The phenomenon of super-resolution in time-reversal acoustics is analyzed theoretically and with numerical simulations. A signal that is recorded and then retransmitted by an array of transducers, propagates back though the medium, and refocuses approximately on the source that emitted it. In a homogeneous medium, the refocusing resolution of the time-reversed signal is limited by diffraction. When the medium has random inhomogeneities the resolution of the refocused signal can in some circumstances beat the diffraction limit. This is super-resolution. A theoretical treatment of this phenomenon is given, and numerical simulations which confirm the theory are presented.

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Super-resolution in time-reversal acoustics

We Discretize in Time With Step-Size h a Stochastic Differential Equation Whose Solution has a Unique Invariant Probability Measure is the Solution of the Discretized System, we Give an Estimate of in Terms of h for Several Discretization Methods. In Particular, Methods Which are of Second Order for the Approximation of in Finite Time are Shown to be Generically of Second Order for the Ergodic Criterion(1).

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Second order discretization schemes of stochastic differential systems for the computation of the invariant law

In time reversal, an array of transducers receives the signal emitted by a localized source, time reverses it and re-emits it into the medium. The emitted waves back-propagate to the source and tend to focus near it. In a homogeneous medium, the cross-range resolution of the refocused field at the source location is λ0L/a, where λ0 is the carrier wavelength, L is the range and a is the array aperture. The refocusing spot size in a homogeneous medium is independent of the bandwidth of the pulse, but broad-band can help in reducing spurious Fresnel zones. In a noisy (random) medium, the cross-range resolution is improved beyond the homogeneous diffraction limit because the array can capture waves that move away from it at the source, but get scattered onto it by the inhomogeneities. We refer to this phenomenon as super-resolution of the time reversal process in random media. Super-resolution implies in particular that, because of multipathing, the array appears to have an effective aperture ae that is greater than a. Since ae depends on the scattering medium, it is not known. In this paper we present a brief review of time reversal theory in a remote sensing regime and a robust procedure for estimating ae from the signals received at the array. Knowing ae permits assessing quantitatively super-resolution in time reversal for applications in spatially localized communications with reduced interference. We also review interferometric imaging and its relation to time reversal and to matched field imaging. We show that ae quantifies in an explicit way the loss of resolution in interferometric array imaging.

/pdf/theory-and-applications-of-time-reversal-and-interferometric-b5k67hvlge.pdf

Theory and applications of time reversal and interferometric imaging

When a signal is emitted from a source, recorded by an array of transducers, time-reversed, and re-emitted into the medium, it will refocus approximately on the source location. We analyze the refocusing resolution in a high frequency remote-sensing regime and show that, because of multiple scattering in an inhomogeneous or random medium, it can improve beyond the diffraction limit. We also show that the back-propagated signal from a spatially localized narrow-band source is self-averaging, or statistically stable, and relate this to the self-averaging properties of functionals of the Wigner distribution in phase space. Time reversal from spatially distributed sources is self-averaging only for broad-band signals. The array of transducers operates in a remote-sensing regime, so we analyze time reversal with the parabolic or paraxial wave equation.

/pdf/statistical-stability-in-time-reversal-5c3og8h4zy.pdf

Statistical stability in time reversal

Spectral methods for identifying scalar diffusions

We present an asymptotic analysis -in the “ white-noise limit”- of a linear parabolic partial differential equation, whose coefficients are perturbed by a wide-band noise. After having studied some ergodic properties of a class of diffusion processes, we prove the convergence in law towards the solution of an Ito stochastic P.D.E. We then establish an expansion in powers of Δ ( 1/Δ being a measure of the bandwith of the driving noise) of the first moment of the solution

R. Bouc

Papers

Asymptotic analysis of P.D.E.s with wide–band noise disturbances, and expansion of the moments