American Institute of Mathematical Sciences

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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 and 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.

[2]

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

[3]

N. Bleistein, J. K. Cohen and J. W. Stockwell Jr, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, Springer-Verlag, New York, 2001.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

N. Bleistein, J. K. Cohen and J. W. Stockwell Jr, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, Springer-Verlag, New York, 2001.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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