We use the same notation as in
(3): the
VTI velocity function parameterization is
, which is equivalent to Thomsen's
parameterization .*V*_{V} is the velocity of a vertical ray, *V*_{H}
is the velocity of a horizontal ray ()and *V*_{N} is the NMO velocity ().

We define the perturbations as a three-component vector = , where each component is a multiplicative factor for each migration velocity. It generates a perturbed velocity field, defined by .

From the analytic expression of the impulse response (derived from figure 1) and some geometric interpretation of the transformation to the angle domain, the first-order derivatives of the imaging depth in the angle domain with respect to perturbations in the anisotropic parameters can be written as follows (7):

(3) |

Similarly, since residual moveout is defined as the difference between the reflector
movement at finite angle and the reflector movement at normal incidence ** z_{0}**,
from equation 3, we can express the first-order derivatives of the residual moveout with
respect to perturbations in the anisotropic parameters:

(4) |

The residual moveout is eventually approximated by the first-order Taylor series expansion about the correct migration velocity (). The linearized expression is the following:

(5) |

1/16/2007