In the next chapter we will address passive sensor imaging of reflectors with ambient noise sources. In order to understand the challenges encountered in sensor array imaging, we first give an overview of conventional sensor array imaging for two basic problems: passive source imaging and active reflector imaging. “Conventional” here means that the sources emit short probing pulses and not stationary random signals; that is, noise.

In the first problem, addressed in Section 4.1, the goal is to image the spatial distribution of sources emitting waves that are recorded by a passive array of receivers. The data set is a vector of N signals recorded by the N receivers. Different imaging functions are introduced. After discussing least-squares imaging we introduce the reverse-time migration imaging function and the Kirchhoff migration imaging function, and we carry out their resolution analysis. The resolution properties are summarized in Section 4.2.3.

In the second problem, addressed in Section 4.3, the goal is to image reflectors buried in the medium from the data collected by an active array of sensors, that can be used both as sources and as receivers. The data set is a matrix of N × N signals, where the (j, l)th signal is recorded by the jth sensor when a short pulse is emitted by the lth sensor. As in the case of passive source imaging, we discuss least-squares imaging, reverse-time migration imaging, and Kirchhoff migration imaging, whose resolution properties are summarized in Section 4.3.9.

Passive array imaging of sources

Here we consider the case of a passive array, in which the sensors are used only as receivers. The goal is to image a source.

Data acquisition

In the configuration described in Figure 4.1, the source y emits a pulse and the sensors (xr)r = 1,…, N record the waves. The data set is the vector of signals (u(t, xr))t ∈ ℝ, r = 1,…,N. The goal of imaging is here to find the source position y. More generally, in the case of distributed sources, the goal is to find the source region.

Imaging function

The goal is to find the spatial source distribution that is supposed to be supported in the region Ω ⊂ ℝ3 (that does not contain the sensor positions (xr)r = 1,…,N).