A Nonlinear Method for Imaging with Acoustic Waves Via Reduced Order Model Backprojection

Druskin, Vladimir; Mamonov, A.; Zaslavsky, Mikhail

doi:10.1137/17m1133580

Cited by 20 publications

(62 citation statements)

References 29 publications

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“…(2) For imaging scattering surfaces in a smooth reference medium, mostly in the context of reflection seismology [51,36] and a related setting in optics [38]. (3) For imaging almost layered media using the so-called Marchenko redatuming method [50], and also for imaging based on data-driven reduced order models [20,22]. This latter work is the foundation of the algorithm in this paper.…”

Section: Introductionmentioning

confidence: 99%

Untangling the nonlinearity in inverse scattering with data-driven reduced order models

et al. 2018

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The motivation of this work is an inverse problem for the acoustic wave equation, where an array of sensors probes an unknown medium with pulses and measures the scattered waves. The goal of the inversion is to determine from these measurements the structure of the scattering medium, modeled by a spatially varying acoustic impedance function. Many inversion algorithms assume that the mapping from the unknown impedance to the scattered waves is approximately linear. The linearization, known as the Born approximation, is not accurate in strongly scattering media, where the waves undergo multiple reflections before they reach the sensors in the array. Thus, the reconstructions of the impedance have numerous artifacts. The main result of the paper is a novel, linear-algebraic algorithm that uses a reduced order model (ROM) to map the data to those corresponding to the single scattering (Born) model. The ROM construction is based only on the measurements at the sensors in the array. The ROM is a proxy for the wave propagator operator, that propagates the wave in the unknown medium over the duration of the time sampling interval. The output of the algorithm can be input into any off-the-shelf inversion software that incorporates state of the art linear inversion algorithms to reconstruct the unknown acoustic impedance.

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Section: Introductionmentioning

confidence: 99%

Untangling the nonlinearity in inverse scattering with data-driven reduced order models

et al. 2018

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“…The second is based on the Gram-Schmidt orthogonalization of the snapshots. 13 3.1.1. Finite differences interpretation.…”

Section: )mentioning

confidence: 99%

Reduced Order Model Approach to Inverse Scattering

Borcea¹,

Druskin²,

Mamonov³

et al. 2020

SIAM J. Imaging Sci.

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We study an inverse scattering problem for a generic hyperbolic system of equations with an unknown coefficient called the reflectivity. The solution of the system models waves (sound, electromagnetic or elastic), and the reflectivity models unknown scatterers embedded in a smooth and known medium. The inverse problem is to determine the reflectivity from the time resolved scattering matrix (the data) measured by an array of sensors. We introduce a novel inversion method, based on a reduced order model (ROM) of an operator called wave propagator, because it maps the wave from one time instant to the next, at interval corresponding to the discrete time sampling of the data. The wave propagator is unknown in the inverse problem, but the ROM can be computed directly from the data. By construction, the ROM inherits key properties of the wave propagator, which facilitate the estimation of the reflectivity. The ROM was introduced previously and was used for two purposes: (1) to map the scattering matrix to that corresponding to the single scattering (Born) approximation and (2) to image i.e., obtain a qualitative estimate of the support of the reflectivity. Here we study further the ROM and show that it corresponds to a Galerkin projection of the wave propagator. The Galerkin framework is useful for proving properties of the ROM that are used in the new inversion method which seeks a quantitative estimate of the reflectivity. * One can also consider truncation of the whole space, as long as the medium is known and non-scattering on one side of the array of sensors. P(q) = cos τ L(q)L(q) T .( 1.9) The ROM is defined by a pair of matrices P ROM (q) ∈ R nm×nm and b ROM ∈ R nm×m , which are proxies of P(q) and the initial wave b(x). These matrices are calculated from the data (1.8) (i.e., the ROM is data-driven) and they capture physical aspects of the wave propagation that are needed for inversion. The ROM was introduced in [12,6] and was used in [13] for imaging, and in [5] for transforming the data (1.8) to that corresponding to the single scattering (Born) approximation. The new results in this paper are:1. We show that the ROM propagator P ROM (q) is a Galerkin projection of the operator (1.9), and use the Galerkin framework to prove properties of the ROM that facilitate the solution of the inverse scattering problem. † The kinematic model (smooth wave speed) appears in the coefficients of the operators L(q) and L(q) T (see [6] and sections 3-4). We suppress the dependence on the known kinematic model in our notation.3

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“…Therefore L andŜ are the same by the uniqueness of the inverse eigenvalue problem, and ĉ = V , from which the result follows. That is, the solution components U j of the difference scheme (15) can be interpreted as coefficients of the spectrally converging Galerkin solution with respect to the orthogonal basis (16). See Figure 2 for an example of an orthogonalized basis with its equivalent staggered finite difference grid.…”

Section: 2mentioning

confidence: 99%

“…If we change the basis to the orthogonalized(16) and form the Galerkin system in this new basis(17) (Ŝ + λM ) ĉ = F…”

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Reduced order models for spectral domain inversion: embedding into the continuous problem and generation of internal data

Borcea¹,

Druskin²,

Mamonov³

et al. 2020

Inverse Problems

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We generate data-driven reduced order models (ROMs) for inversion of the one and two dimensional Schrödinger equation in the spectral domain given boundary data at a few frequencies. The ROM is the Galerkin projection of the Schrödinger operator onto the space spanned by solutions at these sample frequencies. The ROM matrix is in general full, and not good for extracting the potential. However, using an orthogonal change of basis via Lanczos iteration, we can transform the ROM to a block triadiagonal form from which it is easier to extract q. In one dimension, the tridiagonal matrix corresponds to a three-point staggered finite-difference system for the Schrödinger operator discretized on a so-called spectrally matched grid which is almost independent of the medium. In higher dimensions, the orthogonalized basis functions play the role of the grid steps. The orthogonalized basis functions are localized and also depend only very weakly on the medium, and thus by embedding into the continuous problem, the reduced order model yields highly accurate internal solutions. That is to say, we can obtain, just from boundary data, very good approximations of the solution of the Schrödinger equation in the whole domain for a spectral interval that includes the sample frequencies. We present inversion experiments based on the internal solutions in one and two dimensions.

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A Nonlinear Method for Imaging with Acoustic Waves Via Reduced Order Model Backprojection

Cited by 20 publications

References 29 publications

Untangling the nonlinearity in inverse scattering with data-driven reduced order models

Untangling the nonlinearity in inverse scattering with data-driven reduced order models

Reduced Order Model Approach to Inverse Scattering

Reduced order models for spectral domain inversion: embedding into the continuous problem and generation of internal data

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