Evaluating the Stolt-stretch parameter

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Appendix A

In this Appendix, we derive an explicit expression for the Stolt-stretch parameter $W$ 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:

$$t_0(l)={\left.t_0\right\vert _{l=0}}+ {1 \over 2} {\left.{d^2t_0... ...+ {1 \over {4!}} {\left.{d^4t_0}\over {dl^4}\right\vert _{l=0}}l^4+\cdots\;\;,$$ (24)

where $l=x-x_0$ . Expansion (E-1) contains only even powers of $l$ because of the obvious symmetry of $t_0$ as a function of $l$ .

Matching the series expansions term by term is a constructive method for relating different equations to each other. The special choice of parameters $t_v$ , $v_{rms}$ , and $S$ allows Malovichko's equation (13) to provide correct values for the first three terms of expansion (E-1):

$$\left.t_0\right\vert _{l=0} = t_v\;;$$ (25)

$$\left.{d^2t_0}\over {dl^2}\right\vert _{l=0} = {1 \over {t_v v_{rms}^2\left(t_v\right)}}\;;$$ (26)

$$\left.{d^4t_0}\over {dl^4}\right\vert _{l=0} = -{{3 S\left(t_v\right)} \over {t_v^3 v_{rms}^4\left(t_v\right)}}\;\;.$$ (27)

Considering Levin's equation (11) as an implicit definition of the function $t_0\left(t_v\right)$ , we can iteratively differentiate it, following the rules of calculus:

$$\left.{ds}\over {dl}\right\vert _{l=0} = \left. s'\left(t_0\right) {{dt_0}\over {dl}}\right\vert _{l=0} = 0\;;$$

$$\left.{d^2s}\over {dl^2}\right\vert _{l=0} = \left.\left(s'\left(... ...^2t_0}\over {dl^2}}\right\vert _{l=0} = {1\over {v_0^2 s\left(t_v\right)}}\;;$$ (28)

$$\left.{d^3s}\over {dl^3}\right\vert _{l=0} = \left.\left(3 s''\l... ...'\left(t_0\right) \left({dt_0}\over {dl}\right)^3\right)\right\vert _{l=0} = 0$$

$$\left.{d^4s}\over {dl^4}\right\vert _{l=0} = \left(6 s'''\left(t_0\right) \left({dt_0}\over {dl}\right)^2 {{d... ...)^2 + 4s''\left(t_0\right) {{dt_0}\over {dl}} {{d^3t_0}\over {dl^3}}+ \right.$$

$$\left. \left. + s'\left(t_0\right) {{d^4t_0}\over {dl^4}}+ s^{IV}\left(t_0\right) \left({dt_0}\over {dl}\right)^4\right)\right\vert _{l=0} =$$

$$= \left.\left(s''\left(t_v\right) \left({d^2t_0}\over {dl^2}\right... ...}\right)\right\vert _{l=0} = -{{3 W} \over {v_0^4 s^3\left(t_0\right)}}\;\;.$$ (29)

Substituting the definition of Stolt stretch transform (5) into (E-5) produces an equality similar to (E-3), which means that approximation (11) is theoretically accurate in depth-varying velocity media up to the second term in (E-1). 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 fourth-order traveltime derivative in (E-6) coincides with (E-4). Substituting equation (E-4) into (E-6) results in the expression

$${{1-W}\over {v_0^2 s^2\left(t_v\right)}}= {{v^2\left(t_v\right)-... ...right) v_{rms}^2\left(t_v\right)} \over {v_{rms}^4\left(t_v\right) t_v^2}}\;.$$ (30)

It is now easy to derive from equation (E-7) the desired explicit expression for the Stolt stretch parameter $W$ : equation (17) in the main text.

Evaluating the Stolt-stretch parameter