Stable methods for solving the inverse scattering problem for a cylinder

1981 ◽  
Vol 89 (3-4) ◽  
pp. 181-188 ◽  
Author(s):  
David Colton ◽  
Andreas Kirsch
Keyword(s):  
Cross Section ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Initial Approximation ◽  
Constructive Method ◽  
Infinite Cylinder ◽  
Unknown Boundary ◽  
Closed Curves

SynopsisIt is shown that the inverse scattering problem for an infinite cylinder can be stabilized by assuming a priori that the unknown boundary of the cylindrical cross section lies in a compact family of continuously differentiable simple closed curves. A constructive method for determining the shape of this boundary is given under the assumption that an initial approximation is known and that the scattering cross section is known forn distinct incoming plane waves in the resonant region.

TWO VERSIONS OF QUASI-NEWTON METHOD FOR MULTIDIMENSIONAL INVERSE SCATTERING PROBLEM

1993 ◽  
Vol 01 (02) ◽  
pp. 197-228 ◽  
Author(s):  
SEMION GUTMAN ◽  
MICHAEL KLIBANOV
Keyword(s):  
Newton Method ◽  
Inverse Scattering ◽  
Born Approximation ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Circular Cylinders ◽  
Iterative Sequence ◽  
Scattering Field ◽  
Quasi Newton

Suppose that a medium with slowly changing spatial properties is enclosed in a bounded 3-dimensional domain and is subjected to a scattering by plane waves of a fixed frequency. Let measurements of the wave scattering field induced by this medium be available in the region outside of this domain. We study how to extract the properties of the medium from the information contained in the measurements. We are concerned with the weak scattering case of the above inverse scattering problem (ISP), that is, the unknown. spatial variations of the medium are assumed to be close to a constant. Examples can be found in the studies of the wave propagation in oceans, in the atmosphere, and in some biological media. Since the problems are nonlinear, the methods for their linearization (the Born approximation) have been developed. However, such an approach often does not produce good results. In our method, the Born approximation is just the first iteration and further iterations improve the identification by an order of magnitude. The iterative sequence is defined in the framework of a Quasi-Newton method. Using the measurements of the scattering field from a carefully chosen set of directions we are able to recover (finitely many) Fourier coefficients of the sought parameters of the model. Numerical experiments for the scattering from coaxial circular cylinders as well as for simulated data are presented.

Far field patterns and inverse scattering problems for imperfectly conducting obstacles

1989 ◽  
Vol 106 (3) ◽  
pp. 553-569 ◽  
Author(s):  
T. S. Angell ◽  
David Colton ◽  
Rainer Kress
Keyword(s):  
Boundary Condition ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Impedance Boundary Condition ◽  
Far Field ◽  
Scattering Problems ◽  
Impedance Boundary ◽  
Field Patterns

AbstractWe first examine the class of far field patterns for the scalar Helmholtz equation in ℝ2 corresponding to incident time harmonic plane waves subject to an impedance boundary condition where the impedance is piecewise constant with respect to the incident direction and continuous with respect to x ε ∂ D where ∂ D is the scattering obstacle. We then examine the class of far field patterns for Maxwell's equations in subject to an impedance boundary condition with constant impedance. The results obtained are used to derive optimization algorithms for solving the inverse scattering problem.

Far field patterns and the inverse scattering problem for electromagnetic waves in an inhomogeneous medium

1988 ◽  
Vol 103 (3) ◽  
pp. 561-575 ◽  
Author(s):  
David Colton ◽  
Lassi Päivärinta
Keyword(s):  
Inhomogeneous Medium ◽  
Inverse Scattering ◽  
Integral Identity ◽  
Electromagnetic Waves ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Optimal Solution ◽  
Far Field ◽  
Field Patterns

AbstractWe consider the scattering of time harmonic electromagnetic waves by an inhomogeneous medium of compact support. It is first shown that the set of far field patterns of the electric fields corresponding to incident plane waves propagating in arbitrary directions is complete in the space of square-integrable tangential vector fields defined on the unit sphere. We then show that under certain conditions the electric far field patterns satisfy an integral identity involving the unique solution of a new class of boundary value problems for Maxwell's equations called the interior transmission problem for electromagnetic waves. Finally, it is indicated how this integral identity can be used to formulate an optimization scheme yielding an optimal solution of the inverse scattering problem for electromagnetic waves.

Inverse Scattering Related to Cylindrical Bodies Buried in a Lossy Circular Cylinder with Resistive Boundary

Frequenz ◽  
2019 ◽  
Vol 73 (3-4) ◽  
pp. 1-9 ◽  
Author(s):  
Tanju Yelkenci
Keyword(s):  
Circular Cylinder ◽  
Plane Wave ◽  
Cross Section ◽  
Inverse Scattering ◽  
Scattered Field ◽  
Born Approximation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Field Measurements ◽  
Arbitrary Cross Section

Abstract An inverse scattering problem of cylindrical bodies of arbitrary cross section buried in a circular cylinder with resistive boundary is presented. The reconstruction is obtained from the scattered field measurements for a plane wave illumination under the Born approximation. Illustrative examples are presented in order to see the applicability of the method as well as to see the effects of some parameters on the solution.

Inverse scattering for a layered acoustic medium using the first‐order equations of motion

Geophysics ◽  
1983 ◽  
Vol 48 (2) ◽  
pp. 163-170 ◽  
Author(s):  
M. S. Howard
Keyword(s):  
Inverse Scattering ◽  
Equations Of Motion ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Normal Incidence ◽  
Acoustic Medium ◽  
Transmission Losses ◽  
First Order ◽  
Marchenko Equation

The inverse scattering problem for a layered acoustic medium is considered from the first‐order differential equations of motion, resulting in a vector formulation of the problem, and using a vector form of the Schrödinger inverse scattering methods. The result is a vector Marchenko equation. The differentiability constraints on the acoustic impedance are somewhat relaxed compared to the more standard approach of beginning with the wave equation. The solution for plane waves at normal incidence is given along with a good approximate solution which is easily obtainable and takes into account transmission losses not included in the normal WKBJ‐Born approximation. A new solution for extracting separately the velocity and density of the medium using the reflection response for two different angles of incidence is given, which involves a nonlinear integral equation to relate the apparent traveltimes to depth.

Density and compressibility profiles of a layered acoustic medium from precritical incidence data

Geophysics ◽  
1981 ◽  
Vol 46 (9) ◽  
pp. 1244-1246 ◽  
Author(s):  
Shimon Coen
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Normal Incidence ◽  
Refractive Index Profile ◽  
Acoustic Medium ◽  
Index Profile ◽  
Incidence Data ◽  
Layered Fluid

The density and compressibility profiles of a layered fluid are obtained from the reflection coefficient due to plane waves at two precritical angles of incidence and all the frequencies. The inverse scattering problem for a layered fluid, at oblique incidence, is transformed to an equivalent inverse scattering problem for a layered refractive index profile, at normal incidence. The latter inverse scattering problem is transformed to an inverse scattering problem in quantum mechanics whose solution is obtained by the Gelfand‐Levitan theory.

AN ALGORITHM FOR THE FULLY NONLINEAR INVERSE SCATTERING PROBLEM AT FIXED FREQUENCY

2001 ◽  
Vol 09 (03) ◽  
pp. 935-940 ◽  
Author(s):  
F. NATTERER
Keyword(s):  
Helmholtz Equation ◽  
Multiple Scattering ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Scattering Process ◽  
Fixed Frequency ◽  
Fully Nonlinear ◽  
Complex Valued

We reconstruct an object which is described by a complex valued function from the scattered waves generated by irradiating plane waves at fixed frequency. The scattering process is modeled by the Helmholtz equation and includes multiple scattering. We present numerical results from computer generated data.

scholarly journals NDF and PSF Analysis in Inverse Source and Scattering Problems for Circumference Geometries

Electronics ◽  
2021 ◽  
Vol 10 (17) ◽  
pp. 2157
Author(s):  
Ehsan Akbari Sekehravani ◽  
Giovanni Leone ◽  
Rocco Pierri
Keyword(s):  
Closed Form ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Scalar Case ◽  
Far Field ◽  
Numerical Application ◽  
Approximation Accuracy ◽  
Form Evaluation

This paper aims at discussing the resolution achievable in the reconstruction of both circumference sources from their radiated far-field and circumference scatterers from their scattered far-field observed for the 2D scalar case. The investigation is based on an inverse problem approach, requiring the analysis of the spectral decomposition of the pertinent linear operator by the Singular Value Decomposition (SVD). The attention is focused upon the evaluation of the Number of Degrees of Freedom (NDF), connected to singular values behavior, and of the Point Spread Function (PSF), which accounts for the reconstruction of a point-like unknown and depends on both the NDF and on the singular functions. A closed-form evaluation of the PSF relevant to the inverse source problem is first provided. In addition, an approximated closed-form evaluation is introduced and compared with the exact one. This is important for the subsequent evaluation of the PSF relevant to the inverse scattering problem, which is based on a similar approximation. In this case, the approximation accuracy of the PSF is verified at least in its main lobe region by numerical simulation since it is the most critical one as far as the resolution discussion is concerned. The main result of the analysis is the space invariance of the PSF when the observation is the full angle in the far-zone region, showing that resolution remains unchanged over the entire source/investigation domain in the considered geometries. The paper also poses the problem of identifying the minimum number and the optimal directions of the impinging plane waves in the inverse scattering problem to achieve the full NDF; some numerical results about it are presented. Finally, a numerical application of the PSF concept is performed in inverse scattering, and its relevance in the presence of noisy data is outlined.

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