The point source method for inverse scattering in the time domain

Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time‐harmonic scattering, or nearly time‐harmo...

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Published inMathematical methods in the applied sciences Vol. 29; no. 13; pp. 1501 - 1521
Main Authors Russell Luke, D., Potthast, Roland
Format Journal Article
LanguageEnglish
Published Chichester, UK John Wiley & Sons, Ltd 10.09.2006
Wiley
Teubner
Subjects
Applied sciences
Artificial intelligence
Classical and quantum physics: mechanics and fields
Computer science; control theory; systems
Exact sciences and technology
Fourier analysis
General theory of scattering
image reconstruction
inverse problems
Mathematical analysis
Mathematics
Pattern recognition. Digital image processing. Computational geometry
Physics
scattering theory
Sciences and techniques of general use
time-domain scattering
Fourier transformation
Applied mathematics
Inverse scattering method
inverse problems
Numerical method
Time-domain scattering
Algorithm
Image reconstruction
Inverse problem
Point source
Scattering theory
Online AccessGet full text
ISSN0170-4214
1099-1476
1099-1476
DOI10.1002/mma.738

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Abstract Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time‐harmonic scattering, or nearly time‐harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non‐time‐harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time‐dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi‐frequency data. Copyright © 2006 John Wiley & Sons, Ltd.
AbstractList Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time‐harmonic scattering, or nearly time‐harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non‐time‐harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time‐dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi‐frequency data. Copyright © 2006 John Wiley & Sons, Ltd.
Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time-harmonic scattering, or nearly time-harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non-time-harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time-dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi-frequency data.
Author Potthast, Roland
Russell Luke, D.
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Issue 13
Keywords Fourier transformation
Applied mathematics
Inverse scattering method
inverse problems
Numerical method
Time-domain scattering
Algorithm
Image reconstruction
Inverse problem
Point source
Scattering theory
Language English
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References Potthast R. Point Sources and Multipoles in Inverse Scattering Theory. Chapman & Hall: London, 2001.
Ikehata M. Reconstruction of an obstacle from the scattering amplitude at a fixed frequency. Inverse Problems 1998; 14:949-954.
Luke DR. Multifrequency inverse obstacle scattering: the point source method and generalized filtered backprojection. Mathematics and Computers in Simulation 2004; 66(4-5):297-314.
Gilbarg G, Trudinger NS. Elliptic Partial Differential Equations of Second Order. Springer: Berlin, 1998.
Colton D, Monk P. A novel method for solving the inverse scattering problem for time-harmonic waves in the resonance region II. SIAM Journal on Applied Mathematics 1986; 46:506-523.
Ramm AG. Scattering by Obstacles. D. Reidel Publishing: Dordrecht, Holland, 1986.
Colton D, Kirsch A. A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 1996; 12(4):383-393.
Kirsch A. Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Problems 1998; 14:1489-1512.
Morawetz CS. The limiting amplitude principle. Communications on Pure and Applied Mathematics 1962; 15:349-361.
Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory (2nd edn). Springer: New York, 1998.
Koerner TWK. Fourier Analysis. Cambridge University Press: Cambridge, 1988.
Potthast R. A point-source method for inverse acoustic and electromagnetic obstacle scattering problems. IMA Journal of Applied Mathematics 1998; 61:119-140.
Lax PD, Phillips RS. Scattering Theory (revised edn). Academic Press: Toronto, 1989.
Luke DR, Potthast R. The no response test-a sampling method for inverse scattering problems. SIAM Journal on Applied Mathematics 2003; 63(4):1292-1312.
Pawletta S, Westphal A, Drewelow W, Duenow P, Pawletta T, Fink R. Distributed and Parallel Application Toolbox (DP Toolbox). University of Rostock, http://www-at.e-technik.uni-rostock.de/rg_ac/dp/, 2003.
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Leis R. Initial Boundary Value Problems in Mathematical Physics. B.G. Teubner: Stuttgart, 1986.
Potthast R, Sylvester J, Kusiak S. A 'range test' for determining scatterers with unknown physical properties. Inverse Problems 2003; 19(3):533-547.
Colton D, Monk P. A novel method for solving the inverse scattering problem for time-harmonic waves in the resonance region. SIAM Journal on Applied Mathematics 1985; 45:1039-1053.
Potthast R. A fast new method to solve inverse scattering problems. Inverse Problems 1996; 12:731-742.
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– reference: Morawetz CS. The limiting amplitude principle. Communications on Pure and Applied Mathematics 1962; 15:349-361.
– reference: Colton D, Kirsch A. A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 1996; 12(4):383-393.
– reference: Luke DR, Potthast R. The no response test-a sampling method for inverse scattering problems. SIAM Journal on Applied Mathematics 2003; 63(4):1292-1312.
– reference: Luke DR. Multifrequency inverse obstacle scattering: the point source method and generalized filtered backprojection. Mathematics and Computers in Simulation 2004; 66(4-5):297-314.
– reference: Rudin W. Real and Complex Analysis (2nd edn). McGraw-Hill: New York, 1974.
– reference: Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory (2nd edn). Springer: New York, 1998.
– reference: Colton D, Monk P. A novel method for solving the inverse scattering problem for time-harmonic waves in the resonance region II. SIAM Journal on Applied Mathematics 1986; 46:506-523.
– reference: Lax PD, Phillips RS. Scattering Theory (revised edn). Academic Press: Toronto, 1989.
– reference: Potthast R, Sylvester J, Kusiak S. A 'range test' for determining scatterers with unknown physical properties. Inverse Problems 2003; 19(3):533-547.
– reference: Leis R. Initial Boundary Value Problems in Mathematical Physics. B.G. Teubner: Stuttgart, 1986.
– reference: Kirsch A. Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Problems 1998; 14:1489-1512.
– reference: Pawletta S, Westphal A, Drewelow W, Duenow P, Pawletta T, Fink R. Distributed and Parallel Application Toolbox (DP Toolbox). University of Rostock, http://www-at.e-technik.uni-rostock.de/rg_ac/dp/, 2003.
– reference: Ramm AG. Scattering by Obstacles. D. Reidel Publishing: Dordrecht, Holland, 1986.
– reference: Koerner TWK. Fourier Analysis. Cambridge University Press: Cambridge, 1988.
– reference: Colton D, Monk P. A novel method for solving the inverse scattering problem for time-harmonic waves in the resonance region. SIAM Journal on Applied Mathematics 1985; 45:1039-1053.
– reference: Potthast R. Point Sources and Multipoles in Inverse Scattering Theory. Chapman & Hall: London, 2001.
– reference: Ikehata M. Reconstruction of an obstacle from the scattering amplitude at a fixed frequency. Inverse Problems 1998; 14:949-954.
– reference: Potthast R. A point-source method for inverse acoustic and electromagnetic obstacle scattering problems. IMA Journal of Applied Mathematics 1998; 61:119-140.
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  publication-title: SIAM Journal on Applied Mathematics
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  publication-title: Inverse Problems
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Snippet Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is...
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SubjectTerms Applied sciences
Artificial intelligence
Classical and quantum physics: mechanics and fields
Computer science; control theory; systems
Exact sciences and technology
Fourier analysis
General theory of scattering
image reconstruction
inverse problems
Mathematical analysis
Mathematics
Pattern recognition. Digital image processing. Computational geometry
Physics
scattering theory
Sciences and techniques of general use
time-domain scattering
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Title The point source method for inverse scattering in the time domain
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