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Abstract:We develop an efficient algorithm for simulating wave propagation over long distances with both weak and strong scatterers. In domains with weak heterogeneities the wave field is decomposed into forward propagating and back scattered modes using two coupled parabolic equations. In the region near strong scatterers, the Helmholtz equation is used to capture the strong scattering events. The key idea in our method is to combine these two regimes using a combined domain decomposition and wave decomposition method… Show more
“…We see that longitudinal scattering is not captured well leading to poor approximation of the wave heterogeneity and an artificially smooth approximation. In the right plot the dashed line corresponds to an approximation of the system (4-5) that captures the important "right" and "left" coupling [13,14]. The latter approximation corresponds to iterative right and left sweeps implementation of (4-5) when the coupling correction associated with exp(±2ikz/ε q )L 2 is neglected, giving convergence after a few iterations.…”
Section: )mentioning
confidence: 99%
“…The latter approximation corresponds to iterative right and left sweeps implementation of (4-5) when the coupling correction associated with exp(±2ikz/ε q )L 2 is neglected, giving convergence after a few iterations. The boundary condition at the depth of the embedded scatterer is implemented via a domain decomposition approach with the inclusion located in a small domain numerically resolved via a discretization of the Helmholtz equation (see [13] for details). Motivated by this computational example we Fig.…”
Section: )mentioning
confidence: 99%
“…withν(z, κ) the partial Fourier transform of ν(z, x) as defined by (13). This system is supplemented by the initial conditionŝ…”
In this paper we revisit the parabolic approximation for wave propagation in random media by taking into account backscattering. We obtain a system of transport equations for the moments of the components of reflection and transmission operators. In the regime in which forward scattering is strong and backward scattering is weak, we obtain closed form expressions for physically relevant quantities related to the reflected wave, such as the beam width, the spectral width and the mean spatial power profile. In particular, we analyze the enhanced backscattering phenomenon, that is, we show that the mean power reflected from an incident quasi-plane wave has a maximum in the backscattered direction. This enhancement can be observed in a small cone around the backscattered direction and we compute the enhancement factor as well as the shape of the enhanced backscattering cone.
“…We see that longitudinal scattering is not captured well leading to poor approximation of the wave heterogeneity and an artificially smooth approximation. In the right plot the dashed line corresponds to an approximation of the system (4-5) that captures the important "right" and "left" coupling [13,14]. The latter approximation corresponds to iterative right and left sweeps implementation of (4-5) when the coupling correction associated with exp(±2ikz/ε q )L 2 is neglected, giving convergence after a few iterations.…”
Section: )mentioning
confidence: 99%
“…The latter approximation corresponds to iterative right and left sweeps implementation of (4-5) when the coupling correction associated with exp(±2ikz/ε q )L 2 is neglected, giving convergence after a few iterations. The boundary condition at the depth of the embedded scatterer is implemented via a domain decomposition approach with the inclusion located in a small domain numerically resolved via a discretization of the Helmholtz equation (see [13] for details). Motivated by this computational example we Fig.…”
Section: )mentioning
confidence: 99%
“…withν(z, κ) the partial Fourier transform of ν(z, x) as defined by (13). This system is supplemented by the initial conditionŝ…”
In this paper we revisit the parabolic approximation for wave propagation in random media by taking into account backscattering. We obtain a system of transport equations for the moments of the components of reflection and transmission operators. In the regime in which forward scattering is strong and backward scattering is weak, we obtain closed form expressions for physically relevant quantities related to the reflected wave, such as the beam width, the spectral width and the mean spatial power profile. In particular, we analyze the enhanced backscattering phenomenon, that is, we show that the mean power reflected from an incident quasi-plane wave has a maximum in the backscattered direction. This enhancement can be observed in a small cone around the backscattered direction and we compute the enhancement factor as well as the shape of the enhanced backscattering cone.
“…In recent decades the parabolic or paraxial wave equation has emerged as the primary tool to describe small scale scattering situations as they appear in radiowave propagation, radar, remote sensing, propagation in urban environments, and in underwater acoustics [29,32,41], as well as in propagation problems in the earth's crust [9]. The paraxial equation models wave propagation if the dominant scattering occurs in the direction(s) transverse to a principal propagation direction.…”
Abstract. Traditional imaging methods use coherent signals as data. Here, we discuss recent developments in imaging that aim at exploiting as data incoherent noisy signals that are not associated with well-defined arrival times. Indeed, signal constituents that in a classical setting may be regarded as noise may contain important information about the medium to be imaged. We show how it is possible to use the statistics of such noisy signals, specifically, the second-order statistics, for imaging. We consider two particular situations: first, the estimation of an ("empirical") Green's function from noisy signals which can subsequently be used in imaging; second, the localization of a cluster of random sources from noisy signals (passive imaging). The analysis presented here is based on assuming a remote sensing scaling and the paraxial approximation, and it uses in part the results set forth in Papanicolaou, Ryzhik, and Solna [SIAM J. Appl. Math., 64 (2004), pp. 1133-1155] that relate to time-reversal, statistical stability, and superresolution. Robustness with respect to modeling assumptions is illustrated by considering other scaling regimes also. We demonstrate how the estimation problem and its robustness can be considered as a dual to that of time-reversal and stable superresolution. We obtain a novel analysis and foundation for the use of ambient seismic noise in body-wave (tomographic) imaging, motivated by the recent successes of surface-wave tomography using ambient seismic noise.
“…In this paper we address situations in which backscattering is not negligible, which requires more elaborate schemes than the paraxial wave equation. After the pioneering work [5] an iterative two-way paraxial scheme was proposed to solve the full wave equation by an iteration of forward-going and backward-going paraxial wave equations in [12,13]. Comparisons between numerical simulations of the full wave and numerical simulations of the two-way paraxial system have shown very good agreement, with the overall conclusion being that the twoway paraxial scheme has the same numerical complexity as the standard oneway paraxial equation, but it also takes into account random backscattering.…”
This paper presents a probabilistic analysis of an iterative two-way paraxial scheme for the simulation of wave propagation in random media. This scheme has the computational cost of the standard one-way paraxial wave equation but has the accuracy of the full wave equation in a regime beyond the classical paraxial regime. More precisely, it accurately predicts the statistics of the reflected wave field. The accuracy is determined by the order of the iterative scheme and the ratio of the random backscattering intensity over the random forward-scattering intensity.
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