Publications (9)1.88 Total impact
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Article: Imaging using transport models for wave-wave correlations
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ABSTRACT: We consider the imaging of objects buried in unknown heterogeneous media. The medium is probed by using classical (e.g., acoustic or electromagnetic) waves. When heterogeneities in the medium become too strong, inversion methodologies based on a microscopic description of wave propagation (e.g., a wave equation or Maxwell's equations) become strongly dependent on the unknown details of the heterogeneous medium. In some situations, it is preferable to use a macroscopic model for a quantity that is quadratic in the wave fields. Here, such macroscopic models take the form of radiative transfer equations also referred to as transport equations. They can model either the energy density of the propagating wave fields or more generally the correlation of two wave fields propagating in possibly different media. In particular, we consider the correlation of the two fields prop-agating in the heterogeneous medium when the inclusion is absent and present, respectively. We present theoretical and numerical results showing that recon-structions based on this correlation are more accurate than reconstructions based on measurements of the energy density.07/2010; -
Article: Dynamics of wave scintillation in random media
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ABSTRACT: This paper concerns the asymptotic structure of the scintillation function in the sim-plified setting of wave propagation modeled by an Itô-Schrödinger equation. We show that the size of the scintillation function crucially depends on the smoothness of the initial con-ditions for the wave equation and on the size of the "array of detectors" where the wave fields are measured. In many practical settings, we show that the estimates are optimal and devise an equation for the appropriately rescaled scintillation function. The estimates are based on a careful analysis of Wigner transforms and of linear kinetic equations involving oscillatory integrals.04/2010; -
Article: Single scattering estimates for the scintillation function of waves in random media
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ABSTRACT: The energy density of high frequency waves propagating in highly oscillatory random media is well approximated by solutions of deterministic kinetic models. The scintillation function determines the statistical instability of the kinetic solu-tion. This paper analyzes the single scattering term in the scintillation function. This is the term of the scintillation function that is linear in the power spectrum of the random fluctuations. We show that the structure of the scintillation function is already quite complicated in this simplified setting. It strongly depends on the singularity of the initial conditions for the wave field and on the correlation prop-erties of the random medium. We obtain limiting expressions for the scintillation function as the correlation length of the random medium tends to zero.03/2010; -
Article: Small volume expansions for elliptic equations
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ABSTRACT: This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form $\nabla \cdot D^\eps \nabla u^\eps=0$ or $(-\Delta + q^\eps) u^\eps=0$ with prescribed Neumann conditions. The theory is well-known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order $\eps^{2d}$, where $d$ is dimension and $\eps$ is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most $\eps^{d+1}$ rather than the expected $\eps^d$. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables.06/2008; -
Article: Self-averaging of kinetic models for waves in random media
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ABSTRACT: Kinetic equations are often appropriate to model the energy density of high frequency waves propagating in highly heterogeneous media. The limitations of the kinetic model are quantified by the statistical instability of the wave energy density, i.e., by its sensitivity to changes in the realization of the underlying heterogeneous medium modeled as a random medium. In the simplified It\^o-Schr\"odinger regime of wave propagation, we obtain optimal estimates for the statistical instability of the wave energy density for different configurations of the source terms and the domains over which the energy density is measured. We show that the energy density is asymptotically statistically stable (self-averaging) in many configurations. In the case of highly localized source terms, we obtain an explicit asymptotic expression for the scintillation function in the high frequency limit.12/2007; -
Article: Kinetic models for imaging in random media
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ABSTRACT: We derive kinetic models for the correlations and the energy densities of wave fields propagating in random media. These models take the form of radiative transfer and diffusion equations. We use these macroscopic models to address the detection and imaging of small objects buried in highly heterogeneous media. More specifically, we quantify the influence of small objects on (i) the energy density measured at an array of detectors and (ii) the correlation between the wave field measured in the absence of the object and the wave field measured in the presence of the object. We analyze the advantages and disadvantages of such measurements as a function of the level of disorder in the random media. Numerical simulations verify the theoretical predictions.07/2007; -
Article: Time-reversal-based detection in random media
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ABSTRACT: We consider the detection and imaging of inclusions buried in highly heterogeneous media. We assume that only the statistical properties of the heterogeneous media can be observed and that the wave energy density may be modelled by macroscopic equations. The detection and imaging capabilities hinge on ensuring that the measured data are statistically stable, which means that they depend only on the macroscopic statistical parameters of the random media and not on the microscopic statistical realization. In this paper, the macroscopic model is a diffusion equation. In this context, we construct statistical tests to detect inclusions based on macroscopic diffusion measurements and perform asymptotic expansions to image their location and volume. We show that time-reversal measurements enjoy a much larger signal-to-noise ratio in the presence of background noise than do direct wave energy measurements. This is a direct consequence of the enhanced refocusing properties that characterize time reversed waves propagating in heterogeneous media. Finally, we present numerical simulations of acoustic waves propagating in heterogeneous two-dimensional media. The numerical simulations illustrate which factors contribute to 'noise' in the measured data and how they affect the detection and imaging capabilities.Inverse Problems 09/2005; 21(5):1593. · 1.88 Impact Factor -
Article: ANALYSIS OF THE DOUBLE SCATTERING SCINTILLATION OF WAVES IN RANDOM MEDIA
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ABSTRACT: High frequency waves propagating in highly oscillatory media are often modeled by radiative transfer equations that describes the propagation of the energy density of the waves. When the medium is statistically homogeneous, averaging effects occur in such a way that in the limit of vanishing wavelength, the wave energy density solves a deterministic radiative transfer equation. In this paper, we are interested in the remaining stochasticity of the energy density. More precisely, we wish to understand how such stochasticity depends on the statistics of the random medium and on the initial phase-space structure of the propagating wave packets. The analysis of stochasticity is a formidable task involving complicated ana-lytical calculations. In this paper, we consider the propagation of waves modeled by a scalar Schrödinger equation and limit the interaction of the waves with the underlying structure to second order. We calculate the scintillation function (second statistical moment) for such signals, which thus involve fourth-order moments of the random fluctuations, which we assume have Gaussian statistics. Our main result is a detailed analysis of the scintillation function in that setting. This requires the analysis of non-trivial oscillatory integrals, which is carried out in detail. -
Article: Accuracy of transport models for waves in random media
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ABSTRACT: This paper addresses the validity of radiative transfer equations as a model for the energy density of waves propagating in highly heterogeneous media. Comparisons between acoustic wave simulations over domains of size comparable to 500 wavelengths in two space dimensions and Monte Carlo simulations of radiative transfer equations are performed. In the so-called weak coupling regime, the agreement between the energy densities obtained by solving the wave equations and those predicted by solving the radiative transfer equations is remarkable. The domain of validity of the radiative transfer equations is assessed by looking at the fluctuations in the energy density they predict in the presence of small-volume defects in the underlying media.Wave Motion.
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- Inverse Problems (1)
Institutions
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2005
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Columbia University
- Department of Applied Physics and Applied Mathematics
New York City, NY, USA
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