Time reversal of parabolic waves and two-frequency Wigner distribution

Abstract

We consider propagation and time reversal of wave pulses in a random environment. The focus of our analysis is the development of an expression for the two frequency mutual coherence function for the harmonic wave field. This quantity plays a crucial role in the analysis of many wave propagation phenomena and we illustrate by explicitly considering time reversal in the context of time pulses with a high carrier frequency. In a time-reversal experiment the wave received by an active transducer or antenna (receiver-emitter) array, is recorded in a finite time window and then re-emitted into the medium time reversed, that is, the tails of the recorded signals are sent first. The re-emitted wave pulse will focus approximately on the original source location. We use explicit expressions for the mutual coherence functions and their asymptotic approximations in the regime of long or short propagation distance and a high carrier frequency to analyze the refocusing of the wave pulse in the time reversal experiment. A novel aspect of our analysis is that we are able to characterize precisely the decoherence length in temporal frequency. This allows us to analyze for instance the time reversal experiment when the mirror has a finite aperture in time.

A direct imaging algorithm for extended targets

“…Finally, we show how multifrequency information can also be used to obtain robust imaging results in the case with random fluctuations in the background medium. The information at different frequencies decorrelate rapidly with the frequency separation [11]; therefore, using several frequencies stabilizes the imaging result with respect to medium noise. Figure 19 shows the result for the kite shape in a random medium with 10% standard deviation and correlation length 10h when 80 transducers are used.…”

Information transfer in disordered media by broadband time-reversal: stability, resolution and capacity

“…Since the self-averaging takes place in each sub-band and the whole process is linear, stable super-resolution is valid in the full passband. Second, in the case of a finite time-window, the out-put signals, unlike (3), involve a coupling of neighbouring wavenumbers [18]. If the time window is sufficiently large ( β −1 c ) then the coupling takes place only between wavenumbers of separation much smaller than β c and our result carries over without major adjustment.…”

“…. , M, with the constant of proportionality 2 −1 (2π) −1/2 K −1 ( Cβdβ) −1 |C 0 | 2 where C 0 and Cβ are given by (16) and (18), respectively.…”

“…The prescaled version of (13) has been obtained in [14,18]. Several remarks are in order: (i) the Green function for L is of the Gaussian form in x, y; (ii) in the vanishing fluctuation limit D 0 , D 2 → 0 the free-space two-frequency mutual coherence function is recovered; (iii) the apparent singular nature of the limit β → 0 in ( 12) is deceptive.…”