The notation used in Biondi (2005a) is the following: the VTI velocity function parameterization is ,where VV is the velocity of a vertical ray, VH is the velocity of a horizontal ray and VN is the NMO velocity. It is equivalent to Thomsen's parameterization , since and .
We define the perturbations in the VTI velocity function as a three-component vector = , where each component is a multiplicative factor for each migration velocity. The vector generates a perturbed velocity field, defined by .
From the analytic expression of the impulse response (derived from Figure ) and geometric interpretation of the angle-domain transformation, Biondi (2005a) derives the first-order derivatives of the image depth in the angle domain with respect to anisotropic parameter perturbations:
Similarly, because residual moveout is defined as the difference between the reflector movement at finite angle () and the reflector movement at normal incidence (), from equation 5, we can express the first-order derivatives of the residual moveout with respect to anisotropic parameter perturbations:
The residual moveout is eventually approximated by the first-order Taylor series expansion about the correct migration velocity (). The linearized expression is the following: