Relationship between the amplitude and phase of a signal scattered by a point-like acoustic inhomogeneity

Burov, V. A.; Morozov, S. A.

doi:10.1134/1.1418890

Cited by 13 publications

(7 citation statements)

References 1 publication

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“…In addition, µ(x, k E (λ(1 ∓ 0)) = µ ± (x, k E (λ)) for λ ∈ T , where µ ± are continuous in λ ∈ T (whereas a(k E (λ)), b(k E (λ)) are continuous in a neighborhood of T ). 6. If E = |E 1 |, then the functions µ(x, k E (λ)), a(k E (λ)), b(k E (λ)) are singular in λ on the contour…”

Section: Resultsmentioning

confidence: 99%

“…Remark 1 Relations between the absolute value of the scattering amplitude f and its phase for a two-dimensional point-like scatterer was given earlier in [6], see also [2] for further development. To our knowledge no exact formulas for the Faddeev eigenfunctions ψ and related scattering data a(k), b(k) associated with 2D point potentials were given in the literature.…”

Section: The Scattering Data For the Limiting Potentialmentioning

confidence: 88%

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Faddeev eigenfunctions for point potentials in two dimensions

Grinevich

Novikov

2012

Physics Letters A

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Section: Resultsmentioning

confidence: 99%

Section: The Scattering Data For the Limiting Potentialmentioning

confidence: 88%

Faddeev eigenfunctions for point potentials in two dimensions

Grinevich

Novikov

2012

Physics Letters A

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“…Note that already for this simplest case there are no explicit analytic reconstruction formulas for regular potentials. Our numerical results for this case develop studies of [4,2]. These results are presented in details in Subsection 4.2.…”

Section: Introductionmentioning

confidence: 89%

Examples of solution of the inverse scattering problem and the equations of the Novikov–Veselov hierarchy from the scattering data of point potentials

Agaltsov¹,

Novikov²

2019

Russ. Math. Surv.

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The inverse scattering problem is considered for the two- dimensional Schrödinger equation at fixed positive energy. The results include inverse scattering reconstructions from the simplest scattering amplitudes. In particular, a complete analytic solution is given of the phased and phaseless inverse scattering problems for single-point potentials of Bethe– Peierls– Fermi–Zeldovich– Berezin–Faddeev type. Numerical inverse scattering reconstructions from the simplest scattering amplitudes are then studied using the method of the Riemann–Hilbert– Manakov problem in soliton theory. Finally, these numerical inverse scattering results are used to construct corresponding numerical solutions of the non-linear equations of the Novikov–Veselov hierarchy at fixed positive energy. Bibliography: 21 titles.

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“…In this work, the reconstruction of sound speed inhomogeneities of cylindrical shape is considered. The scattering of a cylindrical wave on a cylindrical inhomogeneity has a rigorous analytical solution (see, for example, [33]), that allows to calculate scattering data G(r, x) with very high quality, in comparision with a discretized solution of the Lippmann-Schwinger equation. In the latter case, the question arises about the spatial discretization step of the integral equation, which must decrease as the scatterer strength increases in order to adequately describe the processes of multiple scatterings.…”

Section: Numerical Modelingmentioning

confidence: 99%

Functional-analytical reconstruction of high contrast inhomogeneities

Shurup¹

2023

Preprint

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In practice of acoustic tomography, for example, in medical applications and ocean tomography, the relative deviation of sound speed from its background value usually does not exceed 10-30%. At the same time, in electromagnetic applications, the equivalent contrasts can be noticeably higher than 60%. Since the inverse electromagnetic problem can be reduced in some approximation to Helmholtz equation, a formal comparison of reconstruction results obtained for different "acoustic" contrast and corresponding "dielectric" contrast is possible. In this work examples of such reconstructions are presented, which were obtained by using the functional-analytical algorithm described in works of R.G. Novikov. Previously, the advantages of this algorithm for solving practical problems of acoustic tomography were demonstrated. Results obtained in the present work show that functional-analytical algorithm can also be applied to reconstructing inhomogeneities with high "dielectric" contrast. Moreover, the functional algorithm also perfectly reconstructs very small "dielectric" contrast, recovering of which can be difficult for other approaches due to weak backscattering.

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Relationship between the amplitude and phase of a signal scattered by a point-like acoustic inhomogeneity

Cited by 13 publications

References 1 publication

Faddeev eigenfunctions for point potentials in two dimensions

Faddeev eigenfunctions for point potentials in two dimensions

Examples of solution of the inverse scattering problem and the equations of the Novikov–Veselov hierarchy from the scattering data of point potentials

Functional-analytical reconstruction of high contrast inhomogeneities

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