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Assume that the data *S* are model-consistent, that is
for target offset independent reflectivity *r*^{*}(*t*_{0}) and velocity
*v*^{*}(*t*_{0}). [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.]
Note that

so
(*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 *T*_{0}.
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.

** Next:** Hyperbolic Moveout
** Up:** Symes: Differential semblance
** Previous:** Asymptotic Approximation of Differential
Stanford Exploration Project

4/20/1999