An exact solution of the conversion point for the converted waves from a dipping reflector with homogeneous (isotropic) overburden

For converted waves, stacking traces with a common reflection point, forming gathers, and performing dip moveout (DMO) all require accurately calculating the location of conversion points. Because of the asymmetrical paths of converted waves, even for horizontally layered media, the calculation of a conversion point for converted waves is complicated. Previous authors have obtained analytic solutions for the conversion point for converted waves in a horizontally layered media. We extend those results to the more general case of converted waves from a dipping reflector with a homogeneous, isotropic overburden. By using Snell's law, we derive a quartic equation and solve it uniquely for the conversion point. The resultant analytic expression is a function of offset, compressional-, and shear-wave velocities; normal reflector depth; and dip angle at the conversion point. This solution can be readily used to generate accurate synthetic seismic responses for converted waves based on ray theory. It also can be extended to operators for stacking converted waves and applying DMO correction.