A fundamental problem with inversion is velocity-depth ambiguity. This means that an error in layer velocity can be indistinguishable from an error in reflector geometry. To help resolve the velocity-depth ambiguity, we must use nonzero-offset data to estimate the layer velocities and reflector geometries (Section J.7). We can never claim to completely resolve the velocity-depth ambiguity and obtain the one and the only solution — the true velocity-depth model, from inversion of seismic data. But, we can hope to reduce the many possible solutions to a few geologically plausable velocity-depth models. Of course, if there is ample well control, we may be able to discard all but one of the few solutions and declare that as the solution.
Consider the four geologically plausable velocity-depth models of a salt diapir shown in Figure 9.0-1. Compare these models and note the differences in reflector geometries, which are subtle in some places and distinctively obvious in others. There also are differences in layer velocities from one model to the other.
Compute the zero-offset reflection traveltime responses by way of normal-incidence rays and display them in the form of zero-offset sections as shown in Figure 9.0-2. While there are differences between the velocity-depth models in Figure 9.0-1, differences in the zero-offset traveltime sections are less than marginal. Any differences that we observe in the traveltime sections may be attributed largely to the digitizing errors that have occurred in building the velocity-depth models. We may, within the bounds of ray theory associated with reflectors with reasonable curvature and shape, retain one of the four zero-offset traveltime sections and discard the other three sections shown in Figure 9.0-2. We may also conclude that the zero-offset traveltime section we have kept is consistent with all four of the velocity-depth models shown in Figure 9.0-1.
In reality, we record wavefields that comprise not just traveltimes but also amplitudes. Pretend that the zero-offset traveltime section which we retained in Figure 9.0-2 is equivalent to a zero-offset wavefield section as in Figure 8.2-1, insofar as traveltimes are concerned.
Now, consider the inverse problem. Suppose that the analyst is given the zero-offset section in Figure 8.2-1 and is asked to estimate an earth model in depth from it. As an answer, the analyst may provide any one of the earth models shown in Figure 9.0-1. They all are equally valid answers since they all are consistent with the input zero-offset section in Figure 8.2-1. In fact, there are infinitely many possible earth models, not just the four models in Figure 9.0-1, which are consistent with the zero-offset section. While there are infinite number of solutions, there should only be one valid answer — the geologic model shown in Figure 8.2-1. We will never know which one, if any, of the answers corresponds to the true velocity-depth model.
In practice, the analyst deals with a stacked section and not a zero-offset section. Shown in Figure 9.0-3 are three velocity-depth models estimated from the seismic data associated with the stacked section in the same figure by means of model estimation procedures described in this chapter. The zero-offset traveltimes associated with the layer boundaries included in each of the three models are computed by normal-incidence ray tracing. When modeled traveltimes are repeatedly overlayed on the same stacked section, we note that they all coincide with the observed traveltimes associated with the reflection events that correspond to the layer boundaries included in the velocity-depth models. Which one of the three models best represents the true subsurface structure — that is the question. And this question has a profound implication in practice; in the present case, an accurate delineation of the geometry for the target reflector represented by the deepest layer boundary has an impact on the exploration and development of a potential reservoir.
Mathematically, the nature of any inversion problem with a multiple number of solutions that are all consistent with observed data is characterized as nonuniqueness. This mathematical nonuniqueness is known to the exploration seismologist as velocity-depth ambiguity. Refer back to Figure 9.0-1 and note that in reality only Model 1 is the correct model — the same as the true geologic model shown in Figure 8.2-1. However, any combination of layer velocities and reflector geometries represented by the other three earth models is equally consistent with the input data. This means that errors in layer velocities and errors in reflector geometries are indistinguishable — a restatement of the velocity-depth ambiguity. A robust quantification of velocity-depth ambiguity is provided in Section J.7.
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.
Since we can never obtain the true representation of an earth model from inversion of seismic data, the plausable strategy is to estimate an initial model and then update it to get a final model that may be considered an acceptable approximation to the true model. Model building strategies are discussed in model building, and model updating by reflection tomography is presented in model updating.
The important question in practice is, following the model update, how much velocity-depth ambiguity remains unresolved in a final velocity-depth model. As illustrated in Figures 9.0-1 and 9.0-2, the ambiguity with zero-offset data is infinite. By using nonzero-offset data, we can hope to resolve the velocity-depth ambiguity up to a certain theoretical limit. An important rule to keep in mind is that, for data with good quality, velocity-depth ambiguity for a reflector can be resolved with an acceptable degree of accuracy if the data used in inversion have been recorded with offsets greater than the reflector depth  .
While an earth model can only be estimated with an accuracy that is within the threshold of velocity-depth ambiguity, it does have to be consistent with the input data used in inversion to estimate the model. Consistency is a necessary condition for an earth model to be certified as an acceptable estimate of the true model. A quick way to check for consistency is by ray-theoretical forward modeling of zero-offset traveltimes associated with the reflector boundaries that are in the earth model itself, and then comparing them with the actual traveltimes picked from the stacked data. Any discrepancy between the modeled and actual traveltimes is an indication of errors in the earth model parameters — layer velocities and/or reflector geometries. By using nonzero-offset data, we can hope to reduce the many possible solutions to a few. Furthermore, by introducing constraints, we may be able to converge to a single solution provided the set of constraints are reliable. One set of constraints is the depth information at well locations. This well-top information can be used to calibrate results of inversion of surface seismic data and obtain a single earth model in depth that not only is consistent with the surface data set itself but also with the borehole data.
Aside from consistency, model verification has to include a test of flatness of events on image gathers derived from prestack depth migration. A correct model, again, within the limitations of velocity-depth ambiguity and accuracy of inversion methods, would yield an accurate image from prestack depth migration irrespective of the source-receiver offset. Thus, with the correct model, the resulting image gathers would have flat events. An erroneous earth model, on the other hand, would cause residual moveout on image gathers. The use of image gathers in model estimation, verification, and update is discussed in model with complex overburden structure and illustrated by some of the case studies presented in structural inversion.
- Bickel, 1990, Bickel, S. H., 1990, Velocity-depth ambiguity of reflection traveltimes: Geophysics, 55, 266–276.
- Lines, 1993, Lines, L., 1993, Ambiguity in analysis of velocity and depth: Geophysics, 58. 596–597.
- Introduction to earth modeling in depth
- Inversion methods for data modeling
- Inversion procedures for earth modeling
- Model representation and visualization