The new R

Including causality and viscosity in the definition of R=ik_z (equation 12) it is possible to preserve the energy content for any arbitrary $\epsilon$ value.

Figure 7 presents the modeling-migration results, using the model of synthetic data set in Figure 6, with equations (2) and (12) for two different values of $\epsilon$.Figure 8 shows the respective frequency spectrum.

 

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Figure 6 Synthetic data set

 

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Figure 7 Comparison between modeling-migration using model on Figure 6 using R definition as in equation (2), left, and equation (12), right. The top figures have a damping factor $\epsilon=5$, the bottom figures have a damping factor of $\epsilon=15$

 

mod3_freq
Figure 8 Comparison between the frequency spectrum of the modeling-migration results on Figure 7. The top figures have a damping factor $\epsilon=5$,the bottom figures have a damping factor of $\epsilon=15$

We can assure the stability of the R definition presented in equation (12) based on the results of Figures 7 and 8; besides, this R definition is valid for all values of $\omega$ and k_x.

We also perform the phase-shift migration over a real data set, we present the results on Figures 9 and 10. These results show a comparison between phase-shift migration without considering the damping factor [R definition in equation (1)] and considering the damping factor [R definition in equation (12)].

 

out1
Figure 9 Data Comparison. Left the migration result without considering damping. Right the migration result using the new R with $\epsilon=5$

 

out2
Figure 10 Data Comparison. Left the migration result without considering damping. Right the migration result using the new R with $\epsilon=15$

The image that we obtain with the damping factor is much cleaner than the image without the damping factor; therefore, the events definition and the faults are more clear.