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June  2014, 19(4): 979-998. doi: 10.3934/dcdsb.2014.19.979

High frequency analysis of imaging with noise blending

Ennio Fedrizzi 1,

1. 

Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Received  October 2012 Revised  January 2014 Published  April 2014

We consider sensor array imaging for simultaneous noise blended sources. We study a migration imaging functional and we analyze its sensitivity to singular perturbations of the speed of propagation of the medium. We consider two kinds of random sources: randomly delayed pulses and stationary random processes, and three possible kinds of perturbations. Using high frequency analysis we prove the statistical stability (with respect to the realization of the noise blending) of the scheme and obtain quantitative results on the image contrast provided by the imaging functional, which strongly depends on the type of perturbations.
Keywords: singular perturbations., high-frequency regime, noise blending, Imaging functional, simultaneous sources.
Mathematics Subject Classification: Primary: 35R30, 35R60, 35Q86; Secondary: 86A15, 35L0.
Citation: Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979
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]

A. J. Berkhout, Changing the mindset in seismic data acquisition, The Leading Edge, 27 (2008), 924-938. doi: 10.1190/1.2954035.  Google Scholar

[3]

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

[4]

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

[5]

F. Brenguier, N. M. Shapiro, M. Campillo, A. Nercessian and V. Ferrazzini, 3-D surface wave tomography of the Piton de la Fournaise volcano using seismic noise correlations, Geophysical Research Letters, 34 (2007), L02305. doi: 10.1029/2006GL028586.  Google Scholar

[6]

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

[7]

M. De Hoop, E. Fedrizzi, J. Garnier and K. Sølna, Imaging with noise blending, Contemporary Mathematics, 577 (2012), 105-124. doi: 10.1090/conm/577/11466.  Google Scholar

[8]

M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J.-L. Thomas and F. Wu, Time-reversed acoustics, Reports on Progress in Physics, 63 (2000), 1933-1995. doi: 10.1088/0034-4885/63/12/202.  Google Scholar

[9]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007.  Google Scholar

[10]

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

[11]

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

[12]

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. doi: 10.1111/j.1365-2478.2007.00684.x.  Google Scholar

[13]

G. Hampson, J. Stefani and F. Herkenhoff, Acquisition using simultaneous sources, The Leading Edge, 27 (2008), 918-923. doi: 10.1190/1.2954034.  Google Scholar

[14]

E. Larose, L. Margerin, A. Derode, B. Van Tiggelen, M. Campillo, N. Shapiro, A. Paul, L. Stehly and M. Tanter, Correlation of random wave fields: an interdisciplinary review, Geophysics, 71 (2006), SI11-SI21. doi: 10.1190/1.2213356.  Google Scholar

[15]

A. Mahdad, P. Doulgeris and G. Blacquiere, Separation of blended data by iterative estimation and subtraction of blending interference noise, Geophysics, 76 (2011), Q9-Q17. doi: 10.1190/1.3556597.  Google Scholar

[16]

K. G. Sabra, P. Roux, P. Gerstoft, W. A. Kuperman and M. C . Fehler, Extracting coherent coda arrivals from cross correlations of long period seismic waves during the Mount St. Helens 2004 eruption, Geophysical Research Letters, 33 (2006), L06313. doi: 10.1029/2005GL025563.  Google Scholar

[17]

G. T. Schuster, X. Wang, Y. Huang, W. Dai and C. Boonyasiriwat, Theory of multisource crosstalk reduction by phase-encoded statics, Geophysical Journal International, 184 (2011), 1289-1303. doi: 10.1111/j.1365-246X.2010.04906.x.  Google Scholar

[18]

N. M. Shapiro, M. Campillo, L. Stehly and M. H. Ritzwoller, High-resolution surface-wave tomography from ambient seismic 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, Journal of Geophysical Research, 111 (2006), B10306. doi: 10.1029/2005JB004237.  Google Scholar

[20]

D. J. E. Verschuur and A. J. G. Berkhout, Seismic migration of blended shot records with surface-related multiple scattering, Geophysics, 76 (2011), A7-A13. doi: 10.1190/1.3521658.  Google Scholar

[21]

K. Wapenaar, J. van der Neut and J. Thorbecke, On the relation between seismic interferometry and the simultaneous-source method, Geophysical Prospecting, 60 (2012), 802-823. doi: 10.1111/j.1365-2478.2012.01056.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]

A. J. Berkhout, Changing the mindset in seismic data acquisition, The Leading Edge, 27 (2008), 924-938. doi: 10.1190/1.2954035.  Google Scholar

[3]

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

[4]

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

[5]

F. Brenguier, N. M. Shapiro, M. Campillo, A. Nercessian and V. Ferrazzini, 3-D surface wave tomography of the Piton de la Fournaise volcano using seismic noise correlations, Geophysical Research Letters, 34 (2007), L02305. doi: 10.1029/2006GL028586.  Google Scholar

[6]

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

[7]

M. De Hoop, E. Fedrizzi, J. Garnier and K. Sølna, Imaging with noise blending, Contemporary Mathematics, 577 (2012), 105-124. doi: 10.1090/conm/577/11466.  Google Scholar

[8]

M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J.-L. Thomas and F. Wu, Time-reversed acoustics, Reports on Progress in Physics, 63 (2000), 1933-1995. doi: 10.1088/0034-4885/63/12/202.  Google Scholar

[9]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007.  Google Scholar

[10]

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

[11]

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

[12]

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. doi: 10.1111/j.1365-2478.2007.00684.x.  Google Scholar

[13]

G. Hampson, J. Stefani and F. Herkenhoff, Acquisition using simultaneous sources, The Leading Edge, 27 (2008), 918-923. doi: 10.1190/1.2954034.  Google Scholar

[14]

E. Larose, L. Margerin, A. Derode, B. Van Tiggelen, M. Campillo, N. Shapiro, A. Paul, L. Stehly and M. Tanter, Correlation of random wave fields: an interdisciplinary review, Geophysics, 71 (2006), SI11-SI21. doi: 10.1190/1.2213356.  Google Scholar

[15]

A. Mahdad, P. Doulgeris and G. Blacquiere, Separation of blended data by iterative estimation and subtraction of blending interference noise, Geophysics, 76 (2011), Q9-Q17. doi: 10.1190/1.3556597.  Google Scholar

[16]

K. G. Sabra, P. Roux, P. Gerstoft, W. A. Kuperman and M. C . Fehler, Extracting coherent coda arrivals from cross correlations of long period seismic waves during the Mount St. Helens 2004 eruption, Geophysical Research Letters, 33 (2006), L06313. doi: 10.1029/2005GL025563.  Google Scholar

[17]

G. T. Schuster, X. Wang, Y. Huang, W. Dai and C. Boonyasiriwat, Theory of multisource crosstalk reduction by phase-encoded statics, Geophysical Journal International, 184 (2011), 1289-1303. doi: 10.1111/j.1365-246X.2010.04906.x.  Google Scholar

[18]

N. M. Shapiro, M. Campillo, L. Stehly and M. H. Ritzwoller, High-resolution surface-wave tomography from ambient seismic 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, Journal of Geophysical Research, 111 (2006), B10306. doi: 10.1029/2005JB004237.  Google Scholar

[20]

D. J. E. Verschuur and A. J. G. Berkhout, Seismic migration of blended shot records with surface-related multiple scattering, Geophysics, 76 (2011), A7-A13. doi: 10.1190/1.3521658.  Google Scholar

[21]

K. Wapenaar, J. van der Neut and J. Thorbecke, On the relation between seismic interferometry and the simultaneous-source method, Geophysical Prospecting, 60 (2012), 802-823. doi: 10.1111/j.1365-2478.2012.01056.x.  Google Scholar

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