Space variable damping in conventional imaging condition

Conventional shot profile migration schemes determine the reflection strength at each subsurface point taking into account only the downgoing and the upgoing wavefields at that location. () compares two different imaging conditions  

 118#118 (48)

and  

 119#119 (49)

The first is one commonly used by the industry. It has the advantage of being robust, but has the disadvantage of not computing the correct amplitudes. The second computes the correct amplitudes (except for a damping factor 114#114), but has the disadvantage of being unstable due to zero division. That is why a damping factor 114#114 is needed.

We propose to add a mask function defined as  

 120#120 (50)

where 121#121 can be variable in space.

When 122#122 has enough energy to contribute to the image, the damping factor 114#114 is set to zero. When factor 122#122 is small, the damping factor is kept to avoid zero division. Thus, the imaging condition can be set as  

 123#123 (51)

where the damping is now variable in space.

A simple synthetic was generated to test the preceding idea using wave equation modeling. Figure a shows the downgoing wave, and Figure b the upgoing wave, at a fixed time. Figure  shows the mask 124#124 used in this example.

 

DU
Figure 3 Wavefields at a fixed time. a) Downgoing wave, b) Upgoing wave.

 

ma
Figure 4 Mask used in equation (). Zero at masked area and one out of the masked area.

Figure a shows the reflection strength calculated using the imaging condition stated in equation (). Figure b shows the reflection strength calculated using division of the upgoing wavefield 106#106 by the downgoing wavefield 9#9. Figure c shows the reflection strength calculated using the imaging condition stated in equation (), and Figure d shows the reflection strength calculated using the imaging condition stated in equation (). The advantage of Figure d's result over the others is that it has the correct reflection strength value inside the masked area and doesn't diverge outside it because of the damping factor.

 

comp_1shot
Figure 5 Comparison between four different imaging conditions a) Calculated by wavefield multiplication equation (), b) Calculated by wavefield division (122#122), c) Calculated using constant damping equation (), and d) Calculated using space variable damping equation ().

In Figure  we compare the two imaging conditions stated in equations () and () inside the masked area for two different 114#114. We can see for the imaging condition stated in equation () that the reflection strength inside the masked area doesn't change. This is an important advantage of space variable damping imaging principle, because it let us to build an adaptive mask dependent of the subsurface illumination.

 

comp_im
Figure 6 Comparison between imaging condition stated in equations () and () inside the masked area. a) 125#125, b) 126#126, c) 125#125, d) 126#126

We stack the reflection strength from 11 shots to see how the change observed in one shot affects the final image. The result is shown in Figure . We can see imaging condition from equation () gives the best resolution.

 

comp_stack
Figure 7 Comparison between 11 shot stacks using three different imaging conditions, a) equation (), b) wavefield division (122#122), c) equation (), and d) equation ().