Note that
so (s being the stretch factor, defined above). Thus (s* is the stretch factor belonging to v*) whenceAccording 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.Thus
where 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.