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Derivation of Snell Resampling Operator
In the following appendix, I derive the Snell resampling operation, equation
(
). The graphical basis for the derivation is Figure
. Since the pegleg multiple and primary in the figure have
the same emergence angle,
, the stepout, or spatial derivative, of the
traveltime curves of the two events is the same at x and xp. First we
compute the stepout of the primary event, starting from the standard NMO equation:
| ![\begin{eqnarray}
t^2_p &=& \tau + \frac{x_p^2}{V^2}
\
\frac{d}{dx_p}\left(t^...
...= \frac{2x_p}{V^2}
\ \frac{d t_p}{dx_p} &=& \frac{x_p}{t_p V^2}.\end{eqnarray}](img160.gif) |
(49) |
| (50) |
| (51) |
Using equations (
) and (
), we can similarly compute
the stepout of the corresponding jth-order pegleg multiple:
| ![\begin{displaymath}
\frac{d t_m}{dx} = \frac{x}{t_m V_{eff}^2}.\end{displaymath}](img161.gif) |
(52) |
Finally, we compute xp as a function of x by squaring equations
(
) and (
), setting them equal,
and substituting traveltime equations (
) and (
)
for tm and tp, respectively:
| ![\begin{eqnarray}
\frac{x_p^2}{t_p^2 V^4} &=& \frac{x^2}{t_m^2 V_{eff}^4}.
\ x^...
...{x^2 \tau^2 V^4}{(\tau+j\tau^*)^2 V_{eff}^4 + x^2(V_{eff}^2-V^2)}.\end{eqnarray}](img162.gif) |
(53) |
| (54) |
| (55) |
Next: REFERENCES
Up: Least-squares joint imaging of
Previous: HEMNO Equivalence with Levin
Stanford Exploration Project
5/30/2004