Signal to noise ratio estimation in passive correlation based imaging Adrien SEMIN Joint work with J. Garnier, G. Papanicolaou and C. Tsogka Institute of Applied and Computational Mathematics Foundation for Research and Technology - Hellas P roblèmes I nverses, C ontrôle et O ptimisation de F ormes École Polytechnique, Palaiseau, 2-4 April 2012 A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 1 / 26
Table of contents Introduction 1 Mathematical model 2 Wave equation Cross correlations Migration imaging 3 Numerical results 4 Conclusion 5 A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 2 / 26
Table of contents Introduction 1 Mathematical model 2 Wave equation Cross correlations Migration imaging 3 Numerical results 4 Conclusion 5 A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 3 / 26
Cross-correlations of ambient noise recordings We consider here the problem of imaging using passive incoherent recordings due to ambient noise sources. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 4 / 26
Cross-correlations of ambient noise recordings We consider here the problem of imaging using passive incoherent recordings due to ambient noise sources. Idea : use cross-correlations between pairs of sensors (receivers) to retrieve information about the Green’s function in the background medium. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 4 / 26
Cross-correlations of ambient noise recordings We consider here the problem of imaging using passive incoherent recordings due to ambient noise sources. Idea : use cross-correlations between pairs of sensors (receivers) to retrieve information about the Green’s function in the background medium. achieved either with equi-distribution of sources or through multiple scattering R.L. Weaver and O.I. Lobkis, “Ultrasonics without a source : thermal fluctuation correlations at MHz frequencies”, Phys. Rev. Lett. 87 (13), 134301, 2001. R. Snieder, “Extracting the Green’s function from the correlation of coda waves : a derivation based on stationary phase”, Phys. Rev. E 69, 046610, 2005. C. Bardos, J. Garnier and G. Papanicolaou, “Identification of Green’s functions singularities by cross-correlation of noisy signals”, Inverse Problems, 24, 015011, 2008. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 4 / 26
Our goal Our aim is to use these cross-correlations in order to image reflectors (embedded in clutter). A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 5 / 26
Our goal Our aim is to use these cross-correlations in order to image reflectors (embedded in clutter). To do so we will use coherent imaging methods, such as travel time migration (and coherent interferometry (CINT)). A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 5 / 26
Our goal Our aim is to use these cross-correlations in order to image reflectors (embedded in clutter). To do so we will use coherent imaging methods, such as travel time migration (and coherent interferometry (CINT)). Here we will focus on Signal to Noise Ratio estimation in passive correlation based imaging. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 5 / 26
Our goal Our aim is to use these cross-correlations in order to image reflectors (embedded in clutter). To do so we will use coherent imaging methods, such as travel time migration (and coherent interferometry (CINT)). Here we will focus on Signal to Noise Ratio estimation in passive correlation based imaging. Application : Structural Health Monitoring E. Larose, O.I. Lobkis, and R. L. Weaver, “Passive correlation imaging of a buried scatterer”, J. Acoust. Soc. Am., 119 (6), 2006. K. Sabra et al, “Structural health monitoring by extraction of coherent guided waves from diffuse fields”, J. Acoust. Soc. Am. 123 (1), 2008. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 5 / 26
Table of contents Introduction 1 Mathematical model 2 Wave equation Cross correlations Migration imaging 3 Numerical results 4 Conclusion 5 A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 6 / 26
Mathematical model We consider a domain Ω containing a reflector O A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 7 / 26
Mathematical model We consider a domain Ω containing a reflector O We consider u ( t, x ) solution of the time-dependent wave equation ∂ 2 u ∂t 2 ( t, x ) − c 2 in Ω \ O 0 ∆ u = n ( t, x ) on ∂ O u ( t, x ) = 0 + PML (to model free-space problem) A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 7 / 26
Mathematical model We consider a domain Ω containing a reflector O We consider u ( t, x ) solution of the time-dependent wave equation ∂ 2 u ∂t 2 ( t, x ) − c 2 in Ω \ O 0 ∆ u = n ( t, x ) on ∂ O u ( t, x ) = 0 + PML (to model free-space problem) c 0 is homogeneous propagation speed. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 7 / 26
Mathematical model We consider a domain Ω containing a reflector O We consider u ( t, x ) solution of the time-dependent wave equation ∂ 2 u ∂t 2 ( t, x ) − c 2 in Ω \ O 0 ∆ u = n ( t, x ) on ∂ O u ( t, x ) = 0 + PML (to model free-space problem) c 0 is homogeneous propagation speed. n ( t, x ) models noise sources. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 7 / 26
Noise source term n ( t, x ) is a zero mean stationary (in time) random process : E { n ( t, x ) } = 0 A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 8 / 26
Noise source term n ( t, x ) is a zero mean stationary (in time) random process : E { n ( t, x ) } = 0 n ( t, x ) satisfies the following cross-correlation relation E { n ( t 1 , x 1 ) n ( t 2 , x 2 ) } = F ( t 2 − t 1 ) K ( x 1 ) δ ( x 1 − x 2 ) A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 8 / 26
Noise source term n ( t, x ) is a zero mean stationary (in time) random process : E { n ( t, x ) } = 0 n ( t, x ) satisfies the following cross-correlation relation E { n ( t 1 , x 1 ) n ( t 2 , x 2 ) } = F ( t 2 − t 1 ) K ( x 1 ) δ ( x 1 − x 2 ) F ( t ) is a real-valued even function, has its maximum on t = 0 and has a positive Fourier transform A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 8 / 26
Noise source term n ( t, x ) is a zero mean stationary (in time) random process : E { n ( t, x ) } = 0 n ( t, x ) satisfies the following cross-correlation relation E { n ( t 1 , x 1 ) n ( t 2 , x 2 ) } = F ( t 2 − t 1 ) K ( x 1 ) δ ( x 1 − x 2 ) F ( t ) is a real-valued even function, has its maximum on t = 0 and has a positive Fourier transform K ( x ) characterize the spatial support of noise sources. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 8 / 26
Cross correlations assume that we know u ( t, x 1 ) and u ( t, x 2 ) the time-dependant wave fields recorded by two sensors at x 1 and x 2 on a time interval [0 , T ] 2 1 0 −1 −2 0 20 40 60 80 100 2 1 0 −1 −2 0 20 40 60 80 100 A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 9 / 26
Cross correlations assume that we know u ( t, x 1 ) and u ( t, x 2 ) the time-dependant wave fields recorded by two sensors at x 1 and x 2 on a time interval [0 , T ] 2 1 0 −1 −2 0 20 40 60 80 100 2 1 0 −1 −2 0 20 40 60 80 100 their cross-correlation function over the time interval [0 , T ] , and with time lag τ is given by C T ( τ, x 1 , x 2 ) = 1 � u ( t, x 1 ) u ( t + τ, x 2 ) dt T A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 9 / 26
Properties The expectation of C T (with respect to the distribution of the sources) is independent of T : E { C T ( τ, x 1 , x 2 ) } = C (1) ( τ, x 1 , x 2 ) with C (1) ( τ, x 1 , x 2 ) = 1 � D ( ω, x 1 , x 2 )ˆ ˆ F ( ω ) e − ıωτ dω 2 π � ˆ G ( ω, x 1 , y ) ˆ ˆ D ( ω, x 1 , x 2 ) = G ( ω, x 2 , y ) K ( y ) dy A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 10 / 26
Properties The expectation of C T (with respect to the distribution of the sources) is independent of T : E { C T ( τ, x 1 , x 2 ) } = C (1) ( τ, x 1 , x 2 ) with C (1) ( τ, x 1 , x 2 ) = 1 � D ( ω, x 1 , x 2 )ˆ ˆ F ( ω ) e − ıωτ dω 2 π � ˆ G ( ω, x 1 , y ) ˆ ˆ D ( ω, x 1 , x 2 ) = G ( ω, x 2 , y ) K ( y ) dy The empirical cross correlation C T is a statistical stable self-averaging quantity, i.e. : T →∞ → C (1) ( τ, x 1 , x 2 ) C T ( τ, x 1 , x 2 ) − A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 10 / 26
Table of contents Introduction 1 Mathematical model 2 Wave equation Cross correlations Migration imaging 3 Numerical results 4 Conclusion 5 A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 11 / 26
Assumptions noise sources are spatially localised (gray region) passive sensors ( x j ) 1 � j � J are located between the sources and the reflector A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 12 / 26
Stationary phase analysis Stationary phase analysis done by J. Garnier and G. Papanicolaou shows that the cross correlation C (1) ( τ, x 1 , x 2 ) between two sensors x 1 and x 2 has two peaks J. Garnier and G. Papanicolaou, “Passive sensor imaging using cross correlations of noisy signals in a scattering medium”, SIAM J. Imaging Sciences, Vol. 2, pp. 396–437, 2009. A. SEMIN (IACM FORTH) SNR Estimation April 2-4 2012 13 / 26
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