Isotropic angle-domain elastic reverse-time migration

Angle decomposition

The images shown in the preceding subsection correspond to the conventional imaging conditions from equations  and . We can construct other images using the extended imaging conditions from equations  and , which can be used for angle decomposition after imaging. Then, we can use equation  to compute angle gathers from horizontal space cross-correlation lags.

Figures  and together with Figures  and show, respectively, the PP and PS horizontal lags and angle gathers for the common image gather (CIG) location in the middle of the reflectivity model, given a single source at $x=6.75$ km and $z=0.5$ km. PP and PS horizontal lags are lines dipping at angles that are equal to the incidence angles (real incidence angles for PP reflection and average of incidence and reflection angles for PS reflection) at the CIG location. PP angles are larger than PS angles at all reflectors, as illustrated on the simple synthetic example shown in Figure 2.

Figures 8a and 8c together with Figures 8b and 8d show, respectively, the PP and PS horizontal lags and angle gathers for the same CIG location, given many sources from $x=5.5$ to $7.5$ km and $z=0.5$ km. The horizontal space cross-correlation lags are focused around ${\boldsymbol{\lambda}} =\mathbf{0}$, which justifies the use of conventional imaging condition extracting the cross-correlation of the source and receiver wavefields at zero lag in space and time. Thus, the zero lag of the images obtained by extended imaging condition represent the image at the particular CIG location. The PP and PS gathers for many sources are flat, since the migration was done with correct migration velocity. The PS angle gather, depicted in Figure 8d, shows a polarity reversal at $\theta=0$ as expected.

je-0700-Ecig11,je-0700-Eang11,je-0700-Ecig12,je-0700-Eang12
Figure 7. Horizontal cross-correlation lags for (a) PP and (c) PS reflections for the model in Figures 3a and 3b. The source is at $x=6.75$ km, and the CIG is located at $x=6.5$ km. Panels (b) and (d) depict PP and PS angle gathers decomposed from the horizontal lag gathers in panels (a) and (c), respectively. As expected, PS angles are smaller than PP angles for a particular reflector due to smaller reflection angles.

PPcig,PPcig-ang,PScig,PScig-ang
Figure 8. Horizontal cross-correlation lags for PP (a) and PS (c) reflections for the model in Figures 3a and 3b. These CIGs correspond to 81 sources from $x=5.5$ to $7.5$ km at $z=0.5$ km. The CIG is located at $x=6.5$ km. Panels (b) and (d) depict PP and PS angle gathers decomposed from the horizontal lag gathers in panels (a) and (c), respectively. Since the velocity used for imaging is correct, the PP and PS gathers are flat. The PP angle gathers do not change polarity at normal incidence, but the PS angle gathers change polarity at normal incidence.

Isotropic angle-domain elastic reverse-time migration