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96
Adaptive Covariance Estimation Of Locally Stationary Processes
, 1995
"... this paper so we will not mention this explicitly. The ideas and methods of Calderon and Zygmund [7] in harmonic analysis have shown that although we are not able to find the ..."
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Cited by 87 (8 self)
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this paper so we will not mention this explicitly. The ideas and methods of Calderon and Zygmund [7] in harmonic analysis have shown that although we are not able to find the
A TimeReversal Method for an Acoustical Pulse Propagating in Randomly Layered Media
, 1997
"... In the recent years a considerable amount of mathematical work have been devoted to the study of reflected signals obtained by the propagation of pulses in randomly layered media. We refer to [1] for an extensive survey and applications to inverse problems. The analysis is based on separation of sca ..."
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Cited by 47 (6 self)
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In the recent years a considerable amount of mathematical work have been devoted to the study of reflected signals obtained by the propagation of pulses in randomly layered media. We refer to [1] for an extensive survey and applications to inverse problems. The analysis is based on separation of scales between the correlation scale of the inhomogeneities present in the medium, the typical wavelengths of the pulse and the macroscopic variations of the medium. On the other hand, in the context of ultrasounds, timereversal mirrors have been developed and their effects have been studied experimentally by Mathias Fink and his team at the Laboratoire Ondes et Acoustique (ESPCIParis). We refer to [2]. Our goal is to present a mathematical analysis of a timereversal method for analysing reflected signals in the model described in [1]. We restrict our analysis to the onedimensional case, the threedimensional layered case being the content of a forthcoming paper. It is noticeable that we do ...
Pulse propagation and time reversal in random waveguides
 SIAM J. APPL. MATH
, 2007
"... Mode coupling in a random waveguide can be analyzed with asymptotic analysis based on separation of scales when the propagation distance is large compared to the size of the random inhomogeneities, which have small variance, and when the wavelength is comparable to the scale of the inhomogeneities ..."
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Cited by 19 (8 self)
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Mode coupling in a random waveguide can be analyzed with asymptotic analysis based on separation of scales when the propagation distance is large compared to the size of the random inhomogeneities, which have small variance, and when the wavelength is comparable to the scale of the inhomogeneities. In this paper we study the asymptotic form of the joint distribution of the mode amplitudes at different frequencies. We derive a deterministic system of transport equations that describe the evolution of mode powers. This result is applied to the computations of pulse spreading in a random waveguide. It is also applied to the analysis of time reversal in a random waveguide. We show that randomness enhances spatial refocusing and that diffractionlimited focal spots can be obtained even with smallsize timereversal mirrors. The refocused field is statistically stable for broadband pulses in general. We show here that it is also stable for narrowband pulses, provided that the timereversal mirror is large enough.
Time reversal refocusing for point source in randomly layered media, Wave Motion 42
, 2005
"... Abstract. This paper demonstrates the interest of a timereversal 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 timereversal mirror. The wave is sent back into the medium, either numerically ..."
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Cited by 19 (9 self)
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Abstract. This paper demonstrates the interest of a timereversal 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 timereversal 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 timereveresal refocusing contains information which can be used to this effect and which cannot be obtained by a direct arrivaltime 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
Imaging in randomly layered media by crosscorrelating noisy signals
 SIAM Multiscale Model. Simul
"... Abstract. We consider an active source embedded in a randomly layered medium. We study the crosscorrelation functions of the signals recorded at a series of points located at the surface. We show that this information can be processed to locate the source inside the medium. The analysis is based on ..."
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Cited by 18 (8 self)
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Abstract. We consider an active source embedded in a randomly layered medium. We study the crosscorrelation functions of the signals recorded at a series of points located at the surface. We show that this information can be processed to locate the source inside the medium. The analysis is based on a separation of scales technique and limit theorems for random differential equations. The statistical stability of the imaging method is proved. The analogy with the timereversal of waves is enlightened, but the main difference is also put forward: we propose a passive way of imaging an unknown medium without the use of any active device. We finally extend these ideas for the location of a scatterer illuminated by a controlled source located at the surface or by a set of unknown sources generating random noise. Key words. Acoustic waves, random media, asymptotic theory. AMS subject classifications. 76B15, 35Q99, 60F05. 1. Introduction. Imaging
Ray theory for a locally layered random medium
, 2004
"... We consider acoustic pulse propagation in inhomogeneous media over relatively long propagation distances. Our main objective is to characterize the spreading of the travelling pulse due to microscale variations in the medium parameters. The pulse is generated by a point source and the medium is mode ..."
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Cited by 18 (10 self)
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We consider acoustic pulse propagation in inhomogeneous media over relatively long propagation distances. Our main objective is to characterize the spreading of the travelling pulse due to microscale variations in the medium parameters. The pulse is generated by a point source and the medium is modeled by a smooth three dimensional background that is modulated by stratified random fluctuations. We refer to such media as locally layered. We show that, when the pulse is observed relative to its random arrival time, it stabilizes to a shape determined by the slowly varying background convoluted with a Gaussian. The width of the Gaussian and the random travel time are determined by the medium parameters along the ray connecting the source and the point of observation. The ray is determined by high frequency asymptotics (geometrical optics). If we observe the pulse in a deterministic frame moving with the effective slowness, it does not stabilize and its mean is broader because of the random component of the travel time. The analysis of this phenomenon involves the asymptotic solution of partial differential equations with randomly varying coefficients and is based on a new representation of the field in terms of generalized plane waves that travel in opposite directions relative to the layering.
Pressure Fields Generated By Acoustical Pulses Propagating in Randomly Layered Media
, 1997
"... This paper investigates the pressure field generated at the bottom of a highcontrast randomly layered slab by an acoustical pulse emitted at the surface of the slab. This analysis takes place in the framework introduced by Asch, Kohler, Papanicolaou, Postel and White [1] where the incident pulse wa ..."
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Cited by 16 (3 self)
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This paper investigates the pressure field generated at the bottom of a highcontrast randomly layered slab by an acoustical pulse emitted at the surface of the slab. This analysis takes place in the framework introduced by Asch, Kohler, Papanicolaou, Postel and White [1] where the incident pulse wave length is long compared to the correlation length of the random inhomogeneities, but short compared to the size of the slab. This problem has been studied in the onedimensional case simultaneously by Clouet and Fouque [4] and Lewicki, Burridge and Papanicolaou [6] or for multimode plane wave pulses in Lewicki, Burridge and De Hoop [7]. These situations require only the use of classical diffusionapproximation results whereas the pointsource problem studied in this paper requires a nontrivial combination of diffusionapproximation results with stationary phase methods. The stationary phase method has been used by De Hoop, Chang and Burridge [5] for weakly fluctuating media and in [1] fo...
Coherent interferometry in finely layered random media, Multiscale Model
 Simul
"... Abstract. We study broadband, coherent interferometric array imaging (CINT) in finely layered media in a regime with strong fluctuations. By coherent interferometric imaging we mean the backpropagation of timewindowed cross correlations of the array data. For waves propagating over long distances, ..."
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Cited by 14 (6 self)
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Abstract. We study broadband, coherent interferometric array imaging (CINT) in finely layered media in a regime with strong fluctuations. By coherent interferometric imaging we mean the backpropagation of timewindowed cross correlations of the array data. For waves propagating over long distances, there is statistical stabilization of the traces observed at the array. They have the form of a coherent signal that can be described by the O’Doherty–Anstey (ODA) theory, followed by long and noisy codas. We show that coherent interferometry exploits the time coherence in the data, leading to stable images. Moreover, we prove that in this regime only the ODA behavior plays a role in the imaging, and we quantify explicitly the resolution of CINT in terms of this time coherence and the array aperture. We illustrate the theory with numerical simulations. Key words. O’Doherty–Anstey theory, imaging, layered media
TimeReversal Aperture Enhancement
 SIAM Multiscale Modeling and Simulation
, 2000
"... Timereversal refocusing for waves propagating in inhomogeneous media have recently been observed and studied experimentally in various contexts (ultrasound, underwater acoustics, ...), see for instance [9]. Important potential applications have been proposed in various fields, for instance in imagi ..."
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Cited by 13 (3 self)
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Timereversal refocusing for waves propagating in inhomogeneous media have recently been observed and studied experimentally in various contexts (ultrasound, underwater acoustics, ...), see for instance [9]. Important potential applications have been proposed in various fields, for instance in imaging or communication. However, the full mathematical analysis, meaning both modeling of the physical problem and derivation of the timereversal effect is a deep and complex problem. Two cases that have been considered in depth recently corresponds to one dimensional media and the parabolic approximation regime where the backscattering is negligible. In this paper we give a complete analysis of timereversal of waves emanating from a point source and propagating in a three dimensional randomly layered medium. The wave transmitted through the random medium is recorded on a small timereversal mirror and sent back into the medium, timereversed. Our analysis enables us to contrast the refocusing properties of a homogeneous medium and a random medium. We show that random medium fluctuations actually enhances the spatial refocusing around the initial source position. We consider a regime where the correlation length of the medium is much smaller than the pulse width, which itself is much smaller than the distance of propagation. We derive asymptotic formulas for the refocused pulse which we interpret in terms of an enhanced effective aperture. This interpretation is in fact comparable to the superresolution effect obtained in the other extreme regime corresponding to the parabolic approximation. However, as we discuss, the mechanism that generates the superresolution is very different in these two extreme situations.