# Multiscale analysis of wave propagation in random media. Application to correlation-based imaging

Josselin Garnier^{[1]}

- [1] Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions Université Paris Diderot 75205 Paris Cedex 13 France

Séminaire Laurent Schwartz — EDP et applications (2013-2014)

- page 1-19
- ISSN: 2266-0607

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topGarnier, Josselin. "Multiscale analysis of wave propagation in random media. Application to correlation-based imaging." Séminaire Laurent Schwartz — EDP et applications (2013-2014): 1-19. <http://eudml.org/doc/275749>.

@article{Garnier2013-2014,

abstract = {We consider sensor array imaging with the purpose to image reflectors embedded in a medium. Array imaging consists in two steps. In the first step waves emitted by an array of sources probe the medium to be imaged and are recorded by an array of receivers. In the second step the recorded signals are processed to form an image of the medium. Array imaging in a scattering medium is limited because coherent signals recorded at the receiver array and coming from a reflector to be imaged are weak and dominated by incoherent signals coming from multiple scattering by the medium. If, however, an auxiliary passive (receiver) array can be placed between the reflector to be imaged and the scattering medium then the cross correlations of the incoherent signals on this array can also be used to image the reflector. This situation is important in particular in oil reservoir monitoring when auxiliary receivers can be implemented in wells and its study requires a multiscale analysis of wave propagation in random media. In this review we describe the results obtained in two recent papers using multiscale analysis of wave propagation in random media. In [J. Garnier and G. Papanicolaou, Inverse Problems 28 (2012), 075002] we show that if (i) the source array is infinite, (ii) the scattering medium is modeled by either an isotropic random medium in the paraxial regime or a randomly layered medium, and (iii) the medium between the auxiliary array and the object to be imaged is homogeneous, then imaging with cross correlations completely eliminates the effects of the random medium. It is as if we imaged with an active array, instead of a passive one, near the object. In [J. Garnier and G. Papanicolaou, SIAM J. Imaging Sci. 7 (2014), 1210] we analyze the resolution of the image when both the source array and the passive receiver array are finite. We show that for isotropic random media in the paraxial regime, imaging not only is not affected by the inhomogeneities but the resolution can in fact be enhanced. This is because the random medium can increase the diversity of the illumination. We also show analytically that this does not happen in a randomly layered medium, and there may be some loss of resolution in this case.},

affiliation = {Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions Université Paris Diderot 75205 Paris Cedex 13 France},

author = {Garnier, Josselin},

journal = {Séminaire Laurent Schwartz — EDP et applications},

keywords = {random matrix; wave sensor imaging; Hadamard matrices; singular value decomposition; target localisation and reconstruction; Kolmogorov-Smirnov estimator},

language = {eng},

pages = {1-19},

publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {Multiscale analysis of wave propagation in random media. Application to correlation-based imaging},

url = {http://eudml.org/doc/275749},

year = {2013-2014},

}

TY - JOUR

AU - Garnier, Josselin

TI - Multiscale analysis of wave propagation in random media. Application to correlation-based imaging

JO - Séminaire Laurent Schwartz — EDP et applications

PY - 2013-2014

PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique

SP - 1

EP - 19

AB - We consider sensor array imaging with the purpose to image reflectors embedded in a medium. Array imaging consists in two steps. In the first step waves emitted by an array of sources probe the medium to be imaged and are recorded by an array of receivers. In the second step the recorded signals are processed to form an image of the medium. Array imaging in a scattering medium is limited because coherent signals recorded at the receiver array and coming from a reflector to be imaged are weak and dominated by incoherent signals coming from multiple scattering by the medium. If, however, an auxiliary passive (receiver) array can be placed between the reflector to be imaged and the scattering medium then the cross correlations of the incoherent signals on this array can also be used to image the reflector. This situation is important in particular in oil reservoir monitoring when auxiliary receivers can be implemented in wells and its study requires a multiscale analysis of wave propagation in random media. In this review we describe the results obtained in two recent papers using multiscale analysis of wave propagation in random media. In [J. Garnier and G. Papanicolaou, Inverse Problems 28 (2012), 075002] we show that if (i) the source array is infinite, (ii) the scattering medium is modeled by either an isotropic random medium in the paraxial regime or a randomly layered medium, and (iii) the medium between the auxiliary array and the object to be imaged is homogeneous, then imaging with cross correlations completely eliminates the effects of the random medium. It is as if we imaged with an active array, instead of a passive one, near the object. In [J. Garnier and G. Papanicolaou, SIAM J. Imaging Sci. 7 (2014), 1210] we analyze the resolution of the image when both the source array and the passive receiver array are finite. We show that for isotropic random media in the paraxial regime, imaging not only is not affected by the inhomogeneities but the resolution can in fact be enhanced. This is because the random medium can increase the diversity of the illumination. We also show analytically that this does not happen in a randomly layered medium, and there may be some loss of resolution in this case.

LA - eng

KW - random matrix; wave sensor imaging; Hadamard matrices; singular value decomposition; target localisation and reconstruction; Kolmogorov-Smirnov estimator

UR - http://eudml.org/doc/275749

ER -

## References

top- A. Bakulin and R. Calvert, The virtual source method: Theory and case study, Geophysics, 71 (2006), pp. SI139-SI150.
- C. Bardos, J. Garnier, and G. Papanicolaou, Identification of Green’s functions singularities by cross correlation of noisy signals, Inverse Problems, 24 (2008), 015011. Zbl1145.35307MR2384770
- N. Bleistein, J. K. Cohen, and J. W. Stockwell, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, Springer, New York, 2001. Zbl0967.86001MR1799275
- P. Blomgren, G. Papanicolaou, and H. Zhao, Super-resolution in time-reversal acoustics, J. Acoust. Soc. Am., 111 (2002), pp. 230-248.
- L. Borcea, J. Garnier, G. Papanicolaou, and C. Tsogka, Coherent interferometric imaging, time gating, and beamforming, Inverse Problems, 27 (2011), 065008. Zbl1231.94007MR2805765
- L. Borcea, J. Garnier, G. Papanicolaou, and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging, Inverse Problems, 27 (2011), 085004. Zbl1230.62129MR2819946
- L. Borcea, F. Gonzalez del Cueto, G. Papanicolaou, and C. Tsogka, Filtering deterministic layering effects in imaging, SIAM Multiscale Model. Simul., 7 (2009), pp. 1267-1301. Zbl1187.35274MR2480120
- L. Borcea, G. Papanicolaou, and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), pp. 1419-1460. Zbl1074.62063MR2158118
- L. Borcea, G. Papanicolaou, and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), pp. 1405-1436. Zbl1094.62119MR2249471
- Y. Colin de Verdière, Semiclassical analysis and passive imaging, Nonlinearity, 22 (2009), pp. R45-R75. Zbl1177.86011MR2507319
- M. de Hoop and K. Sølna, Estimating a Green’s function from field-field correlations in a random medium, SIAM J. Appl. Math., 69 (2009), pp. 909-932. Zbl1172.62030MR2476584
- A. Derode, P. Roux, and M. Fink, Robust acoustic time reversal with high-order multiple scattering, Phys. Rev. Lett., 75 (1995), pp. 4206-4209.
- W. Elmore and M. Heald, Physics of Waves, Dover, New York, 1969.
- J.-P. Fouque, J. Garnier, A. Nachbin, and K. Sølna, Time reversal refocusing for point source in randomly layered media, Wave Motion, 42 (2005), pp. 238-260. Zbl1189.76459MR2152315
- J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007. Zbl05194319MR2327824
- J.-P. Fouque, J. Garnier, and K. Sølna, Time reversal super resolution in randomly layered media, Wave Motion, 43 (2006), pp. 646-666. Zbl1231.76270MR2267277
- J. Garnier, Imaging in randomly layered media by cross-correlating noisy signals, SIAM Multiscale Model. Simul., 4 (2005), pp. 610-640. Zbl1089.76054MR2162869
- J. Garnier and G. Papanicolaou, Passive sensor imaging using cross correlations of noisy signals in a scattering medium, SIAM J. Imaging Sci., 2 (2009), pp. 396-437. Zbl1179.35344MR2496063
- J. Garnier and G. Papanicolaou, Resolution analysis for imaging with noise, Inverse Problems, 26 (2010), 074001. Zbl1197.35320MR2608011
- J. Garnier and G. Papanicolaou, Correlation based virtual source imaging in strongly scattering media, Inverse Problems, 28 (2012), 075002. Zbl1260.65090MR2944953
- J. Garnier and G. Papanicolaou, Role of scattering in virtual source array imaging, SIAM J. Imaging Sci., 7 (2014), pp. 1210-1236. Zbl1323.35228MR3213797
- J. Garnier, G. Papanicolaou, A. Semin, and C. Tsogka, Signal-to-noise ratio estimation in passive correlation-based imaging, SIAM J. Imaging Sci., 6 (2013), pp. 1092-1110. Zbl1282.35410MR3063147
- J. Garnier and K. Sølna, Coupled paraxial wave equations in random media in the white-noise regime, Ann. Appl. Probab., 19 (2009), pp. 318-346. Zbl1175.60066MR2498680
- A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, Piscataway, 1997. Zbl0873.65115MR1626707
- G. Papanicolaou, L. Ryzhik, and K. Sølna, Statistical stability in time reversal, SIAM J. Appl. Math., 64 (2004), pp. 1133-1155. Zbl1065.35058MR2068663
- G. Papanicolaou, L. Ryzhik, and K. Sølna, Self-averaging from lateral diversity in the Ito-Schrödinger equation, SIAM Multiscale Model. Simul., 6 (2007), pp. 468-492. Zbl1143.35390MR2338491
- G. T. Schuster, Seismic Interferometry, Cambridge University Press, Cambridge, 2009. Zbl1170.86001
- F. D. Tappert, The Parabolic Approximation Method, in Wave Propagation and Underwater Acoustics, Springer Lecture Notes in Physics, 70 (1977), pp. 224-287. MR475274
- B. J. Uscinski, The Elements of Wave Propagation in Random Media, McGraw Hill, New York, 1977.
- K. Wapenaar, E. Slob, R. Snieder, and A. Curtis, Tutorial on seismic interferometry: Part 2 - Underlying theory and new advances, Geophysics, 75 (2010), pp. 75A211-75A227.

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