Measuring velocity from zero-offset data by image focusing analysis

Zero-offset synthetic-data examples

In this section, I present the application of the method outlined in the previous section to three zero-offset synthetic data sets. These data sets were modeled assuming reflectivity models of increasing complexity. The range of reflector curvature progressively increases from the first model to the third model.

Figure 1 shows the reflectors geometry used to model the three synthetic data sets. I modeled the first data set assuming a ``cloud''of 46 point diffractors (panel a). For the second data set, I added eight curved reflectors with positive radius of curvature of approximately 55 meters (panel b). Finally, for the third data set, I added eight additional curved reflectors with a negative radius of curvature of approximately 55 meters (panel c). I set the maximum amplitude of the curved reflectors to be about 40% of the maximum amplitude of the point diffractors to maintain a balance between the velocity information provided by the point diffractors and that provided by the curved reflectors.

All the figures in this section follow the same pattern established in Figure 1. The left panels correspond to the reflectivity model shown in Figure 1a, the middle panels correspond to the reflectivity model shown in Figure 1b, and the right panels correspond to the reflectivity model shown in Figure 1c.

Figure 2 shows the three data sets modeled from the reflectivity models shown in Figure 1 assuming constant velocity equal to 2 km/s. The data increases in complexity and the texture changes as the curved reflectors are added to the reflectivity model.

Figure 3 shows the migrated sections obtained with the initial (too low) velocity of 1.951 km/s. The crossing of events in these images clearly indicates undermigration. The events corresponding to the reflectors with negative curvature are sufficiently undermigrated that they appear as reflectors with high positive curvature.

Figure 4 shows the results of the focusing analysis on the residual migrated ensembles obtained from the undermigrated images shown in Figure 3. All three panels show the image-focusing semblance spatially averaged in analysis windows defined by the following inequalities along the depth axis: $1.8 \rm {km} \leq$z $\leq 2.1 \rm {km}$, and by the following inequalities along the midpoint axis: $4.85 \rm {km} \leq$x $\leq 5.15 \rm {km}$. These analysis windows are represented in Figure 3 by the inner squares delimited by the grid superimposed onto the images. The panels show the average semblance as a function of the velocity parameter $\rho$ and the radius of curvature ${R}$.

Refl-all-overn
Figure 1. Reflectors geometry used to model the three zero-offset synthetic data sets I used to test the proposed image-focusing velocity-estimation method: (a) a ``cloud'' of point diffractors, (b) point diffractors and curved reflectors with positive curvature, (c) point diffractors, curved reflectors with positive curvature, and curved reflectors with negative curvature. [ER]

Data-all-overn
Figure 2. Zero-offset data modeled from the reflectivity functions shown in Figure 1. [ER]

Mig-all-overn
Figure 3. Migrated sections obtained by migrating the data shown in 2 with the initial (too low) velocity of 1.950 km/s. The inner squares delimited by the grid superimposed onto the images show the analysis windows, where the semblance is spatially averaged to produce the results shown in Figure 4. [ER]

Sembl-all-overn
Figure 4. The image-focusing semblance spatially averaged over the analysis windows shown as a function of velocity parameter $\rho$ and the radius of curvature ${R}$. Panel (a) shows the result corresponding to the point diffractors, panel (b) shows the result corresponding to the point diffractors and curved reflectors with positive curvature, and panel (c) shows the result corresponding to point diffractors, curved reflectors with positive curvature, and curved reflectors with negative curvature. [CR]

The semblance panels show diagonal trends for all cases because of the velocity/curvature ambiguity. When only point diffractors are present, only one trend is visible and the pattern is symmetric around $\rho$=1.025; that is, the correct value of the parameter. The addition of the positive-curvature reflectors adds another trend to the semblance panel (Figure 4b) and breaks downs the symmetry. When reflectors with both negative and positive curvature are present (Figure 4c), the semblance maxima occur around the correct value ($\rho$=1.025) for all the trends in the panel.

In poorly focused images corresponding to $\rho$ values both lower and higher than the correct one, the increase in range of reflector curvatures causes additional crossing events. These crossing events interfere with the local dip estimation (step 2) and consequently with the curvature correction (step 4) Biondi (2009). As a result, the image-focusing semblance that measures dip coherency after the curvature correction is strongly attenuated for poorly focused images.

Measuring velocity from zero-offset data by image focusing analysis