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Next: conclusions Up: Synthetic Data Example Previous: Migration for zero-offset multiple

Source-receiver migration for multiple reflections

For comparison, I migrate the originally recorded data (primaries + multiples) with Fourier finite difference for the Double Square Root equation Zhang and Shan (2001). The migration result is presented in Figure [*]. There are many hyperbolas because the recorded shots are very sparse. I migrated the multiple reflections using both the split-step method for the Double Square Root equation Popovici (1996) and the Fourier finite difference for the Double Square Root equation, in which the average velocity is used as the reference velocity. The migration results are presented in Figure [*] and Figure [*]. The migration result of the multiple data (Figure [*],[*]) is similar to that of the primary data (Figure [*]), although the latter is sharper and less noisy.

 
pffd
Figure 7
Migration result of originally recorded data.
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mmssf
Figure 8
Imaging of multiples by split-step of DSR.
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mmffd
Figure 9
Imaging of multiples by FFD of DSR.
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It is not easy to separate the multiples from the original data in practice. Also, separation costs a lot of computation time. Instead of cross-correlating the primary+multiple with the multiple, I cross-correlate the whole recorded data (primary+multiple) with itself and then run the source-receiver migration. Figure [*] presents the migration result. The cross-correlation between the primary reflection and itself does add some noise to the image. We can see the fake reflector, which is mainly caused by the cross-correlation between the water bottom reflection and the reflection below the water bottom in the primary. Nevertheless, the image is interpretable.

 
ppffd
Figure 10
Migration of cross-correlation between primary+multiple and primary+multiple.
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The amplitude of the primary reflection at different locations and different times varies. Therefore the multiple reflections have sources of different amplitudes. It is important to do amplitude balancing for the pseudo-primary data before migration. I apply deconvolution Claerbout (1999) instead of cross-correlation to the surface data, namely the pseudo-primary data for the source-receiver migration are calculated by  
 \begin{displaymath}
P(x,h,z=0,\omega)=\sum_s \frac{U(x_U,z=0,\omega,s)\bar{D}(x_...
 ...)}
 {D(x_D,z=0,\omega,s)\bar{D}(x_D,z=0,\omega,s)+\epsilon^2}. \end{displaymath} (4)
Figure [*] shows the migration result when deconvolution is used for creating pseudo-primary data. Besides the amplitude balancing, it also has better resolution.

 
mm_dec_ffd
Figure 11
Imaging of multiples after amplitude balance.
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next up previous print clean
Next: conclusions Up: Synthetic Data Example Previous: Migration for zero-offset multiple
Stanford Exploration Project
7/8/2003