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Alkhalifah, T., and I. Tsvankin, 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550-1566.

Beasley, C., W. Lynn, K. Larner, and H. Nguyen, 1988, Cascaded frequency-wavenumber migration - Removing the restrictions on depth-varying velocity: Geophysics, 53, 881-893.

Castle, R. J., 1988, Shifted hyperbolas and normal moveout, in 58th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts: Soc. Expl. Geophys., Session: S9.3.

Claerbout, J. F., 1985, Imaging the Earth's Interior: Blackwell Scientific Publications.

de Bazelaire, E., 1988, Normal moveout revisited - inhomogeneous media and curved interfaces: Geophysics, 53, 143-157.

Fomel, S., 1997, Velocity continuation and the anatomy of prestack residual migration: 67th Ann. Internat. Mtg, Soc. of Expl. Geophys., 1762-1765.

Fomel, S., and V. Grechka, 1996, On nonhyperbolic reflection moveout in anisotropic media, in SEP-92: Stanford Exploration Project, 135-158.

Fomel, S. B., 1994, The method of velocity continuation in the seismic time migration problem: Russian Geology and Geophysics, 35, 100-111.

Gazdag, J., 1978, Wave equation migration with the phase-shift method: Geophysics, 43, 1342-1351.

Larner, K., and C. Beasley, 1987, Cascaded migrations - Improving the accuracy of finite-difference migration: Geophysics, 52, 618-643.
(Errata in GEO-52-8-1165).

Larner, K. L., C. J. Beasley, and W. S. Lynn, 1989, In quest of the flank: Geophysics, 54, 701-717.
(Erratum in GEO-54-7-932-932; Discussion in GEO-54-12-1648-1650 with reply by authors).

Levin, S., 1983, Remarks on two-pass 3-D migration error, in SEP-35: Stanford Exploration Project, 195-200.

----, 1985, Understanding Stolt stretch, in SEP-42: Stanford Exploration Project, 373-374.

Malovichko, A. A., 1978, A new representation of the traveltime curve of reflected waves in horizontally layered media: Applied Geophysics (in Russian), 91, 47-53.

Popovici, A. M., F. Muir, and P. Blondel, 1996, Stolt redux: A new interpolation method: Journal of Seismic Exploration, 5, 341-347.

Sava, P., 2000, Prestack Stolt residual migration for migration velocity analysis, in 70th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts: Soc. Expl. Geophys., 992-995.

Stolt, R. H., 1978, Migration by Fourier transform: Geophysics, 43, 23-48.
(Discussion and reply in GEO-60-5-1583).

----, 1996, Short note - A prestack residual time migration operator: Geophysics, 61, 605-607.

Sword, C. H., 1987, A Soviet look at datum shift, in SEP-51: Stanford Exploration Project, 313-316.

Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954-1966.
(Discussion in GEO-53-04-0558-0560 with reply by author).

Yilmaz, Ö., 2001, Seismic data analysis: Processing, inversion, and interpretation of seismic data: Soc. Expl. Geophys.

Yilmaz, Ö., I. Tanir, C. Gregory, and F. Zhou, 2001, Interpretive imaging of seismic data: The Leading Edge, 20.

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:

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

where $ l=x-x_0$ . Expansion (A-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 (A-1):

$\displaystyle \left.t_0\right\vert _{l=0}$ $\displaystyle =$ $\displaystyle t_v\;;$ (25)
$\displaystyle \left.{d^2t_0}\over {dl^2}\right\vert _{l=0}$ $\displaystyle =$ $\displaystyle {1 \over {t_v v_{rms}^2\left(t_v\right)}}\;;$ (26)
$\displaystyle \left.{d^4t_0}\over {dl^4}\right\vert _{l=0}$ $\displaystyle =$ $\displaystyle -{{3\,S\left(t_v\right)} \over {t_v^3 v_{rms}^4\left(t_v\right)}}\;\;.$ (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:
$\displaystyle \left.{ds}\over {dl}\right\vert _{l=0} =
\left. s'\left(t_0\right)\,{{dt_0}\over {dl}}\right\vert _{l=0} = 0\;;$      

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

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

$\displaystyle \left.{d^4s}\over {dl^4}\right\vert _{l=0}$ $\displaystyle =$ $\displaystyle \left(6\,s'''\left(t_0\right) \left({dt_0}\over {dl}\right)^2
+ 4s''\left(t_0\right)\,{{dt_0}\over {dl}}\,{{d^3t_0}\over {dl^3}}+ \right.$  
    $\displaystyle \left. \left. + s'\left(t_0\right)\,{{d^4t_0}\over {dl^4}}+
s^{IV}\left(t_0\right) \left({dt_0}\over {dl}\right)^4\right)\right\vert _{l=0} =$  
  $\displaystyle =$ $\displaystyle \left.\left(s''\left(t_v\right)\,\left({d^2t_0}\over {dl^2}\right...
...\right)\right\vert _{l=0} =
-{{3\,W} \over {v_0^4 \,s^3\left(t_0\right)}}\;\;.$ (29)

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

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

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

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