Measuring image focusing for velocity analysis

Synthetic-data example

To illustrate the proposed method, I first present its application to a synthetic data set. The model is a medium with constant slowness of .5 s/km and a single reflector with sinusoidal shape. This reflector is shown in Figure 3. I modeled a prestack data set with offsets between -1.5 kilometers and 1.5 kilometers. I then migrated the data with both the correct slowness and a high slowness of .525 s/km; that is, 105% the correct slowness.

I dip-decomposed the image obtained with the correct slowness at zero-subsurface offset and corrected it for curvature according to expression 5. Figure 4a shows the dip-decomposed image at the midpoint of one of the bottoms of the sinusoid ($x$=4.250 km). Because of the curvature, the dips are not aligned and the event is frowning down. Figure 4b shows the panel in Figure 4a corrected for image curvature by applying the shift defined in expression 2. I selected the radius of curvature to be equal to -90 meters. This is consistent with the analytical radius of curvature of the sinusoidal reflector at the same location of -86 meters. I set the reflector local dip to be zero; that is, I set $\bar{\alpha}=0$ in expression 2.

The panels shown in Figure 5 are equivalent to the panels shown in Figure 4, except that the midpoint location is at one of the tops of the sinusoid ($x$=4.750 km). At this location the curvature is positive and thus the uncorrected dip panel (Figure 5a) smiles upward. The corrected panel (Figure 5b) corresponds to a positive radius of curvature of 90 meters.

I computed the conventional semblance over aperture angle and the proposed image-focusing semblance from the migrated image obtained with the high slowness. Figure 6 compares the semblance fields computed by the conventional semblance functional that measures coherency only over aperture angles (Figure 6a), with the semblance cube computed by the proposed image-focusing semblance functional that measures coherency over both aperture angles and structural dips (Figure 6b). The figure shows the semblance fields at $x$=4.750 km, that is at one of the local top of the sinusoidal reflector. The $\rho$-range is the same ( $0.984 \leq \rho \leq 1.134$) for the two panels in the figure. The semblance peak is more sharply defined as a function of the $\rho$ parameter in the result of the new image-focusing functional (right face in Figure 6b) than in the result of conventional method (Figure 6a).

Notice that the semblance peak is located at longer radius of curvature (${R}$=125 meters) than the actual radius of curvature of the reflector (${R}$=86 meters), because residual migration in the angle domain is not exact and does not fully correct for the reflector curvature. This error is inconsequential for the proposed method since the aim is to better estimate $\rho$ not ${R}$.

Refl-sinus-overn Figure 3. Sinusoidal reflector used to generate the synthetic prestack data set. [ER]

ResMig-short-dip-curv-all-X4250-overn Figure 4. The dip-decomposed images at the midpoint of one of the bottoms of the sinusoidal reflector ($x$=4.250 km): without curvature correction (panel a) and after curvature correction (panel b). [CR]

ResMig-short-dip-curv-all-X4750-overn Figure 5. The dip-decomposed images at the midpoint of one of the tops of the sinusoidal reflector ($x$=4.750 km): without curvature correction (panel a) and after curvature correction (panel b). [CR]

Wind-Sembl-short-ang-dip-all-X4750-overn Figure 6. Comparison of the semblance fields computed by the conventional semblance functional that measures coherency only over aperture angles (panel a), with the semblance cube computed by the proposed image-focusing semblance functional that measures coherency over both aperture angles and structural dips (panel b). The figure shows the semblance fields at $x$=4.750 km. [CR]

Measuring image focusing for velocity analysis