October 2017, 11(5): 901-916. doi: 10.3934/ipi.2017042

A direct imaging method for the half-space inverse scattering problem with phaseless data

1. 

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, School of Mathematical Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China

2. 

School of Mathematical Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China

3. 

The Rice Inversion Project, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892, USA

* Corresponding author: Guanghui Huang

Received  December 2016 Published  July 2017

Fund Project: The first author was supported in part by the China NSF under the grant 113211061

We propose a direct imaging method based on the reverse time migration method for finding extended obstacles with phaseless total field data in the half space. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the obstacle is far away from the surface of the half-space where the measurement is taken. Numerical experiments are included to illustrate the powerful imaging quality.

Citation: Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
References:
[1]

G. BaoP. Li and J. Lv, Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30 (2013), 293-299. doi: 10.1364/JOSAA.30.000293.

[2]

P. Bardsley and F. Vasquez, Kirchhoff migration without phases Inverse Problems, 32 (2016), 105006, 17 pp. doi: 10.1088/0266-5611/32/10/105006.

[3]

N. Bleistein, J. Cohen and J. Stockwell, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion Springer, New York, 2001. doi: 10.1007/978-1-4613-0001-4.

[4]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory Applied Mathematical Sciences 188, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.

[5]

A. Chai, M. Moscoso and G. Papanicolaou, Array imaging using intensity-only measurements Inverse Problems 27 (2010), 015005, 16pp. doi: 10.1088/0266-5611/27/1/015005.

[6]

S. N. Chandler-WildeI. G. GrahamS. Langdon and M. Lindner, Condition number estimates for combined potential boundary integral operators in acoustic scattering, J. Integral Equa. Appli., 21 (2009), 229-279. doi: 10.1216/JIE-2009-21-2-229.

[7]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Acoustic waves Inverse Problems 29 (2013), 085005, 17pp. doi: 10.1088/0266-5611/29/8/085005.

[8]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Electromagnetic waves Inverse Problems 29 (2013), 085006, 17pp. doi: 10.1088/0266-5611/29/8/085006.

[9]

Z. Chen and G. Huang, Reverse time migration for extended obstacles: Elastic waves (in Chinese), Sci. Sin. Math., 58 (2015), 1811-1834. doi: 10.1007/s11425-015-5037-x.

[10]

Z. Chen and G. Huang, Reverse time migration for reconstructing extended obstacles in the half space Inverse Problems 31 (2015), 055007, 19pp. doi: 10.1088/0266-5611/31/5/055007.

[11]

Z. Chen and G. Huang, Phaseless imaging by reverse time migration: Acoustic waves, Numer. Math. Theor. Meth. Appl., 10 (2017), 1-21. doi: 10.4208/nmtma.2017.m1617.

[12]

Z. Chen and G. Huang, A direct imaging method for electromagnetic scattering data without phase information, SIAM J. Imag. Sci., 9 (2016), 1273-1297. doi: 10.1137/15M1053475.

[13]

A. J. Devaney, Structure determination from intensity measurements in scattering experiments, Physical Review Letters, 62 (1989), 2385-2388. doi: 10.1103/PhysRevLett.62.2385.

[14]

M. $\acute{\text{D}}$ursoK. BelkebirL. CroccoT. Isernia and A. Litman, Phaseless imaging with experimental data: Facts and challenges, J. Opt. Soc. Am. A, 25 (2008), 271-281.

[15]

G. FranceschiniM. DonelliR. Azaro and A. Massa, Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach, IEEE Trans. Geosci. Remote Sens., 44 (2006), 3527-3539. doi: 10.1109/TGRS.2006.881753.

[16]

L. Grafakos, Classical and Modern Fourier Analysis Pearson, London, 2004.

[17]

O. Ivanyshyn and R. Kress, Identification of sound-soft 3D obstacles from phaseless data, Inverse Problem and Imaging, 4 (2010), 131-149. doi: 10.3934/ipi.2010.4.131.

[18]

O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data, Journal of Computational Physics, 230 (2011), 3443-3452. doi: 10.1016/j.jcp.2011.01.038.

[19]

R. Kress and W. Rundell, Inverse obstacle scattering with modulus of the far field pattern as data, H. W. Engl et al. (eds. ), Inverse Problems in Medical Imaging and Nondestructive Testing, Springer, Vienna, (1997), 75–92.

[20]

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.

[21]

J LiH. LiuZ. 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.

[22]

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

[23]

J LiH. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Modeling and Simulation, 12 (2014), 927-952. doi: 10.1137/13093409X.

[24]

A. NovikovM. Moscoso and G. Papanicolaou, Illumination strategies for intensity-only imaging, SIAM J. Imag. Sci., 8 (2015), 1547-1573. doi: 10.1137/140994617.

[25]

G. N. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, 1995.

[26]

Y. Zhang and J. Sun, Practicle issues in reverse time migration: True amplitude gathers, noise removal and harmonic source encoding, First Break, 26 (2009), 29-35.

[27]

Y. ZhangS. XuN. Bleistein and G. Zhang, True-amplitude, angle-domain, common-image gathers from one-way wave-equation migration, Geophysics, 72 (2007), S49-S58. doi: 10.1190/1.2399371.

show all references

References:
[1]

G. BaoP. Li and J. Lv, Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30 (2013), 293-299. doi: 10.1364/JOSAA.30.000293.

[2]

P. Bardsley and F. Vasquez, Kirchhoff migration without phases Inverse Problems, 32 (2016), 105006, 17 pp. doi: 10.1088/0266-5611/32/10/105006.

[3]

N. Bleistein, J. Cohen and J. Stockwell, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion Springer, New York, 2001. doi: 10.1007/978-1-4613-0001-4.

[4]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory Applied Mathematical Sciences 188, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.

[5]

A. Chai, M. Moscoso and G. Papanicolaou, Array imaging using intensity-only measurements Inverse Problems 27 (2010), 015005, 16pp. doi: 10.1088/0266-5611/27/1/015005.

[6]

S. N. Chandler-WildeI. G. GrahamS. Langdon and M. Lindner, Condition number estimates for combined potential boundary integral operators in acoustic scattering, J. Integral Equa. Appli., 21 (2009), 229-279. doi: 10.1216/JIE-2009-21-2-229.

[7]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Acoustic waves Inverse Problems 29 (2013), 085005, 17pp. doi: 10.1088/0266-5611/29/8/085005.

[8]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Electromagnetic waves Inverse Problems 29 (2013), 085006, 17pp. doi: 10.1088/0266-5611/29/8/085006.

[9]

Z. Chen and G. Huang, Reverse time migration for extended obstacles: Elastic waves (in Chinese), Sci. Sin. Math., 58 (2015), 1811-1834. doi: 10.1007/s11425-015-5037-x.

[10]

Z. Chen and G. Huang, Reverse time migration for reconstructing extended obstacles in the half space Inverse Problems 31 (2015), 055007, 19pp. doi: 10.1088/0266-5611/31/5/055007.

[11]

Z. Chen and G. Huang, Phaseless imaging by reverse time migration: Acoustic waves, Numer. Math. Theor. Meth. Appl., 10 (2017), 1-21. doi: 10.4208/nmtma.2017.m1617.

[12]

Z. Chen and G. Huang, A direct imaging method for electromagnetic scattering data without phase information, SIAM J. Imag. Sci., 9 (2016), 1273-1297. doi: 10.1137/15M1053475.

[13]

A. J. Devaney, Structure determination from intensity measurements in scattering experiments, Physical Review Letters, 62 (1989), 2385-2388. doi: 10.1103/PhysRevLett.62.2385.

[14]

M. $\acute{\text{D}}$ursoK. BelkebirL. CroccoT. Isernia and A. Litman, Phaseless imaging with experimental data: Facts and challenges, J. Opt. Soc. Am. A, 25 (2008), 271-281.

[15]

G. FranceschiniM. DonelliR. Azaro and A. Massa, Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach, IEEE Trans. Geosci. Remote Sens., 44 (2006), 3527-3539. doi: 10.1109/TGRS.2006.881753.

[16]

L. Grafakos, Classical and Modern Fourier Analysis Pearson, London, 2004.

[17]

O. Ivanyshyn and R. Kress, Identification of sound-soft 3D obstacles from phaseless data, Inverse Problem and Imaging, 4 (2010), 131-149. doi: 10.3934/ipi.2010.4.131.

[18]

O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data, Journal of Computational Physics, 230 (2011), 3443-3452. doi: 10.1016/j.jcp.2011.01.038.

[19]

R. Kress and W. Rundell, Inverse obstacle scattering with modulus of the far field pattern as data, H. W. Engl et al. (eds. ), Inverse Problems in Medical Imaging and Nondestructive Testing, Springer, Vienna, (1997), 75–92.

[20]

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.

[21]

J LiH. LiuZ. 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.

[22]

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

[23]

J LiH. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Modeling and Simulation, 12 (2014), 927-952. doi: 10.1137/13093409X.

[24]

A. NovikovM. Moscoso and G. Papanicolaou, Illumination strategies for intensity-only imaging, SIAM J. Imag. Sci., 8 (2015), 1547-1573. doi: 10.1137/140994617.

[25]

G. N. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, 1995.

[26]

Y. Zhang and J. Sun, Practicle issues in reverse time migration: True amplitude gathers, noise removal and harmonic source encoding, First Break, 26 (2009), 29-35.

[27]

Y. ZhangS. XuN. Bleistein and G. Zhang, True-amplitude, angle-domain, common-image gathers from one-way wave-equation migration, Geophysics, 72 (2007), S49-S58. doi: 10.1190/1.2399371.

Figure 1.  Examples 6.1: Imaging results of penetrable obstacles with different shapes from left to right. The top results are imaged with phaseless data by our RTM algorithm, and the bottom one are imaing results with full phase data. The probe wave number is $k=4\pi$, and $N_s=512$, $N_r=512$
Figure 2.  Example 6.2: Imaging results of an elliptic obstacle with boundary conditions as sound soft, sound hard and impedance boundary with $\lambda=1$, respectively. The probe wave number $k=4\pi$, and $N_s=512$, $N_r=512$
Figure 3.  Examples 6.3 (first test): Imaging results of a penetrable obstacle with noise levels $\mu=0.1, 0.2, 0.3, 0.4$ (from left to right). The top row is imaged with single frequency data, and the bottom row is imaged with multi-frequency data
Figure 4.  Example 6.3 (second test): Imaging results of two sound soft obstacles. All the parameters are the same as that in Figure 3
Figure 5.  Example 6.4: Imaging results of two sound soft obstacles overlaid with the true obstacle model. The left column is imaged with the single frequency data, and the right one is imaged with the multi-frequency data
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