August  2013, 7(3): 757-775. doi: 10.3934/ipi.2013.7.757

A direct sampling method for inverse scattering using far-field data

1. 

Faculty of Science, South University of Science and Technology of China, Shenzhen, 518055

2. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Received  June 2012 Revised  November 2012 Published  September 2013

This work is concerned with a direct sampling method (DSM) for inverse acoustic scattering problems using far-field data. Using one or few incident waves, the DSM provides quite reasonable profiles of scatterers in time-harmonic inverse acoustic scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. We shall first present a novel derivation of the DSM using far-field data, then carry out a systematic evaluation of the performances and distinctions of the DSM using both near-field and far-field data. A new interpretation from the physical perspective is provided based on some numerical observations. It is shown from a variety of numerical experiments that the method has several interesting and promising potentials: a) ability to identify not only medium scatterers, but also obstacles, and even cracks, using measurement data from one or few incident directions, b) robustness with respect to large noise, and c) computational efficiency with only inner products involved.
Citation: Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757
References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions,", Dover, (1965).  doi: 10.1119/1.1972842.  Google Scholar

[2]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement,, Proc. Am. Math., 133 (2005), 1685.  doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar

[3]

H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements,", Lecture Notes in Mathematics, (1846).  doi: 10.1007/b98245.  Google Scholar

[4]

G. Bao and P. Li, Inverse medium scattering for the Helmholtz equation at fixed frequency,, Inverse Problems, 21 (2005), 1621.  doi: 10.1088/0266-5611/21/5/007.  Google Scholar

[5]

L. Beilina and M. V. Klibanov, "Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems,", Springer, (2012).  doi: 10.1007/978-1-4419-7805-9.  Google Scholar

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J. Buchanan, R. Gilbert, A. Wirgin and Y. Xu, "Marine Acoustics: Direct and Inverse Scattering of Waves,", Society for Industrial and Applied Mathematics, (2004).  doi: 10.1137/1.9780898717983.  Google Scholar

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F. Cakoni, D. Colton and P. Monk, "The Linear Sampling Method in Inverse Electromagnetic Scattering,", CBMS-NSF Regional Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9780898719406.  Google Scholar

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X. Chen and Y. Zhong, Music electromagnetic imaging with enhanced resolution for small inclusions,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/1/015008.  Google Scholar

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M. Cheney, The linear sampling method and the MUSIC algorithm,, Inverse Problems, 17 (2001), 591.  doi: 10.1088/0266-5611/17/4/301.  Google Scholar

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D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

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D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, (1983).   Google Scholar

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D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Second edition. Applied Mathematical Sciences, (1998).   Google Scholar

[13]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering,, IMA Journal of Applied Mathematics, 31 (1983), 253.  doi: 10.1093/imamat/31.3.253.  Google Scholar

[14]

A. Devaney, Super-resolution processing of multi-static data using time-reversal and music,, J. Acoust. Soc. Am., ().   Google Scholar

[15]

J. Elschner and M. Yamamoto, Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/3/035004.  Google Scholar

[16]

R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/8/085005.  Google Scholar

[17]

F. K. Gruber, E. A. Marengo and A. J. Devaney, Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,, J. Acoust. Soc. Am., 115 (2004), 3042.  doi: 10.1121/1.1738451.  Google Scholar

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M. Hanke, One shot inverse scattering via rational approximation,, SIAM J. Imaging Sciences, 5 (2012), 465.  doi: 10.1137/110823985.  Google Scholar

[19]

T. Hohage, On the numerical solution of a three-dimensional inverse medium scattering problem,, Inverse Problems, 17 (2001), 1743.  doi: 10.1088/0266-5611/17/6/314.  Google Scholar

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S. Hou, K. Solna and H. Zhao, A direct imaging algorithm for extended targets,, Inverse Problems, 22 (2006), 1151.  doi: 10.1088/0266-5611/22/4/003.  Google Scholar

[21]

K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/2/025003.  Google Scholar

[22]

K. Ito, B. Jin and J. Zou, A two-stage method for inverse medium scattering,, J. Comput. Phys., 237 (2013), 211.  doi: 10.1016/j.jcp.2012.12.004.  Google Scholar

[23]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489.  doi: 10.1088/0266-5611/14/6/009.  Google Scholar

[24]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, (2008).   Google Scholar

[25]

R. Kohn, D. Onofrei, M. Vogelius and M. Weinstein, Cloaking via change of variables for the helmholtz equation,, Comm. Pure Appl. Math., 63 (2010), 973.  doi: 10.1002/cpa.20326.  Google Scholar

[26]

J. Li, H. Liu and H. Sun, Enhanced approximate cloaking by SH and FSH lining,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075011.  Google Scholar

[27]

H. Liu and H. Sun, Enhanced near-cloak by FSH lining,, J. Math. Pures Appl. 99 (2013), 99 (2013), 17.  doi: 10.1016/j.matpur.2012.06.001.  Google Scholar

[28]

H. Liu, M. Yamamoto and J. Zou, Reflection principle for the maxwell equations and its application to inverse electromagnetic scattering,, Inverse Problems, 23 (2007), 2357.  doi: 10.1088/0266-5611/23/6/005.  Google Scholar

[29]

H. Liu, H. Zhang and J. Zou, Recovery of polyhedral scatterers by a single electromagnetic far-field measurement,, J. Math. Phy., 50 (2009).  doi: 10.1063/1.3263140.  Google Scholar

[30]

H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers,, Inverse Problems, 22 (2006), 515.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[31]

C. Müller, "Analysis of Spherial Symmetries in Eulidean Spaes,", Springer, (1997).   Google Scholar

[32]

R. Potthast, A survey on sampling and probe methods for inverse problems,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/2/R01.  Google Scholar

[33]

R. Potthast, A study on orthogonality sampling,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/7/074015.  Google Scholar

[34]

R. Schmidt, Multiple emitter location and signal parameter estimation,, IEEE Trans. Antennas Propag., 34 (1986), 276.  doi: 10.1109/TAP.1986.1143830.  Google Scholar

[35]

C. G. Someda, "Electromagnetic Waves,", Boca Raton, (2006).   Google Scholar

[36]

P. M. van den Berg, A. L. van Broekhoven and A. Abubakar, Extended contrast source inversion,, Inverse Problems, 15 (1999), 1325.  doi: 10.1088/0266-5611/15/5/315.  Google Scholar

[37]

G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1944).   Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions,", Dover, (1965).  doi: 10.1119/1.1972842.  Google Scholar

[2]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement,, Proc. Am. Math., 133 (2005), 1685.  doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar

[3]

H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements,", Lecture Notes in Mathematics, (1846).  doi: 10.1007/b98245.  Google Scholar

[4]

G. Bao and P. Li, Inverse medium scattering for the Helmholtz equation at fixed frequency,, Inverse Problems, 21 (2005), 1621.  doi: 10.1088/0266-5611/21/5/007.  Google Scholar

[5]

L. Beilina and M. V. Klibanov, "Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems,", Springer, (2012).  doi: 10.1007/978-1-4419-7805-9.  Google Scholar

[6]

J. Buchanan, R. Gilbert, A. Wirgin and Y. Xu, "Marine Acoustics: Direct and Inverse Scattering of Waves,", Society for Industrial and Applied Mathematics, (2004).  doi: 10.1137/1.9780898717983.  Google Scholar

[7]

F. Cakoni, D. Colton and P. Monk, "The Linear Sampling Method in Inverse Electromagnetic Scattering,", CBMS-NSF Regional Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9780898719406.  Google Scholar

[8]

X. Chen and Y. Zhong, Music electromagnetic imaging with enhanced resolution for small inclusions,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/1/015008.  Google Scholar

[9]

M. Cheney, The linear sampling method and the MUSIC algorithm,, Inverse Problems, 17 (2001), 591.  doi: 10.1088/0266-5611/17/4/301.  Google Scholar

[10]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[11]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, (1983).   Google Scholar

[12]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Second edition. Applied Mathematical Sciences, (1998).   Google Scholar

[13]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering,, IMA Journal of Applied Mathematics, 31 (1983), 253.  doi: 10.1093/imamat/31.3.253.  Google Scholar

[14]

A. Devaney, Super-resolution processing of multi-static data using time-reversal and music,, J. Acoust. Soc. Am., ().   Google Scholar

[15]

J. Elschner and M. Yamamoto, Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/3/035004.  Google Scholar

[16]

R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/8/085005.  Google Scholar

[17]

F. K. Gruber, E. A. Marengo and A. J. Devaney, Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,, J. Acoust. Soc. Am., 115 (2004), 3042.  doi: 10.1121/1.1738451.  Google Scholar

[18]

M. Hanke, One shot inverse scattering via rational approximation,, SIAM J. Imaging Sciences, 5 (2012), 465.  doi: 10.1137/110823985.  Google Scholar

[19]

T. Hohage, On the numerical solution of a three-dimensional inverse medium scattering problem,, Inverse Problems, 17 (2001), 1743.  doi: 10.1088/0266-5611/17/6/314.  Google Scholar

[20]

S. Hou, K. Solna and H. Zhao, A direct imaging algorithm for extended targets,, Inverse Problems, 22 (2006), 1151.  doi: 10.1088/0266-5611/22/4/003.  Google Scholar

[21]

K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/2/025003.  Google Scholar

[22]

K. Ito, B. Jin and J. Zou, A two-stage method for inverse medium scattering,, J. Comput. Phys., 237 (2013), 211.  doi: 10.1016/j.jcp.2012.12.004.  Google Scholar

[23]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489.  doi: 10.1088/0266-5611/14/6/009.  Google Scholar

[24]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, (2008).   Google Scholar

[25]

R. Kohn, D. Onofrei, M. Vogelius and M. Weinstein, Cloaking via change of variables for the helmholtz equation,, Comm. Pure Appl. Math., 63 (2010), 973.  doi: 10.1002/cpa.20326.  Google Scholar

[26]

J. Li, H. Liu and H. Sun, Enhanced approximate cloaking by SH and FSH lining,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075011.  Google Scholar

[27]

H. Liu and H. Sun, Enhanced near-cloak by FSH lining,, J. Math. Pures Appl. 99 (2013), 99 (2013), 17.  doi: 10.1016/j.matpur.2012.06.001.  Google Scholar

[28]

H. Liu, M. Yamamoto and J. Zou, Reflection principle for the maxwell equations and its application to inverse electromagnetic scattering,, Inverse Problems, 23 (2007), 2357.  doi: 10.1088/0266-5611/23/6/005.  Google Scholar

[29]

H. Liu, H. Zhang and J. Zou, Recovery of polyhedral scatterers by a single electromagnetic far-field measurement,, J. Math. Phy., 50 (2009).  doi: 10.1063/1.3263140.  Google Scholar

[30]

H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers,, Inverse Problems, 22 (2006), 515.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[31]

C. Müller, "Analysis of Spherial Symmetries in Eulidean Spaes,", Springer, (1997).   Google Scholar

[32]

R. Potthast, A survey on sampling and probe methods for inverse problems,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/2/R01.  Google Scholar

[33]

R. Potthast, A study on orthogonality sampling,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/7/074015.  Google Scholar

[34]

R. Schmidt, Multiple emitter location and signal parameter estimation,, IEEE Trans. Antennas Propag., 34 (1986), 276.  doi: 10.1109/TAP.1986.1143830.  Google Scholar

[35]

C. G. Someda, "Electromagnetic Waves,", Boca Raton, (2006).   Google Scholar

[36]

P. M. van den Berg, A. L. van Broekhoven and A. Abubakar, Extended contrast source inversion,, Inverse Problems, 15 (1999), 1325.  doi: 10.1088/0266-5611/15/5/315.  Google Scholar

[37]

G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1944).   Google Scholar

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