The linear sampling method seeks to localize the unknown source of an observed, time-dependent field. The unknown source could be, for example, a scatterer embedded within a medium, or an impulsive excitation such as an earthquake or explosion. The source of the observed field is localized by means of solving the so-called near-field equation and mapping the obtained solutions through an indicator functional over a test region assumed to contain the source. In its current formulation, however, the linear sampling method suffers from an ambiguous time parameter that strongly influences its ability to localize the unknown source. Our paper consists of two fundamental results central to the theoretical understanding of the linear sampling method and its numerical implementation. First, we prove the so-called blowup behavior of solutions to the near-field equation for a general source function that is separable in space and time. Second, we show that the linear sampling method can be formulated such that the ambiguous time parameter is irrelevant. We demonstrate that a dependence of the linear sampling method on the time parameter arises from an incorrect implementation of a convolution-type operator found in the near-field equation. When the operator is implemented correctly, the dependence on the time parameter vanishes. We provide detailed algorithms for efficient and proper implementations of the convolutional operator in both the time and frequency domains. The crucial result of the improved implementations is that they allow the linear sampling method to be completely automated, as one does not need to know the space-time dependence of the unknown source. We demonstrate the effectiveness of the improved time-and frequency-domain implementations using several numerical examples applied to imaging scatterers.
Abstract. The memory effect has seen a surge of research into its fundamental properties and applications since its discovery by Feng et al. [Phys. Rev. Lett. 61, 834 (1988)]. While the wave trajectories for which the memory effect holds are hidden implicitly in the diffusion probability function [Phys. Rev. B 40, 737 (1989)], the physical intuition of why these trajectories satisfy the memory effect has often been masked by the derivation of the memory correlation function itself. In this paper, we explicitly derive the specific trajectories through a random medium for which the memory effect holds. Our approach shows that the memory effect follows from a simple conservation argument, which imposes geometrical constraints on the random trajectories that contribute to the memory effect. We illustrate the time-domain effects of these geometrical constraints with numerical simulations of pulse transmission through a random medium. The results of our derivation and numerical simulations are consistent with established theory and experimentation.
We introduce an imaging method based on solving the Lippmann-Schwinger equation of acoustic scattering theory. We compare and contrast the proposed Lippmann-Schwinger inversion with the well-established linear sampling method using numerical examples. We demonstrate that the two imaging methods are physically grounded in different but related wave propagation problems: Lippmann-Schwinger inversion seeks to reconstruct the space and time dependence of a scatterer based on the observed scattered field in a performed physical experiment, whereas the linear sampling method seeks to focus wave fields in a simulated virtual experiment by estimating the space and time dependence of an inverse source function that cancels the effects of the scatterer at a specified focusing point. In both cases, the medium in which the waves propagate is the same; however, neither method requires prior knowledge or assumptions on the physical properties of the unknown scatterer-only knowledge of the background medium is needed. We demonstrate that the linear sampling method is preferable to Lippmann-Schwinger inversion for target-oriented imaging applications, as Lippmann-Schwinger inversion gives nonphysical results when the chosen imaging domain does not contain the scatterer.
We investigate the feasibility of imaging localized velocity contrasts within a nonattenuating acoustic medium using volume-distributed random point sources. We propose a simple, two-step processing flow that utilizes the linear sampling method to invert for the target locations directly from the recorded waveforms. We present several proof-of-concept experiments using Monte Carlo simulations to generate independent realizations of band limited “white noise” sources, which are randomly distributed in both time and space. Despite the unknown and random character of the illumination on the imaging targets, we show that it is possible to image strong velocity contrasts directly from multiply scattered coda waves in the recorded data. We benchmark the images obtained from the random-source experiments with those obtained by a standard application of the linear sampling method to analogous controlled-source experiments.
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