# American Institute of Mathematical Sciences

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

 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.
Citation: Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems and 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. [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. [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. [4] G. Dassios and R. Kleinman, Low Frequency Scattering, Clarendon Press, Oxford, 2000. [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. [6] V. Isakov, Inverse Problems for Partial Differential Equations, Volume 127, Springer, 1998. doi: 10.1007/978-1-4899-0030-2. [7] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [4] G. Dassios and R. Kleinman, Low Frequency Scattering, Clarendon Press, Oxford, 2000. [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. [6] V. Isakov, Inverse Problems for Partial Differential Equations, Volume 127, Springer, 1998. doi: 10.1007/978-1-4899-0030-2. [7] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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