Effects of RIP

Having obtained satisfactory AVA results with my downward continuation migration operator, I moved on to test my geophysically Regularized Inversion with model Preconditioning (RIP). For my first test, I set $\epsilon$ to zero. Although this essentially sets the model styling goal of fitting goals (5) to zero, I am preconditioning the problem so I am still regularizing the inversion through the ${\bf A^{-1}}$ in the data fitting goal. Using this formulation, I ran tests using 5, 10 and 15 iterations. The results after 5 iterations can be seen in Figure 3. The results after 10 iterations are shown in Figure 4 and the results after 15 iterations are shown in Figure 5.

For all three exercises, we notice that the shallowest interface (left panels) has developed an artificial increasing trend. This is partly due to an edge effect at the small p_h, and partially due to the effects of the regularization. It is more pronounced for this interface because the true AVA trend is expected to be almost flat. The edge effect at small p_h is present for the other two interfaces as well.

Looking at the results for the second interface (center panels of Figures 3 through 5), it appears that our results after regularized inversion are more accurate than the result we saw from the migration (center panel of Figure 2). For the second interface, other than the edge effect at small p_h, the AVA trend is quite accurate after 5, 10 and 15 iterations. The best result for the second interface appears to be that after 10 iterations. The result after 15 iterations is deteriorating slightly as the inversion is trying harder to accomodate artifacts that exist in the data.

The results for the deepest interface (right panels of Figures 3 through 5) are also better than the result from the migration (right panel of Figure 2). They have the same problems at large p_h due to survey geometry, and have the edge effect at small p_h seen for the other interfaces, but the AVA trend between these two extremes is close to the expected trend. Due to the known problem at large p_h, I have actually chosen to turn the regularization operator off halfway along the p_h axis to keep the inversion from spending all of its effort trying to correct the sudden decrease in amplitude. Once again, it seems that the AVA trend in the result after 10 iterations is the best.

 

ava.comp.5it0eps
Figure 3 Amplitude variation with p_h for geophysical regularization with 5 iterations and $\epsilon\ =\ 0$.Left panel shows the results for the shallowest interface as seen in Figure 1, center panel for the second interface, and right panel for the deepest interface. The solid line shows the theoretical value and the dots show the values obtained after inversion.

 

ava.comp.10it0eps
Figure 4 Amplitude variation with p_h for geophysical regularization with 10 iterations and $\epsilon\ =\ 0$.Left panel shows the results for the shallowest interface as seen in Figure 1, center panel for the second interface, and right panel for the deepest interface. The solid line shows the theoretical value and the dots show the values obtained after inversion.

 

ava.comp.15it0eps
Figure 5 Amplitude variation with p_h for geophysical regularization with 15 iterations and $\epsilon\ =\ 0$.Left panel shows the results for the shallowest interface as seen in Figure 1, center panel for the second interface, and right panel for the deepest interface. The solid line shows the theoretical value and the dots show the values obtained after inversion.

To examine the effect of a stronger regularization, I next set $\epsilon$ to .002. This means that the data fitting goal still influences the resulting model more than the regularization, but now the regularization is acting through the model styling goal as well as the data fitting goal. This value for $\epsilon$ was selected based on previous trial-and-error experiments in which poor illumination was a problem Clapp (2003). The results after 5 iterations of this test can be seen in Figure 6. The results after 10 and 15 iterations are in Figures 7 and 8, respectively. Since there are no sudden, large amplitude changes along the p_h axis, this stronger regularization should not affect the results much more than the inversion results with $\epsilon\ =\ 0$. As expected, these results are similar to the results with $\epsilon\ =\ 0$. We see the same unfortunate edge effect at small p_h for all of the interfaces in all three experiments (5, 10 and 15 iterations). We also see the effects of the survey geometry on the deepest interface (right panels of Figures 6-8), where I have once again elected to turn off the regularization operator. Overall, the increased strength of regularization has not affected the AVA trends of any of the interfaces any more than the previous inversion experiment.

 

ava.comp.5it002eps
Figure 6 Amplitude variation with p_h for geophysical regularization with 5 iterations and $\epsilon\ =\ 0.002$.Left panel shows the results for the shallowest interface as seen in Figure 1, center panel for the second interface, and right panel for the deepest interface. The solid line shows the theoretical value and the dots show the values obtained after inversion.

 

ava.comp.10it002eps
Figure 7 Amplitude variation with p_h for geophysical regularization with 10 iterations and $\epsilon\ =\ 0.002$.Left panel shows the results for the shallowest interface as seen in Figure 1, center panel for the second interface, and right panel for the deepest interface. The solid line shows the theoretical value and the dots show the values obtained after inversion.

 

ava.comp.15it002eps
Figure 8 Amplitude variation with p_h for geophysical regularization with 15 iterations and $\epsilon\ =\ 0.002$.Left panel shows the results for the shallowest interface as seen in Figure 1, center panel for the second interface, and right panel for the deepest interface. The solid line shows the theoretical value and the dots show the values obtained after inversion.