# Notions of support for far fields

@article{Sylvester2006NotionsOS,
title={Notions of support for far fields},
author={John Sylvester},
journal={Inverse Problems},
year={2006},
volume={22},
pages={1273-1288}
}
• J. Sylvester
• Published 20 June 2006
• Mathematics
• Inverse Problems
In practical remote sensing, faraway sources radiate fields that, within measurement precision, are nearly those radiated by point sources. Algorithms like MUSIC (Devaney J. Acoust. Soc. Am. at press, Kirsch 2002 Inverse Problems 18 1025–40) correctly identify their number, their locations and their strengths based on observations of the near or far fields they radiate. Asymptotic perturbation formulae (Ammari et al 2005 SIAM J. Appl. Math. 65 2107–27, Bruhl et al 2003 Numer. Math. 93 635–54…
30 Citations

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## References

SHOWING 1-10 OF 16 REFERENCES
The Convex Scattering Support in a Background Medium
• Mathematics, Computer Science
SIAM J. Math. Anal.
• 2005
The necessary scattering formalism is introduced and the circular Paley-Wiener theorem is restated as a Picard test, as a tool for inverse scattering in an inverse problem for the Helmholtz equation at fixed energy.
A 'range test' for determining scatterers with unknown physical properties
• Mathematics
• 2003
We describe a new scheme for determining the convex scattering support of an unknown scatterer when the physical properties of the scatterers are not known. The convex scattering support is a subset
The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media
We consider the scattering of time-harmonic plane waves by an inhomogeneous medium. The far field patterns u∞ of the scattered waves depend on the index of refraction 1 + q, the frequency, and
The Convex Back-Scattering Support
• Computer Science, Mathematics
SIAM J. Appl. Math.
• 2005
It is demonstrated that there is an inhomogeneity supported in any neighborhood of the convex back- scattering support which has exactly that back-scattering kernel.
A direct impedance tomography algorithm for locating small inhomogeneities
• Mathematics, Computer Science
Numerische Mathematik
• 2003
This paper considers the case where the goal is to find a number of small objects (inhomogeneities) inside an otherwise known conductor, and uses asymptotic analysis to design a direct reconstruction algorithm for the determination of their locations.
The scattering support
• Mathematics
• 2003
We discuss inverse problems for the Helmholtz equation at fixed energy, specifically the inverse source problem and the inverse scattering problem from a medium or an obstacle. We introduce the
Reconstruction of a Small Inclusion in a Two-Dimensional Open Waveguide
• Mathematics, Computer Science
SIAM J. Appl. Math.
• 2005
A MUSIC (multiple signal classification) type of algorithm for locating the inclusion and illustrating its viability in numerical examples is designed.
Notions of Convexity
The first two chapters of the book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the
Spectral properties of Schrödinger operators and scattering theory
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Inverse Acoustic and Electromagnetic Scattering Theory
• Physics
• 1992
Introduction.- The Helmholtz Equation.- Direct Acoustic Obstacle Scattering.- III-Posed Problems.- Inverse Acoustic Obstacle Scattering.- The Maxwell Equations.- Inverse Electromagnetic Obstacle