Numerical examples of aperture angle along impulse responses

The analytical kinematic results can be verified by numerical computations of impulse responses by wavefield migration and transformation of the resulting prestack image cubes into the angle domain. Figure  shows four zero subsurface-offset sections cut through the impulse responses computed by wavefield-continuation anisotropic migration for three anisotropic rocks and for an isotropic rock. The first anisotropic rock is the Taylor Sand defined above. The second is the Mesa Clay Shale as defined by Tsvankin (2001) using the three Thomsen parameters: $\epsilon=0.189,\;\delta=0.204,\;{\rm and}\; \eta=-.01$.The third is derived from the Green River Shale as described by Tsvankin (2001) by halving the anisotropic parameters ($\epsilon$ and $\delta$); that is, it is defined by the three Thomsen parameters: $\epsilon=0.0975,\;\delta=-0.11,\;{\rm and}\; \eta=.266$.The strong anelliptic nature of the original Green River Shale ($\eta=.74$) causes the group-slowness approximation in equation 6 to break down, and makes the kinematic computations based on ray tracing, and thus on group velocity and angles, inconsistent with wavefield migrations based on the dispersion relation in equation 7. Hereupon I will refer to this rock, for obvious reasons, as the GreenLight River Shale. Notice that the GreenLight River Shale is still strongly elliptical.

The other parameters defining the impulse responses are the same as for Figure ; that is, t_D=.9 seconds, m_D=0 kilometers, and h_D=.4 kilometers, and vertical slowness $S_V=1~{\rm s/km}$.Figure a shows the isotropic case, Figure b shows the Taylor Sand case, Figure c shows the Mesa Clay Shale case, and Figure d shows the GreenLight River Shale case. As in Figure , the line superimposed onto the images represent the impulse response computed using the kinematic expressions in equations 18-24. The kinematic curves perfectly predict the shape of the images even for very steep dips.

Figure  shows two-dimensional slices cut through the cube obtained by the transformation to the angle domain of the impulse responses shown in Figure . The slices are cut at the midpoint and depth corresponding to the expected location of the impulse responses; that is, at the location tracked by the lines shown in Figure . There are three lines superimposed onto the angle-domain images. The solid lines display the numerical computation of $\arctan({\partial z_\xi}/{\partial h_\xi})$ by applying equation 25. They perfectly track, as expected, the results of the transformation of the prestack images to angle domain. The dotted lines display the phase aperture angle $\widetilde{\gamma}$.As expected, they overlap with the solid line around the zero midpoint (i.e. flat reflector), and depart from them at larger midpoints, which correspond to steeper reflections. However, the error introduced by ignoring the difference between $\arctan({\partial z_\xi}/{\partial h_\xi})$and $\widetilde{\gamma}$ is small, and likely to be negligible in most practical situations. Finally, the dashed lines display the group aperture angle $\gamma$.The differences between $\gamma$ and $\widetilde{\gamma}$ are substantial, up to 20% in some cases. Ignoring them might be detrimental to the application of ADCIGs. Notice that in the isotropic case the three lines perfectly overlap and all of them match the image.

 

Quad_hxd_.4-overn
Figure 5 Impulse responses evaluated at zero subsurface offset for four rock types: a) Isotropic, b) Taylor Sand, c) Mesa Clay Shale, and d) GreenLight River Shale. Superimposed onto the images are the impulse responses computed by the kinematic expressions presented in equations 18-24.

 

Quad_Mx-Ang_hxd_.4-overn
Figure 6 Slices of the impulse responses transformed into the angle-domain for four rock types: a) Isotropic, b) Taylor Sand, c) Mesa Clay Shale, and d) GreenLight River Shale. Superimposed onto the images there are the curves computed by applying the kinematic analysis: $\gamma$ (dashed line), $\widetilde{\gamma}$ (dotted line), and $\arctan({\partial z_\xi}/{\partial h_\xi})$ (solid line).