Since the transformation to ADCIGs operates in the image space, I analyze the spreading surfaces (impulse responses) of the generalized prestack migration operator, which are defined in the image space. In homogeneous anisotropic medium the shape of the impulse responses of the generalized integral migration can be easily evaluated analytically as a function of the subsurface offset , in addition to the usual image depth and midpoint .Figure illustrates the geometry used to evaluate this impulse response.
Figure 1 Geometry used for evaluating the impulse response of integral migration generalized to produce a prestack image function of the subsurface offset .
Assuming an arbitrary homogeneous anisotropic medium, simple trigonometry applied to Figure allows us to express the impulse response in parametric form, as a function of the group dip angle and the group aperture angle .If we migrate an impulse recorded at time tD, midpoint mD and surface offset hD, the migration impulse response can be expressed as follows:
In 2-D, The ADCIGs are computed by applying a slant-stack decomposition on the prestack image along the subsurface offset axis, at constant midpoint. The kinematics of the transformation are defined by the following change of variables:
Substituting equation 7 in equations 5 and 6 we obtain
Figure 2 Geometry of the transformation to the angle domain. The image point in the subsurface-offset domain moves to the image point in angle domain .
Figure provides a geometrical interpretation of the transformation to angle domain of an image point with non-zero subsurface offset. The transformation to angle domain moves the image point in the subsurface-offset domain to the image point in angle domain .The depth of the image point in angle domain is determined by the intersection of the lines passing through the points and tilted by with respect to the horizontal. When the migration velocity is correct, and the image is fully focused at zero subsurface offset, the transformation to angle domain does not change the depth of the image point and the reflections are imaged at the same depth for all aperture angles. On the contrary, when the reflections are not focused at zero offset, the transformation to angle domain maps the events at different depths for each different angle. The variability of the depth with the aperture angle is described by the RMO function that we want to measure and quantify as a function of the perturbations in anisotropic parameters encountered along the propagation paths.