Including causality and viscosity in the definition of
*R*=*ik*_{z} (equation 12) it is possible to preserve
the energy content for any arbitrary 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 .Figure 8 shows the respective frequency spectrum.

Figure 6

Figure 7

Figure 8

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 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)].

Figure 9

Figure 10

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.

4/29/2001