


 Evaluating the Stoltstretch parameter  

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Appendix
A
In this Appendix, we derive an explicit expression for the
Stoltstretch parameter
by comparing the accuracy of equations
(11) and (13), which approximate the
traveltime curve in the neighborhood of the vertical ray. It is
appropriate to consider a series expansion of the diffraction
traveltime in the vicinity of the vertical ray:

(24) 
where
.
Expansion (A1) contains only even powers of
because of
the obvious symmetry of
as a function of
.
Matching the series expansions term by term is a constructive method
for relating different equations to each other. The special choice of
parameters
,
, and
allows Malovichko's equation
(13) to provide correct values for the first three terms
of expansion (A1):
Considering Levin's equation (11) as an implicit
definition of the function
, we can iteratively
differentiate it, following the rules of calculus:

(28) 
Substituting the definition of Stolt stretch transform (5)
into (A5) produces an equality similar to
(A3), which means that approximation (11) is
theoretically accurate in depthvarying velocity media up to the
second term in (A1). It is this remarkable property that
proves the validity of the Stolt stretch method
(Claerbout, 1985; Levin, 1983). Moreover, equation
(11) is accurate up to the third term if the value of the
fourthorder traveltime derivative in (A6) coincides with
(A4). Substituting equation (A4) into
(A6) results in the expression

(30) 
It is now easy to derive from equation (A7) the desired
explicit expression for the Stolt stretch parameter
:
equation (17) in the main text.



 Evaluating the Stoltstretch parameter  

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