We analyze the resolution of imaging functions that migrate the matrix of cross correlations of passive sensor arrays. This matrix is obtained by cross correlating the signals generated by ambient noise sources and recorded by the passive sensor array. The cross correlations contain information about reflectors in the surrounding medium as shown in Chapter 5. Therefore, travel time or Kirchhoff migration of the cross correlations can, under favorable circumstances, produce images of such reflectors. However, migration should be carried out appropriately depending on the type of illumination provided by the ambient noise sources.

We present in this chapter a detailed resolution analysis of the imaging functions introduced in Chapter 5 when the background medium is homogeneous. Since the coherence of the noise sources plays an important role, we first revisit the results of the previous chapters when the noise sources are not delta-correlated in space, but have a small correlation length. We can then describe analytically the form of the peaks of the cross correlation of the signals recorded by a pair of receivers in the presence of a reflector (Propositions 6.2– 6.3). The resolution analysis of the backlight and daylight imaging functions introduced in Chapter 5 is carried out in Section 6.4.

The overall result is that resolution depends on the sensor array diameter, the distance from the array to the reflector, and the central frequency, as is the case in active array imaging. When imaging with passive sensor arrays and ambient noise illumination, resolution also depends on the space and time coherence of the noise sources because they determine an effective noise bandwidth.

The detailed analysis shows that the resolution properties of the two imaging functions introduced in Chapter 5 are different. The daylight imaging function has the same properties as those for active array imaging with pulse width proportional to the inverse of the effective noise bandwidth (see Subsection 6.4.1). The backlight imaging function has poor range resolution compared to the daylight imaging function because it is based on a difference of travel times, which is less sensitive to the range than the sum of travel times used in the daylight imaging function (see Subsection 6.4.2).

A comparison of reflector imaging with active and passive arrays

The detection and location of wave reflectors from coherent wave measurements is a central issue in imaging.