Time Reversal Mirrors and Cross Correlation Functions in Acoustic Wave Propagation

Abstract

In time reversal acoustics (TRA), a signal is recorded by an array of transducers, time reversed, and then retransmitted into the configuration. The retransmitted signal propagates back through the same medium and retrofocuses on the source that generated the signal. If the transducer array is a single, planar (flat) surface, then this configuration is referred to as a planar, one‐sided, time reversal mirror (TRM). In signal processing, for example, in active‐source seismic interferometry, the measurement of the wave field at two distinct receivers, generated by a common source, is considered. Cross correlating these two observations and integrating the result over the sources yield the cross correlation function (CCF). Adopting the TRM experiments as the basic starting point and identifying the kinematically correct correspondences, it is established that the associated CCF signal processing constructions follow in a specific, infinite recording time limit. This perspective also provides for a natural rationale for selecting the Green’s function components in the TRM and CCF expressions. For a planar, one‐sided, TRM experiment and the corresponding CCF signal processing construction, in a three‐dimensional homogeneous medium, the exact expressions are explicitly calculated, and the connecting limiting relationship verified. Finally, the TRM and CCF results are understood in terms of the underlying, governing, two‐way wave equation, its corresponding time reversal invariance (TRI) symmetry, and the absence of TRI symmetry in the associated one‐way wave equations, highlighting the role played by the evanescent modal contributions.