Example

Clapp (2004) took nine different realizations of a single linearizion of a complex synthetic model (Figure ). There were several problems with this approach. The most significant problem was that a single non-linear iteration, was far from sufficient. After one iteration, we still have significant move-out that another non-linear iteration of tomography has a chance of using. When doing multiple non-linear iterations we have two choices to make at each iteration. First, should we use the minimum energy model (no random perturbation) or introduce random perturbations? Second, if we are adding random perturbations, how many models should we create at each non-linear iteration?

 

model
Figure 1 The left panels shows the velocity model used as input to the finite difference scheme used to create the data. The right panel is the resulting migrating the data with the correct velocity. The fine structure seen below 1.6km is the reservoir.

For this experiment I decided to create five random perturbed models in the first non-linear iteration. From these five models I generated twenty five models during the second non-linear iteration. I then used these twenty-five models in a conventional migration velocity updating scheme. This gives some measure on the effect of the starting guess on the final solution. Each of the twenty-five models were equally reasonable points from which start a tomographic loop. The difference between the final images gives me some measure of the uncertainty in this updating scheme.

The left panel of Figure  shows my starting guess for the velocity problem. The right panel shows the resulting image. The velocity was created by applying a strong smoother to the correct velocity field then scaling the resulting model by .9. Figure  shows the results after one non-linear iteration. The top panel are the five realizations of ${\bf \gamma}$. The center panels are the resulting five velocity models, and the bottom five panels are the migrated images using these velocity models. The anticline trend is in all of the realizations but we still see significant differences in how the velocity estimate deals with the listric fault.

 

iter0
Figure 2 The left panel shows the initial velocity model. The right panel the resulting migration.

 

iter1
Figure 3 The top panel are the five realizations of ${\bf \gamma}$. The center panels are the resulting five velocity models, and the bottom five panels are the migrated images using these velocity models.

After four iterations, now with twenty-five different models, the differences are more dramatic. Figure  show the twenty-five different gamma panels. We see an overall reduction in the amount move-out (closer to gray), but the realizations still have significantly different character. The twenty-five velocity models (Figure ) also show significant variation, especially as we go deeper in the model. After four iterations we see significant differences in the images (Figure ). In most of the models we have focused the anticline structure, but the images have significant variation below. The basement reflectors are discontinuous in many of the models.

 

iter3_g
Figure 4 The twenty-five gamma panels after the third non-linear iteration. Note how we are overall closer to 1.0 (gray), but we still see differences in the various panels.

 

iter4_v
Figure 5 The final twenty-five velocity models.

 

iter4_i
Figure 6 The twenty-five different images. Note the differences, especially in the reservoir.