The previous results are promising, but it is clear that the 2-D inversion is strongly dependent on the preconditioning operator. Since this operator is constructed from picked reflectors, it is important to know what happens if the reflector is not picked well. To examine this, we created a preconditioning operator from the ``reflectors'' picked in Figure 6. In this figure, note that the picked reflectors cross the correct dips at the depths between 3 and 3.4 kilometers and 3.7 and 4.1 kilometers. They cross themselves at depth 4.5 kilometers. The picked reflector beginning at depth 3.75 km follows the correct dip for the most part, but ignores the slight change in dip at the fault at CMP position 7.2 km. The water bottom has been correctly picked. The picked reflector beginning at depth 4.2 km follows the correct dip, but continues well into the shadow zone where it may or may not be correct. Also within the shadow zone is a completely absurd picked reflector put there to see if any event can be created there. Finally, note that the top and bottom of the salt have not been picked at all. This will leave the preconditioning operator in the salt area to be interpolated from the picked reflectors. In this section, we will refer to the dips and reflectors from the migration result as ``real'' or ``correct'' and the dips and reflectors used for the inversion as ``picked.''
The result of using this preconditioning operator for the 2-D inversion is seen in Figure 7. As expected, the result isn't good. To investigate this closely, we chose 5 areas to look at as defined in Figure 8. In these areas, we calculated the instantaneous energy on both the migrated result and the result of the 2-D badly preconditioned inversion.
Box 1 enclosed an area in which the picked reflector began with an opposite dip to the correct one then changed to the correct one. In Figure 9, we can see that the energy from the migrated result is fairly constant all along the reflectors. In the panel from the inversion result, the energy has almost completely disappeared along the real reflectors where the picked reflector had the wrong dip, then recovers where the picked reflector has the correct dip. This example shows that picking dips that are completely wrong (opposite of the correct one) will cause the inversion to reject what the model styling goal (Eqn. 3) is trying to do.
In Box 2, the energy in the migrated panel is once again fairly constant along all of the reflectors (Fig. 10). In the preconditioned result, one of the most obvious effects is the loss of energy in the upper left corner. This loss of energy is caused by the picked reflector shown in Figure 9 which has the incorrect dip. The picked reflectors in Figure 10 are quite interesting. Both picked reflectors match the real events except where the real events are affected by faulting. For the shallower event, this results in energy loss at the beginning and end of the faulted area but little loss of energy elsewhere because the real dips are very close to the picked dips. For the deeper event, the faulted area has a different dip from the picked reflector, so the entire faulted area has lost energy. This result may mean that this inversion scheme can be modified for fault detection. The deepest event in this box has lost no energy because it is close to the dip of the picked reflector above it. This example shows that the picked dip doesn't have to be totally different from the correct dip for the inversion to reject it.
In Box 3, all of the coherent energy in the migrated result (Fig. 11) ends at CMP=9.5 km. The shallowest picked reflector in this box is the now-familiar incorrect dip reflector that causes the preconditioned result to lose energy in the upper left corner. The next two picked reflectors follow the real events but extend well beyond the coherent energy limit in the migrated result. This has caused the inversion to generate coherent energy well into the shadow zone beneath the salt. The final picked reflector in this box is the silly ``M'' shaped one that is entirely within the shadow zone. This picked reflector has had almost no effect on the energy of the preconditioned result, just as we would hope. This result clearly shows that the reasonably picked reflectors will enhance the result of the preconditioned inversion in shadow zones and that poorly picked reflectors in shadow zones won't generate unreasonable events.
The fourth area (Box 4) examined is very interesting. Here we wanted to test the stability of our inversion if the preconditioning operator contained conflicting dips. Figure 12 shows the energy from the migrated result is somewhat garbled along the left side and coherent with smoothly varying dips otherwise. There is also a bright spot that is in the real model at CMP=9.5 km. The picked reflectors only match the correct dips above the depth of 4.5 km. The energy from the preconditioned result is incoherent and weak except for a small section between the CMPs at 8 km and 9 km and above 4.5 km in depth. The garbled left side energy from the migrated result has been smeared into a large blob and the bright spot has also been smudged. Overall this preconditioned result has managed to destroy almost all of the real information, just as expected. The important point from this experiment is that the preconditioned result did not show unexpected behavior in the area where the picked reflectors cross.
Box 5 contains part of the salt body. The only picked reflector really influencing this box was the water bottom. The energy from the migration shows high energy along the salt top and bottom, with low energy along the water bottom reflection and other layers (Fig. 13). The inversion has attempted to impose the dip of the water bottom on the salt events. The result has eliminated almost all of the energy that belonged to reflectors that were not close to the dip of the water bottom. This example shows that any strong reflectors that seem correct in the migration should be picked to preserve them in the inversion.