Acoustic imaging of layered media

Gibson, Peter C.

doi:10.1016/j.jcp.2018.06.053

Cited by 6 publications

(9 citation statements)

References 21 publications

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“…The Fourier transform of the Helmholtz equation with a source term, as considered in the present paper, leads to a wave equation with a δ forcing term, the corresponding wavefield being a regular function comprised of a superposition of travelling square waves. By contrast, the wavefield in [3] is the convolution of a purely singular superposition of Dirac functions with a smooth source wave form.…”

Section: Connection To Known Resultsmentioning

confidence: 99%

Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

Bugarija

Gibson

et al. 2020

Inverse Problems

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This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies and the jump points of the wave speed are equally spaced with respect to travel time. Numerical examples show that the algorithm works also in the general case of arbitrary wave speed (either with jumps or continuously varying etc.) giving progressively more accurate approximations as the range of recorded frequencies increases. A key underlying theoretical insight is to associate scattering data to compositions of automorphisms of the unit disk, which are in turn related to orthogonal polynomials on the unit circle. The algorithm exploits the three-term recurrence of orthogonal polynomials to reduce the required computation.

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Section: Connection To Known Resultsmentioning

confidence: 99%

Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

Bugarija

Gibson

et al. 2020

Inverse Problems

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“…A line of investigation concerning piecewise constant, or step function, wave speeds has emerged in the last decade, including work by Albeverio et al [3,2] as well as the present author [21,22,23,25]. But this latter work does not allow any continuous variation in wave speed, and is thus in a sense disjoint from results requiring continuous wave speed or local integrability of q.…”

Section: Introductionmentioning

confidence: 93%

“…A version of Algorithm 2 adapted to the physically more realistic situation of a non-Dirac acoustic source is presented in [23], along with numerical experiments that show remarkable accuracy and stability. (For a non-Dirac source it is no longer necessary to require ζ be continuous; one can invert data for essentially arbitrary impedance.)…”

Section: Inverse Scatteringmentioning

confidence: 99%

“…Results of these computations for x 1 = 30 are presented here as a simple illustration (see [23,11] for more comprehensive discussion). Figure 2 depicts the reflection coefficient R, computed using matrix products as described at the beginning of §5.1.…”

Section: Inverse Scatteringmentioning

confidence: 99%

“…provided u and u are continuous at x 1 . The upshot is that the initial value problem (1,3) transforms into an equivalent Sturm-Liouville problem with boundary conditions (22,23), whose unique solution u determines the reflection coefficient via (21). The problem is not regular however, since the left-hand boundary condition ( 22) is affine and not linear.…”

Section: Introductionmentioning

confidence: 99%

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Scattering on the line via singular approximation

Gibson¹

2021

Preprint

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Motivated by applications to acoustic imaging, the present work establishes a framework to analyze scattering for the one-dimensional wave, Helmholtz, Schrödinger and Riccati equations that allows for coefficients which are more singular than can be accommodated by previous theory. In place of the standard scattering matrix or the Weyl-Titchmarsh m-function, the analysis centres on a new object, the generalized reflection coefficient, which maps frequency (or the spectral parameter) to automorphisms of the Poincaré disk. Purely singular versions of the generalized reflection coefficient, which are amenable to direct analysis, serve to approximate the general case. Orthogonal polynomials on both the unit circle and unit disk play a key technical role, as does an exotic Riemannian structure on PSL(2, R). A central role is also played by the newlydefined harmonic exponential operator, introduced to mediate between impedance (or index of refraction) and the reflection coefficient. The approach leads to new, explicit formulas and effective algorithms for both forward and inverse scattering. The algorithms may be viewed as nonlinear analogues of the FFT. In addition, the scattering relation is shown to be elementary in a precise sense at or below the critical threshold of continuous impedance. For discontinuous impedance, however, the reflection coefficient ceases to decay at infinity, the classical trace formula breaks down, and the scattering relation is complicated by the emergence of almost periodic structure.

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Disk polynomials and the one-dimensional wave equation

Gibson

2019

Journal of Approximation Theory

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Acoustic imaging of layered media

Cited by 6 publications

References 21 publications

Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

Scattering on the line via singular approximation

Disk polynomials and the one-dimensional wave equation

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