We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation.

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We present a general method for estimating the location of small, well-separated scatterers in a randomly inhomogeneous environment using an active sensor array. The main features of this method are (i) an arrival time analysis of the echo received from the scatterers, (ii) a singular value decomposition of the array response matrix in the frequency domain, and (iii) the construction of an objective function in the time domain that is statistically stable and peaks on the scatterers. By statistically stable we mean here that the objective function is self-averaging over individual realizations of the medium. This is a new approach to array imaging that is motivated by time reversal in random media, analyzed in detail previously. It combines features from seismic imaging like arrival time analysis with frequency-domain signal subspace methodology like MUltiple SIgnal Classification (MUSIC). We illustrate the theory with numerical simulations for ultrasound.

Abstract. We analyze wave propagation in turbulent media using the Gaussian white-noise approximation. We consider two rigorous Gaussian white-noise models: one for the wave field and the other for the Wigner distribution associated with the wave field. Using the white-noise model for the Wigner distribution we show that the interaction of a wave field with the turbulent medium can be characterized in terms of the turbulence-induced entrance aperture. This aperture is proportional to the turbulence-induced coherence length and inversely proportional to the turbulence-induced spread of the wave energy in the transverse wavevectors. The effect of the turbulent medium is important when the turbulence-induced entrance aperture is smaller than the actual entrance aperture. We also study time reversal of the wave field in a turbulent medium and introduce the notion of a turbulence-induced time-reversal aperture which we show is proportional to the turbulence-induced spread in the transmitted wave energy. When the effect of the turbulent medium is important, the turbulence-induced time-reversal aperture corresponds to a time-reversal resolution much better than the resolution in the absence of the turbulent medium. The propagation and spreading of a wave field can be related to time reversal and refocusing of the wave field by a general duality relation, and we present this duality in terms of the uncertainty principle.

We consider the random Schrödinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the Itô-Schrödinger stochastic partial differential equation (SPDE) which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the self-averaging property of the Wigner transform. It follows easily when the support of the test functions is of the order of the beam width. We also show with a more detailed analysis that the limit is deterministic when the support of the test functions

We establish the self-averaging properties of the Wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance. The main ingredients in the proof are the error estimates for the semiclassical approximation of the Wigner transform by the solution of the Liouville equations, and the limit theorem for two-particle motion along the characteristics of the Liouville equations. The results are applied to a mathematical model of the time-reversal experiments for the acoustic waves, and self-averaging properties of the re-transmitted wave are proved. 1

Abstract. We present a statistical framework for the fixed-frequency computational time-reversal imaging problem assuming point scatterers in a known background medium. Our statistical measurement models are based on the physical models of the multistatic response matrix, the dis-torted wave Born approximation and Foldy-Lax multiple scattering models. We develop maximum likelihood (ML) estimators of the locations and reflection parameters of the scatterers. Using a sim-plified single-scatterer model, we also propose a likelihood time-reversal imaging technique which is suboptimal but computationally efficient and can be used to initialize the ML estimation. We gener-alize the fixed-frequency likelihood imaging to multiple frequencies, and demonstrate its effectiveness in resolving the grating lobes of a sparse array. This enables to achieve high resolution by deploying a large-aperture array consisting of a small number of antennas while avoiding spatial ambiguity. Numerical and experimental examples are used to illustrate the applicability of our results.

Abstract. This paper demonstrates the interest of a time-reversal method for the identification of source in a randomly layered medium. An active source located inside the medium emits a pulse that is recorded on a small time-reversal mirror. The wave is sent back into the medium, either numerically in a computer with the knowledge of the medium, or physically into the real medium. Our goal is to give a precise description of the refocusing of the pulse. We identify and analyze a regime where the pulse refocuses on a ring at the depth of the source and at a critical time. Our objective is to find the location of the source and we show that the time-reveresal refocusing contains information which can be used to this effect and which cannot be obtained by a direct arrival-time analysis. The time reversal technique gives a robust procedure to locate and characterize the source also in the case with ambient noise created by other sources located at the surface. Key words. Acoustic waves, random media, asymptotic theory, time reversal. AMS subject classifications. 76B15, 35Q99, 60F05. 1. Introduction. In

Coherent interferometry is an array imaging method in which we back propagate, or migrate, crosscorrelations of the traces over appropriately chosen space-time windows, rather than the traces themselves. The size of the space-time windows is critical and depends on two parameters. One is the decoherence frequency, which is proportional to the reciprocal of the delay spread in the traces produced by the clutter. The other is the decoherence length, which also depends on the clutter. As is usual, the clutter is modeled by random fluctations in the medium properties. In isotropic clutter the decoherence length is typically much smaller than the array aperture. In layered random media the decoherence length along the layers can be quite large. We show that when the crosscorrelations of the traces are calculated adaptively then coherent interferometry can provide images that are statistically stable relative to small scale clutter in the environment. This means that the images we obtain are not sensitive to the detailed form of the clutter. They only depend on its overall statistical properties. However, clutter does reduce the resolution of the images by

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