Using the proposed recursion 12 and the layer traveltime derivatives in equation 15, we can summarize the steps to accumulate the effects from lateral heterogeneity along a raypath and evaluate the corresponding traveltime derivatives at the surface as follows:
Given a multi-layered medium with known , , and their derivatives in all sublayers, we compute the layer traveltime derivatives (equation 15) for a specified CMP location in the case of reflection traveltime or a specified image-ray escape location in the case of diffraction traveltime. In particular, equation 15 can be rewritten with evaluated reference 1-D traveltime derivatives as follows:
where is the NMO velocity of the -th layer at the specified position. This velocity is constant for an isotropic sublayer but is equal to
for a VTI sublayer, where is the vertical P-wave velocity and is the Thomsen's delta.
We substitute the results from step 1 into the recursion 12 starting from with and end up with
.
We can evaluate the final second-order traveltime derivative at the surface from equation 11 with the layer derivatives of the -th layer and from step 2.
NMO or time-migration velocity can then be found according to equations 6 and 7 by multiplying the the result from step 3 with the total one-way traveltime in the reference 1-D medium
.
Effects of lateral heterogeneity on time-domain processing parameters