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August  2014, 8(3): 645-683. doi: 10.3934/ipi.2014.8.645

Resolution enhancement from scattering in passive sensor imaging with cross correlations

Josselin Garnier 1, and  George Papanicolaou 2,

1. 

Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris Diderot, 75205 Paris Cedex 13, France
2. 

Mathematics Department, Stanford University, Stanford, CA 94305, United States

Received  February 2012 Revised  June 2013 Published  August 2014

It was shown in [Garnier et al., SIAM J. Imaging Sciences 2 (2009), 396] that it is possible to image reflectors by backpropagating cross correlations of signals generated by ambient noise sources and recorded at passive sensor arrays. The resolution of the image depends on the directional diversity of the noise signals relative to the locations of the sensor array and the reflector. When directional diversity is limited it is possible to enhance it by exploiting the scattering properties of the medium since scatterers will act as secondary noise sources. However, scattering increases the fluctuation level of the cross correlations and therefore tends to destabilize the image by reducing its signal-to-noise ratio. In this paper we study the trade-off in passive, correlation-based imaging between resolution enhancement and signal-to-noise ratio reduction that is due to scattering.
Keywords: ambient noise sources, Passive sensor imaging, scattering media..
Mathematics Subject Classification: Primary: 35R30, 35R60; Secondary: 86A15, 78A4.
Citation: Josselin Garnier, George Papanicolaou. Resolution enhancement from scattering in passive sensor imaging with cross correlations. Inverse Problems & Imaging, 2014, 8 (3) : 645-683. doi: 10.3934/ipi.2014.8.645
References:
[1]

C. Bardos, J. Garnier and G. Papanicolaou, Identification of Green's functions singularities by cross correlation of noisy signals, Inverse Problems, 24 (2008), 015011, 26 pp. doi: 10.1088/0266-5611/24/1/015011.  Google Scholar

[2]

J. Berryman, Stable iterative reconstruction algorithm for nonlinear travel time tomography, Inverse Problems, 6 (1990), 21-42. doi: 10.1088/0266-5611/6/1/005.  Google Scholar

[3]

B. L. Biondi, 3D Seismic Imaging, no. 14 in Investigations in Geophysics, Society of Exploration Geophysics, Tulsa, 2006. Google Scholar

[4]

N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals, Dover, New York, 1986.  Google Scholar

[5]

M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139644181.  Google Scholar

[6]

F. Brenguier, N. M. Shapiro, M. Campillo, V. Ferrazzini, Z. Duputel, O. Coutant and A. Nercessian, Towards forecasting volcanic eruptions using seismic noise, Nature Geoscience, 1 (2008), 126-130. doi: 10.1038/ngeo104.  Google Scholar

[7]

T. Callaghan, N. Czink, F. Mani, A. Paulraj and G. Papanicolaou, Correlation-based radio localization in an indoor environment, EURASIP Journal on Wireless Communications and Networking, 2011 (2011), 135p. doi: 10.1186/1687-1499-2011-135.  Google Scholar

[8]

J. F. Claerbout, Imaging the Earth's Interior, Blackwell Scientific Publications, Palo Alto, 1985. Google Scholar

[9]

Y. Colin de Verdière, Semiclassical analysis and passive imaging, Nonlinearity, 22 (2009), R45-R75. doi: 10.1088/0951-7715/22/6/R01.  Google Scholar

[10]

L. Erdös and H.-T. Yau, Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation, Comm. Pure Appl. Math., 53 (2000), 667-735. doi: 10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5.  Google Scholar

[11]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[12]

U. Frisch, Wave Propagation in Random Media, in Probabilistic Methods in Applied Mathematics, edited by A. T. Bharucha-Reid, Academic Press, New York, 1 (1968), 75-198.  Google Scholar

[13]

J. Garnier and G. Papanicolaou, Passive sensor imaging using cross correlations of noisy signals in a scattering medium, SIAM J. Imaging Sciences, 2 (2009), 396-437. doi: 10.1137/080723454.  Google Scholar

[14]

J. Garnier and G. Papanicolaou, Resolution analysis for imaging with noise, Inverse Problems, 26 (2010), 074001, 22pp. doi: 10.1088/0266-5611/26/7/074001.  Google Scholar

[15]

J. Garnier and K. Sølna, Cross correlation and deconvolution of noise signals in randomly layered media, SIAM J. Imaging Sciences, 3 (2010), 809-834. doi: 10.1137/090757538.  Google Scholar

[16]

O. A. Godin, Accuracy of the deterministic travel time retrieval from cross-correlations of non-diffuse ambient noise, J. Acoust. Soc. Am., 126 (2009), EL183-EL189. doi: 10.1121/1.3258064.  Google Scholar

[17]

P. Gouédard, L. Stehly, F. Brenguier, M. Campillo, Y. Colin de Verdière, E. Larose, L. Margerin, P. Roux, F. J. Sanchez-Sesma, N. M. Shapiro and R. L. Weaver, Cross-correlation of random fields: Mathematical approach and applications, Geophysical Prospecting, 56 (2008), 375-393. Google Scholar

[18]

P. A. Martin, Acoustic scattering by inhomogeneous obstacles, SIAM J. Appl. Math., 64 (2003), 297-308. doi: 10.1137/S0036139902414379.  Google Scholar

[19]

P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968. Google Scholar

[20]

G. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Ito-Schroedinger equation, SIAM Journal on Multiscale Modeling and Simulation, 6 (2007), 468-492. doi: 10.1137/060668882.  Google Scholar

[21]

P. Roux, K. G. Sabra, W. A. Kuperman and A. Roux, Ambient noise cross correlation in free space: Theoretical approach, J. Acoust. Soc. Am., 117 (2005), 79-84. doi: 10.1121/1.1830673.  Google Scholar

[22]

L. V. Ryzhik, G. C. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370. doi: 10.1016/S0165-2125(96)00021-2.  Google Scholar

[23]

N. M. Shapiro, M. Campillo, L. Stehly and M. H. Ritzwoller, High-resolution surface wave tomography from ambient noise, Science, 307 (2005), 1615-1618. doi: 10.1126/science.1108339.  Google Scholar

[24]

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, Academic Press, San Diego, 1995. Google Scholar

[25]

R. Snieder, Extracting the Green's function from the correlation of coda waves: A derivation based on stationary phase, Phys. Rev. E, 69 (2004), 046610. doi: 10.1103/PhysRevE.69.046610.  Google Scholar

[26]

L. Stehly, M. Campillo, B. Froment and R. Weaver, Reconstructing Green's function by correlation of the coda of the correlation (C3) of ambient seismic noise, J. Geophys. Res., 113 (2008), B11306. doi: 10.1029/2008JB005693.  Google Scholar

[27]

L. Stehly, M. Campillo and N. M. Shapiro, A study of the seismic noise from its long-range correlation properties, Geophys. Res. Lett., 111 (2006), B10306. doi: 10.1029/2005JB004237.  Google Scholar

[28]

M. C. W. van Rossum and Th. M. Nieuwenhuizen, Multiple scattering of classical waves: Microscopy, mesoscopy, and diffusion, Reviews of Modern Physics, 71 (1999), 313-371. Google Scholar

[29]

K. Wapenaar and J. Fokkema, Green's function representations for seismic interferometry, Geophysics, 71 (2006), SI33-SI46. Google Scholar

[30]

R. Weaver and O. I. Lobkis, Ultrasonics without a source: Thermal fluctuation correlations at MHz frequencies, Phys. Rev. Lett., 87 (2001), 134301. doi: 10.1103/PhysRevLett.87.134301.  Google Scholar

[31]

H. Yao, R. D. van der Hilst and M. V. de Hoop, Surface-wave array tomography in SE Tibet from ambient seismic noise and two-station analysis I. Phase velocity maps, Geophysical Journal International, 166 (2006), 732-744. doi: 10.1111/j.1365-246X.2006.03028.x.  Google Scholar

show all references

References:
[1]

C. Bardos, J. Garnier and G. Papanicolaou, Identification of Green's functions singularities by cross correlation of noisy signals, Inverse Problems, 24 (2008), 015011, 26 pp. doi: 10.1088/0266-5611/24/1/015011.  Google Scholar

[2]

J. Berryman, Stable iterative reconstruction algorithm for nonlinear travel time tomography, Inverse Problems, 6 (1990), 21-42. doi: 10.1088/0266-5611/6/1/005.  Google Scholar

[3]

B. L. Biondi, 3D Seismic Imaging, no. 14 in Investigations in Geophysics, Society of Exploration Geophysics, Tulsa, 2006. Google Scholar

[4]

N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals, Dover, New York, 1986.  Google Scholar

[5]

M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139644181.  Google Scholar

[6]

F. Brenguier, N. M. Shapiro, M. Campillo, V. Ferrazzini, Z. Duputel, O. Coutant and A. Nercessian, Towards forecasting volcanic eruptions using seismic noise, Nature Geoscience, 1 (2008), 126-130. doi: 10.1038/ngeo104.  Google Scholar

[7]

T. Callaghan, N. Czink, F. Mani, A. Paulraj and G. Papanicolaou, Correlation-based radio localization in an indoor environment, EURASIP Journal on Wireless Communications and Networking, 2011 (2011), 135p. doi: 10.1186/1687-1499-2011-135.  Google Scholar

[8]

J. F. Claerbout, Imaging the Earth's Interior, Blackwell Scientific Publications, Palo Alto, 1985. Google Scholar

[9]

Y. Colin de Verdière, Semiclassical analysis and passive imaging, Nonlinearity, 22 (2009), R45-R75. doi: 10.1088/0951-7715/22/6/R01.  Google Scholar

[10]

L. Erdös and H.-T. Yau, Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation, Comm. Pure Appl. Math., 53 (2000), 667-735. doi: 10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5.  Google Scholar

[11]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[12]

U. Frisch, Wave Propagation in Random Media, in Probabilistic Methods in Applied Mathematics, edited by A. T. Bharucha-Reid, Academic Press, New York, 1 (1968), 75-198.  Google Scholar

[13]

J. Garnier and G. Papanicolaou, Passive sensor imaging using cross correlations of noisy signals in a scattering medium, SIAM J. Imaging Sciences, 2 (2009), 396-437. doi: 10.1137/080723454.  Google Scholar

[14]

J. Garnier and G. Papanicolaou, Resolution analysis for imaging with noise, Inverse Problems, 26 (2010), 074001, 22pp. doi: 10.1088/0266-5611/26/7/074001.  Google Scholar

[15]

J. Garnier and K. Sølna, Cross correlation and deconvolution of noise signals in randomly layered media, SIAM J. Imaging Sciences, 3 (2010), 809-834. doi: 10.1137/090757538.  Google Scholar

[16]

O. A. Godin, Accuracy of the deterministic travel time retrieval from cross-correlations of non-diffuse ambient noise, J. Acoust. Soc. Am., 126 (2009), EL183-EL189. doi: 10.1121/1.3258064.  Google Scholar

[17]

P. Gouédard, L. Stehly, F. Brenguier, M. Campillo, Y. Colin de Verdière, E. Larose, L. Margerin, P. Roux, F. J. Sanchez-Sesma, N. M. Shapiro and R. L. Weaver, Cross-correlation of random fields: Mathematical approach and applications, Geophysical Prospecting, 56 (2008), 375-393. Google Scholar

[18]

P. A. Martin, Acoustic scattering by inhomogeneous obstacles, SIAM J. Appl. Math., 64 (2003), 297-308. doi: 10.1137/S0036139902414379.  Google Scholar

[19]

P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968. Google Scholar

[20]

G. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Ito-Schroedinger equation, SIAM Journal on Multiscale Modeling and Simulation, 6 (2007), 468-492. doi: 10.1137/060668882.  Google Scholar

[21]

P. Roux, K. G. Sabra, W. A. Kuperman and A. Roux, Ambient noise cross correlation in free space: Theoretical approach, J. Acoust. Soc. Am., 117 (2005), 79-84. doi: 10.1121/1.1830673.  Google Scholar

[22]

L. V. Ryzhik, G. C. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370. doi: 10.1016/S0165-2125(96)00021-2.  Google Scholar

[23]

N. M. Shapiro, M. Campillo, L. Stehly and M. H. Ritzwoller, High-resolution surface wave tomography from ambient noise, Science, 307 (2005), 1615-1618. doi: 10.1126/science.1108339.  Google Scholar

[24]

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, Academic Press, San Diego, 1995. Google Scholar

[25]

R. Snieder, Extracting the Green's function from the correlation of coda waves: A derivation based on stationary phase, Phys. Rev. E, 69 (2004), 046610. doi: 10.1103/PhysRevE.69.046610.  Google Scholar

[26]

L. Stehly, M. Campillo, B. Froment and R. Weaver, Reconstructing Green's function by correlation of the coda of the correlation (C3) of ambient seismic noise, J. Geophys. Res., 113 (2008), B11306. doi: 10.1029/2008JB005693.  Google Scholar

[27]

L. Stehly, M. Campillo and N. M. Shapiro, A study of the seismic noise from its long-range correlation properties, Geophys. Res. Lett., 111 (2006), B10306. doi: 10.1029/2005JB004237.  Google Scholar

[28]

M. C. W. van Rossum and Th. M. Nieuwenhuizen, Multiple scattering of classical waves: Microscopy, mesoscopy, and diffusion, Reviews of Modern Physics, 71 (1999), 313-371. Google Scholar

[29]

K. Wapenaar and J. Fokkema, Green's function representations for seismic interferometry, Geophysics, 71 (2006), SI33-SI46. Google Scholar

[30]

R. Weaver and O. I. Lobkis, Ultrasonics without a source: Thermal fluctuation correlations at MHz frequencies, Phys. Rev. Lett., 87 (2001), 134301. doi: 10.1103/PhysRevLett.87.134301.  Google Scholar

[31]

H. Yao, R. D. van der Hilst and M. V. de Hoop, Surface-wave array tomography in SE Tibet from ambient seismic noise and two-station analysis I. Phase velocity maps, Geophysical Journal International, 166 (2006), 732-744. doi: 10.1111/j.1365-246X.2006.03028.x.  Google Scholar

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