Kinematics in iterated correlations of a passive acoustic experiment

Conclusions

Using Green's functions under the Born approximation in a homogeneous medium with one scatterer, we show that $C^{(3)}_{B,A}$ constitutes 16 terms, that can be divided into 3 groups. The leading-order terms, group 1, are associated with the correlation of the direct waves recorded at the stations from a source that is generally not at a stationary-phase position. Thus evaluating $C^{(3)}_{B,A}$ directly does not improve the Green's function estimation. Instead we can remove the terms in group 1 from $C^{(3)}_{B,A}$ by muting $C^{(2)}_{A,X}$ and $C^{(2)}_{B,X}$. Group 2 contains the 4 leading-order terms in $C^{(3)}_{B,A}$ after muting $C^{(2)}_{B,X}$ and $C^{(2)}_{B,X}$. A stationary-phase ananysis of the 4 terms tells us that the scatterer must be aligned on ray paths between two stations, outside the station span.

Term 15.6 is non-stationary for all source postitions not aligned with the scatterer and stations $A$ and $B$. The non-stationarity is a function only of source position, not of auxiliary-station position. When we evaluate an ensemble average of multiple sources from different locations, term 15.6 will, in general, interfere destructively. Terms 15.7 and 15.10 are stationary when the source aligns with the scatterer, an auxiliary station and stations A and B. The non-stationarity is a function of source position and of auxiliary-station position. We can exploit this fact by using a network of auxiliary stations positioned randomly, such that if the source position is not at the stationary phase, the contribution from different auxiliary stations stack incoherently. Only term 15.11 remains stationary no matter where the source or auxiliary stations are positioned, so that any stacking of $C^{(3)}_{B,A}$ over auxiliary stations will enhance the contribution of this term.

An additional problem for the utilization of terms 15.6, 15.7 and 15.10 for the improvement of Green's function reconstruction is that for the source position for which these terms have stationary contributions at the correct traveltime, the contribution becomes indistinguishable from the leading-order contribution in $C^{(2)}_{A,X}$ and $C^{(2)}_{B,X}$ that must be removed. This means that stacking is the key to enhancing the contribution of term 15.11 and diminishing the contributions of terms 15.6, 15.7 and 15.10 to the EGF from $C^{(3)}_{B,A}$.

Kinematics in iterated correlations of a passive acoustic experiment