Velocity continuation and the anatomy of residual prestack time migration |

where is replaced by according to equation (35). According to equation (32), the corresponding dynamic equation is

where the function remains to be defined. The simplest case of equal to zero corresponds to Claerbout's velocity continuation equation (Claerbout, 1986), derived in a different way. Levin (1986a) provides the dispersion-relation derivation, conceptually analogous to applying the method of characteristics.

In high-frequency asymptotics, the wavefield can be
represented by the ray-theoretical (WKBJ) approximation,

The general solution of equation (39) follows from the theory of characteristics. It takes the form

where the integral corresponds to the curvilinear integration along the corresponding velocity ray, and corresponds to the starting point of the ray. In the case of a plane dipping reflector, the image of the reflector remains plane in the velocity continuation process. Therefore, the second traveltime derivative in (40) equals zero, and the exponential is equal to one. This means that the amplitude of the image does not change with the velocity along the velocity rays. This fact does not agree with the theory of conventional post-stack migration, which suggests downscaling the image by the ``cosine'' factor (Levin, 1986b; Chun and Jacewitz, 1981). The simplest way to include the cosine factor in the velocity continuation equation is to set the function to be . The resulting differential equation

has the amplitude transport

corresponding to the differential equation

Appendix C proves that the time-and-space solution of the dynamic velocity continuation equation (41) coincides with the conventional Kirchhoff migration operator.

Velocity continuation and the anatomy of residual prestack time migration |

2014-04-01