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Phaseless Imaging by Reverse Time Migration: Acoustic Waves

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  20 February 2017

Zhiming Chen  and
Guanghui Huang
Zhiming Chen*
Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Guanghui Huang*
Affiliation:
The Rice Inversion Project, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892, USA
*
*Corresponding author. Email addresses:zmchen@lsec.cc.ac.cn (Z.-M. Chen), ghhuang@rice.edu (G.- H. Huang)
*Corresponding author. Email addresses:zmchen@lsec.cc.ac.cn (Z.-M. Chen), ghhuang@rice.edu (G.- H. Huang)
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Abstract

We propose a reliable direct imaging method based on the reverse time migration for finding extended obstacles with phaseless total field data. 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 measurement is far away from the obstacle. The imaginary part of the cross-correlation imaging functional always peaks on the boundary of the obstacle. Numerical experiments are included to illustrate the powerful imaging quality

Keywords

Reverse time migrationphaseless datainverse scatteringextended obstacle

MSC classification

Secondary: 78A46: Inverse scattering problems
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 1 , February 2017 , pp. 1 - 21
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Ammari, H., Chow, Y. T., and Zou, J., Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients, arXiv: 1510.03999.Google Scholar
[2] Bao, G., Li, P., and Lv, J., Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30 (2013), pp. 293–299.Google Scholar
[3] Bleistein, N., Cohen, J., and Stockwell, J., Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, Springer, 2001.CrossRefGoogle Scholar
[4] Chen, J., Chen, Z., and Huang, G., Reverse time migration for extended obstacles: acoustic waves, Inverse Problems, 29 (2013), 085005 (17pp).CrossRefGoogle Scholar
[5] Chen, J., Chen, Z., and Huang, G., Reverse time migration for extended obstacles: electromagnetic waves, Inverse Problems, 29 (2013), 085006 (17pp).Google Scholar
[6] Chen, Z. and Huang, G., Reverse time migration for extended obstacles: elastic waves, Science in China Series A: Mathematics (in Chinese), 45 (2015), pp. 1103–1114.Google Scholar
[7] Chen, Z. and Huang, G., Reverse time migration for reconstructing extended obstacles in the half space, Inverse Problems, 31 (2015), 055007 (19pp).CrossRefGoogle Scholar
[8] Chandler-Wilde, S. N., Graham, I. G., Langdon, S., and Lindner, M., Condition number estimates for combined potential boundary integral operators in acoustic scattering, J. Integral Equa. Appli., 21 (2009), pp. 229–279.Google Scholar
[9] Cakoni, F., Colton, D., and Monk, P., The direct and inverse scattering problems for partially coated obstacles, Inverse Problems, 17 (2001), pp. 1997–2015.Google Scholar
[10] Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Problems, Springer, 1998.Google Scholar
[11] Devaney, A. J., Structure determination from intensity measurements in scattering experiments, Physical Review Letters, 62 (1989), pp. 2385–2388.Google Scholar
[12] D’Urso, M., Belkebir, K., Crocco, L., Isernia, T., and Litman, A., Phaseless imaging with experimental data: facts and challenges, J. Opt. Soc. Am. A, 25 (2008), pp. 271–281.Google Scholar
[13] Franceschini, G., Donelli, M., Azaro, R., A., and 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), pp. 3527–3539.Google Scholar
[14] Grafakos, L., Classical and Modern Fourier Analysis, Pearson, 2004.Google Scholar
[15] Hörmander, L., The Analysis of Linear Partial Differential Operators, I, Springer, 1983.Google Scholar
[16] Ivanyshyn, O. and Kress, R., Identification of sound-soft 3D obstacles from phasless data, Inverse Problem and Imaing, 4 (2010), pp. 131–149.CrossRefGoogle Scholar
[17] Ivanyshyn, O. and Kress, R., Inverse scattering for surface impedance from phase-less far field data, Journal of Computational Physics, 230 (2001), pp. 3443–3452.CrossRefGoogle Scholar
[18] Klibanov, M. V., Phaseless inverse scattering problems in threes dimensions, SIAM J. Appl. Math., 74 (2014), pp. 392–410.Google Scholar
[19] Klibanov, M. V., Nguyen, L. H., and Pan, K., Nanostructures imaging via numerical solution of a 3-d inverse scattering problem without the phase information, arXiv: 1404.1183.Google Scholar
[20] Klibanov, M. V. and Romanov, V. G., Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures, Journal of Inverse and Ill-Posed Problems, (23) 2015, pp. 187–193.Google Scholar
[21] Kress, R., Integral equation methods in inverse acoustic and electromagnetic scattering In: Boundary Integral Formulations for Inverse Analysis (Ingham, and Wrobel, , eds) Computational Mechanics Publications Southampton, 1997, pp. 67–92.Google Scholar
[22] Kress, R. and Rundell, W., Inverse obstacle scattering with modulus of the far field pattern as data. Engl, H.W. et al. (eds.), Inverse Problems in Medical Imaging and Nondestructive Testing, Springer, 1997 Google Scholar
[23] Leis, R., Initial Boundary Value Problems in Mathematical Physics, B.G. Teubner, 1986 Google Scholar
[24] Li, L., Zheng, H., and Li, F., Two-diensional constant source inversion method with phaseless data: TM case, IEEE Trans. on Geoscience and remount sensing, 47 (2009), pp. 1719–1736.Google Scholar
[25] Litman, A. and Belkebir, K., Two-dimensional inverse profiling problem using phaseless data, J. Opt. Soc. Am. A, 23 (2006), pp. 2737–2746.Google Scholar
[26] McLean, W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.Google Scholar
[27] Melrose, R. B and Michael, E. T.,Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle, Advances in Mathematics, 55 (1985), pp. 242–315.Google Scholar
[28] Monk, P., Finite Element Methods for Maxwell's Equations, Clarendon Press, 2003.Google Scholar
[29] Novikov, R. G., Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, The Journal of Geometric Analysis, (26) 2016, pp. 346–359.Google Scholar
[30] Novikov, R. G., Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin des Sciences Mathmétiques, (139) 2015, pp. 923–936.Google Scholar
[31] Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer-Verlag, 1973 Google Scholar
[32] Potthast, R., Point-sources and Multipoles in Inverse Scattering Theory, Chapman and Hall/CRC, 2001.Google Scholar
[33] Temme, N. M., Special Functions : An Introduction to the Classical Functions of Mathematical Physics, Wiley, 1996.Google Scholar
[34] Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1995.Google Scholar
[35] Zhang, B., On transmission problems for wave propagation in two locally perturbed half-spaces, Math. Proc. Camb. Phil. Soc., 115 (1994), pp. 545–558.Google Scholar
[36] Zhang, W., Li, L., and Li, F., Inverse scattering from phaseless data in the free space, Science in China Series F: Information Sciences, 52 (2009), pp. 1389–1398.Google Scholar