KINEMATICS OF VELOCITY CONTINUATION

From the kinematic point of view, it is convenient to describe the reflector as a locally smooth surface z = z(x), where z is the depth, and x is the point on the surface (x is a two-dimensional vector in the 3-D problem). The image of the reflector obtained after a common-offset prestack migration with a half-offset h and a constant velocity v is the surface z = z(x;h,v). Appendix A provides the derivations of the partial differential equation describing the image surface in the depth-midpoint-offset-velocity space. The purpose of this section is to discuss the laws of kinematic transformations implied by the velocity continuation equation. Later in this paper, I obtain dynamic analogues of the kinematic relationships in order to describe continuation of migrated sections in the velocity space.

The kinematic equation for prestack velocity continuation, derived in Appendix A, takes the following form:  

$${{\partial \tau} \over {\partial v}} = v\,\tau\,\left({{\pa... ..., \left({{\partial \tau} \over {\partial h}}\right)^2\right)\;.$$ (1)

Here $\tau$ denotes the one-way vertical traveltime $\left(\tau = {z \over v}\right)$. The right-hand side of equation (1) consists of three distinctive terms. Each has its own geophysical meaning. The first term is the only one remaining when the offset h equals zero. It corresponds to the procedure of zero-offset residual migration. Setting the reflector dip to zero eliminates the first and third terms, leaving the second, dip-independent one. We can associate the second term with the process of residual normal moveout. The third term is both dip- and offset- dependent. The process that it describes is residual dip moveout. It is convenient to analyze each of the three processes separately, evaluating their contributions to the cumulative process of prestack velocity continuation.