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The goal of this chapter is to show that it is possible to image reflectors from the cross correlations of the signals generated by ambient noise sources and recorded by a passive array of receivers.
We first describe in Section 5.1 the different configurations of noise sources, receiver arrays and reflectors that can be considered. There are mainly two types of configurations: the backlight configuration, in which the reflectors are between the noise sources and the receiver array, and the daylight configuration, in which the receiver array is between the noise sources and the reflectors to be imaged. In Section 5.2 we extend the stationary phase analysis of Chapter 3 to passive sensor imaging of reflectors. The main result, described in Proposition 5.1, is that, in the presence of reflectors, the cross correlations between any two sensors have, in addition to the main peaks at the travel times between them, other peaks at lag times related to travel times from the sensors to the reflectors. The analysis identifies the relation between the secondary peaks in the cross correlations and the travel times between sensors and reflectors for the different imaging configurations. With this information we show in Section 5.3 how to image the reflectors by migrating cross correlations. Two imaging functions are proposed. They involve travel time migration with sums or differences of travel times between the array sensors and the search point, and the appropriate version to be chosen depends on the type – daylight or backlight – of noise source configurations, as described in Subsections 5.3.1–5.3.3.
We assume in this chapter that the background medium is homogeneous or smoothly varying. The case of a scattering medium will be addressed in the following chapters.
Imaging configurations of noise sources, sensors, and reflectors
In this section we discuss the different configurations of noise sources, sensors, and reflectors. The stationary phase analysis in the following section indicates that we should distinguish three types of configurations of sources, sensors, and reflectors (Figure 5.1).We use terminology from analogous situations in photography but it should be kept in mind that imaging is coherent here, which means that the recorded signals are time-resolved amplitudes and not just intensities, as is the case in photography.