Kinematics in iterated correlations of a passive acoustic experiment

Iterated correlation dependance on source position

For the geometry in Figure 6, the source, stations A and B, auxiliary stations and scatterer are not aligned at a stationary phase of terms 15.6, 15.7 and 15.10. We will investigate the retrieved result of evaluating $C^{(3)}_{B,A}$ after muting $C^{(2)}_{A,X}$ and $C^{(2)}_{B,X}$, and summing over all auxiliary stations and multiplying with the phase-modifying factor as in equation 18. The sources are positioned on a circle with radius $800$ m centered between stations $A$ and $B$ in the geometry described as before; see Figure 6. Evaluating $C^{(3)}_{B,A}$ and summation over the auxiliary stations for terms 15.6 15.7, 15.10 and 15.11, and then evaluating equation 18 for each source contribution gives the correlogram in Figure 12(a).

This correlogram confirms that when the source, stations $A$ and $B$, an auxiliary station, and the scatterer are aligned, each term has a stationary phase. We also see how term 15.11 is stationary with respect to source postion. The behavior of term 15.6 is similar to that of the leading term in $C^{(2)}_{B,A}$; see Figure 4(a). This can be expected from the constraint on $t$ in condition 25, which is equal to condition 11 on $t$ for $C^{(2)}_{B,A}$. We can expect that when we time-average the $C^{(3)}_{B,A}$ of multiple sources at different angular positions, term 15.6 interferes destructively.

Figure 12(a) also tells us that terms 15.7 and 15.10 are also non-stationary with respect to source position. However, the arrival time of non-stationary positions is dependent upon scatterer position (see condition 29); this implies that in a medium with randomly positioned scatterers, terms 15.7 and 15.10 would interfere destructively. Last we investigate whether muting $C^{(2)}$ before evaluating $C^{(3)}_{B,A}$ can work for the source postions located at stationary phases for terms 15.6 15.7 and 15.10; see Figure 12(b). We see how muting the $C^{(2)}_{A,X}$ and $C^{(2)}_{B,X}$ for source positions at and close to $\phi=\pi$ radians also would remove the energy associated with the scatterer. This is expected, because the scatterer is directly behind the source as seen from both stations $A$ and $B$; thus the contribution arrives simultaneously with the direct event from the source.

corrC4ABSa,corrC4ABSb
Figure 12. a) Gather showing time-domain equivalents of 15.6, 15.7, 15.10 and 15.11 after summing over auxiliary stations as a function of source position angle $\phi$. b) Gather of $\vert\tilde{C}^3_{B,A}(t)\vert$ as a function of source position angle $\phi$. $\mathbf{[ER]}$

Kinematics in iterated correlations of a passive acoustic experiment