A scattering‐based algorithm for wave propagation in one dimension

Gibson, Peter C.

doi:10.1002/num.22207

Cited by 4 publications

(7 citation statements)

References 17 publications

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“…Proposition 2. A technical problem relating to absolutely continuous impedance functions was mentioned already in §1.1.6: to extend the trace formula (24) to α ∈ L 2 R (X). More generally, what is the maximal domain of validity of the singular trace formula of Theorem 12?…”

Section: Discussionmentioning

confidence: 99%

“…A singular trace formula. The classical trace formula (24) is not valid for R 2 = g ζ (0) when ζ ∈ Step + (X) is a non-constant step function, since in this case g ω ζ (0) is almost periodic and fails to decay as |ω| → ∞. For step functions the trace formula has a singular counterpart derived in [26], as follows.…”

Section: 3mentioning

confidence: 99%

“…The algorithm requires O(n 2 ) flops, and is inherently stable (see [24]). It turns out also to be remarkably accurate in practice.…”

Section: Effective Computationmentioning

confidence: 99%

“…The present work therefore avoids reliance on [37]. All that is needed for the inverse scattering formula is the nonlinear Bessel's inequality for (24),…”

Section: Introductionmentioning

confidence: 99%

“…Thus the almost periodic part of the reflection coefficient g ζ (0) is determined by ζ 2 ; in particular, the almost periodic part of g ζ (0) is zero if and only if ζ is continuous. The classical trace formula (24) breaks down if R 2 has a non-zero almost periodic part, since in this case the left-hand integral diverges and α acquires a purely singular distributional part.…”

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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Section: Discussionmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

“…The algorithm requires O(n 2 ) flops, and is inherently stable (see [24]). It turns out also to be remarkably accurate in practice.…”

Section: Effective Computationmentioning

confidence: 99%

“…The present work therefore avoids reliance on [37]. All that is needed for the inverse scattering formula is the nonlinear Bessel's inequality for (24),…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Scattering on the line via singular approximation

Gibson¹

2021

Preprint

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Acoustic imaging of layered media

Gibson

2018

Journal of Computational Physics

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The refined impedance transform for 1D acoustic reflection data

Gibson¹

2018

Inverse Problems

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The one dimensional wave equation serves as a basic model for imaging modalities such as seismic which utilize acoustic data reflected back from a layered medium. In 1955 Peterson et al. described a single scattering approximation for the one dimensional wave equation that relates the reflection Green's function to acoustic impedance. The approximation is simple, fast to compute and has become a standard part of seismic theory. The present paper reexamines this classical approximation in light of new results concerning the (exact) measurement operator for reflection imaging of layered media, and shows that the classical approximation can be substantially improved. We derive an alternate formula, called the refined impedance transform, that retains the simplicity and speed of computation of the classical estimate, but which is qualitatively more accurate and applicable to a wider range of recorded data. The refined impedance transform can be applied to recorded data directly (without the need to deconvolve the source wavelet), and solves exactly the inverse problem of determining the value of acoustic impedance on the far side of an arbitrary slab of unknown structure. The results are illustrated with numerical examples.

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A scattering‐based algorithm for wave propagation in one dimension

Cited by 4 publications

References 17 publications

Scattering on the line via singular approximation

Scattering on the line via singular approximation

Acoustic imaging of layered media

The refined impedance transform for 1D acoustic reflection data

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