American Institute of Mathematical Sciences

Advanced
May  2015, 9(2): 591-600. doi: 10.3934/ipi.2015.9.591

Determining an obstacle by far-field data measured at a few spots

Qi Wang 1, and  Yanren Hou 2,

1. 

Department of Computing Sciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049
2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an,710049

Received  January 2014 Revised  May 2014 Published  March 2015

We consider the inverse scattering problem of determining an acoustic sound-soft obstacle by using the far-field data. It is shown that if the shape of the obstacle is known in advance, then the far-field data measured at four different spots can uniquely determine the location and size of the obstacle. If the shape of the obstacle is unknown, we show that the location of the obstacle can be approximately determined by using the far-field data measured at four appropriately chosen spots.
Keywords: Helmholtz equation, locating unknown scatterer., uniqueness, Inverse scattering.
Mathematics Subject Classification: Primary: 78A46; Secondary: 81U4.
Citation: Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems & Imaging, 2015, 9 (2) : 591-600. doi: 10.3934/ipi.2015.9.591
References:
[1]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9780898719406.  Google Scholar

[2]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $2^{nd}$ edition, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[3]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259. doi: 10.1093/imamat/31.3.253.  Google Scholar

[4]

G. Dassios and R. Kleinman, Low Frequency Scattering, Clarendon Press, Oxford, 2000.  Google Scholar

[5]

G. Hu, X. Liu and B. Zhang, Unique determination of a perfectly conducting ball by a finite number of electric far field data, J. Math. Anal. Appl., 352 (2009), 861-871. doi: 10.1016/j.jmaa.2008.09.016.  Google Scholar

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V. Isakov, Inverse Problems for Partial Differential Equations, Volume 127, Springer, 1998. doi: 10.1007/978-1-4899-0030-2.  Google Scholar

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A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.  Google Scholar

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J. Li, H. Y. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746. doi: 10.1137/130907690.  Google Scholar

[9]

J. Li, H. Y. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309. doi: 10.1137/130920356.  Google Scholar

[10]

H. Y. Liu and J. Zou, Zeros of Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering, IMA J. Appl. Math., 72 (2007), 817-831. doi: 10.1093/imamat/hxm013.  Google Scholar

[11]

J. Li, H. Y. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309. doi: 10.1137/130920356.  Google Scholar

[12]

J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Problems and Imaging, 7 (2013), 757-775. doi: 10.3934/ipi.2013.7.757.  Google Scholar

[13]

J. Li, H. Y. Liu and J. Zou, Multilevel linear sampling method for inverse scattering problems, SIAM Journal on Scientific Computing, 30 (2008), 1228-1250. doi: 10.1137/060674247.  Google Scholar

[14]

J. Li, H. Y. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM Journal on Scientific Computing, 31 (2009), 4013-4040. doi: 10.1137/080734170.  Google Scholar

[15]

J. Li, H. Y. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM Multiscale Modeling and Simulations, 12 (2014), 927-952. doi: 10.1137/13093409X.  Google Scholar

[16]

H. Y. Liu and J. Zou, On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems, Journal of Physics: Conference Series, 124 (2008), 012006. doi: 10.1088/1742-6596/124/1/012006.  Google Scholar

[17]

R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory, Chapman & Hall/CRC Research Notes in Mathematics, 427, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035483.  Google Scholar

show all references

References:
[1]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9780898719406.  Google Scholar

[2]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $2^{nd}$ edition, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[3]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259. doi: 10.1093/imamat/31.3.253.  Google Scholar

[4]

G. Dassios and R. Kleinman, Low Frequency Scattering, Clarendon Press, Oxford, 2000.  Google Scholar

[5]

G. Hu, X. Liu and B. Zhang, Unique determination of a perfectly conducting ball by a finite number of electric far field data, J. Math. Anal. Appl., 352 (2009), 861-871. doi: 10.1016/j.jmaa.2008.09.016.  Google Scholar

[6]

V. Isakov, Inverse Problems for Partial Differential Equations, Volume 127, Springer, 1998. doi: 10.1007/978-1-4899-0030-2.  Google Scholar

[7]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.  Google Scholar

[8]

J. Li, H. Y. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746. doi: 10.1137/130907690.  Google Scholar

[9]

J. Li, H. Y. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309. doi: 10.1137/130920356.  Google Scholar

[10]

H. Y. Liu and J. Zou, Zeros of Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering, IMA J. Appl. Math., 72 (2007), 817-831. doi: 10.1093/imamat/hxm013.  Google Scholar

[11]

J. Li, H. Y. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309. doi: 10.1137/130920356.  Google Scholar

[12]

J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Problems and Imaging, 7 (2013), 757-775. doi: 10.3934/ipi.2013.7.757.  Google Scholar

[13]

J. Li, H. Y. Liu and J. Zou, Multilevel linear sampling method for inverse scattering problems, SIAM Journal on Scientific Computing, 30 (2008), 1228-1250. doi: 10.1137/060674247.  Google Scholar

[14]

J. Li, H. Y. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM Journal on Scientific Computing, 31 (2009), 4013-4040. doi: 10.1137/080734170.  Google Scholar

[15]

J. Li, H. Y. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM Multiscale Modeling and Simulations, 12 (2014), 927-952. doi: 10.1137/13093409X.  Google Scholar

[16]

H. Y. Liu and J. Zou, On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems, Journal of Physics: Conference Series, 124 (2008), 012006. doi: 10.1088/1742-6596/124/1/012006.  Google Scholar

[17]

R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory, Chapman & Hall/CRC Research Notes in Mathematics, 427, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035483.  Google Scholar

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