for target offset independent reflectivity r*(t0) and velocity v*(t0). [Since differential semblance does not depend to leading order on the amplitude, as noted above, I set the amplitude to 1 in the following, for simplicity - it can be reintroduced with almost no change in the results to follow.]
(s being the stretch factor, defined above). Thus
(s* is the stretch factor belonging to v*) whence
According to the calculus of pseudodifferential operators,
where you get from the next to the last line to the last by substituting for , and using previously derived formulas for the partial derivatives of T0.
is independent of v, i.e. depending only on v* and .
In the next section I introduce the so called hyperbolic moveout approximation to traveltime. Note that up to this point the development is entirely independent of this approximation. In particular the formulas worked out in this section have precise analogues for versions of differential semblance based on multidimensional seismic models.