Angle-domain illumination gathers by wave-equation-based methods

Reducing the cost by phase encoding

As mentioned in previous sections, equations 13 or 17 is intuitive to implement, however, it requires performing offset to angle transform for each component of the sensitivity kernel. This can be potentially expensive, since the cost is now proportional to the number of sources, receivers and frequencies. In this section, we show that the computational cost can be significantly reduced by using the phase-encoding technique, which was first introduced into wave-equation shot-record migration (Romero et al., 2000), and then adapted to Hessian computation by Tang (2009).

The basic idea behind phase encoding is simple, i.e., instead of computing the Green's functions sequentially with point sources as source functions, we now compute them simultaneously with encoded areal source as the source function. Thanks to the linearity of the wave equation with respect to the sources, the resultant wavefield computed using the encoded areal source can be expressed as the sum of the wavefields computed using the point sources. Therefore, instead of performing many wavefield propagations with the number of propagations being proportional to the number of point sources, by using the encoded areal sources, we reduce the number of wavefield propagations to just one, drastically reducing the computational cost. The encoding can be done in the receiver domain, which is suitable for any acquisition geometry, or in both receiver and source domain, which is suitable for land or ocean-bottom-node acquisition geometry (Tang, 2009). One drawback of the phase-encoding method, however, it that it generates crosstalk artifacts (Tang, 2009). The crosstalk can be attenuated by carefully choosing the encoding functions, such as plane-wave phase-encoding function, random phase-encoding function or a mix of the two (Tang, 2008a).

Figure 9 shows the exact scattering-angle-domain illumination for the same constant velocity model and the $1$ -shot-$401$ -receiver configuration (Figure 6). It is obtained using the point source Green's function and is very expensive, and we show it here for bench-marking purposes. Figure 10 shows the result obtained by assembling the receiver-side Green's functions but without any phase-encoding function applied. The result is apparently dominated by crosstalk artifacts and has a completely wrong illumination pattern. The crosstalk artifacts can be effectively attenuated using the random phase-encoding function, as shown in Figures 11, 12 and 13, which are computed using different number of random realizations. Although not shown here, similar results can be obtained for the dip-dependent scattering-angle-domain illumination.

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Figure 9. Exact scattering-angle-domain illumination. (a) shows the illumination for scattering angle $18.75^{\circ }$ ; (b), (c) and (d) shows the illumination angle gathers for spatial locations $-1000$ m, 0 m and $1000$ m, respectively. [CR]

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Figure 10. Scattering-angle-domain illumination with crosstalk artifacts. View descriptions are the same as Figure 9. [CR]

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Figure 11. Receiver-side encoded scattering-angle-domain illumination with $1$ random realization. View descriptions are the same as Figure 9. [CR]

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Figure 12. Receiver-side encoded scattering-angle-domain illumination with $5$ random realizations. View descriptions are the same as Figure 9. [CR]

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Figure 13. Receiver-side encoded scattering-angle-domain illumination with $20$ random realizations. View descriptions are the same as Figure 9. [CR]

Angle-domain illumination gathers by wave-equation-based methods