It is desirable to estimate the relationship between the signal-to-noise ratio (SNR, defined in Appendix B) enhancement and the number of shot gathers ( ) for iterative least-squares migration of supergathers. While it affords no simple analytical expression for the dependence of SNR on the number of iterations of least-squares migration, I focus on how the SNR is reduced by iterative stacking (multiple migrations of all shots) of Romero et al. (2000), where all the shots in a survey are phase-encoded and blended together to form a supergather before migration (Figure ). Here, I assume the data are noise free and the noise is defined to be the crosstalk noise only. In other words, in equation the `standard migration image' term is assumed to be noise free whereas the `crosstalk' term is responsible for all the noise. For convenience, the terms in equation are regrouped here as follows,
In equation , I further assume the signal term and noise terms in the parentheses are of comparable energy, and that those noise terms are incoherent. Consequently the SNR is roughly for , the image associated with i-th sources . After summation over all the sources, the SNR of is , assuming the signal term from all the sources are coherent.
This SNR analysis is summarized in Figure . Here, shots in Figure (a) are encoded and stacked together to form a supergather, which is noise free, in Figure (b). The supergather is then migrated times--once for each of the source locations--to produce images as shown in Figure (c). Every image contains one signal image from a correctly decoded and migrated shot and noisy images from the rest shots being migrated with wrong source locations and wrong time shifts. As analyzed before, every image in Figure (c) has a SNR approximately . After stacking all the images together in Figure (d), the SNR becomes .
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Here the key assumptions are:
(1) The correctly decoded and migrated shots from all the
images give coherent signal, which will constructively stack after stacking. In addition, geometrical spreading effects can be ignored;
(2) The incorrectly decoded and migrated shots generate random noise with the same strength due to random encoding, which will destructively stack after stacking;
(3) The crosstalk noise from each migration at each iteration is uncorrelated.
When is small, e.g. , the SNR of a conventional Kirchhoff migration image is often large enough, because is large in this case. Several studies (Beasley, 2008; Hampson et al., 2008; Berkhout, 2008; Fromyr et al., 2008) have shown that conventional stacking and migration of simultaneously acquired supergathers can effectively suppress the interference of reflections from different sources, i.e., crosstalk. However, if is large, the crosstalk noise is intolerable due to the decrease of the number of supergathers ( ). In the next section, multisource least-squares migration is applied to synthetic multishot supergathers to suppress the crosstalk and improve the SNR.