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In the general case, modeling multiples becomes more expensive. Equation
(
) is not valid anymore (except for smoothly varying
media), and the convolution becomes nonstationary (shot
gathers are different from one location to another).
Hence, the wavefield is not only a function of offset, h, but also
depends on another spatial coordinate such as shot location s. In
3-D, the integral in equation
spans the entire
acquisition plane van Dedem (2002), which makes the prediction very expensive.
Introducing the nonstationary convolution, equation (
) can be
written as
| ![\begin{displaymath}
u_1(h,s) = \int u_0(h-h',s+h') \; u_0(h',s) \; dh'.\end{displaymath}](img364.gif) |
(149) |
Now, following Dragoset and Jericevic (1998) for 2-D prediction, we introduce some
amplitude corrections in the previous equation:
| ![\begin{eqnarray}
u_0(h-h',s+h') & = & F_{t\rightarrow \omega}[\sqrt{t}u_0(h-h',s...
...{\omega}{4\pi}}
F_{t\rightarrow \omega}[\sqrt{t}u_{0g}(h',s,t)].\end{eqnarray}](img365.gif) |
(150) |
| |
Replacing u0 by the data with primaries and multiples, equation
(
) with the amplitude correction
is used throughout this thesis to model surface-related multiples in 2-D.
Next: Limitations of the multiple
Up: A surface-related multiple prediction
Previous: One-dimensional earth and impulsive
Stanford Exploration Project
5/5/2005