Measuring image focusing for velocity analysis

Field-data example

I applied the proposed method to a 2D marine line extracted from a 3D data set. The images shown in Figure 1 were produced from this 2D line. I will focus on the analysis of the results for a small window of the image that contains both convex and concave reflectors. In contrast with the previous synthetic-data example, I performed the curvature correction defined in equation 5 by using a field of local dips estimated numerically. I applied the Seplib program Sdip to the ensemble of sections obtained by stacking along the aperture-angle axis the residual migrated images for each value of $\rho$.

Figure 7a shows the migrated stack of the analysis window for a particular choice of the $\rho$ parameter ($\rho$=1.04) that maximizes flatness in the aperture-angle gather at the midpoint location corresponding to the black line superimposed onto the stack; that is for $x$=5.646 km. Figure 7b shows the aperture-angle gather and Figure 7c the corresponding semblance panel.

Starting from the prestack images, I computed dip-decomposed images that are function of both the aperture angle $\gamma$ and the structural dip $\alpha$. Figure 8 shows the 3D cube of the dip-decomposed image at the same midpoint location as the previous figure; that is for $x$=5.646 kilometers. The convex reflector of interest, at depth of 950 meters, shows an upward-smiling moveout in the structural-dips panel, consistently with the result observed when discussing the synthetic-data example in the previous section. Figure 9 displays the image-focusing semblance cube at that same midpoint location. The left panel in the cube displays semblance as a function of depth and radius of curvature (${R}$) at $\rho$=1.04; the right panel displays semblance as a function of depth and $\rho$ at ${R}$=125 meters. The location of the semblance peak in the cube at depth of 950 meters is consistent with the location of the semblance peak in the conventional $\rho$ scan shown in Figure 7c. The semblance peak in the image-focusing cube is slightly tighter than in the conventional scan, but the differences are not substantial.

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Figure 7. The migrated stack of the analysis window for $\rho$=1.04 (panel a), the aperture-angle gather at $x$=5.5646 km for $\rho$=1.04 (panel b), and the aperture-angle semblance section at $x$=5.646 km (panel c). [CR]

Wind-ResMig-dip-ang-X5646-overn Figure 8. The dip-decomposed image at $x$=5.646 kilometers. The convex reflector of interest, at depth of 950 meters, shows an upward-smiling moveout in the structural-dips panel. [CR]

Wind-Sembl-dip-ang-X5646-overn Figure 9. The image-focusing semblance cube at $x$=5.646 kilometers. The location of the semblance peak in the cube at depth of 950 meters is consistent with the location in the conventional $\rho$ scan shown in Figure 7c. The peak is slightly tighter than in the conventional scan. [CR]

Figures 10-12 shows similar analysis of the migrated images presented above, but at the midpoint location corresponding to the reflector with negative curvature; that is for $x$=5.539 kilometers. The reflector is locally dipping with negative dip of approximately 45 degrees. The stationary point in the dip-decomposed image shown in the right panel of Figure 11 is located at that value of the structural dip, and it is frowning instead of smiling because of the negative local curvature. The value of $\rho$ for which the reflector is the flattest along the aperture-angle axis ($\rho$=.95), is substantially lower than for the previous reflector ($\rho$=1.04). This substantial difference in apparent velocity, notwithstanding the proximity of the two midpoint locations, is probably related to the fact that the wavefronts that illuminate the two events propagate through different zones of the velocity model due to the dip of the second reflector.

The semblance peak in the image-focusing cube (right panel in Figure 12) is now substantially better defined than in the conventional semblance panel shown in Figure 10c, suggesting a potential resolution benefit for velocity estimation. Further analysis of this potential benefit is needed before drawing definitive conclusions.

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Figure 10. The migrated stack of the analysis window for $\rho$=0.95 (panel a), the aperture-angle gather at $x$=5.539 km for $\rho$=0.95 (panel b), and the aperture-angle semblance section at $x$=5.539 km (panel c). [CR]

Wind-ResMig-dip-ang-X5539-overn Figure 11. The dip-decomposed image at $x$=5.539 kilometers. The concave reflector of interest, at depth of 1200 meters and dip of 45 degrees, shows a downward-frowning moveout in the structural-dips panel. [CR]

Another potential advantage of explicitly taking into account, and correcting for, reflectors' curvature in the semblance analysis, is that it automatically enables the simultaneous measurements of coherency for several structural dips, in addition to the stationary dip, at each analysis point. The semblance measurements are thus automatically averaged along the reflector, following both its local dip and its local curvature. To test this hypothesis, I computed a modified version of the conventional semblance functional along the aperture-angle axis according to the following expression:

$$S_\gamma \left({\bf x},\rho,{R}\right)= \frac{ \sum_\alpha \left[... ...m_\gamma {\bf R}_{{\rm Curv}}\left({\bf x},\gamma ,\alpha ,\rho,{R}\right)^2 },$$ (A-10)

that averages both numerator and denominator along the structural-dip axis. Figure 13 compares the result of conventional semblance with the result of computing the semblance functional defined in 10. Figure 13c displays conventional semblance, and it is the same panel shown in Figure 10c. Figure 13a displays the constant $\rho$ section of the semblance cube computed using 10 and Figure 13b displays the constant curvature (${R}$=-75 meters) section of this semblance cube. Although both panel b) and panel c) are computed by measuring coherency only along the aperture-angle axis, the semblance peak corresponding to the concave reflector is clearly better focused and more easily pickable in panel b) than in panel c). This example suggests that there is an advantage on averaging semblance over structural dips. On the other hand, there is the additional cost of computing the dip-decomposed images and the additional complexity of picking a higher dimensionality semblance cube.

Wind-Sembl-dip-ang-X5539-overn Figure 12. The image-focusing semblance cube at $x$=5.539 kilometers. The location of the semblance peak in the cube at depth of 1200 meters is consistent with the location in the conventional $\rho$ scan shown in Figure 10c, but is substantially better defined than in the conventional scan. [CR]

Wind-Sembl-curv-all-X5539-overn
Figure 13. Comparison of the result of computing the semblance functional defined in 10 (panels a and b) with the result of conventional semblance (panel a), at $x$=5.539 kilometers. The semblance peak corresponding to the concave reflector is clearly better focused and more easily pickable in panel b) than in panel c). [CR]

Measuring image focusing for velocity analysis