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June  2017, 11(3): 477-500. doi: 10.3934/ipi.2017022

Ambient noise correlation-based imaging with moving sensors

Mathias Fink 1, and  Josselin Garnier 2,,

1. 

Institut Langevin, ESPCI and CNRS, PSL Research University, 1 rue Jussieu, 75005 Paris, France
2. 

Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France

1 Corresponding author

Received  March 2016 Revised  February 2017 Published  April 2017

Waves can be used to probe and image an unknown medium. Passive imaging uses ambient noise sources to illuminate the medium. This paper considers passive imaging with moving sensors. The motivation is to generate large synthetic apertures, which should result in enhanced resolution. However Doppler effects and lack of reciprocity significantly affect the imaging process. This paper discusses the consequences in terms of resolution and it shows how to design appropriate imaging functions depending on the sensor trajectory and velocity.

Keywords: Passive sensor imaging, correlation-based imaging, ambient noise sources, moving sensors.
Mathematics Subject Classification: Primary: 35R30, 35R60; Secondary: 78A46.
Citation: Mathias Fink, Josselin Garnier. Ambient noise correlation-based imaging with moving sensors. Inverse Problems & Imaging, 2017, 11 (3) : 477-500. doi: 10.3934/ipi.2017022
References:
[1]

H. Ammari, J. Garnier, W. Jing, H. Kang, M. Lim, K. Sølna and H. Wang, Mathematical and Statistical Methods for Multistatic Imaging, Lecture Notes in Mathematics, Vol. 2098, Springer, Berlin, 2013. doi: 10.1007/978-3-319-02585-8.  Google Scholar

[2]

V. Bacot, M. Labousse, A. Eddi, M. Fink and E. Fort, Time reversal and holography with spacetime transformations, Nature Physics, 12 (2016), 972-977.  doi: 10.1038/nphys3810.  Google Scholar

[3]

A. Badon, G. Lerosey, A. C. Boccara, M. Fink and A. Aubry, Retrieving time-dependent Green's functions in optics with low-coherence interferometry, Phys. Rev. Lett., 114 (2015), 023901.  doi: 10.1364/CLEO_QELS.2015.FW1C.1.  Google Scholar

[4]

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

M. Campillo and A. Paul, Long-range correlations in the diffuse seismic coda, Science, 299 (2003), 547-549.  doi: 10.1126/science.1078551.  Google Scholar

[8]

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

[9]

M. Davy, M. Fink and J. de de Rosny, Green's function retrieval and passive imaging from correlations of wideband thermal radiations, Phys. Rev. Lett., 110 (2013), 203901.  doi: 10.1103/PhysRevLett.110.203901.  Google Scholar

[10]

A. Derode, E. Larose, M. Campillo and M. Fink, How to estimate the Green's function of a heterogeneous medium between two passive sensors? Application to acoustic waves, Appl. Phys. Lett., 83 (2003), 3054-3056.  doi: 10.1063/1.1617373.  Google Scholar

[11]

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

[12]

J. Garnier and M. Fink, Super-resolution in time-reversal focusing on a moving source, Wave Motion, 53 (2015), 80-93.  doi: 10.1016/j.wavemoti.2014.11.005.  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 G. Papanicolaou, Passive Imaging with Ambient Noise, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316471807.  Google Scholar

[16]

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

[17]

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

[18]

K. G. Sabra, Influence of the noise sources motion on the estimated Green's functions from ambient noise cross-correlations, J. Acoust. Soc. Am., 127 (2010), 3577-3589.  doi: 10.1121/1.3397612.  Google Scholar

[19]

G. T. Schuster, Seismic Interferometry, Cambridge University Press, Cambridge, 2009. Google Scholar

[20]

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

[21]

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

[22]

K. Wapenaar, Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation, Phys. Rev. Lett., 93 (2004), 254301.  doi: 10.1103/PhysRevLett.93.254301.  Google Scholar

[23]

K. Wapenaar, E. Slob, R. Snieder and A. Curtis, Tutorial on seismic interferometry: Part 2 -Underlying theory and new advances, Geophysics, 75 (2010), 75A211-75A227.  doi: 10.1190/1.3463440.  Google Scholar

[24]

R. Weaver and O. I. Lobkis, Ultrasonics without a source: Thermal fluctuation correlations at MHz frequencies, Phys. Rev. Lett., 87 (2011), 134301.  doi: 10.1103/PhysRevLett.87.134301.  Google Scholar

[25]

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

[26]

Throughout the paper, symbols of scalar quantities are printed in italic type and symbols of vectors are printed in bold italic type. Google Scholar

show all references

References:
[1]

H. Ammari, J. Garnier, W. Jing, H. Kang, M. Lim, K. Sølna and H. Wang, Mathematical and Statistical Methods for Multistatic Imaging, Lecture Notes in Mathematics, Vol. 2098, Springer, Berlin, 2013. doi: 10.1007/978-3-319-02585-8.  Google Scholar

[2]

V. Bacot, M. Labousse, A. Eddi, M. Fink and E. Fort, Time reversal and holography with spacetime transformations, Nature Physics, 12 (2016), 972-977.  doi: 10.1038/nphys3810.  Google Scholar

[3]

A. Badon, G. Lerosey, A. C. Boccara, M. Fink and A. Aubry, Retrieving time-dependent Green's functions in optics with low-coherence interferometry, Phys. Rev. Lett., 114 (2015), 023901.  doi: 10.1364/CLEO_QELS.2015.FW1C.1.  Google Scholar

[4]

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

M. Campillo and A. Paul, Long-range correlations in the diffuse seismic coda, Science, 299 (2003), 547-549.  doi: 10.1126/science.1078551.  Google Scholar

[8]

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

[9]

M. Davy, M. Fink and J. de de Rosny, Green's function retrieval and passive imaging from correlations of wideband thermal radiations, Phys. Rev. Lett., 110 (2013), 203901.  doi: 10.1103/PhysRevLett.110.203901.  Google Scholar

[10]

A. Derode, E. Larose, M. Campillo and M. Fink, How to estimate the Green's function of a heterogeneous medium between two passive sensors? Application to acoustic waves, Appl. Phys. Lett., 83 (2003), 3054-3056.  doi: 10.1063/1.1617373.  Google Scholar

[11]

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

[12]

J. Garnier and M. Fink, Super-resolution in time-reversal focusing on a moving source, Wave Motion, 53 (2015), 80-93.  doi: 10.1016/j.wavemoti.2014.11.005.  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 G. Papanicolaou, Passive Imaging with Ambient Noise, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316471807.  Google Scholar

[16]

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

[17]

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

[18]

K. G. Sabra, Influence of the noise sources motion on the estimated Green's functions from ambient noise cross-correlations, J. Acoust. Soc. Am., 127 (2010), 3577-3589.  doi: 10.1121/1.3397612.  Google Scholar

[19]

G. T. Schuster, Seismic Interferometry, Cambridge University Press, Cambridge, 2009. Google Scholar

[20]

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

[21]

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

[22]

K. Wapenaar, Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation, Phys. Rev. Lett., 93 (2004), 254301.  doi: 10.1103/PhysRevLett.93.254301.  Google Scholar

[23]

K. Wapenaar, E. Slob, R. Snieder and A. Curtis, Tutorial on seismic interferometry: Part 2 -Underlying theory and new advances, Geophysics, 75 (2010), 75A211-75A227.  doi: 10.1190/1.3463440.  Google Scholar

[24]

R. Weaver and O. I. Lobkis, Ultrasonics without a source: Thermal fluctuation correlations at MHz frequencies, Phys. Rev. Lett., 87 (2011), 134301.  doi: 10.1103/PhysRevLett.87.134301.  Google Scholar

[25]

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

[26]

Throughout the paper, symbols of scalar quantities are printed in italic type and symbols of vectors are printed in bold italic type. Google Scholar

Figure 1.  Experimental set-up for passive Green's function estimation in Section 2. The circles are noise sources (at the surface $\partial B$), the triangle is a receiver at ${\boldsymbol{x}}_{\rm r}(t)$ on a circular trajectory (with radius $R_0$), and the shaded area is a complex medium
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Figure 2.  Experimental set-up for passive reflector imaging in Section 3. The circles are noise sources (at the surface $\partial B$), the triangle is a receiver at ${\boldsymbol{x}}_{\rm r}(t)$ on a circular trajectory (with radius $R_0$), andthe diamond is a reflector at ${\boldsymbol{y}}_{\rm ref}$
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Figure 3.  xperimental set-up for passive reflector imaging in Section 4. The circles are noise sources (at the surface $\partial B$), the triangle is a receiver on a linear trajectory (with length $a$), andthe diamond is a reflector
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Figure 4.  Experimental set-up for passive Green's function estimation in Section 5. The circle is the trajectory of the moving source ${\boldsymbol{x}}_{\rm s}(t)$ and the two triangles are two observation points at $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$
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Figure 5.  Experimental set-up for the time-reversal experiment in Appendix A. The source xs(t) is moving on a circular trajectory (with radius R0) and the triangles are the sources/receivers of the time-reversal mirror (on ∂B)
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