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.

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.

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.

scholarly journals An injective far-field pattern operator and inverse scattering problem in a finite depth ocean

1991 ◽  
Vol 34 (2) ◽  
pp. 295-311 ◽  
Author(s):  
Yongzhi Xu
Keyword(s):  
Inverse Scattering ◽  
Acoustic Waves ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Finite Depth ◽  
Field Pattern ◽  
Far Field ◽  
Optimal Scheme ◽  
Loss Of Information ◽  
Far Field Pattern

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R2 and R3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L2(∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

scholarly journals Fast Localization of Small Inhomogeneities from Far-Field Pattern Data in the Limited-Aperture Inverse Scattering Problem

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2087
Author(s):  
Won-Kwang Park
Keyword(s):  
Inverse Scattering ◽  
Random Noise ◽  
Scattering Problem ◽  
Synthetic Data ◽  
Inverse Scattering Problem ◽  
Mathematical Structure ◽  
Indicator Function ◽  
Field Pattern ◽  
Far Field ◽  
Far Field Pattern

In this study, we consider a sampling-type algorithm for the fast localization of small electromagnetic inhomogeneities from measured far-field pattern data in the limited-aperture inverse scattering problem. For this purpose, we designed an indicator function based on the structure of left- and right-singular vectors of a multistatic response matrix, the elements of which were measured far-field pattern data. We then rigorously investigated the mathematical structure of the indicator function in terms of purely dielectric permittivity and magnetic permeability contrast cases by establishing a relationship with an infinite series of Bessel functions of an integer order of the first kind and a range of incident and observation directions before exploring various intrinsic properties of the algorithm, including its feasibility and limitations. Simulation results with synthetic data corrupted by random noise are presented to support the theoretical results.

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.

Dense sets and far field patterns for acoustic waves in an inhomogeneous medium

1988 ◽  
Vol 31 (3) ◽  
pp. 401-407 ◽  
Author(s):  
David Colton
Keyword(s):  
Inhomogeneous Medium ◽  
Inverse Scattering ◽  
Acoustic Waves ◽  
Scattering Problem ◽  
Plane Waves ◽  
Related Result ◽  
Wave Number ◽  
Far Field ◽  
Field Patterns ◽  
Time Harmonic

In this paper, we shall obtain two results on the class of far field patterns corresponding to the scattering of time harmonic acoustic plane waves by an inhomogeneous medium of compact support. Although the problem of characterizing the class of far field patterns is of basic importance in inverse scattering theory, very little is known about this class other than the fact that the far field patterns are entire functions of their independent (complex) variables for each positive fixed value of the wave number. In particular, the class of far field patterns is not all of L2(∂Ω) where ∂Ω is the unit sphere and this implies that the inverse scattering problem is improperly posed since the far field patterns are, in practice, determined from inexact measurements. The purpose of this paper is to show that while the class of far field patterns corresponding to the scattering of time harmonic plane waves by an inhomogeneous medium is not all of L2(∂Ω), it is dense in L2(∂Ω) for sufficiently small values of the wave number. In addition, a related result will be obtained for a special translation of the class of far field patterns. Analogous results for the scattering of time harmonic acoustic waves by a homogeneous scattering obstacle have recently been obtained by Colton [1], Colton and Kirsch [2], Colton and Monk [3, 4] and Kirsch [8].

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.

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