# Iteration with zero-offset data

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

Depth migration of the zero-offset section in Figure 8.2-1 using Model A shown in Figure 8.2-11 produces an accurate image of the salt diapir and yields the same output model as the input model in a single iteration. This happened because the input data set is the true zero-offset section and the input velocity-depth model (Model A in Figure 8.2-11) is the same as the true velocity-depth model (Figure 8.2-1).

Now consider Model B in Figure 8.2-12 in which the salt velocity and the base-salt boundary are specified incorrectly. There are practical reasons behind these deliberate errors introduced into the model. In practice, often the top-salt boundary is determined with reasonable accuracy by way of time migration. However, accurate delineation of the base-salt boundary is almost impossible with time migration. Additionally, the salt velocity may vary because of dolomite or shale intrusions, and these variations are often difficult to determine.

Start with Model B as the initial velocity-depth model and perform depth migration (Figure 8.2-12). The result indicates that the geometry of the base-salt boundary and the flat reflector below has changed (top of the right column). Interpret all the layer boundaries — top-salt, base-salt, and the deeper reflector, from the result of depth migration and create an updated velocity-depth model (center of the left column). Use this new model and perform depth migration once more (center of the right column). Interpret the new image from depth migration, again, for all the three layer boundaries and obtain an updated velocity-depth model (bottom of the left column). Finally, perform depth migration for the third time (bottom of the right column) and interpret for the three layer boundaries, once more. After this third iteration, we find that the velocity-depth model does not change from the previous iteration; hence, convergence is achieved. Nevertheless, the final solution has converged to a velocity-depth model that is different from the true model. We conclude from this experiment that convergence and consistency, albeit required for a model to be certified as a valid solution from depth migration, do not guarantee that the solution is the true model. The result of iterative depth migration obviously is dictated by the parameters of the initial velocity-depth model.

Consider now an initial model in which the geometry of only one reflector — that associated with the base-salt boundary, is in error while layer velocities are specified correctly (Figure 8.2-13). This may be possible in practice if there is an abundance of well data in the area of investigation that enables a reliable specification of layer velocities.

**Figure 8.2-1**(Top) An earth model in depth with a salt diapir; (middle) CMP-stacked section derived from the modeled prestack data in Figure 8.2-3; (bottom) the modeled zero-offset section. The stacked and zero-offset sections are appropriately aligned in the lateral direction with respect to the earth model above. Trace spacings in the stacked and zero-offset sections are 25 m and 50 m, respectively. No amplitude scaling has been applied to the sections. The aspect ratio of the horizontal and vertical axes in the velocity-depth model is 1.**Figure 8.2-11**Iterative depth migration using the zero-offset section as in Figure 8.2-1 and Model A (same as the true velocity-depth model as in Figure 8.2-1).**Figure 8.2-12**Iterative depth migration using the zero-offset section as in Figure 8.2-1 and Model B that contains errors in salt velocity and base-salt reflector geometry. For comparison, the true velocity-depth model (Model A in Figure 8.2-11) is superimposed on the final velocity-depth model (bottom of left column).**Figure 8.2-13**Iterative depth migration using the zero-offset section as in Figure 8.2-1 and Model C that contains errors in base-salt reflector geometry. For comparison, the true velocity-depth model (Model A in Figure 8.2-11) is superimposed on the final velocity-depth model (bottom of left column).**Figure 8.2-14**Iterative depth migration using the zero-offset section as in Figure 8.2-1 and Model D that contains errors in salt velocity and base-salt reflector geometry. For comparison, the true velocity-depth model (Model A in Figure 8.2-11) is superimposed on the final velocity-depth model (bottom of left column).

It often is preferable to specify a simple, in this case a flat reflector, geometry for the base-salt boundary. We shall let iterative depth migration produce a final geometry for the base-salt boundary. Now, start with Model C as the initial velocity-depth model and perform iterative depth migration (Figure 8.2-13). Convergence is achieved after three iterations. The final solution has converged to a velocity-depth model that is fairly close to the true model. We conclude that true geometry of reflectors can be recovered by iterative depth migration provided layer velocities are correctly specified in the initial velocity-depth model.

The case of Model D shown in Figure 8.2-14 is similar to that of Model B (Figure 8.2-12). While the layer velocity for the salt diapir in Model B has been specified erroneously too low, it has been specified in Model D erroneously too high. As a result, the final velocity-depth model, once again, departs from the true model, significantly.

Sometimes time migration may yield an inaccurate image of the top-salt boundary because of the incorrect overburden velocity field. Such is the case in Model E (Figure 8.2-15) where not only the overburden velocity, and therefore, the top-salt boundary are incorrect, but also the salt velocity and the base-salt boundary are in error. This model has too many errors in layer velocities and reflector geometries. It has taken four iterations to achieve convergence. The final solution radically departs from the true model.

If the initial model contains significant errors but only in reflector geometries, and layer velocities are specified correctly and thus are not altered from one iteration to the next, then convergence to the true velocity-depth model can be achievable (Figure 8.2-16).

To recap the results of the analysis of the six models (Figures 8.2-11 through 8.2-16), refer to Figure 8.2-17. Starting with an initial velocity-depth model (top left), which contains some errors in model parameters, perform iterative depth migration. After *n* iterations, obtain an earth image in depth (top right) that corresponds to an earth model in depth (center left) that is different from the true model (center right). Nevertheless, forward modeling of zero-offset traveltimes (bottom left) using the estimated model (center left) from iterative depth migration yields results that are consistent with the traveltimes on the zero-offset wavefield section (bottom right) used as input to iterative depth migration. We have met the convergence and consistency criteria for the final solution from iterative depth migration, but we have only obtained *a solution*, and not *the solution* — the true model (Figure 8.2-11).

**Figure 8.2-15**Part 1: Iterative depth migration using the zero-offset section as in Figure 8.2-1 and Model E that contains errors in overburden and salt velocities, and top-salt and base-salt reflector geometries. For comparison, the true velocity-depth model (Model A in Figure 8.2-11) is superimposed on the final velocity-depth model (bottom of left column).**Figure 8.2-15**Part 2: Iterative depth migration using the zero-offset section as in Figure 8.2-1 and Model E that contains errors in overburden and salt velocities, and top-salt and base-salt reflector geometries. For comparison, the true velocity-depth model (Model A in Figure 8.2-11) is superimposed on the final velocity-depth model (bottom of left column).**Figure 8.2-16**Iterative depth migration using the zero-offset section as in Figure 8.2-1 and Model F that contains errors in top-salt and base-salt reflector geometries. For comparison, the true velocity-depth model (Model A in Figure 8.2-11) is superimposed on the final velocity-depth model (bottom of left column).**Figure 8.2-17**Top left: initial velocity-depth model for iterative depth migration; top right: final image from iterative depth migration; center left: final model from iterative depth migration; center right: true model; bottom left: zero-offset traveltime section created by forward modeling using normal-incidence rays through the final model (center left) from iterative depth migration; bottom right: zero-offset wavefield section input to prestack depth migration.**Figure 8.2-18**Initial velocity-depth models used in iterative depth migrations illustrated in Figures 8.2-11 through 8.2-16.**Figure 8.2-19**Final velocity-depth models from iterative depth migrations illustrated in Figures 8.2-11 through 8.2-16.

Figure 8.2-18 shows the initial velocity-depth models (Figures 8.2-11 through 8.2-16) that contain various types of errors in model parameters. Figure 8.2-19 shows the final solutions derived from iterative depth migration using the initial models in Figure 8.2-18. Note that those initial models with correctly specified layer velocities but incorrectly defined reflector geometries (Models A, C, and F) converge to the true model (Model A). Those initial models with incorrect layer velocities (Models B, D, and E), however, do not converge to the true model.

Zero-offset traveltime sections computed from the final velocity-depth models are shown in Figure 8.2-20. Compare these traveltime sections with the input zero-offset wavefield section (Figure 8.2-11), and note that they all are consistent with the latter. Consistency, however, does not guarantee that the final solution from iterative depth migration yields the true model. In practice, we will never know which one of the final solutions shown in Figure 8.2-19 corresponds to the true model.

## See also

- 2-D poststack depth migration
- Image rays and lateral velocity variations
- Time versus depth migration
- Iterative depth migration
- Iteration with CMP-stacked data
- Iteration with prestack data
- Iteration in practice