Figure (a), where a nicely focused point at $26.6^\circ$ is obtained, because the local offset gather in (a) is linear. (b) ADCIG computed from Figure (b), where a downward-frowning curve instead of a point is obtained, because the local-offset gather in (b) is non-linear.