Limitations in resolving velocity-depth ambiguity by tomography
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| Series | Investigations in Geophysics |
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| Author | Öz Yilmaz |
| DOI | http://dx.doi.org/10.1190/1.9781560801580 |
| ISBN | ISBN 978-1-56080-094-1 |
| Store | SEG Online Store |
Reflection traveltime tomography can be surprisingly successful in updating an initial model with significant errors. Consider, for instance, the velocity-depth model in Figure 9.5-25 with constant layer velocities. This model is indeed practically consistent with the input seismic data as verified by the depth image in Figure 9.5-25b and the modeled zero-offset traveltimes overlayed on the unmigrated stacked section in Figure 9.5-25c.
Perform prestack depth migration using the initial model to derive the image section in Figure 9.5-26a and compute the residual moveout semblance spectra of the selected image gathers in Figure 9.5-27. Note the significant moveout errors manifested by the semblance peaks which are off the zero moveout centerline on the spectra. Now, compute the horizon-consistent residual moveout semblance spectra along the depth horizons shown in Figure 9.5-26b and note the significant departures from the zero moveout line (Figure 9.5-28). Pick the residual moveout profiles from these spectra and perform a tomographic update to obtain a new set of interval velocity profiles and depth horizons as shown in Figure 9.5-29.
Combine the updated interval velocities and reflector geometries to create the updated velocity-depth model shown in Figure 9.5-30a. Note that this model is fairly close to the updated models derived from the applications of time-to-depth conversion (Figure 9.5-16a) and layer-by-layer inversion (Figure 9.5-23a) strategies. Model verification tests (Figures 9.5-30b,c) and residual moveout semblance spectra (Figures 9.5-31 and 9.5-32) demonstrate the feasibility and accuracy of the updated model in Figure 9.5-30a.
As demonstrated by the extreme case of an erroneous initial model (Figure 9.5-25), a tomographic update can steer the model toward an acceptable final model. How far, though, can we go with tomography to obtain a finely accurate model? In the following model experiments, we shall examine this fundamentally important question regarding tomography.
Figure 9.5-33 shows horizon-consistent residual moveout semblance spectra derived from image gathers that were created by prestack depth migration of the synthetic data as in Figure 9.1-2a using the initial velocity-depth model shown in Figure 9.1-5b. Shown in Figure 9.5-34 are the results of tomographic updating — the interval velocity profiles before (thin curves) and after (thick curves) the tomographic update, and the updated velocity-depth model itself. Compare with the initial velocity-depth model shown in Figure 9.1-5b and the true velocity-depth model shown in Figure 9.1-1b. Compare with the true model (Figure 9.1-1) and the initial model (Figure 9.1-5) and note that tomographic update has produced a result fairly close to the true model. A close-up of the center portions of the initial model, the final model, and the true model after the tomographic update is shown in Figure 9.5-35.
Figure 9.5-16 (a) The velocity-depth model in Figure 9.5-12a after model updating by reflection tomography, (b) prestack depth migration with depth horizons as in (a), (c) modeled zero-offset traveltimes overlayed on the unmigrated stacked section.
Figure 9.5-23 (a) The velocity-depth model in Figure 9.4-27a after model updating by reflection tomography, (b) prestack depth migration with depth horizons as in (a), (c) modeled zero-offset traveltimes overlayed on the unmigrated stacked section.
Figure 9.5-26 (a) Prestack depth migration using the velocity-depth model in Figure 9.5-25a, (b) the same image section with depth horizons as in Figure 9.5-25a.
Figure 9.5-27 Selected image gathers associated with the prestack depth-migrated section in Figure 9.5-26b and residual moveout semblance spectra computed from these gathers.
Figure 9.5-28 Residual moveout semblance spectra computed from the image gathers as in Figure 9.5-27 along the depth horizons shown in Figure 9.5-26b. The velocity-depth model used in prestack depth migration is shown in Figure 9.5-26a.
Figure 9.1-1 An earth model that comprises six flat layers: (a) the interval velocity profiles for the six horizons H1-H6; (b) true velocity-depth model created from the profiles in (a).
Figure 9.1-2 (a) The CMP-stacked section derived from the modeled common-shot gathers using the earth model shown in Figure 9.1-1b; (b) the stacking velocity section.
Figure 9.1-5 (a) The interval velocity profiles as in Figure 9.1-4a displayed by the thick curves and smoothed interval velocity profiles displayed by the thin curves; (b) estimated velocity-depth model using the smoothed interval velocity profiles in (a). Compare with the true velocity-depth model shown in Figure 9.1-1b and the model derived from the unsmoothed interval velocities shown in Figure 9.1-4b.
Iterate the updating process until the residual moveout for all the layers is minimized. Specifically, perform prestack depth migration using the tomographically updated velocity-depth model in Figure 9.5-34 and generate a new set of image gathers. Compute a new set of residual moveout semblance spectra from the image gathers as shown in Figure 9.5-36. Compare with the residual moveout spectra in Figure 9.5-33 before the tomographic update and note that the residual moveout after the update has been reduced significantly.
We now test the tomographic update in the presence of large errors in the initial velocity-depth model. Consider the velocity-depth model shown in Figure 9.1-1 based on the parameters listed in Table 9-2. We shall deliberately introduce errors to this model by setting the velocities for layer H3 to a constant value of 2550 m/s and for layer H4 to a constant value of 3250 m/s. The erronenous velocity-depth model is shown in Figure 9.5-37.
| Layer | Velocity (m/s) | Depth (m) |
| H1 | 1500 | 100 |
| H2 | 2000 | 1000 |
| H3 | 2400 – 2700 | 1500 |
| H4 | 3000 – 3500 | 1800 |
| H5 | 4500 | 2250 |
| H6 | 3000 | 2700 |
By adopting the distorted model in Figure 9.5-37 as the initial velocity-depth model, perform prestack depth migration, generate image gathers and compute residual moveout semblance spectra as shown in Figure 9.5-38. Note the large moveouts caused by the velocity errors associated with layers H3 and H4. The tomographic update shown in Figure 9.5-39a has modified the velocities for layers H3 and H4, and for the underlying layers. Specifically, the constant-velocities assigned to layers H3 and H4 have been replaced with laterally varying velocity profiles that are consistent with the actual velocity profiles shown in Figure 9.1-1. Nevertheless, compare the profiles in Figures 9.1-1a and 9.5-39a and note that the velocity estimate for the underlying layer H5 has been adversely influenced by the tomographic update applied to the layers above.
A close-up of the central portions of true velocity-depth model (Figure 9.1-1b), the initial velocity-depth model (Figure 9.5-37b), and the updated velocity-depth model (Figure 9.5-39b) is shown in Figure 9.5-40. Although the errors in the initial velocity-depth model for the shallow layers have been reduced, note that, in general, starting with an initial velocity-depth model with large errors, tomographic updating does not always produce satisfactory results.
The fact that the tomographic update for one layer is influenced by the errors in the parameters for the layers above is best illustrated by the model in Figure 9.1-18. This is the model with horizontal layers (Table 9-2) but with a near-surface layer H1 with laterally varying velocities between 800 m/s and 1500 m/s. Results of earth model estimation using Dix conversion are shown in Figure 9.1-21c. Consider this model as the initial velocity-depth model and perform prestack depth migration of the data associated with the stacked section in Figure 9.1-19a, generate image gathers, and compute the residual moveout semblance spectra shown in Figure 9.5-41. Pick the spectra to obtain the residual moveout profiles by tracking the semblance peaks, then use these profiles in a tomographic update to get the new model shown in Figure 9.5-42. Use the resulting model as input to a second iteration for model updating. Compare the resulting residual moveout spectra shown in Figure 9.5-43 with those from the first iteration shown in Figure 9.5-41 and note that the second iteration has reduced the residual moveout, significantly. The updated model from the second iteration is shown in Figure 9.5-44. A close-up of the central portions of the intermediate velocity-depth model (Figure 9.5-42b), the velocity-depth model updated for the second time (Figure 9.5-44b), and the true velocity-depth model (Figure 9.1-18b) is shown in Figure 9.5-45. Compare these estimates with the close-up view (Figure 9.1-22b) of the initial velocity-depth model shown in Figure 9.1-21c. The lesson to learn from this experiment is not to start with an initial model (Figure 9.1-21c) with large errors and expect a tomographic update to correct for these errors.
Figure 9.1-18 An earth model that comprises six flat layers: (a) the interval velocity profiles for the six horizons H1-H6; (b) true velocity-depth model created from the profiles in (a). This model is the same as the model in Figure 9.1-1 except that H1 has laterally varying velocities.
Figure 9.1-19 (a) The CMP-stacked section associated with the earth model shown in Figure 9.1-18; (b) stacking velocity section.
Figure 9.1-21 (a) The interval velocity profiles derived from Dix conversion of the horizon-consistent stacking velocity profiles picked from the semblance spectra shown in Figure 9.1-20, (b) the interval velocity profiles of (a) after lateral smoothing, (c) estimated velocity-depth model. Compare with the true velocity-depth model shown in Figure 9.1-18b.
Figure 9.1-22 Central portions of (a) the true velocity-depth model shown in Figure 9.1-18b, (b) the velocity-depth model shown in Figure 9.1-21c estimated using the smoothed interval velocity profiles shown in Figure 9.1-21b.
Figure 9.1-24 (a) The interval velocity profiles derived from coherency inversion semblance spectra shown in Figure 9.1-23; (b) estimated velocity-depth model. Compare with the true velocity-depth model shown in Figure 9.1-18b.
Figure 9.4-28 (a) A time-migrated stacked section with a set of interpreted horizons, (b) the unmigrated stacked section with the zero-offset traveltimes modeled from the time horizons in (a).
Finally, we shall update the model shown in Figure 9.1-24 which was estimated by coherency inversion. Consider this model as the initial velocity-depth model and perform prestack depth migration, generate image gathers, and compute the residual moveout spectra shown in Figure 9.5-46. Pick the spectra, again, by tracking the semblance peaks to obtain the residual moveout profiles. Then use these profiles in a tomographic update to get the new model shown in Figure 9.5-47. A close-up of the central portions of the initial velocity-depth model (Figure 9.1-24b), the updated velocity-depth model (Figure 9.5-47b), and the true velocity-depth model (Figure 9.1-18b) is shown in Figure 9.5-48.
Based on the results of the experiments using Dix conversion and coherency inversion applied to the model with a near-surface layer that has lateral velocity variations (Figure 9.1-18), again, the lesson to learn is that, whatever the strategy used in the model building, if you start with an initial model with large errors, do not expect tomographic updating to correct for these errors.
Figure 9.5-29 Model updating by reflection tomography: (a) The interval velocity profiles before (thin curves) and after (thick curves) tomographic update, (b) the depth horizons in Figure 9.5-25a after update, with the velocity-depth model before the update as in Figure 9.5-25a shown in the background.
Figure 9.5-30 (a) The velocity-depth model in Figure 9.5-25a after model updating by reflection tomography, (b) prestack depth migration with depth horizons as in (a), (c) modeled zero-offset traveltimes overlayed on the unmigrated stacked section.
Figure 9.5-31 Selected image gathers associated with the prestack depth-migrated section in Figure 9.5-30b and residual moveout semblance spectra computed from these gathers.
Figure 9.5-32 Residual moveout semblance spectra computed from the image gathers as in Figure 9.5-31 along the depth horizons shown in Figure 9.5-30b. The velocity-depth model used in prestack depth migration is shown in Figure 9.5-30a.
Figure 9.5-33 Horizon-consistent residual moveout semblance spectra derived from image gathers that were created by prestack depth migration of the synthetic data as in Figure 9.1-2a using the initial velocity-depth model shown in Figure 9.1-5b.
Figure 9.5-34 (a) The interval velocity profiles before (thin curves) and after (thick curves) tomographic update; (b) the updated velocity-depth model. Compare with the initial velocity-depth model shown in Figure 9.1-5b and the true velocity-depth model shown in Figure 9.1-1b.
Figure 9.5-35 Central portions of (a) the estimated initial velocity-depth model shown in Figure 9.1-5b using the smoothed interval velocity profiles shown in Figure 9.1-5a, (b) the updated velocity-depth model shown in Figure 9.5-34b, and (c) the true velocity-depth model shown in Figure 9.1-1b.
Figure 9.5-36 Horizon-consistent residual moveout semblance spectra derived from image gathers that were created by prestack depth migration of the synthetic data as in Figure 9.1-2a using the updated velocity-depth model shown in Figure 9.5-34b.
Figure 9.5-37 (a) The velocity profiles as in Figure 9.1-1a except for horizons H3 and H4 which deliberately have been assigned wrong velocities; (b) the velocity-depth model created from the interval velocity profiles in (a).
Figure 9.5-38 Horizon-consistent residual moveout semblance spectra derived from image gathers that were created by prestack depth migration of the synthetic data as in Figure 9.1-2a using the initial velocity-depth model shown in Figure 9.5-37b.
Figure 9.5-39 (a) The interval velocity profiles before (thin curves) and after (thick curves) tomographic update; (b) the updated velocity-depth model. Compare with the initial velocity-depth model shown in Figure 9.5-37b and the true velocity-depth model shown in Figure 9.1-1b.
Figure 9.5-40 Central portions of (a) the true velocity-depth model shown in Figure 9.1-1b, (b) the initial velocity-depth model shown in Figure 9.5-37b, and (c) the updated velocity-depth model shown in Figure 9.5-39b.
Figure 9.5-41 Horizon-consistent residual moveout semblance spectra derived from image gathers that were created by prestack depth migration of the synthetic data as in Figure 9.1-19a using the initial velocity-depth model shown in Figure 9.1-21c.
We now examine reflection traveltime tomography for its ability to update an initial model in the presence of a complex overburden. Figure 9.5-49a shows the rms velocity profiles along the time horizons in Figure 9.4-28a. Dix conversion yields the interval velocity profiles shown in Figure 9.5-49b. Admittedly, some editing and smoothing before and after Dix conversion were done to tame the interval velocities to behave in a structurally sensible manner.
By using the interval velocity profiles shown in Figure 9.5-49b, the time horizons interpreted from the time-migrated section shown in Figure 9.4-28a were converted to depth by way of image rays. Then, these depth horizons were combined with the interval velocity profiles (Figure 9.5-49b) to create an initial velocity-depth model. This was followed by depth migration of the stacked section in Figure 9.4-28b using the initial velocity-depth model, which in turn, was revised based on the interpretation of the depth-migrated section. A second iteration of poststack depth migration (Figure 9.5-50a) produced a velocity-depth model (Figure 9.5-49c) that is consistent with the reflection times observed on the unmigrated stacked section (Figure 9.5-50b).
Actually, the now consistent velocity-depth model in Figure 9.5-49c is what we shall update by way of reflection tomography. To begin with, perform prestack depth migration and generate image gathers as in Figure 9.5-51. Compute the residual moveout semblance spectra and observe that there are semblance peaks that are off the center line, indicating the presence of residual moveout. Stacking of the image gathers yields the image section from prestack depth migration shown in Figure 9.5-52. The significant improvement in imaging the diapiric structure and the base-salt reflector (horizon H8 as in Figure 9.5-50a) becomes obvious when this section is compared with the poststack depth-migrated section in Figure 9.5-50a. Nevertheless, the fact that the semblance spectra (Figure 9.5-51) exhibit residual moveout along the events associated with the layer boundaries included in the velocity-depth model suggests that the image from prestack depth migration may be improved by updating the initial velocity-depth model.
Figure 9.5-42 (a) The interval velocity profiles before (thin curves) and after (thick curves) the first tomographic update; (b) the updated velocity-depth model. Compare with the initial velocity-depth model shown in Figure 9.1-21c and the true velocity-depth model shown in Figure 9.1-18b.
Figure 9.5-43 Horizon-consistent residual moveout semblance spectra derived from image gathers that were created by prestack depth migration of the synthetic data as in Figure 9.1-19a using the updated velocity-depth model shown in Figure 9.5-42b.
Figure 9.5-44 (a) The interval velocity profiles before (thin curves) and after (thick curves) the second tomographic update; (b) the velocity-depth model updated for the second time. Compare with the initial velocity-depth model shown in Figure 9.1-21c, the intermediate velocity-depth model shown in Figure 9.5-42b, and the true velocity-depth model shown in Figure 9.1-18b.
Figure 9.5-45 Central portions of (a) the intermediate velocity-depth model shown in Figure 9.5-42b, (b) the velocity-depth model updated for the second time shown in Figure 9.5-44b, and (c) the true velocity-depth model shown in Figure 9.1-18b.
Figure 9.5-46 Horizon-consistent residual moveout semblance spectra derived from image gathers that were created by prestack depth migration of the synthetic data as in Figure 9.1-19a using the initial velocity-depth model shown in Figure 9.1-24b.
Figure 9.5-47 (a) The interval velocity profiles before (thin curves) and after (thick curves) the first tomographic update; (b) the updated velocity-depth model. Compare with the initial velocity-depth model shown in Figure 9.1-24b and the true velocity-depth model shown in Figure 9.1-18b.
Figure 9.5-48 Central portions of (a) the initial velocity-depth model shown in Figure 9.1-24b using the interval velocity profiles shown in Figure 9.1-24a, (b) the updated velocity-depth model shown in Figure 9.5-47b using the interval velocity profiles shown in Figure 9.5-47a, and (c) the true velocity-depth model shown in Figure 9.1-18b.
Now here is a dilemma that we have to live with when updating velocity-depth models. As a result of the inevitable muting of the image gathers (Figure 9.5-51), there is often insufficient offset range to accurately measure residual moveouts for shallow horizons. Because of the very poor residual moveout spectra, shallow horizons can be unreliable in model updating. Errors in the model parameters for the shallow layers will of course adversely affect the estimation and updating of the model parameters for the deeper layers. At greater depths, although there is sufficient cable length, faster velocities prevent recognition of velocity or depth errors with adequate resolution.
There is of course some mid-range of depths and velocities for which residual moveout semblance spectra can be used for model updating. In the present example, residual moveout semblance spectra associated with horizons H5 to H8 (as denoted in Figure 9.5-50a) were computed and picked as input to the reflection traveltime tomography solver (Section J.6). When picking the residual moveout profiles from the semblance spectra in Figure 9.5-52, keep in mind not to track all the rapid fluctuations that are much less than a cable length. These are the manifestations of the lateral velocity variations in the layers above as was demonstrated by the experiments conducted using synthetic data in models with horizontal layers.
Figure 9.5-49 (a) RMS velocity profiles associated with the data shown in Figure 9.4-28a, (b) the interval velocity profiles computed from (a) by way of Dix conversion, and (c) an initial velocity-depth model derived from the combination of the interval velocity profiles in (b) with the depth horizons that were created by image-ray depth conversion of the time horizons shown in Figure 9.4-28a.
Figure 9.5-50 (a) Poststack depth migration of the stacked section in (b) using the initial velocity-depth model in Figure 9.4-49c, (b) zero-offset traveltimes modeled from the depth horizons in (a) and superimposed on the stacked section.
Figure 9.5-51 Part 1: Selected image gathers associated with the prestack depth-migrated section in Figure 9.5-52 and residual moveout semblance spectra computed from these gathers.
Figure 9.5-51 Part 2: Selected image gathers associated with the prestack depth-migrated section in Figure 9.5-52 and residual moveout semblance spectra computed from these gathers.
Figure 9.5-52 Prestack depth migration of the data associated with the stacked section shown in Figure 9.5-50b using the initial velocity-depth model shown in Figure 9.5-49c, and (b) the residual moveout semblance spectra computed from the image gathers associated with the prestack depth-migrated section in (a) along Horizons H5-H8 labeled as in Figure 9.5-50a.
Figure 9.5-53 (a) The interval velocity profiles before (thin curves) and after (thick curves) the second tomographic update, and (b) the depth horizons after the update superimposed on the intermediate velocity-depth model derived from the first iteration of tomographic update.
Figure 9.5-54 (a) The final velocity-depth model after the second tomographic update, and (b) prestack depth migration of the data associated with the stacked section shown in Figure 9.5-50b using the final velocity-depth model in (a).
The workflow for tomographic updating — prestack depth migration to generate image gathers, computing residual moveout along events associated with the layer boundaries, picking residual moveout profiles and using them as input to the tomography to derive the updates for layer velocities (Figure 9.5-53a) and reflector geometries (Figure 9.5-53b), were repeated twice to produce the final model and image shown Figure 9.5-54. In each iteration, the first four layers were kept unchanged because of poor estimates for residual moveout along the horizons that correspond to the base of these layers (H1 through H4). Compare the images from prestack depth migrations shown in Figures 9.5-52a and 9.5-54a obtained from the initial and final velocity-depth models of Figures 9.5-52b and 9.5-54b, respectively, and note that the differences are marginal.
Iterative tomographic updating, as demonstrated by the model experiments in this section, does not necessarily result in convergence of an initial model to a final model with zero residual moveout. Instead, the solution may just wobble and never converge. This scenario is especially likely for cases of complex overburden structures as in the present example. In such cases, additional information; such as well control, check-shot velocities, and geologic constraints, is required to derive an acceptable final model.
See also
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