Iteration in practice
In this section, we tested iterative depth migration of zero-offset, CMP-stack and prestack data using six different initial velocity-depth models. Aside from the true model, each model was corrupted by errors in layer velocities and/or reflector geometries. We summarize the results of the model experiments as follows:
- When will the image from iterative depth migration converge to the true velocity-depth model? The answer to this question is many-fold, and it depends on the type of input data and errors in the initial velocity-depth model (Figure 8.2-27). If the input data set is zero-offset section and if the input velocity-depth model is the true model (an impossible case in practice), then the answer is yes. For prestack data, the answer also is yes, whereas for stacked data, the answer is nearly. If the initial velocity-depth model is in error of reflector geometries only, the answer is most likely, very likely, and likely for zero-offset, prestack and stacked data, respectively. Finally, if the initial velocity-depth model is in error of layer velocities, no matter what the input data set is, the answer is invariably no. Needless to say, iterative depth migration never guarantees that the solution converges to a true velocity-depth model.
- It is wrong to discontinue an iterative application of depth migration without achieving convergence — the intermediate output does not represent a valid earth image in depth and neither does it infer a valid earth model in depth. Zero-offset traveltimes associated with this earth model are not consistent with the traveltimes in the input section.
- In iterative depth migration, only one set of parameters — either layer velocities or reflector geometries, should be modified from one iteration to the next. Changing the parameter type at some intermediate iteration will only divert the solution to a different end result and will cause the convergence to that end result to take longer. It is advisable to keep layer velocities unaltered from one iteration to the next, and only modify reflector geometries by interpreting the output image from depth migration.
- Number of iterations depends on how much the initial model departs from the true model. If the initial model contains errors in layer velocities, only, fewer iterations are required than if the model contains errors in reflector geometries. If the model contains errors both in layer velocities and reflector geometries, a large number of iterations is required to achieve convergence.
- Iterative depth migration converges to an answer, which almost never corresponds to the true velocity-depth model. Therefore, the final velocity-depth model estimated from iterative depth migration needs to be calibrated to well data. Specifically, layer velocities are adjusted so as to match at well locations the depth values of the layer boundaries included in the final velocity-depth model with the well tops corresponding to those layer boundaries.
Figure 8.2-27 Performance of iterative depth migration for different types of input data and errors in the initial velocity-depth model.
Figure 8.2-28 (a) Portion of a CMP-stacked section, and (b) the image from time migration.
Figure 8.2-29 Part 1: Iterative poststack depth migration applied to field data shown in Figure 8.2-28. See text for details.
Figure 8.2-29 Part 2: Iterative poststack depth migration applied to field data shown in Figure 8.2-28. See text for details.
Figure 8.2-28 shows a portion of a CMP-stacked section and its time migration. The deepest layer with an abundance of diffractions represents salt with anhydrite-dolomite rafts. The objective is to obtain an accurate geometry for the base-salt boundary represented by the strong, deepest reflection.
To illustrate poststack iterative depth migration, start with a simple, horizontally layered earth model with constant layer velocities as the initial velocity-depth model (Figure 8.2-29a) and migrate the stacked section to obtain the depth image shown in Figure 8.2-28b. Superimpose the flat depth horizons associated with the initial velocity-depth model on this image section, and note that the reflector geometries implied by the depth image show discrepancy with the flat depth horizons. Discard the latter and interpret the image section to derive a set of structurally consistent depth horizons (Figure 8.2-28c). This completes the first iteration of poststack depth migration and model updating.
Next, keep the layer velocities the same as for the initial velocity-depth model (Figure 8.2-28a), but use the updated depth horizons (Figure 8.2-28c) to build a new velocity-depth model as shown in Figure 8.2-28d. Now, perform poststack depth migration once more and note that the reflector geometries implied by the depth image (Figure 8.2-28e) are still in discrepancy with the intermediate velocity-depth model (Figure 8.2-28d). Discard the depth horizons and reinterpret them from the image section (Figure 8.2-28f). This completes the second iteration of poststack depth migration and model updating.
Repeat the steps for model updating, poststack depth migration and re-interpreting the depth horizons for the third (Figures 8.2-28g,h,i), and fourth time (Figures 8.2-28j,k,l) until convergence is achieved; that is, the velocity-depth model input to depth migration is consistent with the reflector geometries inferred by the depth image. The final velocity-depth model and the depth image derived from it are shown in Figures 8.2-30a,b. Convergence criterion can be verified by normal-incidence modeling of the zero-offset traveltimes that correspond to the reflectors associated with the layer boundaries included in the velocity-depth model. Superimpose the modeled traveltimes on the unmigrated stacked section and note that the modeled and the actual traveltimes are in good agreement.
While the consistency of the modeled and actual zero-offset traveltimes verifies that the final velocity-depth model (Figure 8.2-30a) from poststack iterative depth migration meets the convergence criterion, the model is not guaranteed to be accurate. In fact, as demonstrated by the synthetic data examples in this section, there exists not just one but a multiple number of velocity-depth models that are consistent with the stacked data. An acceptable model is that which also is consistent with prestack data; thus, one strategic reason for prestack imaging is resolving the uncertainty in the acceptable velocity-depth models and reducing the many possible models to a few that are geologically plausible.
- 2-D poststack depth migration
- Image rays and lateral velocity variations
- Time versus depth migration
- Iterative depth migration
- Iteration with zero-offset data
- Iteration with CMP-stacked data
- Iteration with prestack data