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1. A pulse propagates toward a time reversal array of size a. The propagation distance L is large compared to a. The ambient medium has a randomly varying index of refraction with a typical correlation length that is small compared to a. The signal is time reversed at the array and sent back into the medium. The back propagated signal refocuses with spot size λL/ae, where ae is the effective aperture of the array (Section 3.3).
Source publication
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 i...
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Citations
... To quantify statistical stability, variance calculations are required, which are based on high-order moment analysis. (2) Motivated by statistical stability analysis and time-reversal experiments for waves in random media [21,49], new methods for communication and imaging have been introduced that are based on wavefield correlations. The understanding and analysis of these methods again require high-order moments calculations. ...
... We can substitute (56-57) into (48)(49) in order to obtain the first-order system of coupled random differential equations satisfied by the mode amplitudes (55): ...
... (1) In the regime ε ≪ 1 the evanescent mode amplitudes, that satisfy (49), can be expressed to leading order in closed forms as functions of the guided and radiating mode amplitudes (60)(61). Indeed it is possible to invert the operator ∂ 2 z + γ in (49) for γ < 0 by using the Green's function that satisfies the radiation condition and to obtain: ...
In this paper we review some aspects of wave propagation in random media. In the physics literature the picture seems simple: for large propagation distances, the wavefield has Gaussian statistics, mean zero, and second-order moments determined by radiative transfer theory. The results for the first two moments can be proved under general circumstances by multiscale analysis. The Gaussian conjecture for the statistical distribution of the wavefield can be proved in some propagation regimes, such as the white-noise paraxial regime that we address in the first part of this review. It may, however, be wrong in other regimes, such as in randomly perturbed open waveguides, that we address in the second part of this review. In the third and last part, we reconcile the two results by showing that the Gaussian conjecture is restored in randomly perturbed open waveguides in the high-frequency regime, when the number of propagating modes increases.
... It moreover turns out that refocusing is enhanced when the medium is randomly scattering, and that the time-reversed refocused wave is statistically stable, in the sense that its shape depends on the statistical properties of the random medium, but not on its particular realization. The phenomenon of focusing enhancement has been analyzed quantitatively [3,9,24,26]. Statistical stability of time-reversal refocusing for broadband pulses is usually qualitatively proved by invoking the fact that the time-reversed refocused wave is the superposition of many independent frequency components, which gives the selfaveraging property in the time domain [3,26]. ...
... The phenomenon of focusing enhancement has been analyzed quantitatively [3,9,24,26]. Statistical stability of time-reversal refocusing for broadband pulses is usually qualitatively proved by invoking the fact that the time-reversed refocused wave is the superposition of many independent frequency components, which gives the selfaveraging property in the time domain [3,26]. However, so far, there has not been a fully satisfactory analysis of the statistical stability phenomenon, because it involves the evaluation of a fourth-order moment of the Green's function of the random wave equation. ...
When waves propagate through a complex medium like the turbulent atmosphere the wave field becomes incoherent and the wave intensity forms a complex speckle pattern. In this paper we study a speckle memory effect in the frequency domain and some of its consequences. This effect means that certain properties of the speckle pattern produced by wave transmission through a randomly scattering medium is preserved when shifting the frequency of the illumination. The speckle memory effect is characterized via a detailed novel analysis of the fourth-order moment of the random paraxial Green's function at four different frequencies. We arrive at a precise characterization of the frequency memory effect and what governs the strength of the memory. As an application we quantify the statistical stability of time-reversal wave refocusing through a randomly scattering medium in the paraxial or beam regime. Time reversal refers to the situation when a transmitted wave field is recorded on a time-reversal mirror then time reversed and sent back into the complex medium. The reemitted wave field then refocuses at the original source point. We compute the mean of the refocused wave and identify a novel quantitative description of its variance in terms of the radius of the time-reversal mirror, the size of its elements, the source bandwidth and the statistics of the random medium fluctuations.
... We will show that, despite the spatial correlations induced by the periodicities of the system, it is possible to reproduce the original experiment of Derode et al. (2003), yet, with precise control of the scattering medium. Previous studies (Blomgren et al., 2002;Derode et al., 2000Derode et al., , 2001bPapanicolaou et al., 2004) have proved that TR focusing is a statistically stable phenomenon, i.e., it does not depend on a particular realization of the heterogeneous medium. However, when the medium changes in between the forward and backward stages of the TR, the focusing is destroyed for changes that are larger than the average wavelength (Vigo et al., 2004). ...
Time reversal (TR) focusing of acoustical waves is a widely studied phenomenon that usually requires a chaotic cavity or disordered scattering medium to achieve spatial and frequency decorrelation of the acoustic field when using a single channel. On the other hand, sonic crystals were disregarded as scattering media for the TR process because of their periodic structure and previous results showing poor spatial focusing when compared to a disordered medium. In this paper, an experimental realization of a tunable sonic crystal, which can achieve single-channel TR focusing amplitudes in the audible range comparable to those obtained in a disordered scattering medium, is presented. Furthermore, the tunable nature of the system allows it to switch the time-reversed pulse on and off by changing its geometrical configuration. A robustness analysis with respect to the perturbations in the sonic crystal configurations is also presented, showing that the time-reversed pulses with high temporal and spatial contrasts are preserved only for configurations that are close to the original one.
... In each iteration step, we use the boundary source f n and its response Λf n to compute the boundary source f n+1 for the next iteration step. The iteration algorithm in this paper was inspired by time reversal methods, see [2,3,9,8,16,17,22,35,36,37]. We note that when the traditional time-reversal algorithms are used in imaging, one typically needs to assume that the medium contains some point-like scatterers. ...
We study the wave equation on a bounded domain of $\mathbb R^m$ and on a compact Riemannian manifold $M$ with boundary. We assume that the coefficients of the wave equation are unknown but that we are given the hyperbolic Neumann-to-Dirichlet map $\Lambda$ that corresponds to the physical measurements on the boundary. Using the knowledge of $\Lambda$ we construct a sequence of Neumann boundary values so that at a time $T$ the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. Such waves are called an artificial point source. The convergence of the wave takes place in the function spaces naturally related to the energy of the wave. We apply the results for inverse problems and demonstrate the focusing of the waves numerically in the 1-dimensional case.
... Qualitatively, the mechanisms responsible for the good properties of the CCF are similar to the ones that ensure efficient pulse refocusing during time-reversal experiments. Time-reversal refocusing properties are well understood mathematically not only for three-dimensional waves in randomly layered media [70] but also, for instance, for paraxial waves [10,22,108] or classical waves with weak fluctuations and in the high frequency limit [11]. ...
The present research is aimed at developing coherent interferometric (CINT) imaging algorithms to localize sources and reflectors in applications involving fluid flows. CINT imaging has been shown to be efficient and statistically stable in quiescent cluttered media where classical imaging techniques, such as Kirchhoff’s migration, may possibly fail due to their lack of statistical robustness. We aim at extending these methods to inhomogeneous moving media, for it has relevance to aero-acoustics, atmospheric and underwater acoustics, infrasound propagation, or even astrophysics. In this thesis report we address both the direct problem of modeling the propagation of acoustic waves in a randomly heterogeneous ambient flow, and the inverse problem of finding the position of sources or reflectors by the CINT algorithm implemented with the traces of the acoustic waves that have travelled through the flow.
... However, this approach only works for the near-field source. The time reversal method [20][21][22][23][24] inverses the measured signal in time and reinject it back into the same medium. This approach is able to refocus the source and return a super-resolution result in a medium with multiple reflections, scattering and refractions [23]. ...
... The CINT imaging function is given by the superposition of crosscorrelations backpropagated in the reference medium. Its mathematical expression resembles that of the time reversal function analyzed in [36,37], and its statistical stability and resolution are studied in [29,27]. Unlike in time reversal, where super-resolution of focusing occurs, the stability of CINT comes at the expense of resolution, which is determined by two characteristic scales in the random medium: the decoherence frequency and length. ...
... + k 2 e x (x)e y (x)(1 + 4πη 0 (x) + 4πη (1) 2k) 2 e x (x)e y (x)(1 + 4πη 0 (x) + 4πη (1) (x)), (D.35)where e x (x) and e y (x) are defined by .36) Here V PMLx and V PML y each contain two PML layers that we surround V with in x and y directions respectively. ...
We consider the problem of optical imaging of small nonlinear scatterers in random media. We propose an extension of coherent interferometric imaging (CINT) that applies to scatterers that emit second-harmonic light. We compare this method to a nonlinear version of migration imaging and find that the images obtained by CINT are more robust to statistical fluctuations. This finding is supported by a resolution analysis that is carried out in the setting of geometrical optics in random media. It is also consistent with numerical simulations for which the assumptions of the geometrical optics model do not hold.
... To this end, a variant of the classical beamforming is employed to estimate the sound source, in which the closed form of Green's function is replaced by a numerical version. Another source localization method in the time domain that is frequently used in both acoustic and elastic media is time reversal [14][15][16][17][18][19]. It is based on the symmetric nature of the wave equation with respect to the time variable in non-dissipative media. ...
... The Itô-Schrödinger model can be derived rigorously from the Helmholtz equation by a separation of scales technique in the high-frequency regime [8,9,10]. It is physically relevant and it models many situations, for instance laser beam propagation [1,21], time reversal in random media [2,18], or underwater acoustics [22]. The Itô-Schrödinger model allows for the use of Itô's stochastic calculus, which in turn enables the closure of the hierarchy of moment equations [6,14]. ...
When waves propagate through a complex or heterogeneous medium the wave field is corrupted by the heterogeneities. Such corruption limits the performance of imaging or communication schemes. One may then ask the question: is there an optimal way of encoding a signal so as to counteract the corruption by the medium? In the ideal situation the answer is given by time reversal: for a given target or focusing point, in a first step let the target emit a signal and then record the signal transmitted to the source antenna, time reverse this and use it as the source trace at the source antenna in a second step. This source will give a sharply focused wave at the target location if the source aperture is large enough. Here we address this scheme in the more practical situation with a limited aperture, time-harmonic signal, and finite-sized elements in the source array. Central questions are then the focusing resolution and signal-to-noise ratio at the target, their dependence on the physical parameters, and the capacity to focus selectively in the neighborhood of the target point and therefore to transmit images. Sharp results are presented for these questions.
... The Itô-Schrödinger model can be derived rigorously from (1) by a separation of scales technique in the high-frequency regime (see [1] in the case of a randomly layered medium and [21][22][23] in the case of a three-dimensional random medium). It models many situations, for instance, laser beam propagation [42], time reversal in random media [4,35], underwater acoustics [43], or migration problems in geophysics [7]. The Itô-Schrödinger model allows for the use of Itô's stochastic calculus, which in turn enables the closure of the hierarchy of moment equations [17,27]. ...
... In the context of design of imaging techniques this insight is important to make proper balance in between noise and resolution in the image. We remark that in certain regimes one may be able to prove statistical stability, that is, that the signal-to-noise ratio goes to infinity in the scaling limit [35,36]. The results we present here are more general in the sense that we can actually describe a finite signal-to-noise ratio and how the parameters of the problem determine this. ...
... The second-order moments play an important role, as they give the mean intensity profile and the correlation radius of the transmitted beam [14,23] can be used to analyze time reversal experiments [4,35] and wave imaging problems [9,10], and we will need them to compute the scintillation index of the transmitted beam and the variance of the Wigner transform. We describe them in detail in this section. ...
In this paper we consider the Ito-Schrodinger model for wave propagation in
random media in the paraxial regime. We solve the equation for the fourth-order
moment of the field in the regime where the correlation length of the medium is
smaller than the initial beam width. As applications we derive the covariance
function of the intensity of the transmitted beam and the variance of the
smoothed Wigner transform of the transmitted field. The first application is
used to explicitly quantify the scintillation of the transmitted beam and the
second application to quantify the statistical stability of the Wigner
transform.
