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Synthetic Example

To test oriented velocity continuation and its velocity resolution we generate a synthetic dataset using a model with a constant velocity gradient beginning with a $2.0 $$\frac{km}{s}$ surface velocity. Diffractors are created as reflectivity spikes within the model with random spatial and magnitude distributions. Kirchhoff forward modeling is used to generate 24 offsets with a $50 m$ interval.

Zero-offset data are shown in Figure 6a, and $1.0 km$ common-offset data appear in Figure 6b. A time shift between the data is noticeable.

Both zero and common-offset data are warped to squared time, slope decomposed, and migrated with a $2.0 $$\frac{km}{s}$ initial velocity. The initially migrated slope decomposed images are propagated through a range of plausible migration velocities using oriented velocity continuation. Common-offset partial images are then stacked over offset for each continuation velocity.

Figure 9 illustrates slope gathers for zero and $1.15 km$ offsets generated for the image location $x = 2.32 km$ with migration velocities $2.1, 3.0$, and $3.9 $$\frac{km}{s}$. $3.0 $$\frac{km}{s}$ is the correct velocity for the diffractor at $1.4 s$ located directly underneath the midpoint of this gather. When a different velocity is used, the shape of the event deviates from planar. We perform velocity analysis by testing the semblance, or flatness, of diffraction events in slope gathers over the range of velocities. Because velocity does not vary laterally in our synthetic model, we average the semblance across midpoints to generate semblance panels for the zero-offset and common-offset cases (Figures 7a and 7b respectively).

As seen from the slope gathers (Figure 9), for the zero-offset case, there is a stationary point corresponding to diffraction energy with zero slope which does not shift vertically under velocity perturbations. For the $1.15 km$ common-offset case, perturbing velocity changes the slope decomposed diffraction shape and shifts it vertically. Slope gathers with incorrect velocities (Figure 9b) are time shifted with respect to those generated for the zero-offset case (Figure 9a). When the correct migration velocity is used, horizontal common-offset diffraction energy appears at the same time as for the zero-offset case. The vertical shift of incorrectly migrated common-offset data leads to a sharper change in estimated flatness values while converging on the correct migration velocity and therefore improves velocity resolution. Therefore, common-offset semblance panel appears to have higher spatial and vertical resolution than the zero-offset case. This higher spatial resolution can be attributed to the improved illumination of scattering objects with the full range of offsets.

Final images for zero-offset and stacked common-offset cases using migration velocities estimated from the semblance panels are shown in Figure 8. Differences between the two images are too small to easily detect in this example. However, due to the higher velocity resolution visible in the semblance panel resulting from the consideration of multiple offsets, Figure 7b, we expect the stacked common-offset image to be better resolved and less prone to noise than zero-offset case when applied to field datasets.

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Figure 6.
Modeled data sections: (a) zero-offset; (b) $1.0 km$ offset section.
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Figure 7.
Velocity scan semblance panels calculated for: (a) zero-offset; (b) all 24 offsets ($50 m$ interval).
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Figure 8.
Images: (a) zero-offset OVC (velocity from Figure 7a); (b) stacked common-offset OVC (velocity from Figure 7b)
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Figure 9.
Slope gathers: (a) zero-offset; (b) $1.15 km$ offset
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Next: Field Data Example Up: Examples Previous: Toy Model Example

2017-04-20