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Kinematic imaging of pegleg multiples in a laterally-homogeneous earth
In a ``1-D Earth'' (horizontally-stratified, v(z) medium), the normal-moveout
(NMO) equation Taner and Koehler (1969) describes the prestack traveltime curve
of a primary reflection at relatively short source-receiver offset:
| ![\begin{displaymath}
t = \sqrt{ \tau + \frac{x^2}{V_{\rm rms}^2(\tau)} }.\end{displaymath}](img71.gif) |
(16) |
Applied as an offset-dependent vertical time shift to a CMP gather, the NMO
equation flattens an arbitrary primary to its zero-offset traveltime
,
where (half) offset is denoted x and the root-mean-square (RMS) velocity,
, is defined in a laterally-homogeneous earth as:
| ![\begin{displaymath}
V_{\rm rms}^2 = \frac{1}{\tau} \sum_{i=1}^{n_{\tau}} v_i^2 \Delta \tau\end{displaymath}](img73.gif) |
(17) |
The earth is parameterized by
layers of time thickness
,
with constant interval velocity vi in each layer.
Analogously, a modified NMO equation can image pegleg multiples in a 1-D Earth,
as motivated graphically by Figure
. From the figure, we see that
kinematically, a first-order pegleg can be conceptualized as a ``pseudo-primary''
with the same offset, but with an additional two-way zero-offset traveltime to
the multiple generator,
. In equation form, let us extend this intuition
to the general case of a
-order pegleg to write an NMO equation for
peglegs:
| ![\begin{displaymath}
t = \sqrt{ (\tau+j\tau^{*})^2 + \frac{x^2}{V_{\rm eff}^2} }.\end{displaymath}](img78.gif) |
(18) |
is the effective RMS velocity of the pseudo-primary shown in Figure
. To derive an expression for
, we modify the
definition of RMS velocity, equation (
), to reflect a
-order pegleg multiple's additional travel between the surface and
multiple generator:
| ![\begin{displaymath}
V_{\rm eff}^2 = \frac{1}{\tau+j\tau^*}
\left( j \sum_{i=1}...
...Delta \tau
+ \sum_{i=1}^{n_{\tau}} v_i^2 \Delta \tau \right).\end{displaymath}](img80.gif) |
(19) |
Analogously,
is the number of assumed layers between the earth's
surface and the multiple generator. Notice that the two terms inside the
parentheses of equation (
) are simply the definition of RMS
velocity at
and
, respectively. We can substitute equation
(
) accordingly to derive the final expression for
:
| ![\begin{displaymath}
V_{\rm eff}^2 = \frac{ \left( j\tau^* V_{\rm rms}^2(\tau^*) + \tau V_{\rm rms}^2(\tau) \right)}
{\tau+j\tau^*}.\end{displaymath}](img83.gif) |
(20) |
Wang (2003) derives a similar series of expressions.
schem
Figure 3 Pegleg multiples ``S201G'' and ``S102G'' have the
same traveltimes as ``pseudo-primary'' with the same offset and an extra
zero-offset traveltime . |
| ![schem](../Gif/schem.gif) |
Next: Amplitude corrections for pegleg
Up: Particular Implementation of LSJIMP
Previous: Particular Implementation of LSJIMP
Stanford Exploration Project
5/30/2004