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Let's define u0 as the primary wavefield and u1
the surface-related, first-order multiple wavefield recorded at the surface.
If the earth varies only as a function of depth, then u will not
depend on both s and g, but only on the offset, h=g-s.
In this one-dimensional case, equation (
) becomes
| ![\begin{eqnarray}
u_1(g-s) & = & \int u_0(g-g') \; u_0(g'-s) \; dg' \\ u_1(h) & = & \int u_0(h-h') \; u_0(h') \; dh',\end{eqnarray}](img362.gif) |
(144) |
| (145) |
where h'=g'-s.
Equation (
) clearly represents a convolution, so can
be computed by multiplication in the Fourier domain such that
where Ui(kh) is the Fourier transform of ui(h) defined by
| ![\begin{displaymath}
U_i(k_h) = \int u_i(h) \; e^{-2 \pi i k_h h} dh.\end{displaymath}](img363.gif) |
(147) |
Equation (
) can be extended to deal with a
smoothly varying earth by considering common shot-gathers (or common
midpoint gathers) independently, and assuming the earth is locally
one-dimensional in the vicinity of the shot, e.g., Rickett and Guitton (2000):
|
U1(kh,s) = U0(kh,s)2.
|
(148) |
In practice, however, the primaries are not known and u0 is
replaced by the data with primaries and multiples.
A similar approach has been used by Kelamis and Verschuur (2000) for attenuating
surface-related multiples on land data for relatively flat geology.
Next: 2- and 3-D earth
Up: A surface-related multiple prediction
Previous: A surface-related multiple prediction
Stanford Exploration Project
5/5/2005