# Migration of PS data

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

We learned in migration principles that migration of the stacked PP data can be performed by summation of amplitudes (equation 4-5) along a hyperbolic traveltime trajectory (equation 4-4) associated with a coincident source-receiver pair at the surface and a point diffractor at the subsurface. The amplitude from the resulting summation is placed at the apex time of the diffraction hyperbola. Prior to summation (equation 4-5), amplitude and phase factors inferred by the Kirchhoff integral solution to the scalar wave equation are applied to the stacked data (migration principles).

Similarly, migration of the stacked PS data can be conceptualized as a summation along a traveltime trajectory associated with a coincident source-receiver pair at the surface and a point diffractor at the subsurface. The difference between the zero-offset PP and the PS diffraction summation trajectories is in the velocities. Set x = 0 in equation (76) to get the traveltime equation for the zero-offset PS diffraction summation trajectory as

 ${\displaystyle t_{0}=\left({\frac {1}{\alpha }}+{\frac {1}{\beta }}\right)z.}$ (75)

 ${\displaystyle t={\frac {1}{\alpha }}{\sqrt {x_{P}^{2}+{\frac {\alpha ^{2}}{(\gamma +1)^{2}}}t_{0}^{2}}}+{\frac {\gamma }{\alpha }}{\sqrt {(x-x_{P})^{2}+{\frac {\alpha ^{2}}{(\gamma +1)^{2}}}t_{0}^{2}}},}$ (76)

 ${\displaystyle t^{2}=t_{0}^{2}+\left({\frac {\gamma +1}{\alpha }}\right)^{2}x_{P}^{2},}$ (81)

where t is the time associated with the unmigrated PS-stack, and t0 is the time associated with the migrated PS-stack given by equation (75).

Equation (81) is of the same form as equation (4-4) that describes the zero-offset PP diffraction summation trajectory. In fact, setting γ = 1 in equation (81) yields equation (4-4) for the case of constant velocity. Note from equation (81) that the migration velocity for the PS data is given by α/(γ + 1), which is neither the P-wave velocity α nor the S-wave velocity β. The PS-velocity α/(γ + 1) can be obtained from the PP-velocity α and the velocity ratio γ, both estimated from the velocity analysis described earlier in this section. Once the velocity field is estimated, the PS-stack can, in principle, be migrated using any of the algorithms described in migration.