Waves propagating through heterogeneous media experience scattering that can convert a coherent pulse into small incoherent fluctuations. This may appear as attenuation for the transmitted front pulse. The classic O’Doherty–Anstey theory... more
Waves propagating through heterogeneous media experience scattering that can convert a coherent pulse into small incoherent fluctuations. This may appear as attenuation for the transmitted front pulse. The classic O’Doherty–Anstey theory describes such a transformation for scalar waves in finely layered media. Recent observations for seismic waves in the earth suggest that this theory can explain a significant component of seismic attenuation. An important question to answer is then how the O’Doherty–Anstey theory generalizes to seismic waves when several wave modes, possibly with the same velocity, interact. An important aspect of the O’Doherty–Anstey theory is the statistical stability property, which means that the transmitted front pulse is actually deterministic and depends only on the statistics of the medium but not on the particular medium realization when the medium is modeled as a random process. It is shown in this paper that this property generalizes in the case of elast...
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: Wave propagation in disordered (random) media is the underlying theme. Many types of wave propagation problems can most conveniently be analyzed in this framework. Acoustic waves in the earth's crust and shallow water (or surface... more
: Wave propagation in disordered (random) media is the underlying theme. Many types of wave propagation problems can most conveniently be analyzed in this framework. Acoustic waves in the earth's crust and shallow water (or surface gravity) waves are two examples. The interaction of sound waves with the heterogeneities in the earth's crust is important in seismology. The interaction of surface gravity waves with rough submerged obstructions has been analyzed extensively because of its importance in oceanography. It is also important in the development of new techniques for analyzing scattering problems in fluid mechanics. The theory to be outlined in this paper had its first application in seismic exploration. The earth is strongly heterogeneous, also on small scales, and it is important to describe when and how fine scale heterogeneities interact with a traveling seismic pulse. We are interested in pulse shaped waves that interact with the rapidly varying features (i.e. mic...
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Our aim in this chapter is to reconstruct shape perturbations of an extended inclusion from MSR measurements. As for small volume inclusions, we present direct imaging algorithms and analyze their resolution and stability. Our algorithms... more
Our aim in this chapter is to reconstruct shape perturbations of an extended inclusion from MSR measurements. As for small volume inclusions, we present direct imaging algorithms and analyze their resolution and stability. Our algorithms are based on an asymptotic expansion for the perturbations in the data due to small shape perturbations. A concept equivalent to the polarization tensor for small volume targets is introduced.
In this paper we consider resolution estimates in both the linearized conductivity problem and the wave imaging problem. Our purpose is to provide explicit formulas for the resolving power of the measurements in the presence of... more
In this paper we consider resolution estimates in both the linearized conductivity problem and the wave imaging problem. Our purpose is to provide explicit formulas for the resolving power of the measurements in the presence of measurement noise. We show that the low-frequency regime in wave imaging and the inverse conductivity problem are very sensitive to measurement noise, while high frequencies increase stability in wave imaging.
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Research Interests: Engineering, Mathematics, Computer Science, Algorithms, Computational Physics, and 13 moreNumerical Simulation, Mathematical Sciences, Physical sciences, Long Range, Efficient Algorithm for ECG Coding, Parabolic Wave Equation, Wave propagation, Calculation, Parabolic Equations, Domain Decomposition, Long Distance, Helmholtz Equation, and Decomposition method
ABSTRACT
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We present a theory for wave scintillation in the situation with a timedependent partially coherent source and a time-dependent randomly heterogeneous medium. Our objective is to understand how the scintillation index of the measured... more
We present a theory for wave scintillation in the situation with a timedependent partially coherent source and a time-dependent randomly heterogeneous medium. Our objective is to understand how the scintillation index of the measured intensity depends on the source and medium parameters. We deduce from an asymptotic analysis of the random wave equation a general form of the scintillation index and we evaluate this in various scaling regimes. The scintillation index is a fundamental quantity that is used to analyze and optimize imaging and communication schemes. Our results are useful to quantify the scintillation index under realistic propagation scenarios and to address such optimization challenges. ∗Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France (josselin.garnier@polytechnique.edu) †Department of Mathematics, University of California, Irvine CA 92697 (ksolna@math.uci.edu) 1 ar X iv :2 20 1. 06 67 1v 1 [ ph ys ic s. op tic s] 1 8 Ja n 20 22
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A theory for the characterization of the fourth moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian-Schell model is used for the partially coherent random source. The white-noise... more
A theory for the characterization of the fourth moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian-Schell model is used for the partially coherent random source. The white-noise paraxial regime is considered, which holds when the wavelength is much smaller than the correlation radius of the source, the beam radius of the source, and the correlation length of the medium, which are themselves much smaller than the propagation distance. The complex wave amplitude field can then be described by the Itô-Schrödinger equation. This equation gives closed evolution equations for the wave field moments at all orders and here the fourth order equations are considered. The general fourth moment equations are solved explicitly in the scintillation regime (when the correlation radius of the source is of the same order as the correlation radius of the medium, but the beam radius is much larger) and the result gives a characterization of ...
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In this paper we analyze an imaging technique based on intensity speckle correlations over incident field position proposed in [J. A. Newmann and K. J. Webb, Phys. Rev. Lett. 113, 263903 (2014)]. Its purpose is to reconstruct a field... more
In this paper we analyze an imaging technique based on intensity speckle correlations over incident field position proposed in [J. A. Newmann and K. J. Webb, Phys. Rev. Lett. 113, 263903 (2014)]. Its purpose is to reconstruct a field incident on a strongly scattering random medium. The thickness of the complex medium is much larger than the scattering mean free path so that the wave emerging from the random section forms an incoherent speckle pattern. Our analysis clarifies the conditions under which the method can give a good reconstruction and characterizes its performance. The analysis is carried out in the white-noise paraxial regime, which is relevant for the applications in optics that motivated the original paper.
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We present an analysis of acoustic daylight imaging in an Earth-like model assuming a random distribution of noise sources spatially supported in an annulus located away from the surface. We assume a situation with scalar wave propagation... more
We present an analysis of acoustic daylight imaging in an Earth-like model assuming a random distribution of noise sources spatially supported in an annulus located away from the surface. We assume a situation with scalar wave propagation and that the measurements are of the wave field at the surface. Then, we obtain a relation between the autocorrelation function of the measurements and the trace of the scattered field generated by an impulsive source localized just below the surface. From this relation it is, for example, clear that the eigenfrequencies can be recovered from the autocorrelation. Moreover, the complete scattering operator can be extracted under the additional assumption that the annulus is close to the surface and has a thickness smaller than the typical wavelength.
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We prove the convergence of the solutions of the parabolic wave equation to that of the Gaussian white-noise model widely used in the physical literature. The random medium is isotropic and is assumed to have integrable correlation... more
We prove the convergence of the solutions of the parabolic wave equation to that of the Gaussian white-noise model widely used in the physical literature. The random medium is isotropic and is assumed to have integrable correlation coefficient in the propagation direction. We discuss the limits of vanishing inner scale and divergent outer scale of the turbulent medium.
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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,... more
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.
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We present a study of sound wave propagation in a time dependent random medium and an application to imaging. The medium is modeled by small temporal and spatial random fluctuations in the wave speed and density, and it moves due to an... more
We present a study of sound wave propagation in a time dependent random medium and an application to imaging. The medium is modeled by small temporal and spatial random fluctuations in the wave speed and density, and it moves due to an ambient flow. We develop a transport theory for the energy density of the waves, in a forward scattering regime, within a cone (beam) of propagation with small opening angle. We apply the transport theory to the inverse problem of estimating a stationary wave source from measurements at a remote array of receivers. The estimation requires knowledge of the mean velocity of the ambient flow and the second-order statistics of the random medium. If these are not known, we show how they may be estimated from additional measurements gathered at the array, using a few known sources. We also show how the transport theory can be used to estimate the mean velocity of the medium. If the array has large aperture and the scattering in the random medium is strong, ...
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Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the... more
Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatili...
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Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long-range... more
Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long-range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the option price in the regime where the volatility process is fast mean-reverting. The volatility process is modeled as a smooth and bounded function of a fractional Ornstein-Uhlenbeck process. We give the expression for the implied volatility, which has a fractional term structure.
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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... more
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 prove that the centered fourth-order moments of the field satisfy the Gaussian summation rule, 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 second application is used to explicitly quantify the scintillation of the transmitted beam and the third application to quantify the statistical stability of the Wigner transform.
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We analyze the time reversal of waves in a turbulent medium using the parabolic Markovian model. We prove that the time reversal resolution can be a nonlinear function of the wavelength and independent of the aperture. We establish a... more
We analyze the time reversal of waves in a turbulent medium using the parabolic Markovian model. We prove that the time reversal resolution can be a nonlinear function of the wavelength and independent of the aperture. We establish a duality relation between the turbulence-induced wave spread and the time-reversal resolution which can be viewed as an uncertainty inequality for random media. The inequality becomes an equality when the wave structure function is Gaussian.
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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... more
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 physica...
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We study the problem of constructing sparse and fast mean reverting portfolios. The problem can be motivated by convergence trading and formulated as a generalized eigenvalue problem with a cardinality constraint [6]. We use a proxy of... more
We study the problem of constructing sparse and fast mean reverting portfolios. The problem can be motivated by convergence trading and formulated as a generalized eigenvalue problem with a cardinality constraint [6]. We use a proxy of mean reversion coefficient, the direct Ornstein-Uhlenbeck (OU) estimator, which can be applied to both stationary and nonstationary data. In addition, we introduce three different methods to enforce the sparsity of the solutions. One method uses the ratio of l1 and l2 norms and the other two use l1 norm. We analyze various formulations of the resulting nonconvex optimization problems and develop efficient algorithms to solve them for portfolio sizes as large as hundreds. By adopting a simple convergence trading strategy, we test the performance of our sparse mean reverting portfolios on both synthetic and historical real market data. In particular, the l1 regularization method, in combination with quadratic program formulation as well as This work was...
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A central question in free-space optical communications is how to improve the transfer of information between a transmitter and receiver. The capacity of the communication channel can be increased by multiplexing of independent modes... more
A central question in free-space optical communications is how to improve the transfer of information between a transmitter and receiver. The capacity of the communication channel can be increased by multiplexing of independent modes using either: (1) the MIMO (Multiple-Input-Multiple- Output) approach, where the communication is done with modes obtained from the singular value decomposition of the transfer matrix from the transmitter array to the receiver array, or (2) the OAM (Orbital Angular Momentum) approach, which uses vortex beams that carry angular momenta. In both cases, the number of usable modes is limited by the finite aperture of the transmitter and receiver, and the effect of the turbulent atmosphere. The goal of this paper is twofold: First, we show that the MIMO and OAM multiplexing schemes are closely related. Specifically, in the case of circular apertures, the usable singular modes of the transfer matrix are essentially the same as the commonly used Laguerre-Gauss...
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and Overview of the Book.- Waves in Homogeneous Media.- Waves in Layered Media.- Effective Properties of Randomly Layered Media.- Scaling Limits.- Asymptotics for Random Ordinary Differential Equations.- Transmission of Energy Through a... more
and Overview of the Book.- Waves in Homogeneous Media.- Waves in Layered Media.- Effective Properties of Randomly Layered Media.- Scaling Limits.- Asymptotics for Random Ordinary Differential Equations.- Transmission of Energy Through a Slab of Random Medium.- Wave-Front Propagation.- Statistics of Incoherent Waves.- Time Reversal in Reflection and Spectral Estimation.- Applications to Detection.- Time Reversal in Transmission.- Application to Communications.- Scattering by a Three-Dimensional Randomly Layered Medium.- Time Reversal in a Three-Dimensional Layered Medium.- Application to Echo-Mode Time Reversal.- Other Layered Media.- Other Regimes of Propagation.- The Random Schrodinger Model.- Propagation in Random Waveguides.
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In this paper, we study method for identification of targets in a cluttered environment using the inter-element response, the response matrix, of an active array. We examine the properties of the response matrix and its singular value... more
In this paper, we study method for identification of targets in a cluttered environment using the inter-element response, the response matrix, of an active array. We examine the properties of the response matrix and its singular value decomposition (SVD). In particular we analyze the patterns of singular values for scatterers of different sizes and use it for detection and characterization of scatterers. We use the corresponding singular vectors to image the scatterer and present a novel approach using the SVD of the response matrix for imaging the shape of extended targets. We form the difference matrix between response matrices measured at two different times and use its SVD to detect significant changes in a cluttered medium. We also demonstrate that by measuring the response matrix at consecutive times we can track the motion of a target in a cluttered environment. Numerical experiments are presented to illustrate and validate our approach.
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Abstract : In the project we have derived new results for wave propagation in random and complex media and looked at specific applications associated with imaging and communication through a cluttered medium. The main new theoretical... more
Abstract : In the project we have derived new results for wave propagation in random and complex media and looked at specific applications associated with imaging and communication through a cluttered medium. The main new theoretical result is an explicit expression for the forth moment of the wave field in the scintillation regime which is relevant for instance for laser beam propagation through the turbulent atmosphere. This is important because it allows us to analyze statistical stability of imaging and communication schemes. We have used the results on the fourth moment to analyze wavefront correction schemes and obtained novel theoretical results that characterize the performance of these. We have also analyzed precursors that emerge in random media, in the beam regime, and how the frequency contents of the source governs the evolution of these. Moreover, we have analyzed how one can use the statistics of reflected (or transmitted) signals to infer information about the microstructure of the medium through which the signals has propagated. We have also generalized the concept of a Brewster angle associated with optimal transmission to the case with a random slab.