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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

 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.
Citation: Josselin Garnier, Knut Solna. Filtered Kirchhoff migration of cross correlations of ambient noise signals. Inverse Problems and 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. [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. [3] B. L. Biondi, "3D Seismic Imaging," no. 14 in Investigations in Geophysics, Society of Exploration Geophysics, Tulsa, 2006. [4] N. Bleistein, J. K. Cohen and J. W. Stockwell Jr, "Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion," Springer Verlag, New York, 2001. [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. [6] L. Borcea, G. Papanicolaou and C. Tsogka, Coherent interferometric imaging, Geophysics 71 (2006), S1165-S1175. doi: 10.1190/1.2209541. [7] M. Born and E. Wolf, "Principles of Optics," Cambridge University Press, Cambridge, 1999. [8] J. F. Claerbout, "Imaging the Earth's Interior," Blackwell Scientific Publications, Palo Alto, 1985. [9] Y. Colin de Verdière, Semiclassical analysis and passive imaging, Nonlinearity, 22 (2009), R45-R75. doi: 10.1088/0951-7715/22/6/R01. [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. [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. [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. [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. [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. [15] F. Natterer and F. Wubbeling, "Mathematical Methods in Image Reconstruction," Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324. [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. [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. [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. [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. [20] K. Wapenaar and J. Fokkema, Green's function representations for seismic interferometry, Geophysics, 71 (2006), SI33-SI46. doi: 10.1190/1.2213955. [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.

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##### 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. [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. [3] B. L. Biondi, "3D Seismic Imaging," no. 14 in Investigations in Geophysics, Society of Exploration Geophysics, Tulsa, 2006. [4] N. Bleistein, J. K. Cohen and J. W. Stockwell Jr, "Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion," Springer Verlag, New York, 2001. [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. [6] L. Borcea, G. Papanicolaou and C. Tsogka, Coherent interferometric imaging, Geophysics 71 (2006), S1165-S1175. doi: 10.1190/1.2209541. [7] M. Born and E. Wolf, "Principles of Optics," Cambridge University Press, Cambridge, 1999. [8] J. F. Claerbout, "Imaging the Earth's Interior," Blackwell Scientific Publications, Palo Alto, 1985. [9] Y. Colin de Verdière, Semiclassical analysis and passive imaging, Nonlinearity, 22 (2009), R45-R75. doi: 10.1088/0951-7715/22/6/R01. [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. [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. [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. [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. [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. [15] F. Natterer and F. Wubbeling, "Mathematical Methods in Image Reconstruction," Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324. [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. [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. [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. [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. [20] K. Wapenaar and J. Fokkema, Green's function representations for seismic interferometry, Geophysics, 71 (2006), SI33-SI46. doi: 10.1190/1.2213955. [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.
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