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Time-reversal algorithms in viscoelastic media†

Published online by Cambridge University Press:  03 April 2013

HABIB AMMARI ,
ELIE BRETIN ,
JOSSELIN GARNIER  and
ABDUL WAHAB
HABIB AMMARI
Affiliation:
Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d'Ulm, 75005 Paris, France email: habib.ammari@ens.fr
ELIE BRETIN
Affiliation:
Centre de Mathématiques Appliquées, CNRS UMR 7641, École Polytechnique, 91128 Palaiseau, France email: bretin@cmap.polytechnique.fr
JOSSELIN GARNIER
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, 75205 Paris Cedex 13, France email: garnier@math.jussieu.fr
ABDUL WAHAB
Affiliation:
Department of Mathematics, COMSATS Institute of Information Technology, G.T. Road, Wah Cantt. 47040, Pakistan email: wahab@ciitwah.edu.pk
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Abstract

In this paper we consider the problem of reconstructing sources in a homogeneous viscoelastic medium from wavefield measurements. We first present a modified time-reversal imaging algorithm based on a weighted Helmholtz decomposition and justify mathematically that it provides a better approximation than by simply time reversing the displacement field, where artifacts appear due to the coupling between the pressure and shear waves. Then we investigate the source inverse problem in an elastic attenuating medium. We provide a regularized time-reversal imaging which corrects the attenuation effect at the first order. The results of this paper yield the fundamental tools for solving imaging problems in elastic media using cross-correlation techniques.

Keywords

Elastic wave propagationTime-reversal algorithmsAttenuation correctionWeighted Helmholtz decompositionHelmholtz–Kirchhoff identity

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 4 , August 2013 , pp. 565 - 600
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

†

This work was supported by the ERC Advanced Grant Project MULTIMOD–267184.

References

[1]Aki, K. & Richards, P. G. (1980) Quantitative Seismology, Vol. 1, W. H. Freeman, San Francisco, CA.Google Scholar
[2]Ammari, H. (2008) An Introduction to Mathematics of Emerging Biomedical Imaging, Mathematics & Applications, Vol. 62, Springer-Verlag, Berlin, Germany.Google Scholar
[3]Ammari, H., Asch, M., Jugnon, V., Guadarrama Bustos, L. & Kang, H. (2011) Transient imaging with limited-view data. SIAM J. Imaging Sci. 4, 1097–1121.CrossRefGoogle Scholar
[4]Ammari, H., Bossy, E., Jugnon, V. & Kang, H. (2010) Mathematical modelling in photo-acoustic imaging of small absorbers. SIAM Rev. 52, 677–695.CrossRefGoogle Scholar
[5]Ammari, H., Bretin, E., Garnier, J. & Wahab, A. (2011) Time reversal in attenuating acoustic media. In: Mathematical and Statistical Methods for Imaging, Contemporary Mathematics series, Vol. 548, American Mathematical Society, Providence, RI, pp. 151–163.CrossRefGoogle Scholar
[6]Ammari, H., Bretin, E., Garnier, J. & Wahab, A. (2012) Noise source localization in an attenuating medium. SIAM J. Appl. Math. 72, 317–336.CrossRefGoogle Scholar
[7]Ammari, H., Bretin, E., Jugnon, V. & Wahab, A. (2011) Photoacoustic imaging for attenuating acoustic media. In: Mathematical Modeling in Biomedical Imaging II, Lecture Notes in Mathematics, Vol. 2035, Springer-Verlag, Berlin, Germany, pp. 57–84.Google Scholar
[8]Ammari, H., Capdeboscq, Y., Kang, H. & Kozhemyak, A. (2009) Mathematical models and reconstruction methods in magneto-acoustic imaging. Euro. J. Appl. Math. 20, 303–317.CrossRefGoogle Scholar
[9]Ammari, H., Garapon, P., Guadarrama Bustos, L. & Kang, H. (2010) Transient anomaly imaging by the acoustic radiation force. J. Diff. Equ. 249, 1579–1595.CrossRefGoogle Scholar
[10]Ammari, H., Garapon, P., Kang, H. & Lee, H. (2008) A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements. Quart. Appl. Math. 66, 139–175.CrossRefGoogle Scholar
[11]Ammari, H., Guadarrama-Bustos, L., Kang, H. & Lee, H. (2011) Transient elasticity imaging and time reversal. Proc. R. Soc. Edinburgh Math. 141, 1121–1140.CrossRefGoogle Scholar
[12]Ammari, H. & Kang, H. (2007) Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences series, Vol. 162, Springer-Verlag, New York.Google Scholar
[13]Bercoff, J., Tanter, M., Muller, M. & Fink, M. (2004) The role of viscosity in the impulse diffraction field of elastic waves induced by the acoustic radiation force. IEEE Trans. Ultrason. Ferro. Freq. Control 51, 1523–1536.CrossRefGoogle ScholarPubMed
[14]Borcea, L., Papanicolaou, G. & Tsogka, C. (2003) Theory and applications of time reversal and interferometric imaging. Inverse Probl. 19, 134–164.CrossRefGoogle Scholar
[15]Borcea, L., Papanicolaou, G. & Tsogka, C. (2005) Interferometric array imaging in clutter. Inverse Probl. 21, 1419–1460.CrossRefGoogle Scholar
[16]Borcea, L., Papanicolaou, G., Tsogka, C. & Berrymann, J. G. (2002) Imaging and time reversal in random media. Inverse Probl. 18, 1247–1279.CrossRefGoogle Scholar
[17]Bretin, E., Guadarrama Bustos, L. & Wahab, A. (2011) On the Green function in visco-elastic media obeying a frequency power-law. Math. Meth. Appl. Sci. 34, 819–830.CrossRefGoogle Scholar
[18]Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. (1987) Spectral Methods in Fluid Dynamics, Springer-Verlag, New York.Google Scholar
[19]Catheline, S., Benech, N., Brum, J. & Negreira, C. (2008) Time-reversal of elastic waves in soft solids. Phys. Rev. Lett. 100, 064301.CrossRefGoogle ScholarPubMed
[20]Catheline, S., Gennisson, J. L., Delon, G., Sinkus, R., Fink, M., Abdouelkaram, S. & Culioli, J. (2004) Measurement of visco-elastic properties of solid using transient elastography: An inverse problem approach. J. Acous. Soc. Am. 116, 3734–3741.CrossRefGoogle Scholar
[21]Chen, Z. & Zhang, X. (preprint) An anisotropic perfectly matched layer method for three-dimensional elastic scattering problems.Google Scholar
[22]de Rosny, J., Lerosey, G., Tourin, A. & Fink, M. (2007) Time reversal of electromagnetic waves. In: Lecture Notes in Computer Science and Engineering, Vol. 59, Springer-Verlag, New York.Google Scholar
[23]Fink, M. (1997) Time-reversed acoustics. Phys. Today 50, 34.CrossRefGoogle Scholar
[24]Fink, M. & Prada, C. (2001) Acoustic time-reversal mirrors. Inverse Probl. 17, R1–38.CrossRefGoogle Scholar
[25]Fouque, J.-P., Garnier, J. & Nachbin, A. (2004) Time reversal for dispersive waves in random media. SIAM J. Appl. Math. 64, 1810–1838.Google Scholar
[26]Fouque, J.-P., Garnier, J., Nachbin, A. & Sølna, K. (2005) Time reversal refocusing for point source in randomly layered media. Wave Motion 42, 238–260.CrossRefGoogle Scholar
[27]Fouque, J.-P., Garnier, J., Papanicolaou, G. & Sølna, K. (2007) Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York.Google Scholar
[28]Fouque, J.-P., Garnier, J. & Sølna, K. (2006) Time reversal super resolution in randomly layered media. Wave Motion 43, 646–666.CrossRefGoogle Scholar
[29]Galdi, G. P. (1994) An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I, Linearized Steady Problems, Springer-Verlag, New York.Google Scholar
[30]Greenleaf, J. F., Fatemi, M. & Insana, M. (2003) Selected methods for imaging elastic properties of biological tissues. Annu. Rev. Biomed. Eng. 5, 57–78.CrossRefGoogle ScholarPubMed
[31]Hastings, F., Schneider, J. B. & Broschat, S. L. (1996) Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation. J. Acoust. Soc. Am. 100, 3061–3069.CrossRefGoogle Scholar
[32]Hörmander, L. (2003) The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Classics in Mathematics, Springer-Verlag, Berlin, Germany.Google Scholar
[33]Kalimeris, K. & Scherzer, O. (to appear) Photoacoustic imaging in attenuating acoustic media based on strongly causal models. Math. Meth. Appl. Sci. (arXiv:1211.1516v1) doi:10.1002/mma.2756.Google Scholar
[34]Kowar, R. & Scherzer, O. (2011) Photoacoustic imaging taking into account attenuation. In: Mathematical Modeling in Biomedical Imaging II, Lecture Notes in Mathematics, Vol. 2035, Springer-Verlag, Berlin, Germany, pp. 85–130.Google Scholar
[35]Kowar, R., Scherzer, O. & Bonnefond, X. (2011) Causality analysis of frequency dependent wave attenuation. Math. Meth. Appl. Sci. 34, 108–124.CrossRefGoogle Scholar
[36]Larmat, C., Montagner, J. P., Fink, M., Capdeville, Y., Tourin, A. & Clévédé, E. (2006) Time-reversal imaging of seismic sources and application to the great Sumatra earthquake. Geophys. Res. Lett. 33, L19312.CrossRefGoogle Scholar
[37]Lerosey, G., de Rosny, J., Tourin, A., Derode, A., Montaldo, G. & Fink, M. (2005) Time-reversal of electromagnetic waves and telecommunication. Radio Sci. 40, RS6S12.CrossRefGoogle Scholar
[38]Näsholm, S. P. & Holm, S. (2013) On a fractional zener elastic wave equation. Fract. Calcul. Appl. Anal. 16, 26–50.CrossRefGoogle Scholar
[39]Norville, P. D. & Scott, W. R. (2005) Time-reversal focusing of elastic surface waves. J. Acoust. Soc. Am. 118, 735–744.CrossRefGoogle Scholar
[40]Phung, K. D. & Zhang, X. (2008) Time reversal focusing of the initial state for Kirchhoff plate. SIAM J. Applied Math. 68, 1535–1556.CrossRefGoogle Scholar
[41]Prada, C., Kerbrat, E., Cassereau, D. & Fink, M. (2002) Time reversal techniques in ultrasonic nondestructive testing of scattering media. Inverse Probl. 18, 1761–1773.CrossRefGoogle Scholar
[42]Pujol, J. (2003) Elastic Wave Propagation and Generation in Seismology, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
[43]Sarvazyan, A. P., Rudenko, O. V., Swanson, S. C., Fowlkers, J. B. & Emelianovs, S. V. (1998) Shear wave elasticity imaging: A new ultrasonic technology of medical diagnostics. Ultrasound Med. Biol. 24, 1419–1435.CrossRefGoogle ScholarPubMed
[44]Strang, G. (1968) On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517.CrossRefGoogle Scholar
[45]Szabo, T. L. and Wu, J.A model for longitudinal and shear wave propagation in viscoelastic media. J. Acous. Soc. Am. 107, 2437–2446.CrossRefGoogle Scholar
[46]Tanter, M. and Fink, M.Time reversing waves for biomedical Applications, In: Mathematical Modeling in Biomedical Imaging I, Lecture Notes in Mathematics vol. 1983, Springer-Verlag, 2009, pp. 73–97.CrossRefGoogle Scholar
[47]Teng, J. J., Zhang, G. & Huang, S. X. (2007) Some theoretical problems on variational data assimilation. Appl. Math. Mech. 28, 581–591.CrossRefGoogle Scholar
[48]Wapenaar, K. (2004) Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation. Phys. Rev. Lett. 93, 254301.CrossRefGoogle ScholarPubMed
[49]Wapenaar, K. & Fokkema, J. (2006) Green's function representations for seismic interferometry. Geophysics 71, SI33–I46.CrossRefGoogle Scholar
[50]Wiegmann, A. (June 1999), Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds. Technical Report LBNL-43565, Lawrence Berkeley National Laboratory, Berkeley CA.CrossRefGoogle Scholar
[51]Xu, Y. & Wang, L. V. (2004) Time reversal and its application to tomography with diffraction sources. Phys. Rev. Lett. 92, 033902.CrossRefGoogle ScholarPubMed