Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T10:36:13.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Near-field Imaging Point-like Scatterers and Extended Elastic Solid in a Fluid

Part of: Equations of mathematical physics and other areas of application Waves Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  17 May 2016

Tao Yin ,
Guanghui Hu  and
Liwei Xu
Tao Yin*
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 400044, P.R. China
Guanghui Hu*
Affiliation:
Beijing Computational Science Research Center, Beijing 100094, P.R. China
Liwei Xu*
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 400044, P.R. China Institute of Computing and Data Science, Chongqing University, Chongqing 400044, P.R. China
*
Corresponding author., Email addresses:, taoyincqu@163.com(T. Yin), hu@csrc.ac.cn(G. Hu), xul@cqu.edu.cn(L. Xu)
Corresponding author., Email addresses:, taoyincqu@163.com(T. Yin), hu@csrc.ac.cn(G. Hu), xul@cqu.edu.cn(L. Xu)
Corresponding author., Email addresses:, taoyincqu@163.com(T. Yin), hu@csrc.ac.cn(G. Hu), xul@cqu.edu.cn(L. Xu)
Get access

Abstract

Consider the time-harmonic acoustic scattering from an extended elastic body surrounded by a finite number of point-like obstacles in a fluid. We assume point source waves are emitted from arrayed transducers and the signals of scattered near-field data are recorded by receivers not far away from the scatterers (compared to the incident wavelength). The forward scattering can be modeled as an interaction problem between acoustic and elastic waves together with a multiple scattering problem between the extend solid and point scatterers. We prove a necessary and sufficient condition that can be used simultaneously to recover the shape of the extended elastic solid and to locate the positions of point scatterers. The essential ingredient in our analysis is the outgoing-to-incoming (OtI) operator applied to the resulting near-field response matrix (or operator). In the first part, we justify the MUSIC algorithm for locating point scatterers from near-field measurements. In the second part, we apply the factorization method, the continuous analogue of MUSIC, to the two-scale scattering problem for determining both extended and point scatterers. Numerical examples in 2D are demonstrated to show the validity and accuracy of our inversion algorithms.

Keywords

MUSCInear-field imagingpoint sourcesfactorization methodinverse scatteringfluid-solid interaction problem

MSC classification

Secondary: 74J25: Inverse problems 74J20: Wave scattering 35R30: Inverse problems 35Q74: PDEs in connection with mechanics of deformable solids
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1317 - 1342
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Albeverio, S. and Kurasov, P., Singular Perturbations of Differential Operators, in Solvable Schrödinger Type Operators, London Math. Soc. Lecture Note Ser. 271, Cambridge University Press, Cambridge, UK, 2000.Google Scholar
[2]Albeverio, S. and Pankrashkin, K., A remark on Krein's resolvent formula and boundary conditions J. Phys. A, 38 (2005), 4859–4864.Google Scholar
[3]Ammari, H. and Kang, H., Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities, J. Math. Anal. Appl., 296 (2004), 190–208.CrossRefGoogle Scholar
[4]Ammari, H. and Kang, H., Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics 1846, Springer-Verlag, Berlin, 2004.Google Scholar
[5]Bao, G., Huang, K., Li, P. and Zhao, H., A direct imaging method for inverse scattering using the Generalized Foldy-Lax formulation, Contemp. Math., 615 (2014), 49–70.Google Scholar
[6]Bao, G. and Li, P., Numerical solution of inverse scattering for near-field optics, Optics Letters, 32 (2007), 1465–1467.CrossRefGoogle ScholarPubMed
[7]Cheney, M., The linear sampling method and the music algorithm, Inverse Problems, 17 (2001), 591–596.Google Scholar
[8]Challa, D. P. and Sini, M., Inverse scattering by point-like scatterers in the Foldy regime, Inverse Problems, 28 (2012), 125006.Google Scholar
[9]Challa, D. P. and Sini, M., On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul., 12 (2014), 55–108.Google Scholar
[10]Challa, D. P., Hu, G. and Sini, M., Multiple scattering of electromagnetic waves by a finite number of point-like obstacles, Mathematical Models and Methods in Applied Sciences, 24 (2014), 863–899.Google Scholar
[11]Colton, D., and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, Berlin, Springer, 1998.CrossRefGoogle Scholar
[12]Devaney, A.J., Super-resolution processing of multi-static data using time reversal and MUSIC, available at http://www.ece.neu.edu/faculty/devaney (2000).Google Scholar
[13]Elschner, J., Hsiao, G. C. and Rathsfeld, A., An inverse problem for fluid-solid interaction, Inverse Problems and Imaging, 2 (2008), 83–120.Google Scholar
[14]Elschner, J., Hsiao, G. C. and Rathsfeld, A., An optimization method in inverse acoustic scattering by an elastic obstacle, SIAM J. Appl. Math., 70 (2009), 168–187.Google Scholar
[15]Foldy, L. L., The multiple scattering of waves. I., General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev., 2 (1945), 107–119.Google Scholar
[16]Gruber, F. K., Marengo, E. A. and Devaney, A.J., Time-reversal imaging with multiple signal classification considering multiple scattering between the targets, J. Acoust. Soc. Am., 115 (2004), 3042–3047.Google Scholar
[17]Hou, S., Solna, K. and Zhao, H., A direct imaging algorithm for extended targets, Inverse Problems, 22 (2006), 1151–1178.CrossRefGoogle Scholar
[18]Hsiao, G. C., Kleinman, R. E. and Roach, G. F., Weak solution of fluid-solid interaction problem, Math. Nachr., 218 (2000), 139–163.Google Scholar
[19]Hu, G., Li, J., Liu, H. and Sun, H., Inverse elastic scattering for multiscale rigid bodies with a single far-field measurement, SIAM Journal on Imaging Sciences, 6 (2014), 2285–2309.Google Scholar
[20]Hu, G., Li, J. and Liu, H., Recovering complex elastic scatterers by a single far-field pattern, Journal of Differential Equation, 257 (2014), 469–489.Google Scholar
[21]Hu, G., Yang, J., Zhang, B. and Zhang, H., Near-field imaging of scattering obstacles via the factorization method, Inverse Problems, 30,095005.Google Scholar
[22]Hu, G., Mantile, A. and Sini, M., Direct and inverse acoustic scattering by a collection of extended and point-like scatterers, Multiscale Modeling and Simulation, 12 (2014), 996–1027.Google Scholar
[23]Huang, K. and Li, P., A two-scale multiple scattering problem, Multiscale Model. Simul., 8 (2010), 1511–1534.Google Scholar
[24]Huang, K., Li, P. and Zhao, H., An efficient algorithm for the generalized Foldy- Lax formulation, J. Comput. Phys., 234 (2013), 376–398.CrossRefGoogle Scholar
[25]Ito, K., Jin, B. and Zou, J., A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003.Google Scholar
[26]Jones, D. S., Low-frequency scattering by a body in lubricated contact, Quart. J. Mech. Appl. Math., 36 (1983), 111–137.Google Scholar
[27]Kirsch, A., The music algorithm and the factorization method in inverse scattering theory for inhomogenious media, Inverse Problems, 18 (2002), 1025–1040.Google Scholar
[28]Kirsch, A., Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489–1512.Google Scholar
[29]Kirsch, A. and Grinberg, N., The Factorization Method for Inverse Problems, New York, Oxford Univ. Press, 2008.Google Scholar
[30]Kirsch, A. and Ruiz, A., The factorization method for an inverse fluid-solid interaction scattering problem, Inverse Problems and Imaging, 6 (2012), 681–695.Google Scholar
[31]Kurasov, P. and Posilicano, A., Finite speed of propagation and local boundary conditions for wave equations with point interactions, Proc. Amer. Math. Soc., 133 (2005), 3071–3078.CrossRefGoogle Scholar
[32]Li, J., Liu, H. and Wang, Q., Locating multiple multi-scale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285–2309.Google Scholar
[33]Luke, C. J. and Martin, P. A., Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle, SIAM Appl. Math., 55 (1995), 904–922.Google Scholar
[34]Monk, P. and Selgas, V., An inverse fluid-solid interaction problem, Inverse Probl. Imaging, 3 (2009), 173–198.CrossRefGoogle Scholar
[35]Monk, P. and Selgas, V., Near field sampling type methods for the inverse fluid-solid interaction problem, Inverse Problems and Imaging, 5 (2011), 465–483.Google Scholar
[36]Natroshvili, D., Sadunishvili, G., Sigua, I., Some remarks concerning Jones eigenfraquencies and Jones modes, Georgian Mathematical Journal, 12 (2005), 337–348.Google Scholar
[37]Yin, T., Hu, G., Xu, L. and Zhang, B., Near-field imaging of scattering obstacles with the factorization method II: Fluid-solid interaction, submitted. Available at: http://www.wias-berlin.de/people/hu/home.htmlGoogle Scholar
[38]Zhao, H., Analysis of the response matrix for an extended target, SIAM Appl. Math., 64 (2004), 725–745.Google Scholar