Signal-to-noise Ratio Analysis

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 ($S$ ) 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,

$$\textbf{m}_{mig}=\sum_{i=1}^{S}\textbf{m}_{mig,i}= \sum_{i=1}^{S}... ...f{L}_{j}^{T}\textbf{P}_{j}^{T}\textbf{P}_{i}\textbf{L}_{i}\textbf{m}}^{noise}).$$ (9)

In equation , I further assume the signal term and $S-1$ noise terms in the parentheses are of comparable energy, and that those $S-1$ noise terms are incoherent. Consequently the SNR is roughly $1/\sqrt{S-1}$ for $\textbf{m}_{mig,i}$ , the image associated with i-th sources . After summation over all the $S$ sources, the SNR of $\textbf{m}_{mig}$ is $\sqrt{S}/\sqrt{S-1}$ , assuming the signal term from all the $S$ sources are coherent.

This SNR analysis is summarized in Figure . Here, $S$ 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 $S$ times--once for each of the $S$ source locations--to produce $S$ images as shown in Figure (c). Every image contains one signal image from a correctly decoded and migrated shot and $S-1$ noisy images from the rest $S-1$ shots being migrated with wrong source locations and wrong time shifts. As analyzed before, every image in Figure (c) has a SNR approximately $1/\sqrt{S-1}$ . After stacking all the $S$ images together in Figure (d), the SNR becomes $\sqrt{S}/\sqrt{S-1}$ .

Figure: (a) Time-shifted shot gathers, (b) blended supergather created by blending $S$ time-shifted shot gathers, (c) migration images after migrating the supergather for each shot position with SNR approximately $\frac{1}{\sqrt{S-1}}$ , (d) final image after summing $S$ migration images. The final SNR is $\frac{\sqrt{S}}{\sqrt{S-1}}$ .

Here the key assumptions are:
(1) The correctly decoded and migrated shots from all the $S$ 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.

$$\textrm{SNR} \approx \sqrt{SI}/\sqrt{S-1}.$$ (10)

In the case that there are $N$ supergathers in the survey, the SNR is proportional to

$$\textrm{SNR} \approx \sqrt{NSI}/\sqrt{S-1}$$

$$\approx \sqrt{NI}, ~\textrm{when}~S\gg1$$ (11)

where $N$ is the number of supergathers and $I$ is the number of iterations. The total number of shots is $N\times S$ . When $S$ is equal to 1 for the conventional sources situation, there will be no crosstalk noise. Since I assume there is no noise in the original shot gathers, the SNR of the migration image is infinity, and when $S$ is much greater than 1, the SNR is independent of $S$ . Equations  and  will be validated with numerical examples for $S\gg1$ . In the case of iterative least-squares migration, the crosstalk noise in the gradient or conjugate direction from each iteration is correlated with each other for static encoding; moreover, after being scaled with the step length, the variance of the crosstalk noise would be different for every iteration, where early iterations receive large weight. Therefore, I expect the SNR enhancement to be less than the prediction from equation , where crosstalk noise is assumed to be of comparable energy.

When $S$ is small, e.g. $S=2$ , the SNR of a conventional Kirchhoff migration image is often large enough, because $N$ 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 $S$ is large, the crosstalk noise is intolerable due to the decrease of the number of supergathers ($N$ ). In the next section, multisource least-squares migration is applied to synthetic multishot supergathers to suppress the crosstalk and improve the SNR.