Measuring image focusing for velocity analysis

Zero-offset synthetic-data example

The synthetic-data and field-data examples discussed in the two previous sections applied the image-focusing semblance to prestack data sets, where useful velocity information is provided by the data redundancy over offsets. In this section, I present experiments on two simple zero-offset synthetic data sets. The only velocity information contained in the migrated images obtained from zero-offset data is the focusing and unfocusing of reflections.

Figure 14 shows the reflectors' geometry assumed to model the two synthetic data sets. I modeled the first data set assuming a "cloud" of point diffractors (panel a), whereas I modeled the second data set assuming a "cloud" of convex reflectors (panel b). In both cases the velocity was assumed to be constant and equal to 2 km/s and the data were migrated assuming a high slowness of .5125 s/km; that is, 102.5% of the correct slowness.

Figure 15 summarizes the main result of this section. All three panels show the image-focusing semblance spatially averaged in an inner rectangle of the image space defined by the following inequalities along the depth axis: $1.850 \rm {km} \leq z \leq 2.150 \rm {km}$, and by the following inequalities along the midpoint axis: $4.875 \rm {km} \leq$x $\leq 5.125 \rm {km}$. The panel shows the average semblance as a function of the velocity parameter $\rho$ and the radius of curvature ${R}$. Figure 15a shows the result corresponding to the point diffractors and Figure 15b shows the result corresponding to the convex reflectors. In both cases, I applied the curvature correction defined in 5 by using a field of local dips ( $\bar{\alpha}$) estimated numerically by applying the Seplib program Sdip to the ensemble of residual migrated images for each value of $\rho$.

The important observation supported by this figure is that, in both Figure 15a and Figure 15b, the semblance energy is concentrated in a relatively narrow interval that includes the correct value of $\rho$; that is $\rho =1.025$. This result indicates that we can extract useful velocity information from zero-offset data by using the image-focusing semblance.

The third panel in Figure 15, shows the semblance average computed from the images of the convex reflectors when I applied the curvature correction defined in 5 by using a constant local dip equal to zero; that is, when I uniformly set $\bar{\alpha}=0$. As predicted by expression 9, there is strong ambiguity between the reflector curvature and the velocity parameter and the semblance is high also for values of $\rho$ that are far away from the correct one. We can consequently conclude that the velocity information contained in panel a) and b) derives from the inconsistency between the focusing information extracted using the image-focusing semblance and the local dip estimation. This inconsistency occurs when the image is sufficiently unfocused that the local dip estimation becomes unreliable. The following figures illustrate this concept.

Figures 16-20 provide a graphical explanation of the results shown in Figure 15. Figure 16 shows the migrated images of the point-diffractors data corresponding to the values of $\rho$ at the edges of the semblance peak in Figure 15a. The inner rectangle delimited by the grid superimposed to the images shows where the semblance is spatially averaged to produce the results shown in Figure 15. The image in Figure 16a is undermigrated and corresponds to $\rho =1.0125$, whereas the image in Figure 16b is overmigrated and corresponds to $\rho =1.0375$. In both of these images the unfocusing starts to cause crossing of events in the inner rectangle delimited by the grid superimposed to the images. The local dips are then multivalued and the automatic estimation of the local dips becomes unreliable and inconsistent with the more global behavior of the dips. Therefore, outside the interval $1.0125 \leq \rho \leq 1.0375$ the semblance average drops substantially in value.

Similar behavior is displayed by the migrated images of the convex-reflectors data corresponding to the values of $\rho$ at the edges of the semblance peak in Figure 15a. These images are shown in Figure 17, and correspond to $\rho =1.01$ (Figure 17a), and to $\rho=1.07$ (Figure 17b). In this case, the $\rho$ range is wider than in the previous case because the convex-reflectors' density is lower than the point-diffractors' density, and thus a larger velocity error is needed before poorly focused events start crossing.

Figures 18-20 show sections cut through the image-focusing semblance cubes at constant value of $\rho$ and ${R}$ before spatial averaging. Figure 18a shows semblance for the point-diffractors data for $\rho =1.025$ and ${R}=0$ meters; that is, the values of $\rho$ and ${R}$ for which the data are best focused. Figure 18a shows semblance for $\rho =1.0125$ and ${R}=40$ meters. This value of $\rho$ is the one corresponding to the undermigrated image in Figure 16a. Because of undermigration, the image from the point diffractors appears to have a positive radius of curvature approximately equal to 40 m. However, because of inconsistency between the focusing information and the local dip estimation, semblance is in average lower in the panel on the right than in the panel on the left.

Similar behavior is displayed by the image-focusing semblance cubes computed from the images of the convex-reflectors data. We find the ``best focused'' semblance panel (Figure 19a) still at infinite curvature (${R}=0$ meters), but at a wrong value of $\rho$; that is, at $\rho =1.04$. However, the important result is that the interval with relative high semblance still includes the correct value of $\rho$. The section shown in Figure 19b corresponds to undermigrated image shown in Figure 19b, and it is taken for $\rho =1.01$ and ${R}=120$ meters. The apparent curvature is lower than for the point diffractors because the actual curvature of the reflector is lower.

Finally, Figure 20 shows sections through the image-focusing semblance cubes for the convex-reflectors data when the local dip is uniformly set equal to zero. These panels correspond to the average semblance shown in Figure 15c, and are sections taken for the same values of $\rho$ and ${R}$ as the sections shown in Figure 17. Because of the ambiguity between velocity and curvature, both panels show well-focused and high value semblance peaks.

Refl-all-overn Figure 14. Reflectors' geometry assumed to model the two zero-offset synthetic data sets I used to test the proposed image-focusing velocity-estimation method: a) a "cloud" of point diffractors, and b) a "cloud" of convex reflectors. [ER]

Wind-Stack-all-overn
Figure 15. The image-focusing semblance spatially averaged in an inner rectangle of the image space as a function of velocity parameter $\rho$ and the radius of curvature ${R}$. Panel a) shows the result corresponding to the point diffractors, and panel b) shows the result corresponding to the convex reflectors when the curvature correction was applied by using a field of local dips estimated numerically from the migrated images. Panel c) shows the result corresponding to the convex reflectors when the curvature correction was applied by using a constant local dip equal to zero (i.e. $\bar{\alpha}=0$). [CR]

ResMig-all-scatter-overn Figure 16. Migrated images of the point-diffractors data corresponding to the values of $\rho$ at the edges of the semblance peak in Figure 15a; that is, $\rho =1.0125$ for panel a), and $\rho =1.0375$ for panel b). [CR]

ResMig-all-repl-bump-overn Figure 17. Migrated images of the convex-reflectors data corresponding to the values of $\rho$ at the edges of the semblance peak in Figure 15b; that is, $\rho =1.01$ for panel a), and $\rho =1.7$ for panel b). [CR]

Wind-Sembl-scatter-all-overn Figure 18. Sections cut through the image-focusing semblance cubes at constant value of $\rho$ and ${R}$ before spatial averaging. These panels were computed from the point-diffractors data. Panel a) shows semblance for $\rho =1.025$ and ${R}=0$ meters, and panel b) shows semblance for for $\rho =1.0125$ and ${R}=40$ meters. [CR]

Wind-Sembl-repl-bump-all-overn Figure 19. Sections cut through the image-focusing semblance cubes at constant value of $\rho$ and ${R}$ before spatial averaging. These panels were computed from the convex-reflectors data. Panel a) shows semblance for $\rho =1.04$ and ${R}=0$ meters, and panel b) shows semblance for for $\rho =1.01$ and ${R}=120$ meters. [CR]

Wind-Sembl-repl-bump-dip-0-all-overn Figure 20. Sections cut through the image-focusing semblance cubes at constant value of $\rho$ and ${R}$ before spatial averaging. These panels were computed from the convex-reflectors data. The local dip was set to be constant and equal to zero when applying the curvature correction. In contrast, the local dips were numerically estimated when computing the semblance panels shown in Figure 19. Panel a) shows semblance for $\rho =1.04$ and ${R}=0$ meters, and panel b) shows semblance for for $\rho =1.01$ and ${R}=120$ meters. [CR]

Measuring image focusing for velocity analysis