Reflection traveltime tomography
Reflection traveltime tomography is based on perturbing the initial model parameters by a small amount and then matching the change in traveltimes to the traveltime measurements made from residual moveout analysis of image gathers  . A mathematical treatment of the subject is given in Section J.6. Here, we remind ourselves of the underlying assumptions, outline the theory, and examine the performance of the method with model experiments. We shall complement the discussion on tomography with a field data example.
We must do the best we can in building an accurate earth model in depth (model building) so that only small changes remain to be made to the model by tomography. Specifically, a tomographic update can be expected to work provided the changes to be made to the initial earth model parameters in terms of slownness and depths at layer boundaries are small compared to the model parameters themselves. Additionally, we shall assume that the initial model is made up of horizontal layers with laterally invariant model parameters.
In the usual implementation of reflection traveltime tomography, the model parameters are perturbed while preserving the offset values of the seismic data (Section J.6). The tomographic update Δp to the model parameters that comprise the changes in the slowness and depths to layer boundaries is given by the generalized linear inversion (GLI) solution (equation J-88 in Section J.6)
where Δt denotes the column vector that represents the residual moveout times measured from the image gathers, L is a sparse matrix — its elements are in terms of the slowness and depth parameters associated with the initial model, and T denotes matrix transposition.
Figure 9.5-2 (a) The velocity-depth model created by using a combination of Dix conversion to estimate layer velocities (Figure 9.5-1c) and image-ray depth conversion of the time horizons interpreted from the time-migrated stacked section (Figure 9.5-1a), (b) poststack depth migration with depth horizons as in (a), (c) modeled zero-offset traveltimes overlayed on the unmigrated stacked section.
Figure 9.5-3 (a) Prestack depth migration using the velocity-depth model in Figure 9.5-2a, (b) the same image section with depth horizons as in Figure 9.5-2a.
Figure 9.5-5 (a) An image gather associated with the prestack depth-migrated section in Figure 9.5-3b, (b) residual moveout spectrum computed from (a), (c) the image gather in (a) after residual moveout correction, (d) the residual moveout section derived from the semblance spectra shown in Figure 9.5-6.
Figure 9.5-6 Residual moveout semblance spectra computed from the image gathers as in Figure 9.5-4 along the depth horizons shown in Figure 9.5-3b. The velocity-depth model used in prestack depth migration is shown in Figure 9.5-2a.
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.
Consider the earth model with horizontal layers shown in Figure 9.1-1. We shall make an attempt to update the initial estimate shown in Figure 9.1-5 using reflection tomography.
- Generate a set of image gathers (Figure 9.5-4) from prestack depth migration (Figure 9.5-3) using an initial velocity-depth model (Figure 9.5-2a).
- Convert the image gathers from depth to time using the interval velocity functions extracted from the initial velocity-depth model at the image-gather locations.
- Compute the horizon-consistent residual moveout for all offsets along events on image gathers that correspond to the layer boundaries included in the model. Figure 9.5-6 shows the residual moveout spectra along the six horizons of the model in Figure 9.5-2a. The vertical axis represents the residual moveout measured at a reference offset, usually the maximum offset. The horizontal axis represents the CMP locations along the line. Since residual moveout can be either negative or positive, the vertical axis is in both the positive and negative directions.
- Pick the residual moveout profiles for all the horizons by tracking the semblance peaks. Any departure from the horizontal axis indicates a nonzero value for residual moveout.
- Build the traveltime error vector Δt using the residual moveout times. As an example, you may have 10 layers, 1000 CMPs with a fold of 30. This means that the length of the traveltime error vector Δt is 300 000.
- Define the initial model by a set of slowness and depth parameters and construct the coefficient matrix L in equation (10) (Section J.6). As an example, you may have 10 layers and each layer may be defined by 50 pairs of slowness and depth values in the lateral direction. This means you would have 1000 parameters in your model space. It also means that for the example given in step (b), you have 300 000 equations to solve for 1000 parameters. The solution for this overdetermined system is given by equation (10).
- Estimate the change in parameters vector Δp, by way of the GLI solution given by equation (10).
- Update the parameter vector p + Δp. Figure 9.5-11a shows the interval velocity profiles before and after the tomographic update and Figure 9.5-11b shows the updated depth horizons superimposed on the initial velocity-depth model of Figure 9.5-2a. In a tomographic update, you may wish to perturb a subset of layer velocities and/or reflector geometries. This depends on your confidence in the initial model parameters for the layers and the quality of the residual moveout profiles to be used in inversion.
Figure 9.5-7 Model updating by residual moveout corrections: (a) The interval velocity profiles as in Figure 9.5-1c before (thin curves) and after (thick curves) residual moveout corrections, (b) the depth horizons in Figure 9.5-2a after an update with the velocity-depth model before update as in the same figure shown in the background.
Figure 9.5-10 Residual moveout semblance spectra computed from the image gathers as in Figure 9.5-9 along the depth horizons shown in Figure 9.5-8b. The velocity-depth model used in prestack depth migration is shown in Figure 9.5-8a.
By combining the updated interval velocity profiles (Figure 9.5-11a) with the new depth horizons (Figure 9.5-11b), we obtain the velocity-depth model after the first iteration of tomographic update (Figure 9.5-12a). Following the update, we check for consistency of the new model with the input seismic data. Overlay the depth horizons from the updated model in Figure 9.5-12a onto the image section derived from prestack depth migration shown in Figure 9.5-12b and note that they coincide with the reflectors associated with the layer boundaries included in the model. Also, the modeled zero-offset reflection traveltimes using the updated model coincide with the observed traveltimes of the events on the unmigrated stacked section (Figure 9.5-12c) that are associated with the layer boundaries included in the velocity-depth model of Figure 9.5-12a.
Next, compute the residual moveout semblance spectra from the image gathers at selected locations along the line after the model update (Figure 9.5-13) and compare them with those before the update (Figure 9.5-4). While most of the semblance peaks are now aligned with the vertical axis of the spectra, the tomographic update may be repeated to further remove any remaining residual moveout errors. The horizon-consistent residual moveout semblance spectra after the first iteration of the tomographic model update are shown in Figure 9.5-14. The residual moveout profiles that were picked from the spectra before the update (Figure 9.5-6) have been overlayed on the spectra after the update to observe the extent of the removal of residual moveout errors by the tomographic update.
Proceed to a second iteration of tomographic update by picking a new set of residual moveout profiles from the spectra shown in Figure 9.5-14. Then, follow the updating procedure described above to obtain a new set of interval velocity profiles (Figure 9.5-15a) and depth horizons (Figure 9.5-15b). Note that the changes in interval velocities and depth horizons that result from the second iteration are smaller compared to those from the first iteration (Figure 9.5-11).
Figure 9.1-11 Coherency inversion for layer H6 as in Figure 9.1-2a: (a) CMP raypaths, (b) the CMP gather at the analysis location, (c) the data window that includes the color-marked event in the CMP gather in (b) with moveouts associated with the different trial velocities assigned to layer H6. See text for details.
Figure 9.1-12 Semblance spectra derived from coherency inversion applied to the central CMP gather associated with the stacked data in Figure 9.1-1a using maximum offsets of 2400 m, 1400 m, and 400 m. See text for details.
Figure 9.1-13 Horizon-consistent coherency inversion semblance spectra computed from the CMP gathers associated with the stacked data in Figure 9.1-2a. No smoothing has been applied to the velocity profiles picked from these semblance spectra.
Figure 9.1-14 (a) The interval velocity profiles derived from coherency inversion semblance spectra shown in Figure 9.1-13; (b) estimated velocity-depth model. Compare with the true velocity-depth model shown in Figure 9.1-1b.
Figure 9.1-15 Horizon-consistent coherency inversion semblance spectra computed from the CMP gathers associated with the stacked data in Figure 9.1-2a. When computing the semblance spectrum for a given layer, smoothing has been applied to the velocity profiles picked from the semblance spectra associated with the layers above.
Figure 9.1-16 (a) The interval velocity profiles derived from coherency inversion semblance spectra shown in Figure 9.1-15; (b) estimated velocity-depth model. Compare with the true velocity-depth model shown in Figure 9.1-1b.
Combine the new set of interval velocity profiles and depth horizons to create the next update of the velocity-depth model shown in Figure 9.5-16a. Verify the updated model by performing prestack depth migration to generate an image section (Figure 9.5-16b) and zero-offset modeling of traveltimes associated with the layer boundaries included in the model (Figure 9.5-16c). Next, compute the residual moveout semblance spectra to examine any remaining moveout errors (Figure 9.5-17). Finally, compute the horizon-consistent residual moveout semblance spectra and overlay the residual moveout profiles from the first iteration (Figure 9.5-18). Note that the changes from the first to the second iteration are marginal. At this point, you may wish to end the iterations for tomographic update.
The extent to which an initial velocity-depth model is perturbed by a tomographic update depends on the accuracy of that initial model, which in turn, depends on how it has been estimated. Consider the two initial velocity-depth models shown in Figures 9.4-27a and 9.5-2a, which, for convenience, will be referred to as Models 1 and 2. Model 1 in Figure 9.4-27a was built by applying the layer-by-layer inversion strategy with a combination of coherency inversion and poststack depth migration to estimate the layer velocities and delineate the reflector geometries. Model 2 in Figure 9.5-2a was built by applying the time-to-depth conversion strategy with a combination of Dix conversion to estimate the layer velocities and image-ray depth conversion to delineate the reflector geometries.
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 9.1-23 Horizon-consistent coherency inversion semblance spectra computed from the CMP gathers associated with the stacked data in Figure 9.1-19a. When computing the semblance spectrum for a given layer, smoothing has been applied to the velocity profiles picked from the semblance spectra associated with the layers above.
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 9.4-27 (a) Final velocity-depth model derived from the layer-by-layer application of coherency inversion and poststack depth migration, (b) depth migration using the velocity-depth model in (a), (c) modeled zero-offset traveltimes using the velocity-depth model in (a) overlayed on the unmigrated stacked section as in Figure 9.4-15.
As with the model in Figure 9.5-2a, perform prestack depth migration using Model 1 to obtain the image section shown in Figure 9.5-19a. Then, compute the residual moveout semblance spectra for selected image gathers (Figure 9.5-20). By comparing these spectra with those in Figure 9.5-4, it may not be obvious as to whether Model 1 or Model 2 requires more perturbation. However, a comparison of the horizon-consistent residual moveout semblance spectra shown in Figures 9.5-6 and 9.5-21 reveals that Model 1 gives rise to less residual moveout on image gathers than does Model 2. As such, the tomographic update of Model 1 yields changes in interval velocities and reflector geometries (Figure 9.5-22) that are smaller than those for Model 2 (Figure 9.5-7).
Irrespective of the strategy followed to derive the initial model, the results of model updating need to be verified for consistency with the input seismic data (Figure 9.5-23) and examined for any remaining residual moveouts (Figure 9.5-24) to decide whether or not to continue with the iterations of tomographic update.
- Sherwood et al., 1986, Sherwood, J. W. C., Chen, K. C., and Wood, M., 1986, Depths and interval velocities from seismic reflection data for low-relief structures: Proc. Offshore Tech. Conf., 103–110.
- Kosloff et al., 1996, Kosloff, D., Sherwood, J. W. C., Koren, Z., Machet, E., and Falkovitz, Y., 1996, Velocity and interface depth determination by tomography of depth migrated gathers: Geophysics, 61, 1511–1523.
- Model updating
- Residual moveout analysis
- Limitations in resolving velocity-depth ambiguity by tomography
- Turning-ray tomography