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As a start, it is convenient also to limit impedance to a layered model,
and to impose axisymmetry on the velocity functions. Conventional
first-pass velocity analysis often makes a like assumption.
The surface kinematics of reflection wavefronts in such a (not necessarily
isotropic) layered earth may be exactly and completely described by two
functions of the ray parameter, *p*. These are the **travel-time
function**, *t*(*p*), and the **offset function**, *x*(*p*). The
kinematics of a set of such layers is simply and exactly formed because these
functions are elements of Abelian groups. That is, the reflection response
from the base of *n* such layers arranged in any order is given by

and
If we now define *v*^{2}(*p*) = *x*(*p*)/*pt*(*p*) then *v*^{2}(*p*)*t*(*p*) = *x*(*p*)/*p*.
But this latter is addable for constant *p* and so then must be *v*^{2}(*p*)*t*(*p*).
That is,
where *v*(*p*) is some as yet unspecified velocity.

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Stanford Exploration Project

11/16/1997