Incorporating the two sources of heterogeneity (equations 13 and 14) into equation 4 and evaluating the integral, we can derive the traveltime that includes the heterogeneity effects. Twice differentiating the result with respect to and and evaluating at the vertical reference ( and ), we arrive at the following layer traveltime derivatives:
where denotes the traveltime of the -th layer in the reference 1-D horizontally-layered anisotropic media with constant elastic parameters within each layer. Therefore, the terms with derivatives are the usual results ones get under the 1-D medium assumption. The additional heterogeneous terms (H) that combine the effects from curved interfaces and laterally varying velocity are given by
H
(16)
H
H
The expression for
is similar to that of
with shifted indices. If a single horizontal layer is considered, equation 11 becomes reminiscent of the result by Grechka and Tsvankin (1999):
but with the second derivative on group slowness as opposed to group velocity. is the usual normal-moveout velocity in the reference 1-D medium, which translates to in the case of diffraction traveltime.
Effects of lateral heterogeneity on time-domain processing parameters