Time-reversing array retrofocusing in noisy environments

Acoustic time reversal is a robust means of retrofocusing acoustic energy, in both time and space, to the original sound-source location. However, noise may limit the performance of a time-reversing array (TRA) at long source-array ranges, or when the original-source or TRA-element power levels are low. The operation of a TRA requires two steps (reception and transmission) so both TRA-broadcast noise and ambient noise must be taken into account. In this paper, predictions are made for how a simple omnidirectional noise field influences the probability that the signal amplitude from a narrow-band TRA will exceed the noise at the TRA's retrofocus. A general formulation for the probability of TRA retrofocusing, which can be used for TRA design, is developed that includes: the variance of the noise field, the original source strength, the TRA's element output power, the number of TRA elements (N), and the propagation characteristics of the environment. This formulation predicts that a TRA's array gain (in dB) at the retrofocus may be as high as + 10log10(N) to + 20 log10(N) depending on the relative strengths of the original source and the TRA's elements. Monte Carlo simulations in both a free-space environment and a shallow-ocean sound-channel environment compare well to this probability formulation even when simple approximate parametric relationships for the appropriate Green's functions are used. The dominant deviation between theory and simulation in the sound channel is caused by acoustic absorption.