next up previous [pdf]

Next: Conclusion Up: Phase encoding with Gold Previous: Gold codes

Examples

We illustrate the use of the encoding methods on a simple model of a flat reflector, 0.5 km deep, embedded in a medium with a constant velocity of 2 km/s. The original data is migrated with a 5$ \%$ slower velocity. We used the same slower velocity to perform the modeling. For the areal shot migration, we show examples of migrating Gold code phase-encoded data with the slower velocity. For comparison of the crosstalk behavior, when migrating with a velocity different from that used to model, we also show images migrated with a 5$ \%$ faster velocity. Super-areal data are comprised of the collection of 10 modeling experiments initiated at every 10th CMP coordinate. We computed the prestack migrated image with 61 subsurface offsets to observe how crosstalk is shifted when using Gold codes. It should have been reasonable to use a much smaller number of offsets, as the moveout information is restricted to the 21 central offsets.

Figure 4 shows the areal shot migration of data generated by the prestack exploding-reflector modeling without combining the modeling experiments into super-areal shots. The panel on the left is the zero-subsurface-offset section, and the panel on the right is a SODCIG. This result represents the ideal image we would like to obtain if the crosstalk could be eliminated. Our objective in phase encoding the modeling experiments is to achieve satisfactory crosstalk attenuation in such a way that the moveout information is not altered.

perm0
perm0
Figure 4.
Areal shot migration of synthesized data with no combination of the modeling experiments into super-areal data. $ \bf {[CR]}$
[pdf] [png]

Figure 5 shows the areal shot migration of data generated by the prestack exploding-reflector modeling with no phase encoding applied. The super-areal data, input to areal shot migration, are comprised of modeling experiments initiated at every tenth SODCIG. The SODCIGs resulting from the areal shot migration, show strong crosstalk at subsurface offsets different from zero. The crosstalk is periodic with a period half of the spacing of the modeling experiments in a super-areal shot.

perma
perma
Figure 5.
Areal shot migration of synthesized data with no phase encoding applied. The super-areal data comprises 10 modeling experiments. Notice the crosstalk in the SODCIG. $ \bf {[CR]}$
[pdf] [png]

Figure 6 shows areal shot migration of one realization of phase encoding modeling with conventional random codes. The strong crosstalk observed in Figure 4 is now dispersed throughout the image. The dispersed crosstalk can be further attenuated by migrating more random realizations, but this increases the cost of migration.

conv1r
conv1r
Figure 6.
Areal shot migration of one realization of synthesized data with conventional random-phase encoding. $ \bf {[CR]}$
[pdf] [png]

Figure 7 shows the migration of 4 realizations of conventional random encoding modeling.

conv5r
conv5r
Figure 7.
Areal shot migration of four realizations of synthesized data with conventional random-phase encoding. $ \bf {[CR]}$
[pdf] [png]

Crosstalk attenuation is incomplete when using conventional random codes because their autocorrelations are not a perfect spike, nor are their cross-correlations zero. Gold codes partially satisfy these requirements: the autocorrelation is almost a perfect spike, except for -1's at lags different from zero, and, similarly, the cross-correlations are -1 everywhere except where they peak. Therefore, to obtain good results when using Gold codes, it is critical to select the codes which provide the best crosstalk attenuation. That is because the cross-correlation functions have peaks with the same magnitude as those of the autocorrelation function.

In Figure 8, the areal shot migration was performed on encoded data with Gold codes that have cross-correlation peaks at every 5th (Figure 8a), 10th (Figure 8b) and, 20th lags (Figure 8c). This means that when applying the multi-offset imaging condition, unrelated wavefields, delayed in time by the phase functions, will cross-correlate at depths different from that of the related wavefields, which were encoded with the same phase. The crosstalk of the Figure 5 is now shifted in depth according to the selected set of Gold codes. In Figure 8a, the apexes of the crosstalk are displaced in depth at a constant spacing of, approximately, 0.1 km; in Figure 8b the spacing is approximately 0.2 km, and in Figure 8c, 0.4 km. Notice that in spite of the crosstalk is still present in Figure 8c, its complete elimination can be achieved with a simple depth-windowing of the image.

gold1x
Figure 8.
Areal shot migration of one realization of synthesized data with Gold code phase encoding. Gold codes have cross-correlation peaks at every 5th (a), 10th (b) and, 20th lag (c), and the depth shifts are, respectively, 0.1 km, 0.2 km, and 0.4 km. $ \bf {[CR]}$
gold1x
[pdf] [png]

The amount of depth shift of the crosstalk is defined by the lag where the cross-correlation of the Gold codes peaks. For the simple case of constant velocity, the depth shifts, $ \delta z$, are given by

$\displaystyle \delta z = \frac{n_{\lambda}v}{2n_{\omega}d_{\omega}}$ (A-14)

where $ n_{\omega}$ is the number of frequencies, $ d_{\omega}$ is the frequency interval and $ n_{\lambda}$ is the lag where the cross-correlation of the Gold codes peaks. For the present example, this amounts to $ \delta z = 0.021*n_{\lambda}$ km.

One possibility to statistically attenuate the crosstalk is to randomly select the Gold codes. Figure 9 shows the areal shot migration of one realization of encoded data with randomly selected Gold codes. The crosstalk shows different patterns than that of the sequentially selected Gold codes.

gold1r
gold1r
Figure 9.
Areal shot migration of one realization of synthesized data with Gold code phase encoding. Gold codes are randomly selected. $ \bf {[CR]}$
[pdf] [png]

As before, migrating more realizations of randomly selected Gold-encoded data further attenuates crosstalk. Figure 10 shows the migration of 4 realizations of randomly selected Gold-encoded data. Comparison with Figure 9 shows that much of the remaining crosstalk energy has been attenuated. However, this approach does not exploit the statistical properties of the Gold phase functions and the crosstalk shows up locally coherent, at random positions. This strategy is definitely not suited to provide a kinematically reliable image.

gold5r
gold5r
Figure 10.
Areal shot migration of four realizations of synthesized data with Gold code phase encoding. Gold codes are randomly selected. $ \bf {[CR]}$
[pdf] [png]

Considering that the crosstalk is shifted in depth, as observed in Figure 8, and that the amount of shift is determined by the lag where the cross-correlation of the Gold codes peaks (equation 14), one can choose the Gold codes according to a suitable interval that completely shifts the crosstalk away from the zone of interest. This strategy shares similar idea as the linear-phase encoding Romero et al. (2000), which aims to shift the crosstalk out of the migration domain by using a linear function of frequency. In the Appendix, we show that phase encoding with Gold codes is equivalent to linear phase encoding. Figure 11 shows the areal shot migration of data encoded by selecting every 50th Gold code, meaning that the depth shifts are multiples of 1 km, as predicted by equation 14, which should be adequate to completely push the crosstalk away from the SODCIG. Contrary to what was expected, the crosstalk is still present. Of course, its complete elimination can be achieved by windowing around the central traces (Figure 12).

gold1o
gold1o
Figure 11.
Areal shot migration of one realization of synthesized data with Gold phase encoding. Gold codes are selected such that the crosstalk is shifted away from the reflector. A simple windowing should completely eliminate the crosstalk. $ \bf {[CR]}$
[pdf] [png]

goldwd
goldwd
Figure 12.
Crosstalk of Figure 11 can be completely eliminated by windowing the SODCIG. Compare with Figure 5. $ \bf {[CR]}$
[pdf] [png]

To understand the origin of the crosstalk in Figure 11, we migrated the data without adding the results of individual migrations of the super areal data. Figure 13 shows three of these migrations. The three panels show the same spatial position. The top panel shows the migration of super areal data whose modeling includes a SODCIG coincident with that displayed. Remember that we are using a ``comb'' function to select every 10th SODCIG to initiate the modeling experiments comprised by a super areal data. The central panel shows the migration of the super areal data modeled from the initial image shifted five CMP positions away from that of the uppermost panel, and the panel on the bottom shows the migration of the super areal data modeled from the initial image shifted nine CMP positions away from that of the uppermost panel. The modeling experiments which generated the super areal data for Figure 13c are actually separated one CMP position from the ones for Figure 13a, given that the modeling experiments in super areal data are separated every 10th CMP. Figure 13a and Figure 13c show that much of the reflector energy comes from the migration of modeling experiments initiated at CMP positions close to the considered image point. Figure 13b shows that much of the crosstalk energy of Figure 11 is related to migration of super areal shots whose modeling experiments are initiated at SODCIGs shifted five CMP positions with respect to the considered SODCIG.

goldsp
Figure 13.
Areal shot migration of three super areal data. The results were not stacked together. Notice that much of the crosstalk energy of Figure 11 is related to the migration of super areal shots ($ b$) other than the one which modeling was initiated at that CMP position ($ a$ and $ c$). $ \bf {[CR]}$
goldsp
[pdf] [png]

In migration velocity analysis, at every velocity iteration, data is migrated with an updated velocity. To verify how the crosstalk behaves when migrating with a different velocity, we used a 5$ \%$ faster velocity to migrate the Gold encoded data modeled with the initial 5$ \%$ slower velocity. We used sets of Gold codes with cross-correlation peaks at every 5th (Figure 14a), 10th (Figure 14b) and, 20th lag (Figure 14c) lags. The depth shift of the crosstalk is proportional to the increase in velocity. The behavior is similar to that of migration with a slower velocity, regarding that the interference with the region of interest decreases with the increase of the lag of the cross-correlation peaks.

gold2x
Figure 14.
Areal shot migration of one realization of synthesized data with Gold phase encoding. Gold codes have cross-correlation peaks at every 5th (a), 10th (b) and, 20th lag (c). $ \bf {[CR]}$
gold2x
[pdf] [png]

The use of Gold codes can be less costly than conventional random codes, given that just one realization is necessary to achieve an almost perfect result, while even using more realizations of conventional random encoding does not produce an image with similar quality. In addition, it is suited to be used in a horizon based strategy to migration velocity analysis, where a few important reflectors are chosen to do velocity update.

However, an inadequate choice of the Gold codes is potentially more dangerous than using conventional random codes. Because the crosstalk may not be shifted out of the region of interest, coherent artifacts can coincide with reflectors and obliterate the moveout information. Conventional random codes, in turn, randomizes the crosstalk. This can lead to a noisy residual moveout scan or a noisy gradient. In a companion paper Tang et al. (2008) show that the prestack exploding-reflector modeled data, phase encoded with conventional random codes, is able to provide a reasonable direction for velocity update.


next up previous [pdf]

Next: Conclusion Up: Phase encoding with Gold Previous: Gold codes

2009-04-13