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It was shown in Chapters 5–6 that it is possible to image reflectors by back-propagating cross correlations of signals generated by ambient noise sources and recorded at passive sensor arrays. The resolution of the image depends on the directional diversity of the noise signals relative to the location of the sensor array and the reflector, and it can be poor when the directional diversity of the illumination is limited. In this chapter we show that it is possible to enhance directional diversity by exploiting the scattering properties of the medium because scatterers can act as secondary noise sources. However, scattering also increases the fluctuation level of the cross correlations and therefore tends to destabilize the image by reducing its signal-to-noise ratio. We study the trade-off in passive, correlation-based imaging between resolution enhancement and signal-to-noise ratio reduction that is due to scattering.
In Section 8.2 we analyze the case of a point-like reflector embedded in a weakly scattering medium and we calculate the first and second moments of the cross correlations of signals recorded by passive sensors (Propositions 8.1 and 8.2) in order to discuss quantitatively the trade-off between resolution enhancement and signal-to-noise ratio reduction. This analysis also gives a mathematical interpretation of the migration images presented in Subsections 8.2.7 and 8.4.2. In order to clarify the role of scattering we analyze in Section 8.3 the cases of deterministic flat interfaces and randomly scattering interfaces, which are configurations in which scattering has very different directional properties. Finally, in Section 8.4 we show that migration of special fourth-order cross correlations can achieve a better trade-off between resolution enhancement and signal-to-noise ratio reduction compared to migration of the standard cross correlation matrix.
Role of scattering in correlation-based imaging
It was shown in Chapters 5–6 that it is possible to image a reflector from the cross correlations of signals generated by ambient noise sources and recorded by a passive receiver array. This is possible provided that the illumination by the noise sources is appropriate, in the sense that the ray joining a receiver and the reflector should intersect the source region. When the receivers are between the sources and the reflector (daylight illumination), the daylight imaging function (6.20) should be used, which has good resolution.