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Examples

We demonstrate the imaging condition introduced in this paper with the Sigsbee 2A synthetic model (Paffenholz et al., 2002). Figure 7 shows the correct migration velocity and the image created by shot-record migration with wavefield extrapolation using the time-shift imaging condition introduced in this paper. The image in the bottom panel of Figure 7 is extracted at ${ \tau}=0$.

IMGSLO0t
IMGSLO0t
Figure 7.
Sigsbee 2A model: correct velocity (top) and migrated image obtained by shot-record wavefield extrapolation migration with time-shift imaging condition (bottom).
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The top row of Figure 8 shows common-image gathers at locations $x=\{7,9,11,13,15,17\}$ km obtained by time-shift imaging condition. As in the preceding synthetic example, we can observe events with linear trends at slopes corresponding to local migration velocity. Since the migration velocity is correct, the strongest events in common-image gathers correspond to ${ \tau}=0$. For comparison, the bottom row of Figure 8 shows common-image gathers at the same locations obtained by space-shift imaging condition. In the later case, the strongest events occur at ${ \bf h}=0$. The zero-offset images (${ \tau}=0$ and ${ \bf h}=0$) are identical.

alloff
alloff
Figure 8.
Imaging gathers at positions $x=\{7,9,11,13,15,17\}$ km. Time-shift imaging condition (top row), and space-shift imaging condition (bottom row).
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Figure 9 shows the angle-decomposition for the common-image gather at location $x=7$ km. From left to right, the panels depict the migrated image, a common-image gather resulting from migration by wavefield extrapolation with time-shift imaging, the common-image gather after slant-stacking in the $z-{ \tau}$ plane, and an angle-gather derived from the slant-stacked panel using equation equation (23).

For comparison, Figure 10 depicts a similar process for a common-image gather at the same location obtained by space-shift imaging. Despite the fact that the offset gathers are completely different, the angle-gathers are comparable showing similar trends of angle-dependent reflectivity.

SRt0-7
SRt0-7
Figure 9.
Time-shift imaging condition gather at $x=7$ km. From left to right, the panels depict the image, the time-shift gather, the slant-stacked time-shift gather and the angle-gather.
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SRx0-7
SRx0-7
Figure 10.
Space-shift imaging condition gather at $x=7$ km: From left to right, the panels depict the image, the space-shift gather, the slant-stacked space-shift gather and the angle-gather.
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The top row of Figure 11 shows angle-domain common-image gathers for time-shift imaging at locations $x=\{7,9,11,13,15,17\}$ km. Since the migration velocity is correct, all events are mostly flat indicating correct imaging. For comparison, the bottom row of Figure 11 shows angle-domain common-image gathers for space-shift imaging condition at the same locations in the image.

allang
allang
Figure 11.
Angle-gathers at positions $x=\{7,9,11,13,15,17\}$ km. Time-shift imaging condition (top row), and space-shift imaging condition (bottom row). Compare with Figure 8.
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Finally, we illustrate the behavior of time-shift imaging with incorrect velocity. The top panel in Figure 12 shows an incorrect velocity model used to image the Sigsbee 2A data, and the bottom panel shows the resulting image. The incorrect velocity is a smooth version of the correct interval velocity, scaled by $10\%$ from a depth $z=5$ km downward. The uncollapsed diffractors at depth $z=7$ km clearly indicate velocity inaccuracy.

IMGSLO2t
IMGSLO2t
Figure 12.
Sigsbee 2A model: incorrect velocity (top) and migrated image obtained by shot-record wavefield extrapolation migration with time-shift imaging condition. Compare with Figure 7.
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Figures 13 and 14 show imaging gathers and the derived angle-gathers for time-shift and space-shift imaging at the same location $x=7$ km. Due to incorrect velocity, focusing does not occur at ${ \tau}=0$ or ${ \bf h}=0$ as in the preceding case. Likewise, the reflections in angle-gathers are non-flat, indicating velocity inaccuracies. Compare Figures 9 and 13, and Figures 10 and 14. Those moveouts can be exploited for migration velocity analysis (Clapp et al., 2004; Sava and Biondi, 2004b,a; Biondi and Sava, 1999).

SRt2-7
SRt2-7
Figure 13.
Time-shift imaging condition gather at $x=7$ km. From left to right, the panels depict the image, the offset-gather, the slant-stacked gather and the angle-gather. Compare with Figure 9.
[pdf] [png] [scons]

SRx2-7
SRx2-7
Figure 14.
Space-shift imaging condition gather at $x=7$ km. From left to right, the panels depict the image, the offset-gather, the slant-stacked gather and the angle-gather. Compare with Figure 10.
[pdf] [png] [scons]


next up previous [pdf]

Next: Discussion Up: Time-shift imaging condition in Previous: Time-shift imaging in Kirchhoff

2007-04-08