Signal to noise ratio estimation in passive correlation based - - PowerPoint PPT Presentation

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

Problèmes Inverses, Contrôle et Optimisation de Formes École Polytechnique, Palaiseau, 2-4 April 2012

• A. SEMIN

(IACM FORTH) SNR Estimation April 2-4 2012 1 / 26

Table of contents

1

Introduction

2

Mathematical model Wave equation Cross correlations

3

Migration imaging

4

Numerical results

5

Conclusion

• A. SEMIN

(IACM FORTH) SNR Estimation April 2-4 2012 2 / 26

Table of contents

1

Introduction

2

Mathematical model Wave equation Cross correlations

3

Migration imaging

4

Numerical results

5

Conclusion

• A. SEMIN

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Cross-correlations of ambient noise recordings

We consider here the problem of imaging using passive incoherent recordings due to ambient noise sources.

• A. SEMIN

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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

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Our goal

Our aim is to use these cross-correlations in order to image reflectors (embedded in clutter).

• A. SEMIN

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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

1

Introduction

2

Mathematical model Wave equation Cross correlations

3

Migration imaging

4

Numerical results

5

Conclusion

• A. SEMIN

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Mathematical model

We consider a domain Ω containing a reflector O

• A. SEMIN

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Mathematical model

We consider a domain Ω containing a reflector O We consider u(t, x) solution of the time-dependent wave equation        ∂2u ∂t2 (t, x) − c2

0∆u = n(t, x)

in Ω \ O u(t, x) = 0

• n ∂O

+ 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        ∂2u ∂t2 (t, x) − c2

0∆u = n(t, x)

in Ω \ O u(t, x) = 0

• n ∂O

+ PML (to model free-space problem) c0 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        ∂2u ∂t2 (t, x) − c2

0∆u = n(t, x)

in Ω \ O u(t, x) = 0

• n ∂O

+ PML (to model free-space problem) c0 is homogeneous propagation speed. n(t, x) models noise sources.

• A. SEMIN

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Noise source term

n(t, x) is a zero mean stationary (in time) random process : E {n(t, x)} = 0

• A. SEMIN

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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(t1, x1)n(t2, x2)} = F(t2 − t1)K(x1)δ(x1 − x2)

• 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(t1, x1)n(t2, x2)} = F(t2 − t1)K(x1)δ(x1 − x2)

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(t1, x1)n(t2, x2)} = F(t2 − t1)K(x1)δ(x1 − x2)

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

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Cross correlations

assume that we know u(t, x1) and u(t, x2) the time-dependant wave fields recorded by two sensors at x1 and x2 on a time interval [0, T]

20 40 60 80 100 −2 −1 1 2 20 40 60 80 100 −2 −1 1 2

• A. SEMIN

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Cross correlations

assume that we know u(t, x1) and u(t, x2) the time-dependant wave fields recorded by two sensors at x1 and x2 on a time interval [0, T]

20 40 60 80 100 −2 −1 1 2 20 40 60 80 100 −2 −1 1 2

their cross-correlation function over the time interval [0, T], and with time lag τ is given by CT (τ, x1, x2) = 1 T

• u(t, x1)u(t + τ, x2)dt
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Properties

The expectation of CT (with respect to the distribution of the sources) is independent of T : E {CT (τ, x1, x2)} = C(1)(τ, x1, x2) with C(1)(τ, x1, x2) = 1 2π

• ˆ

D(ω, x1, x2)ˆ F(ω)e−ıωτdω ˆ D(ω, x1, x2) =

• ˆ

G(ω, x1, y) ˆ G(ω, x2, y)K(y)dy

• A. SEMIN

(IACM FORTH) SNR Estimation April 2-4 2012 10 / 26

Properties

The expectation of CT (with respect to the distribution of the sources) is independent of T : E {CT (τ, x1, x2)} = C(1)(τ, x1, x2) with C(1)(τ, x1, x2) = 1 2π

• ˆ

D(ω, x1, x2)ˆ F(ω)e−ıωτdω ˆ D(ω, x1, x2) =

• ˆ

G(ω, x1, y) ˆ G(ω, x2, y)K(y)dy The empirical cross correlation CT is a statistical stable self-averaging quantity, i.e. : CT (τ, x1, x2)

T →∞

− → C(1)(τ, x1, x2)

• A. SEMIN

(IACM FORTH) SNR Estimation April 2-4 2012 10 / 26

Table of contents

1

Introduction

2

Mathematical model Wave equation Cross correlations

3

Migration imaging

4

Numerical results

5

Conclusion

• A. SEMIN

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Assumptions

noise sources are spatially localised (gray region) passive sensors (xj)1jJ are located between the sources and the reflector

• A. SEMIN

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Stationary phase analysis

Stationary phase analysis done by J. Garnier and G. Papanicolaou shows that the cross correlation C(1)(τ, x1, x2) between two sensors x1 and x2 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

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Stationary phase analysis

Stationary phase analysis done by J. Garnier and G. Papanicolaou shows that the cross correlation C(1)(τ, x1, x2) between two sensors x1 and x2 has two peaks

• ne peak (direct arrival) at the travel time between x1 and x2

|x1 − x2| c0

• 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

Stationary phase analysis

Stationary phase analysis done by J. Garnier and G. Papanicolaou shows that the cross correlation C(1)(τ, x1, x2) between two sensors x1 and x2 has two peaks

• ne peak (direct arrival) at the travel time between x1 and x2

|x1 − x2| c0

• ne peak (reflected arrival) at the sum of the travel times between xi and the

reflector |x1 − zO| c0 + |zO − x2| c0

• 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

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Daylight imaging functional

To keep only the interesting peak that concerns the reflector, the image at a search point z is computed using the following daylight imaging functional : ID(z) =

J

• j,l=1

C(1),sym

coda

(τ(z, xl) + τ(z, xj), xj, xl),

• A. SEMIN

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Daylight imaging functional

To keep only the interesting peak that concerns the reflector, the image at a search point z is computed using the following daylight imaging functional : ID(z) =

J

• j,l=1

C(1),sym

coda

(τ(z, xl) + τ(z, xj), xj, xl), τ(z, xj) is the travel time between z and xj : τ(z, xj) = |z − xj| c0

• A. SEMIN

(IACM FORTH) SNR Estimation April 2-4 2012 14 / 26

Daylight imaging functional

To keep only the interesting peak that concerns the reflector, the image at a search point z is computed using the following daylight imaging functional : ID(z) =

J

• j,l=1

C(1),sym

coda

(τ(z, xl) + τ(z, xj), xj, xl), τ(z, xj) is the travel time between z and xj : τ(z, xj) = |z − xj| c0 Csym

T,coda is defined by

C(1),sym

coda

(t, xj, xl) =

• C(1)(t, xj, xl) + C(1)(−t, xj, xl)
• 1]τ(xj,xl),+∞[(t).
• A. SEMIN

(IACM FORTH) SNR Estimation April 2-4 2012 14 / 26

Daylight imaging functional

To keep only the interesting peak that concerns the reflector, the image at a search point z is computed using the following daylight imaging functional : ID(z) =

J

• j,l=1

C(1),sym

coda

(τ(z, xl) + τ(z, xj), xj, xl), Csym

T,coda is defined by

C(1),sym

coda

(t, xj, xl) =

• C(1)(t, xj, xl) + C(1)(−t, xj, xl)
• 1]τ(xj,xl),+∞[(t).

Proposition

If we consider that the reflector is far enough from the receivers and T is large enough, then we can approximate ID(z) by ID(z) ≃ 2

J

• j,l=1

CT (τ(z, xl) + τ(z, xj), xj, xl),

• A. SEMIN

(IACM FORTH) SNR Estimation April 2-4 2012 14 / 26

Table of contents

1

Introduction

2

Mathematical model Wave equation Cross correlations

3

Migration imaging

4

Numerical results

5

Conclusion

• A. SEMIN

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Simulation setup

wave equation on the rectangle [0, 50λ] × [−15λ, 15λ], with a reflector located on [44λ, 46λ] × [−λ, λ], random distribution of sources has support on the rectangle ΩS = [0, 4λ] × [−15λ, 15λ], we record the solution u of the wave equation at J recievers located at xj = (5λ, (j − (J + 1)/2)λ/2), for 1 j J, c0 = 3km s−1,

Figure: Geometry of the passive sensor imaging problem for a daylight illumination.

• A. SEMIN

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Source term

Dependence in space : K(x) is given by K(x) = 1 |ΩS|1ΩS(x)

• A. SEMIN

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Source term

Dependence in space : K(x) is given by K(x) = 1 |ΩS|1ΩS(x) Dependence in time : F(t) is given by F(t) = 1 3T

• F(τ)F(t + τ)dτ

F(t) = sin(B(t − T/2)) B(t − T/2) cos(2πf(t − T/2)) exp

• −(t − T/2)2

2c2

t

• B = 0.3 Hz is the bandwidth

f = 0.3 Hz is the central frequency ct = 2.5 s is the correlation time

• A. SEMIN

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Computation

Space discretization is done by 8-th order mixed spectral finite elements with Gauss-Lobatto points Time discretization is done by 4-th order Runge-Kutta. We implement the source term by n(t, x) = 1 √Ns

Ns

• s=1

δ(x − xs) ˆ F−1 (rsF(F)) (t)

• A. SEMIN

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Computation

Space discretization is done by 8-th order mixed spectral finite elements with Gauss-Lobatto points Time discretization is done by 4-th order Runge-Kutta. We implement the source term by n(t, x) = 1 √Ns

Ns

• s=1

δ(x − xs) ˆ F−1 (rsF(F)) (t) rs(ω) is a random distribution satisfying rs(−ω) = rs(ω) and following expectations : E {rs(ω)} = 0 and E {rs(ω1)rs(ω2)} = 1 3δ(ω1 + ω2)

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Daylight imaging functional

Figure: Daylight imaging functional for the homogeneous medium. J = 21

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Daylight imaging functional

Figure: Daylight imaging functional for the homogeneous medium. J = 31

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Daylight imaging functional

Figure: Daylight imaging functional for the homogeneous medium. J = 41

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Daylight imaging functional

Figure: Daylight imaging functional for the homogeneous medium. J = 51

• A. SEMIN

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SNR ratio

A good question that naturally arise is “Under what conditions do we obtain such a good image” ?

• A. SEMIN

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SNR ratio

A good question that naturally arise is “Under what conditions do we obtain such a good image” ? In other words, “What are the parameters that control the quality of the image, and how” ?

• A. SEMIN

(IACM FORTH) SNR Estimation April 2-4 2012 20 / 26

SNR ratio

A good question that naturally arise is “Under what conditions do we obtain such a good image” ? In other words, “What are the parameters that control the quality of the image, and how” ? A partial answer to this question is that the signal to noise ratio (SNR) of the image increases with the number of the receivers, where SNR = |ID|(z∗) maxz=z∗|ID|(z) where z∗ is the point where the image admits its maximal value and z = z∗ means that squares of size 2λ × 2λ centered at z and z∗ do not intersect.

• A. SEMIN

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SNR ratio versus number of receivers

20 25 30 35 40 45 50 55 60 65 1.5 2 2.5 3 3.5 4 4.5

Figure: SNR computation versus number of receivers (B = 0.3 Hz, T = 800 s)

• A. SEMIN

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SNR ratio versus time

2000 4000 6000 8000 10000 12000 14000 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure: SNR computation versus time (J = 21, B = 0.3 Hz)

• A. SEMIN

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SNR ratio versus scaled bandwidth

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Figure: SNR computation versus scaled bandwith (J = 61, T = 800 s)

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SNR interpretation

SNR ratio is linear with respect to number of receivers

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SNR interpretation

SNR ratio is linear with respect to number of receivers SNR ratio is linear with respect to square root of time

• A. SEMIN

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SNR interpretation

SNR ratio is linear with respect to number of receivers SNR ratio is linear with respect to square root of time SNR ratio is linear with respect to square root of bandwidth

• A. SEMIN

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SNR interpretation

SNR ratio is linear with respect to number of receivers SNR ratio is linear with respect to square root of time SNR ratio is linear with respect to square root of bandwidth We can summarize these results by SNR ∼ J √ BT

• J. Garnier, G. Papanicolaou, A. Semin and C. Tsogka, “Signal to Noise Ratio

estimation in passive correlation based imaging”, preprint, 2011.

• A. SEMIN

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Table of contents

1

Introduction

2

Mathematical model Wave equation Cross correlations

3

Migration imaging

4

Numerical results

5

Conclusion

• A. SEMIN

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Conclusion and perspectives

SNR analysis consistent with theory

• A. SEMIN

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Conclusion and perspectives

SNR analysis consistent with theory Exploit higher order cross-correlations to improve SNR ?

• J. Garnier and G. Papanicolaou, “Fluctuation theory of ambient noise

imaging”, C. R. Geoscience, 2011.

• B. Froment, M. Campillo and P. Roux, “Reconstructing the Green’s

function trough iteration of correlations”, C.R. Geoscience, 2011.

• A. SEMIN

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Conclusion and perspectives

SNR analysis consistent with theory Exploit higher order cross-correlations to improve SNR ?

• J. Garnier and G. Papanicolaou, “Fluctuation theory of ambient noise

imaging”, C. R. Geoscience, 2011.

• B. Froment, M. Campillo and P. Roux, “Reconstructing the Green’s

function trough iteration of correlations”, C.R. Geoscience, 2011. Perspective : SNR analysis for inhomogeneous media.

• A. SEMIN

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