The procedure for estimating layer velocities from coherency inversion requires CMP gathers at analysis locations and horizon times picked from unmigrated stacked data. Alternatively, time horizons picked from time-migrated data can be forward-modeled to derive the zero-offset traveltimes needed for coherency inversion. As for Dix conversion, the velocity estimate from coherency inversion is local, independent of data away from the analysis location.
A procedure for velocity-depth model estimation that includes coherency inversion is conducted layer-by-layer starting from the surface. As for Dix conversion, consider the synthetic data set associated with the model with horizontal layers shown in Figure 9.1-1. We shall adopt the interval velocity profile for the first layer H1 estimated from Dix conversion and start the application of coherency inversion with layer H2. Assume that the velocity-depth model for the first n − 1 layers already has been estimated. For the nth layer, follow the steps below for coherency inversion:
- For a trial constant velocity assigned to the nth layer, perform normal-incidence traveltime inversion to convert the time horizon corresponding to the base-layer boundary to a trial depth horizon.
- Given the geometry of the CMP gather at the analysis location, assign a trial velocity to the half space that includes the layer yet to be determined and compute the CMP traveltimes using the known overburden velocity-depth model. The modeled CMP traveltime trajectory that corresponds to the base of the layer under consideration is in general nonhyperbolic, for the ray tracing used to compute the CMP traveltimes accounts for ray bending at layer boundaries and incorporates vertical velocity gradients within layers above. Shown in Figures 9.1-7a through 9.1-11a are the CMP ray-paths at an analysis location through the unknown and half space that includes the layer under consideration and the known overburden part of the model. The corresponding CMP traveltime trajectories are shown in color in Figures 9.1-7b through 9.1-11b overlayed on the CMP gather at the analysis location.
- Extract data window along the modeled traveltime trajectory. Shown in Figures 9.1-7c through 9.1-11c are the data windows for seven trial velocities around the optimum velocity that best flattens the event within the data window.
- Compute semblance using the data within the selected window as a measure of discrepancy between the modeled and the actual traveltimes. Figure 9.1-12 shows the semblance spectra computed from the CMP gather in Figures 9.1-7b through 9.1-11b from the reflection events associated with layers H2 through H6.
- Repeat all the steps above for a range of constant velocities.
- Pick the constant trial velocity as the layer velocity for which the semblance is the maximum.
Figure 9.1-7 Coherency inversion for layer H2 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 H2. See text for details.
Figure 9.1-8 Coherency inversion for layer H3 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 H3. See text for details.
Figure 9.1-9 Coherency inversion for layer H4 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 H4. See text for details.
Figure 9.1-10 Coherency inversion for layer H5 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 H5. See text for details.
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.
The results of coherency inversion are used to pick velocity nodes at analysis locations. Specifically, the layer velocity at a given location is selected based on the semblance curve and the data window along the modeled traveltime trajectory making sure that the estimated velocities are geologically plausable. Data windows along modeled traveltime trajectories can be examined for flatness criterion to pick an optimum velocity node. Specifically, the events in Figures 9.1-7c through 9.1-11c can be considered as events on a moveout-corrected CMP gather. A flat event on a moveout-corrected CMP gather suggests that the stacking velocity associated with that event is optimum. Likewise, a flat event in the data window panels in Figures 9.1-7c through 9.1-11c suggests that the layer velocity from coherency inversion is optimum. An erroneously too low or too high velocity causes residual moveout which can be observed on the event within the data window.
The semblance spectra in Figure 9.1-12 were computed using three different maximum offsets — 2400 m, 1400 m, and 400 m. Note that, for a given layer, the longer the cable length, the sharper the semblance peak — the higher the velocity resolution. Also, the deeper the event and the higher the velocity, the broader the semblance peak — the poorer the velocity resolution.
Figures 9.1-7 through 9.1-11 show the results of coherency inversion at one CMP location. In practice, for 2-D data, coherency inversion often is applied continuously along the line. As for horizon-consistent stacking velocity analysis, for each layer, a horizon-consistent semblance spectrum is computed using coherency inversion. For 3-D data, as for conventional velocity analysis, coherency inversion normally is applied at uniformly spaced grid points over the survey area.
Figure 9.1-13 shows the semblance spectra derived from coherency inversion for layers H2 through H6 of the model shown in Figure 9.1-1. The interval velocity profiles shown in Figure 9.1-14a have been picked by tracking the semblance peaks. Compare with the profiles associated with the true model shown in Figure 9.1-1a, and note that by coherency inversion the interval velocities for layers H2 and H3 have been estimated accurately. But the semblance spectra in Figure 9.1-13 for the underlying layers H4, H5, and H6 exhibit oscillations akin to those associated with the interval velocities derived from Dix conversion of stacking velocities (Figure 9.1-4). Specifically, when there are lateral velocity variations within a cable length in one layer, in this case layer H3, the interval velocity estimation for the layers below is adversely affected as shown in Figure 9.1-13. This phenomenon also was observed by Sorin .
Shown in Figure 9.1-14b is the velocity-depth model derived from coherency inversion. The model was constructed layer-by-layer starting from the top. Coherency inversion was used to estimate the layer velocities (Figure 9.1-14a), and normal-incidence traveltime inversion of the time horizons picked from the stacked section (Figure 9.1-1) was used to obtain the depth horizons. Note the distortions caused by the oscillations of the interval velocity profiles in the estimated velocity-depth model. As in Dix conversion, these oscillations must be smoothed out for one layer before estimating the interval velocity for the next layer. The resulting semblance spectra exhibit a reduced degree of oscillations (Figure 9.1-15). The velocity-depth model based on the revised interval velocity profiles (Figure 9.1-16a), which have been picked from the new set of semblance spectra, is shown in Figure 9.1-16b. A closer look at the central portion of the estimated model using coherency inversion is shown in Figure 9.1-17. Provided that the oscillations shorter than a cable length are eliminated from the interval velocity profiles, coherency inversion seems to produce an acceptable estimate of the layer velocities.
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.
Model with low-relief structure
The results of coherency inversion for the model with low-relief structures (Figure 9.2-1) are shown in Figure 9.2-6. Note that, for shallow layers, the inversion yields accurate velocity estimates. It also has detected the velocity variations within the deltaic sequence over the full-fold CMP range, correctly — note the increase in velocity for Horizon 6 from around 3000 m/s on the left to 3200 m/s in the middle and back to 3000 m/s on the right, and compare with Figure 9.2-1. Nevertheless, note the very short-wavelength variations in the velocity curves derived from picking the maxima of the semblance plots. As was demonstrated by the coherency inversion tests applied to the model with horizontal layers (models with horizontal layers), this observation has an important practical implication with regard to evaluation and use of the results of velocity estimation. Specifically, lateral velocity variations of very short-wavelengths that are much less than a cable length should not be incorporated into a velocity-depth model. Instead, some lateral smoothing of velocity estimates is almost always needed.
Shown in Figure 9.2-7a is a CMP gather used in coherency inversion to obtain a velocity estimate for the layer above Horizon 7b. Following the procedure described in the previous section, use the normal-incidence rays to convert the time horizon picked from the stacked data (Horizon 7a in Figure 9.2-2) at the analysis location to a trial depth horizon. The normal-incidence depth conversions are shown in Figure 9.2-8 for the three trial velocities as in Figure 9.2-7. The normal-incidence rays in Figure 9.2-8 are overlayed on the true velocity-depth model. Note that, in case of 3000 m/s, the trial depth horizon is shallower than the actual layer boundary for Horizon 7a; it coincides with the actual boundary in case of 3500 m/s (the true velocity of the layer above Horizon 7a); and, it is deeper in the case of 4000 m/s velocity.
Figure 9.2-3 (a) True velocity-depth model associated with the stacked section in Figure 9.2-2; (b) result of stacking velocity inversion; (c) result of coherency inversion.
Figure 9.2-5 Stacking velocity inversion for the model in Figure 9.2-1. Input to inversion are horizon-consistent stacking velocities (Figure 9.2-4) and the horizon times picked from stacked data (Figure 9.2-2). Posted velocities are the true layer velocities as in Figure 9.2-1.
For each trial constant velocity assigned to the layer above Horizon 7a, perform modeling of CMP traveltimes at the analysis location. The raypaths associated with the CMP traveltimes for the three trial velocities are also displayed in Figure 9.2-8. Note that the velocity contrast at layer boundaries and the geometry of these boundaries within the overburden has caused reflection point smearing at Horizon 7a.
The CMP data window that includes the reflection event for Horizon 7a is displayed in Figure 9.2-7c along the modeled traveltime trajectories. The flatness criterion suggests that the optimum layer velocity at the analysis location is 3500 m/s. The maximum of the semblance curve derived from coherency inversion coincides with the optimum choice for the layer velocity (Figure 9.2-7b).
- Sorin et al. (1996), Sorin, V., 1996, Velocity estimation in homogenous media: 56th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1515–1517.
- Models with horizontal layers
- Dix conversion
- Near-surface layer with laterally varying velocities
- Model with low-relief structure
- Stacking velocity inversion
- Velocity resolution