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May  2011, 5(2): 371-390. doi: 10.3934/ipi.2011.5.371

Filtered Kirchhoff migration of cross correlations of ambient noise signals

Josselin Garnier 1, and  Knut Solna 2,

1. 

Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, site Chevaleret, case 7012, 75205 Paris Cedex 13
2. 

Department of Mathematics, University of California at Irvine, Irvine, CA 92697

Received  June 2010 Revised  December 2010 Published  May 2011

In this paper we study passive sensor imaging with ambient noise sources by suitably migrating cross correlations of the recorded signals. We propose and study different imaging functionals. A new functional is introduced that is an inverse Radon transform applied to a special function of the cross correlation matrix. We analyze the properties of the new imaging functional in the high-frequency regime which shows that it produces sharper images than the usual Kirchhoff migration functional. Numerical simulations confirm the theoretical predictions.
Keywords: noise sources, Passive sensor imaging, cross correlation..
Mathematics Subject Classification: Primary: 35R30, 35R60; Secondary: 86A1.
Citation: Josselin Garnier, Knut Solna. Filtered Kirchhoff migration of cross correlations of ambient noise signals. Inverse Problems & Imaging, 2011, 5 (2) : 371-390. doi: 10.3934/ipi.2011.5.371
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. doi: 10.1088/0266-5611/24/1/015011.  Google Scholar

[2]

G. Beylkin, Imaging of discontinuities in the inverse scattering problem by the inversion of a causal Radon transform, J. Math. Phys., 26 (1985), 99-108. doi: 10.1063/1.526755.  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, J. K. Cohen and J. W. Stockwell Jr, "Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion," Springer Verlag, New York, 2001.  Google Scholar

[5]

L. Borcea, G. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460. doi: 10.1088/0266-5611/21/4/015.  Google Scholar

[6]

L. Borcea, G. Papanicolaou and C. Tsogka, Coherent interferometric imaging, Geophysics 71 (2006), S1165-S1175. doi: 10.1190/1.2209541.  Google Scholar

[7]

M. Born and E. Wolf, "Principles of Optics," Cambridge University Press, Cambridge, 1999. 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]

A. Curtis, P. Gerstoft, H. Sato, R. Snieder and K. Wapenaar, Seismic interferometry - turning noise into signal, The Leading Edge, 25 (2006), 1082-1092. doi: 10.1190/1.2349814.  Google Scholar

[11]

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), 909-932. doi: 10.1137/070701790.  Google Scholar

[12]

J. Garnier, Imaging in randomly layered media by cross-correlating noisy signals, SIAM Multiscale Model. Simul., 4 (2005), 610-640. doi: 10.1137/040613226.  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]

O. I. Lobkis and R. L. Weaver, On the emergence of the Green's function in the correlations of a diffuse field, J. Acoustic. Soc. Am., 110 (2001), 3011-3017. doi: 10.1121/1.1417528.  Google Scholar

[15]

F. Natterer and F. Wubbeling, "Mathematical Methods in Image Reconstruction," Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324.  Google Scholar

[16]

P. Roux and M. Fink, Green's function estimation using secondary sources in a shallow water environment, J. Acoust. Soc. Am., 113 (2003), 1406-1416. doi: 10.1121/1.1542645.  Google Scholar

[17]

K. G. Sabra, P. Gerstoft, P. Roux and W. Kuperman, Surface wave tomography from microseisms in Southern California, Geophys. Res. Lett., 32 (2005), L14311. doi: 10.1029/2005GL023155.  Google Scholar

[18]

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

[19]

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. Google Scholar

[20]

K. Wapenaar and J. Fokkema, Green's function representations for seismic interferometry, Geophysics, 71 (2006), SI33-SI46. doi: 10.1190/1.2213955.  Google Scholar

[21]

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. doi: 10.1088/0266-5611/24/1/015011.  Google Scholar

[2]

G. Beylkin, Imaging of discontinuities in the inverse scattering problem by the inversion of a causal Radon transform, J. Math. Phys., 26 (1985), 99-108. doi: 10.1063/1.526755.  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, J. K. Cohen and J. W. Stockwell Jr, "Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion," Springer Verlag, New York, 2001.  Google Scholar

[5]

L. Borcea, G. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460. doi: 10.1088/0266-5611/21/4/015.  Google Scholar

[6]

L. Borcea, G. Papanicolaou and C. Tsogka, Coherent interferometric imaging, Geophysics 71 (2006), S1165-S1175. doi: 10.1190/1.2209541.  Google Scholar

[7]

M. Born and E. Wolf, "Principles of Optics," Cambridge University Press, Cambridge, 1999. 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]

A. Curtis, P. Gerstoft, H. Sato, R. Snieder and K. Wapenaar, Seismic interferometry - turning noise into signal, The Leading Edge, 25 (2006), 1082-1092. doi: 10.1190/1.2349814.  Google Scholar

[11]

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), 909-932. doi: 10.1137/070701790.  Google Scholar

[12]

J. Garnier, Imaging in randomly layered media by cross-correlating noisy signals, SIAM Multiscale Model. Simul., 4 (2005), 610-640. doi: 10.1137/040613226.  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]

O. I. Lobkis and R. L. Weaver, On the emergence of the Green's function in the correlations of a diffuse field, J. Acoustic. Soc. Am., 110 (2001), 3011-3017. doi: 10.1121/1.1417528.  Google Scholar

[15]

F. Natterer and F. Wubbeling, "Mathematical Methods in Image Reconstruction," Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324.  Google Scholar

[16]

P. Roux and M. Fink, Green's function estimation using secondary sources in a shallow water environment, J. Acoust. Soc. Am., 113 (2003), 1406-1416. doi: 10.1121/1.1542645.  Google Scholar

[17]

K. G. Sabra, P. Gerstoft, P. Roux and W. Kuperman, Surface wave tomography from microseisms in Southern California, Geophys. Res. Lett., 32 (2005), L14311. doi: 10.1029/2005GL023155.  Google Scholar

[18]

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

[19]

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. Google Scholar

[20]

K. Wapenaar and J. Fokkema, Green's function representations for seismic interferometry, Geophysics, 71 (2006), SI33-SI46. doi: 10.1190/1.2213955.  Google Scholar

[21]

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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