Seismic image mispositioning in the depth-angle domain

From the 2D phase-shift extrapolation of two oriented waves, down-going and up-going, using the relation between $k_h$, the horizontal offset wavenumber, and $k_z$, the vertical one, I compute the true position of the image point lying on a reflecting surface with dip angle $alpha$. The resulting coordinates take into account the orientation of the slowness vectors $mathbfp_s$ and $mathbfp_r$ with scattering angle $2theta$, one describing the down-going wave and the other the up-going one. In this framework, I can show that a wrong estimate of the migration velocity causes the displacement and, then, the mispositioning of the image point along the normal to the reflector. I derive the well-known expression of the vertical residual moveout in the $(z,theta)$-domain with respect the true coordinate. Similarly, the image point displacement happens when the time of flight $t_0$ is not correctly identified as a primary echo. Assuming a planar reflector and substituting $t_0$ with $t_m$, the time of flight of a multiple event of order $m$, I derive in the $(z,theta)$-domain the general residual moveout expression.

@TechReport{Bon06,
  author       = {Bonomi, E.},
  title        = {Seismic image mispositioning in the depth-angle domain},
  institution  = {CRS4, Center for Advanced Studies, Research and Development in Sardinia},
  number       = {06/17},
  month        = {september},
  year         = {2006},
  address      = {Cagliari, Italy},
  type         = {techreport},
  keywords     = {Seismic wavefield imaging, angle-domain common-image gathers, velocity analysis, multiple interface reflections},
  url          = {https://publications.crs4.it/pubdocs/2006/Bon06},
}