Green’s function retrieval through cross-correlations …hestia.lgs.jussieu.fr/~boschil/publications/colombi14.pdfcross-correlation are symmetric with respect to zero time (Fan and

Green’s function retrieval through cross-correlationsin a two-dimensional complex reverberating medium

Andrea Colombia)

ISTerre, Universit�e Joseph Fourier, Grenoble, BP 53 38041 Grenoble CEDEX 9, France

Lapo BoschiISTEP, Institut des Sciences de la Terre de Paris, UPMC, Universit Pierre et Marie Curie,4 place Jussieu F-75005, Paris, France

Philippe Roux and Michel CampilloISTerre, Universit�e Joseph Fourier, Grenoble, BP 53 38041 Grenoble CEDEX 9, France

(Received 17 July 2013; revised 2 January 2014; accepted 27 January 2014)

Cross-correlations of ambient noise averaged at two receivers lead to the reconstruction of the

two-point Green’s function, provided that the wave-field is uniform azimuthally, and also

temporally and spatially uncorrelated. This condition depends on the spatial distribution of the

sources and the presence of heterogeneities that act as uncorrelated secondary sources. This study

aims to evaluate the relative contributions of source distribution and medium complexity in the

two-point cross-correlations by means of numerical simulations and laboratory experiments in a

finite-size reverberant two-dimensional (2D) plate. The experiments show that the fit between the

cross-correlation and the 2D Green’s function depends strongly on the nature of the source used to

excite the plate. A turbulent air-jet produces a spatially uncorrelated acoustic field that rapidly builds

up the Green’s function. On the other hand, extracting the Green’s function from cross-correlations

of point-like sources requires more realizations and long recordings to balance the effect of the most

energetic first arrivals. When the Green’s function involves other arrivals than the direct wave,

numerical simulations confirm the better Green’s function reconstruction with a spatially uniform

source distribution than the typical contour-like source distribution surrounding the receivers that

systematically gives rise to spurious phases. VC 2014 Acoustical Society of America.

[http://dx.doi.org/10.1121/1.4864485]

PACS number(s): 43.20.Gp, 43.40.Dx, 43.60.Cg [RKS] Pages: 1034–1043

I. INTRODUCTION

Over the last decade, the average of cross-correlations

over long times and multiple sources that was initiated in

helioseismology (Duvall et al., 1993) has become a powerful

concept in seismic and acoustic imaging (Lobkis and

Weaver, 2001; Shapiro and Campillo, 2004; Sabra et al.,2005; Roux et al., 2004). Focusing solely in geophysics,

where impressive imaging results have been obtained since

the first papers in 2004, cross-correlation observations man-

aged to illuminate regions of the Earth that were previously

invisible to tomographers (Mordret et al., 2013; Gao et al.,2011; Ritzwoller et al., 2007). As most ambient noise

observed on Earth propagates in the form of short-period

surface waves (with average period �10 s) (Hillers et al.,2012), seismic-noise correlation is particularly relevant to

crust-lithosphere imaging; however, recent studies suggest

that the same techniques can be applied to lower-frequency

signals sampling the entire mantle down to the core (Poli

et al., 2012; Nishida, 2013).

The cross-correlation of the signal recorded at two

stations consists of a so-called “causal” contribution that

contains energy that travels from one reference station to

another, and an “anti-causal” contribution that travels from

the second station toward the reference station. If the noise

sources are uniformly distributed or just sample the

stationary-phase region (Snieder, 2004; Roux et al., 2005;

Curtis et al., 2012), the causal and anti-causal parts of the

cross-correlation are symmetric with respect to zero time

(Fan and Snieder, 2009; Hillers et al., 2012), and the

Green’s function is well approximated. As shown in recent

studies in geophysics, non-uniformities in the geographic

distribution of noise sources greatly complicate amplitude

measurements, but still allow the estimation of the wave

velocity, as the bias introduced in the time of flight is classi-

cally small (Froment et al., 2010; Tsai, 2010, 2009). The

length of the cross-correlation time window is often regarded

as alternative to the uniformity of the source spatial distribu-

tion, but the validity of this assumption and the trade-off

between the two are still under study.

In the last 10 years, laboratory experiments have been

conducted to explore the effects of source distribution and

medium complexities. Using a solid reverberant body and

diffuse thermal noise, Weaver and Lobkis (2001) wrote a

seminal paper where the time-averaged cross-correlation

between two receivers perfectly converges to the Green’s

function. The same authors explored then the same idea with

point-like ultrasonic sources (Lobkis and Weaver, 2001).

They showed that the Green’s function extraction was more

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

1034 J. Acoust. Soc. Am. 135 (3), March 2014 0001-4966/2014/135(3)/1034/10/$30.00 VC 2014 Acoustical Society of America

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 134.157.39.120 On: Mon, 28 Apr 2014 14:10:53

http://dx.doi.org/10.1121/1.4864485

mailto:[email protected]

http://crossmark.crossref.org/dialog/?doi=10.1121/1.4864485&domain=pdf&date_stamp=2014-03-01

complicated, even if the cross-correlation was averaged over

strongly reverberated signals obtained from sources at differ-

ent locations. Moreover, small unexplained wavelets

between the Green’s function and the cross-correlation func-

tion remained. A similar experiment was recently imple-

mented with a non-reverberating solid that contains point

scatterers (Mikesell et al., 2012). Using the signal generated

by sources placed on a contour surrounding the target area,

the Green’s function is reconstructed from the average of a

large set of cross-correlations. The discrepancy between the

averaged cross-correlation and the Green’s function, which

was also found in this study, might suggest that in such com-

plex media relaxing the source distribution requirements

(Wapenaar and Fokkema, 2006) might not be a valid

approach. Similar to these laboratory experiments, but at a

larger scale, retrieval of reflections from seismic noise

recorded at the surface in horizontally layered sediments was

also polluted by the presence of unexpected arrivals

(Draganov et al., 2007). Finally, at the global Earth scale,

the reconstruction of reflected body waves using noise

records from seismometers proved to be strongly dependent

on the noise distribution and the reverberation generated by

discontinuities/inhomogeneities within the medium (Poli

et al., 2012; Boue et al., 2013)

The present paper deals with flexural waves in a rever-

berating two-dimensional (2D) plate (Larose et al., 2007),

for which it was previously verified both numerically and

experimentally that the direct path of the dispersive Green’s

function can be satisfactorily approximated by stacked

cross-correlations. Focusing now on the retrieval of rever-

berated phases, it is shown that the fit between the Green’s

function and the cross-correlation depends on the nature and

distribution of the contributing sources. In particular, numer-

ical simulations first show that point sources uniformly

distributed over the surface, even if limited in extent, effi-

ciently participate in the Green’s function extraction when

compared to point sources only located along a curve sur-

rounding the receivers. Using laboratory experiments, such

source configuration corresponds to the difference between

striking a small portion of the plate with an air-jet coming

from an air nozzle, and deploying a set of point sources

along a line around the receiving points.

This study is organized in two parts. In the first part,

acoustic numerical modeling is used in two dimensions to

test the effects of the source distribution on the reconstruc-

tion of the direct wave and on one reflected wave. A large

number of sources deployed in two different geometries are

considered; i.e., (1) a dense distribution of sources along

circles of different radii; and (2) an even distribution of sour-

ces on the surface. It is shown that non-physical arrivals are

produced with a circular distribution. These spurious arrivals

are well explained by simple geometrical considerations on

the correlation terms between the direct and reflected paths.

When the distribution of the sources is spatially uniform, the

reconstruction of the Green’s function is improved. This

illustrates the importance of source�receiver diversity in the

framework of the cross-correlation theorem. In the second

part, the problem of dispersive flexural waves that propagate

in a finite plate is considered. The wave-field is made much

more complex than in the previous case, because of the

numerous reverberations on the plate edges. The numerical

results indicate that for both the contour and surface distribu-

tion of the sources, the correlation is a better approximation

of the Green’s function when long time series, including

reverberations, are considered. Again, the surface distribu-

tion leads to better results in terms of reconstruction quality.

Finally, laboratory experiments are performed in a plate in

which different types of acoustic sources are compared. For

the contour source distribution, the experiments confirm the

presence of spurious arrivals in the cross-correlation, as

observed with numerical simulation. To improve the spatial

diversity, an air-jet was used as a spatially distributed inco-

herent source. The reconstruction is clearly improved com-

pared to the stack of individual point sources.

A. Spatially versus azimuthally uniform sourcedistribution

The theory of diffuse wave-field interferometry is based

on the finding that by (1) cross-correlation of the signal gen-

erated by a point source at two receivers, (2) iteration over a

sufficiently large set of more or less uniformly distributed

sources, and (3) stacking (or averaging) of the cross-

correlations, the resulting trace approximates the Green’s

function associated with the locations of the two receivers

(Sabra et al., 2005; Wapenaar, 2004; Boschi et al., 2013).

Mathematically, this is expressed by the following expres-

sion at frequency x and valid for a 2D medium:

G12 � G�21 ¼4ixj

c

ðS

G1xG�2xd2x

þþ

CðG1xrG�2x �rG1xG�2xÞn̂dx; (1)

where Gij is the (scalar) Green’s function associated with

source location i and receiver location j, c is the phase veloc-

ity, and j is the attenuation. Note that Eq. (1) has been dem-

onstrated in a 3D complex medium (Snieder et al., 2008;

Campillo and Roux, 2014) and is applied here in a 2D con-

figuration. After Fourier transformation back to the time

domain, the two terms on the left-hand side of Eq. (1) are the

“causal” and “anti-causal” parts of the Green’s function; i.e.,

the Green’s function between a source at the location of

receiver 1, recorded at receiver 2, and the reciprocal Green’s

function (a source at receiver 2, recorded at 1), respectively.

S is the surface occupied by the sources, and C the boundary

of this surface (Wapenaar, 2004). In Eq. (1), the surface inte-

gral corresponds to a homogeneous spatial distribution of

sources inside the contour C. The other integral can be inter-

preted as the intensity flux through the contour C. In the case

where the contour is in the far-field of the receivers and the

medium heterogeneities, the line integral can be formulated

as the correlation performed from sources at every point

along the contour C (Wapenaar, 2004). In practice, the spa-

tial source averaging of cross-correlations is the processing

strategy (usually known as stacking) that is followed in seis-

mological applications, and it is the strategy also chosen

here, to evaluate the Green’s function retrieval of reverber-

ated echoes.

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Colombi et al.: Green’s function retrieval in complex media 1035


Many sources (or alternatively, many scattering

obstacles) are needed for the wave-field to become diffuse,

which is essential for the Green’s function to be retrieved

from a stack of cross-correlations. In practice, if the com-

plexities of the medium (heterogeneities, interfaces, or scat-

terers) are confined within C and attenuation j is neglected,

a simple layout of point sources, such as a circle or an ellipse

enclosing the receivers, might be sufficient to reproduce

G12 � G�21 (Snieder, 2004; Froment et al., 2010).

In this approach, the surface integral in Eq. (1) can be

neglected. If the interest is simply in the reconstruction of

the “first arrival,” i.e., the direct path of the Green’s function,

it might be enough to have sources in the so-called

“stationary-phase regions,” i.e., aligned with the azimuth

defined by the receiver pair (Fan and Snieder, 2009; Roux

et al., 2005; Curtis et al., 2012). If the medium is more com-

plex, including scatterers, boundaries, and/or smooth hetero-

geneities, the Green’s function results from the superposition

of a variety of arrivals, and it is harder to reconstruct. In this

case, it becomes necessary to illuminate the medium with a

uniform, dense distribution of sources, to achieve the com-

plete reconstruction of the Green’s function (Sato, 2010).

In the literature, numerical and laboratory experiments

have described the Green’s function recovery in three

dimensions through correlation (Snieder and Fleury, 2010;

Fleury et al., 2010; Mikesell et al., 2012) with sources

located along a surface that surrounds the receiver pair, but

never in the volume of the sample. In two dimensions, this

would be equivalent to neglecting the surface integral on the

right-hand side of Eq. (1). Using a simple 2D numerical

example, it is demonstrated how the presence of only one

reflecting boundary, responsible for a second arrival in the

Green’s function, requires the source distribution to be uni-

form over the surface for the Green’s function to be accu-

rately reconstructed. Figures 1(a) and 1(b) show the uniform

and circular source layouts, respectively. The receivers,

FIG. 1. (Color online) (a), (b) The sources (triangle)�receiver (dot) layout for the two cases. The size of the domain is 6 � 6 m in both examples, and the

receivers are 0.6 m apart and aligned with the center. The impulse used has a central frequency of 18 kHz. The N(north)-side is the boundary responsible for

the reflections. (c) Cross-correlation gather calculated using ray theory and the geometrical spreading factor for the configuration in (a) ordered along the yaxis upon the azimuth h. The cross-correlation is averaged over 0.5� bins. (d) Same as (c), but for the configuration in (b), ordered along the y axis upon the

azimuth h for increasing radii starting from the bottom, indicated with the innermost circle as numbers in light gray. (e), (f) Stacks of the cross-correlations

shown in the gathers (c) and (d). Dashed black lines show the approximate position of the sources and their associated cross-correlations. Note the different

positions of the stationary phase for positive and negative time in (c). Dotted lines in (d) and (f) associate stationary phases in the gather to the corresponding

physical or spurious arrival. Note that no spurious arrival is visible in (e).

1036 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Colombi et al.: Green’s function retrieval in complex media


respectively R1 and R2, are 60 cm away from one another,

while the boundary is at a distance of 3 m from both R1 and

R2, in the “north” direction. In this first numerical experi-

ment, the wave propagation is modeled via a simple

ray-theory approximation; i.e., a Gaussian pulse centered at

18 kHz that propagates along a straight ray path with an am-

plitude weighted according to the geometrical spreading.

The cross-correlation patterns in Figs. 1(c) and 1(d)

show the arrival times of different wave packets that depend

on the azimuth of the source with respect to the two receiver

positions. Each cross-correlation in the gathers corresponds

to an average cross-correlation over a set of sources that are

grouped in 0.5� bins, according to their azimuth h. The

cross-correlation patterns are dominated by the “ballistic”

term that corresponds to the direct wave recorded at the two

receivers, and for which the arrival time in the cross-

correlation changes smoothly with the azimuth. The stack in

Fig. 1(e) results from the uniform source distribution of

Fig. 1(c). As expected from the surface integral in Eq. (1),

this reveals the “direct” arrival, which propagates from one

receiver to the other, and the “reflected” arrival, which prop-

agates from one receiver to the boundary and, after reflec-

tion, to the other receiver. These two wavelets result from

constructive interference due to sources located in the

stationary-phase areas (Roux et al., 2005; Snieder, 2004)

that are clearly visible in both the causal (positive-time) and

anti-causal (negative-time) cross-correlations, at 60.7 and

66 ms, respectively. The wavelet associated with the

reflected wave in the cross-correlation gather in Fig. 1(c) is

characterized by a sudden amplitude drop outside the

stationary-phase region [marked by arrows in Fig. 1(c)]. On

the other hand, Fig. 1(d) shows that a number of stationary-

phase points also occur with the circular source distribution,

which in the stacked cross-correlation [Fig. 1(f)] gives rise

to a set of wavelets where their arrival times depend on the

radius ri of each source circle. These arrivals are not part of

the Green’s function associated with the receiver locations,

which depends only on the medium properties and the inter-

face geometry. These wavelets are called “spurious” (non-

physical). Increasing the distance of the sources with respect

to the receiver separation is not sufficient to remove the spu-

rious arrivals. Note that the total number of sources used is

similar in Figs. 1(a) and 1(b), which means that the spurious

arrivals do not depend on the sampling of the circular pat-

terns. If the exact Green’s function is to be retrieved by inter-

ferometry, these spurious arrivals have to be eliminated by

improving the source distribution. In Fig. 1(c), no stationary-

phase points are observed, other than those associated with

the arrivals in the true Green’s function, and the stacked

cross-correlation [Fig. 1(e)] consequently does not include

any spurious wavelets. Therefore spurious arrivals are

cancelled more efficiently by a spatially, rather than azimu-thally, uniform source distribution.

With the simple example of a two-wave problem (direct

and reflected), this numerical experiment demonstrates the

contributions from the cross terms in the cross-correlation

that leads to spurious arrivals. This is a general feature for

multi-wave problems, as classically encountered in elastic

media with longitudinal P waves and transversal S waves.

Similar arguments can be drawn that indicate the importance

of the source�receiver spatial diversity. In the following,

numerical simulations are used to further compare the spa-

tially uniform and contour-like source distributions in more

complex and realistic scenarios that include dispersion and

multiple reverberations.

II. NUMERICAL SIMULATIONS OF FLEXURAL WAVES

Flexural waves generated in a thin 2D plate are a good

example of dispersive waves (Larose et al., 2007).

Propagation of A0 Lamb waves (Lamb, 1904) are numeri-

cally modeled using the spectral element method (SEM)

(Padovani et al., 1994) in the 1.5� 1.5 m thin plate (2 mm

thickness) with free boundaries depicted in Fig. 8. The plate

is made of aluminum, which is characterized by a density

q¼ 2710 kg/m3, a compressional velocity Vp¼ 6300 m/s,

and a shear velocity Vs¼ 3100 m/s. The plate is assumed to

be isotropic and homogeneous.

The present approach differs from other studies where

numerical simulation has been used (Snieder et al., 2008), in

that it allows for a much longer propagation time within a

closed, reverberating cavity; there is time for multiple border

reflections, which results in a very complex field.

A. Simulation setup

To simulate the propagation of elastic waves on the

plate, the latest release of SPECFEM3D (Peter et al., 2011)

is used in combination with the meshing tool CUBIT. After

decomposition of the mesh in a load-balanced fashion, the

simulation is executed in parallel to calculate a �25-ms-long

accelerogram. In such a short time, a wave at 10 kHz boun-

ces back and forth from the sides of the plate more than

20 times. As the computational cost of each simulation is

considerable, it is possible to swap the roles played by the

sources and receivers in the wave equation by using the reci-

procity theorem (Aki and Richards, 1980; Curtis et al.,2012). To simulate a complex reverberated field, only two

simulations need to be conducted, each initiated by a single

source placed at the location of one of the receivers, R1 and

R2. The signal is then “recorded” at source locations S,

which in analogy with Sec. I A above, are alternatively dis-

tributed uniformly over the surface of the plate [Fig. 2(a)], or

along an ellipse that surrounds R1 and R2 [Fig. 2(b)]. This

strategy does not affect the results, and it is common practice

in this type of numerical experiment (Mikesell et al., 2012).

The source signal emitted respectively in R1 and R2 is a

Gaussian time function that acts in the vertical direction and

excites the 10�25 kHz frequency band (Komatitsch and

Tromp, 2002). Note that the excited plate waves are disper-

sive A0 Lamb waves, which have a minimum phase velocity

of �600 m/s and a maximum of �1400 m/s in this band. The

receivers are located at the surface and record the z-compo-

nent of the acceleration. Importantly, attenuation is

neglected in the simulations. The external coupling of the

plate with the air and with the supporting structure is like-

wise not modeled. Therefore, these numerical simulations

are not expected to perfectly match experimental laboratory

results.



For comparison purposes, the plate response is also com-

puted between R1 and R2; i.e., the Green’s function that

besides the first arrival, contains other secondary phases due to

reverberation. Figure 3 describes the first wave packets as a set

of reflections from the four edges of the plate. The main diffi-

culty in this exercise is to isolate single phases when the me-

dium is dispersive and reverberant. Due to dispersion, a series

of wave packets are partially distinguishable after the direct ar-

rival, each of which contains reflected phases. From the travel-

time of the direct arrival, a group velocity of �1200 m/s is

obtained that corresponds to the central frequency of the sig-

nal. Three other travel-times are derived easily from a geomet-

ric analysis of each source�boundary�receiver configuration,

using the blueprint in Fig. 2. Any arrivals after these four

phases result from interfering paths, because of multiple reflec-

tions at the plate boundaries.

B. Elliptical versus uniform distribution of sources

Figures 4–7 show the results of the averaging of the

cross-correlations of the complex wave-fields from the

source distributions of Figs. 2(a) and 2(b). The choice of an

ellipse instead of a circle for the source distribution in Fig.

2(b) is motivated by the stack of cross-correlation functions

with the same amount of dispersion. Indeed, the effect of dis-

persion (and attenuation whenever considered) classically

depends on the path length. In the correlation process, it can

be shown that the addition of the path lengths between each

source(s) and the receivers in R1 and R2, [namely

sR1þ sR2; see Fig. 2(b)] is independent of the source posi-

tion along each ellipse with foci in R1 and R2 (Roux et al.,2005). The cross-correlation gather and the corresponding

stack in Fig. 4 are obtained after the selection of a 2-ms time

window starting from 0 time, which contains both the direct

arrival and a few of the later arrivals, as discussed in Fig. 3.

Note that prior to stacking, the cross-correlations are aver-

aged over 0.5� bins according to the azimuth h.

The cross-correlation pattern in Fig. 4(a) is qualitatively

analogous to that of Fig. 1(c), and is dominated by the ballis-

tic arrivals in each correlation function. The constructive

interference that results in the direct arrival in the cross-

correlation stack corresponds to the stationary-phase region

at �60.5 ms (Fan and Snieder, 2009). The stack matches

relatively well the portion of the reference Green’s function

that corresponds to the direct arrival, but the fit deteriorates

before and after.

By cross-correlation over a longer time window (Fig. 5),

the gather is no longer characterized by the ballistic signa-

ture, but rather by the constant-arrival-time stripes that can

be associated to the predicted arrival times: (1) at �60.5 ms,

the direct phase from R1 to R2 and vice versa; and (2) at

�61 ms, the phases traveling from R1 (and, respectively,

R2) to the two closest reflecting sides, and then, after

FIG. 2. (Color online) Blueprint of the plate showing the two receivers R1 and R2 and the two source distributions used during the numerical examples. The

receivers are intentionally misaligned from the center of the plate to avoid symmetry. (a) The plate with the sources uniformly distributed over the surface at

0.06 m spacing. The azimuth (h) is computed according to the convention [-p, p], and it grows counter-clockwise, as indicated by the arrow. (b) The plate with

sources arranged along an elliptical contour surrounding the receivers. The dotted lines sR1 and sR2 indicate the source�receiver paths for a given source s.

The same convention is used for the azimuth. The two distributions contain the same number of sources (�600) to avoid any bias in the comparison.

FIG. 3. (Color online) Temporal representation of the first 2 ms of the nor-

malized Green’s function that contains direct arrivals, and the first reflec-

tions from the plate edges labeled after a travel-time analysis according the

source�boundary�receiver distances.



reflection, to R2 (and, respectively, R1). Importantly, the

stacked cross-correlation practically coincides with the refer-

ence Green’s function, except for 0 time, where (weak) spu-

rious oscillations (i.e., non-physical wiggles generated by

the cross-correlation of the border reflections) pollute the

trace.

Some authors (Fan and Snieder, 2009; Wapenaar et al.,2010) use virtual or actual rings of sources to generate com-

plex wave fields and to study the role of azimuthal source

density on the stacked cross-correlations (Froment et al.,2010). Following this approach, Figs. 6 and 7 show the

cross-correlation results with the source�receiver configura-

tion of Fig. 2(b). The cross-correlations in Fig. 6 were com-

puted over a short time window (2 ms), as in Fig. 4;

however, the gather plot is much more complicated, with the

superimposed ballistic signature of numerous phases other

than the obvious, “direct” phase. By considering just the

direct arrival plus the four boundary reflections, a total of 25

cross-correlated terms are expected, of which 10 are physical

while 15 are due to spurious oscillations. In analogy with

Fig. 5, the ballistic signature is lost by increasing the length

of the cross-correlation window. The gather in Fig. 7(a) is

characterized by causal and anti-causal arrivals that sum up

constructively in the stack [Fig. 7(b)], which correspond to

the causal and anti-causal direct waves and the edge reflec-

tions also seen in Fig. 5. However, vertical stripes of high

FIG. 4. (Color online) (a) Cross-correlations gather for the source distribu-

tion in Fig. 2(a) computed over a 2-ms time window, which includes only the

first arrival and the secondary reflections. The traces are ordered according to

the azimuth and have a frequency content between 10 and 25 kHz. The aver-

aging procedure used in Fig. 1(d) defined over 0.5� bins has been applied

here too. The signal coming from sources located within a wavelength (at

10 kHz) has been muted to avoid near-field effects. (b) Stacked cross-

correlation (black) compared to the reference Green’s function (light gray).

FIG. 5. (Color online) Same as Fig. 4, but for a time window of 25 ms start-

ing from 0 time.

FIG. 6. (Color online) (a) Cross-correlations gather for the elliptical source

distribution in Fig. 2(b) computed over a 2-ms time window, which includes

only the first arrival and the secondary reflections. The traces are ordered

according to the azimuth and have a frequency content between 10 and

25 kHz. (b) Stacked cross-correlation (black) compared to the reference

Green’s function (light gray). Note that the total number of sources remains

the same as in Fig. 4.



cross-correlation in the gather are less visible than in the

analogous pattern in Fig. 5, and the fit between the stack and

the reference Green’s function is accordingly not as good.

This effect is explained on the basis of Fig. 1 and the

corresponding discussion in Sec. I. However, in Fig. 1(c),

the gather plot is relatively simple, because the medium (a

uniform half space) only includes a single obstacle (the

reflecting boundary at the north side of the plate). In Fig. 6,

where flexural waves are reflected from four edges, the num-

ber of reflections quickly grows, and so does the number of

spurious arrivals in the cross-correlation stack. The contour-

like source distribution is not ideal for extracting the Green’s

function if arrivals other than the direct one are expected.

This can be understood by the approximation made in

Eq. (1), as the presence of boundaries outside the contour Cis in contradiction with the hypothesis that allows the surface

integral in Eq. (1) to be neglected.

III. LABORATORY EXPERIMENTS

In practical applications of the cross-correlation process,

the spatial-temporal complexity of the noise source is one of

the key factors to reconstruct the Green’s function. One limit

of numerical simulation is the difficulty (if it is possible at

all) of implementing an actual spatially diverse and tempo-

rally uncorrelated source signal (Peter et al., 2011). On the

contrary, this can be achieved in the laboratory; e.g., by

means of air-jet forcing (Larose et al., 2007). Compared to a

deterministic point source that is typical of laboratory

experiments (Mikesell et al., 2012), an air-jet flow is

turbulent, which results in spatially and temporally random

forcing in the target area.

A. Experimental setup

Flexural waves are excited on the 1.5� 1.5� 0.002 m

aluminum plate described in Fig. 8(a). The coupling between

the plate and its supporting frame is damped by means of

small absorbing-elastic patches that prevent resonance

effects. Two broadband Brel & Kjaer accelerometers are

attached at the center of the plate, at 60 cm apart and with a

relative azimuth that is roughly parallel to the one plate edge

(Fig. 8). The accelerometers ensure a flat frequency response

between 500 Hz and 15 kHz. The sampling frequency of

both of the accelerometers is set at 100 kHz. Using either a

piezoelectric transducer (around 10 kHz) or an air-jet gener-

ated by an air nozzle (around 2 kHz), A0 Lamb waves

(Lamb, 1904) are generated in the plate.

B. Excitation by air-jet sources

Acting at 10�20 cm above the plate, the air-jet produces

a wide (tens of cm) turbulent air flow that creates a random-

like pressure field at the plate surface. Moreover the air-jet

was also moved a few centimeters around a central position

[see Fig. 8(a)], which further contributes to the creation of a

continuous incoherent extended source. The total time of the

continuous recording is about 250 s, which is divided into 5-

s-long time windows, with each 5-s interval corresponding

to a different position of the air nozzle, roughly along the

inner ellipse in Fig. 8. The recorded signal is band-passed in

the 0.5�5 kHz frequency range, and no further processing is

applied (i.e., no whitening or “one-bit” filtering). Each 5-s-

long continuous record is used in the same way as the point

source signals presented in Sec. II. Following common prac-

tice in ambient noise seismology (Poli et al., 2012), the

cross-correlations are computed over the largest available

time window for which continuous recordings are available

(i.e., 5 s), regardless of the expected duration of the Green’s

function. The gather in Fig. 9, which is obtained by cross-

correlation of each 5-s-long record, is quite different from

that seen so far in this study. Unlike Figs. 1(c), 1(d), 4, and

6, no ballistic signature is visible, although maxima with

constant arrival times appear, as in Figs. 5 and 7, where late

arrivals were considered. These maxima can be associated

to the direct (61 ms) and reflected arrivals of the Green’s

function. We infer that a single realization of the experiment

(a single cross-correlation in the gather) might be sufficient

to generate a wave-field that is sufficiently spatially and tem-

porally complex for the Green’s function to emerge.

The stack in Fig. 9 shows that the causal and anti-causal

contributions of the correlation function are close to being

symmetric. Symmetry with respect to time is an essential

property of the averaged correlation function, and it is used

here as a heuristic measure of the Green’s function retrieval.

Indeed, measurement of the actual Green’s function would

require a low-frequency piezo-transducer, which was not

available at the time of the experiment. From Fig. 9, it

appears that not only has the direct arrival of the Green’s

function been extracted, but also that later arrivals (63 ms)

FIG. 7. (Color online) Same as Fig. 6, except for the time window of 25 ms

starting from 0 time. Note that the total number of sources remains the same

as in Fig. 5.



are visible. With an analysis analogous to that of Fig. 3, it is

possible to attribute each peak to a boundary reflection. The

results from this experiment highlight the robustness of the

cross-correlation measurements when they are used with a

spatially uncorrelated source and long time recordings.

C. Point-source results

In the point-source experiment, piezoelectric transducers

are coupled with the plate at every position of the two ellipses

shown in Fig. 8(a). A broadband Gaussian pulse centered at

10 kHz is used as the source function. After each pulse, a

�60-ms signal is recorded, as shown in Fig. 8(c), which is

sufficient for flexural waves to propagate �40 times back and

forth in the 1.5 -m-wide plate. The signal recorded was aver-

aged over 100 realizations at the same location, to further

improve the signal-to-noise ratio. The transducer is then

shifted step-by-step along two contours of approximately el-

liptical shape [Fig. 8(a)], and the correlation process iterates

over each source. As in the air-jet example, the symmetry of

the causal and anti-causal contributions is used to benchmark

the quality of the reconstructed Green’s function.

In Fig. 10, the cross-correlation is shown for a 3-ms

time window that includes direct arrival and first border

reflections only, while 40 ms of signals (i.e., including the

late reverberation) are cross-correlated in Fig. 11. Using the

shorter time window, the cross-correlation results (Fig. 10)

are dominated by the ballistic arrival (and other not well

identifiable patterns), as confirmed by the two early maxima

that are visible in the stack [Fig. 10(b)]. Similar to other

point-source results illustrated above with numerical simula-

tions, the arrival times of the causal and anti-causal direct

wave at 60.7 ms correspond to the stationary phase regions

in the gather. In this short window case, the stack is not sym-

metric. For the longer time window case, the stack is of

greater quality, although spurious terms do not disappear

completely, which leaves a residual asymmetry [Fig. 11(b)].

Later arrivals are not clearly visible in either the gather or

the stack.

As expected, in the case of point-like sources, cross-

correlations strongly depend on the azimuth of the source. A

single shot from the transducer does not generate a wave-

field that is diffuse enough, despite the numerous reverbera-

tions from the plate boundaries. The recovery of the direct

arrival is only possible by stacking a large number of cross-

correlations, which means that a sufficient source diversity

(each cross-correlation in the gather) is necessary. The re-

covery of secondary arrivals appears to be difficult even

with the use a long reverberation, as spurious oscillations do

not cancel out, thus polluting the stacked correlation.

FIG. 8. (Color online) (a) The blueprint of the plate already presented in Fig. 2. R1 and R2 are the receivers, intentionally misaligned from the center of the

plate, to avoid symmetry. During the laboratory experiment, the piezoelectric transducer (the source) is placed along the two ellipses that are visible in the

Figure. The light gray circles indicate the area excited by the air-jet when it moves along the ellipse. (b) The signal recorded when the air-jet is used as a

source, filtered in the 0.5�5 kHz band and as discussed in Sec. III. (c) The signal and the long coda recorded at R2 induced by a point source located along the

inner ellipse in the laboratory case.



IV. CONCLUSIONS

Using flexural waves on a thin 2D plate, we proposed an

analysis supported by laboratory and numerical experiments

that aims to separate the relative roles of the reverberation

and the source distribution that gives rise to a complex

wave-field. A wave-field can be considered as “sufficiently

diffuse” when the stacked cross-correlation of the recorded

signals approximates well the Green’s function, including

phases other than the direct arrival.

The way the plate is excited is critical to understand the

spatial-temporal complexity of the generated wave-field. For

individual point sources, the ballistic signature is dominant

in the cross-correlation gather when short time windows are

considered, and the wave-field is clearly not sufficiently

complex for the Green’s function to be adequately extracted

by a single or a stack of a few cross-correlations. The

Green’s function reconstruction is nearly complete only after

a large number of averages, with the associated point sources

uniformly covering the target area, and using a long time

window for computing the cross-correlation that includes

many reflections from the edges. In general, increasing the

duration of the recorded time window, and hence including

long reverberations, drastically improves the cross-

correlation results for both the direct and secondary arrivals.

For the cross-correlation of continuous ambient noise to

approximate well the Green’s function, it is also essential

that a broad area is covered. It is shown experimentally that

an air-jet exciting a finite area of an aluminum plate is much

FIG. 10. (Color online) Same as Figs. 4 and 6, except that records from the

laboratory experiment are used where sources are placed on two ellipses, as

in Fig. 8(a). (a) Starting from the top of the cross-correlation gather, the val-

ues of the angle correspond to the outer ellipse, while the second loop corre-

sponds to the inner ellipse. The cross-correlation time window used is 3 ms,

starting from 0 time. (b) The black line depicts the stacked cross-correlation,

while the light gray shows its version as reversed about the 0-time axis.

FIG. 11. (Color online) Same as Fig. 10, except for a cross-correlation time

window of 40 ms starting from 0 time.

FIG. 9. (Color online) (a) Cross-correlation gather as a function of time

using the air-jet forcing, for a signal filtered between 0.5 kHz and 5 kHz. (b)

Stack (black) of the 50, 5-s-long records, giving a �250-s-long cumulated

signal; (light gray) the same stack, reversed with respect to 0 time. The sym-

metry of the stacked cross-correlation can be seen as a measure of how well

it approximates the Green’s function. The time window is sufficiently long to

contain the first arrival and the first reverberations from the plate boundaries.



more effective than the combination of point sources in the

reconstruction of the Green’s function from the correlation

process. Such continuous incoherent extended source recalls

the excitation of seismic noise by oceanic storms, which

leads to great results for surface-wave extraction in seismol-

ogy (Shapiro and Campillo, 2004; Ritzwoller et al., 2007).

This experimental demonstration is also demonstrated

numerically, as a spatially uniform distribution of point-like

sources is accordingly more effective than an azimuthallyuniform source distribution along a close line, even if the

total number of sources is the same. In its most general

form, the correlation theorem (Wapenaar, 2004) [Eq. (1)]

states that the 2D Green’s function associated with the loca-

tion of two receivers is obtained by combining a surface and

a line integral over the region populated by sources and its

boundary. In the case of complex 2D media, the surface inte-

gral has a dominant role in the reconstruction of the com-

plete Green’s function.

ACKNOWLEDGMENT

This work has been supported from the EU network

QUEST through a Marie Curie grant, and the Perspective

Research Scholarship PBEZP2_145960 from the Swiss

National Science Foundation.

Aki, K. and Richards, P. G. (1980). Quantitative Seismology: Theory andMethods (W. H. Freeman & Co., San Francisco, CA), pp. 932–934.

Boschi, L., Weemstra, C., Verbeke, J., Ekstr€om, G., Zunino, A., and

Giardini, D. (2013). “On measuring surface wave phase velocity from sta-

tionstation cross-correlation of ambient signal,” Geophys. J. Int. 192(1),

346–358.

Boue, P., Poli, P., Campillo, M., Pedersen, H., Briand, X., and Roux, P.

(2013). “Teleseismic correlations of ambient seismic noise for deep global

imaging of the earth,” Geophys. J. Int. 194(2), 844–848.

Campillo, M., and Roux, P. (2014). “Seismic imaging and monitoring with

ambient noise correlations,” Treatise on Geophysics, 2nd Ed., edited

by B. Romanowicz and A. Dziewonski (Elsevier, Amsterdam), Vol. 1,

pp. 256–271.

Curtis, A., Behr, Y., Entwistle, E., Galetti, E., Townend, J., and Bannister,

S. (2012). “The benefit of hindsight in observational science:

Retrospective seismological observations,” Earth Planet. Sci. Lett.

345–348, 212–220.

Draganov, D., Wapenaar, K., Mulder, W., Singer, J., and Verdel, A. (2007).

“Retrieval of reflections from seismic background-noise measurement,”

Geophys. Res. Lett. 34, L05307-1–L05307-5, doi:10.1029/2006GL028735.

Duvall, T. L., Jefferies, S. M., Harvey, J. W., and Pomerantz, M. A. (1993).

“Time distance helioseismology,” Nature 362(6419), 430–432.

Fan, Y., and Snieder, R. (2009). “Required source distribution for interfer-

ometry of waves and diffusive fields,” Geophys. J. Int. 179(2), 1232–1244.

Fleury, C., Snieder, R., and Larner, K. (2010). “General representation theo-

rem for perturbed media and application to green’s function retrieval for

scattering problems,” Geophys. J. Int. 183(3), 1648–1662.

Froment, B., Campillo, M., Roux, P., Gou�edard, P., Verdel, A., and Weaver,

R. (2010). “Estimation of the effect of nonisotropically distributed energy

on the apparent arrival time in correlations,” Geophysics 75(5),

SA85–SA93.

Gao, H., Humphreys, E. D., Yao, H., Van der Hilst, R. D. (2011). “Crustal

and lithosphere structure of the Pacific Northwest with ambient noise

tomography,” Earth Planetary Sci. Lett. 304, 202�211.

Hillers, G., Graham, N., Campillo, M., Kedar, S., Lands, M., and Shapiro,

N. (2012). “Global oceanic microseism sources as seen by seismic arrays

and predicted by wave action models,” Geochem., Geophys., Geosyst.

13(1), Q01021, doi:10.1029/2011GC003875

Komatitsch, D., and Tromp, J. (2002). “Spectral-element simulations of

global seismic wave propagation—I. Validation,” Geophys. J. Int. 149,

390–412.

Lamb, H. (1904). “On the propagation of tremors over the surface of an elas-

tic solid,” Philos. Trans. R. Soc. London 203, 1–42.

Larose, E., Roux, P., and Campillo, M. (2007). “Reconstruction of

Rayleigh-Lamb dispersion spectrum based on noise obtained from an air-

jet forcing,” J. Acoust. Soc. Am. 122(6), 3437–3444.

Lobkis, O. I., and Weaver, R. L. (2001). “On the emergence of the Green’s

function in the correlations of a diffuse field,” J. Acoust. Soc. Am. 110(6),

3011–3017.

Mikesell, T. D., Wijk, K. van, Blum, T. E., Snieder, R., and Sato, H. (2012).

“Analyzing the coda from correlating scattered surface waves,” J. Acoust.

Soc. Am. 131, EL275–EL281.

Mordret, A., Landes, M., Shapiro, N. M., Singh, S. C., Roux, P., and Barkved,

O. I. (2013). “Near-surface study at the Valhall oil field from ambient noise

surface wave tomography,” Geophys. J. Int. 193(3), 1627–1643.

Nishida, K. (2013). “Global propagation of body waves revealed by cross corre-

lation analysis of seismic hum,” Geophys, Res. Lett. 40(9), 1691–1696.

Padovani, E., Priolo, E., and Seriani, G. (1994). “Low- and high-order finite-

element method: Experience in seismic modeling,” J. Comp. Acoust. 2,

371–422.

Peter, D., Komatitsch, D., Luo, Y., Martin, R., Le Goff, N., Casarotti, E., Le

Loher, P., Magnoni, F., Liu, Q., Blitz, C., Nissen-Meyer, T., Basini, P.,

and Tromp, J. (2011). “Forward and adjoint simulations of seismic wave

propagation on fully unstructured hexahedral meshes,” Geophys. J. Int.

186(2), 721–739.

Poli, P., Pedersen, H. A., Campillo, M., and the POLENET/LAPNET

Working Group. (2012). “Body-wave imaging of earth’s mantle disconti-

nuities from ambient seismic noise,” Science 338(6110), 1063–1065.

Ritzwoller, M. H., Yang, Y., Richmond, R., Pasyanos, M. E., Villasenor, A.,

Levin, V., Hofstetter, R., Pinsky, V., Kraeva, N., and Lerner-Lam, A.

(2007). “Short period surface wave dispersion across the Mediterranean

region: Improvements using regional seismic networks,” Proceedings ofthe 29th Monitoring Research Review of Ground-Based NuclearExplosion Monitoring Technologies, pp. 226–235.

Roux, P., Kuperman, W. A., and the NPAL Group. (2004). “Extracting

coherent wave fronts from acoustic ambient noise in the ocean,” J. Acoust.

Soc. Am. 116(1), 1995–2003.

Roux, P., Sabra, K. G., Kuperman, W. A., and Roux, A. (2005). “Roux am-

bient noise cross-correlation in free space: Theoretical approach,”

J. Acoust. Soc. Am. 117(1), 79–84.

Sabra, K. G., Gerstoft, P., Roux, P., Kuperman, W. A., and Fehler, M. C.

(2005). “Surface wave tomography from microseisms in southern California,”

Geophys. Res. Lett. 32(14), SI23–SI31, doi:10.1029/2005GL023155.

Sato, H. (2010). “Retrieval of green’s function having coda waves from the

cross-correlation function in a scattering medium illuminated by a ran-

domly homogeneous distribution of noise sources on the basis of the first-

order born approximation,” Geophys. J. Int. 180(2), 759–764.

Shapiro, N. M., and Campillo, M. (2004). “Emergence of broadband

Rayleigh waves from correlations of the ambient seismic noise,” Geophys.

Res. Lett. 31(7), L07614, doi:10.1029/2004GL019491.

Snieder, R. (2004). “Extracting the Green’s function from the correlation of

coda waves: a derivation based on stationary phase,” Phys. Rev. E 69, 046610.

Snieder, R., and Fleury, C. (2010). “Cancellation of spurious arrivals in

Green’s function retrieval of multiple scattered waves,” J. Acoust. Soc.

Am. 128(4), 1598–1605,

Snieder, R., Wijk, K. van, Haney, M., and Calvert, R. (2008). “Cancellation

of spurious arrivals in Green’s function extraction and the generalized op-

tical theorem,” Phys. Rev. E 78, 036606.

Tsai, V. C. (2009). “On establishing the accuracy of noise tomography

travel-time measurements in a realistic medium,” Geophys. J. Int. 178(3),

1555–1564.

Tsai, V. C. (2010). “The relationship between noise correlation and the

Green’s function in the presence of degeneracy and the absence of equi-

partition,” Geophys. J. Int. 182(3), 1509–1514.

Wapenaar, K. (2004). “Retrieving the elastodynamic Green’s function of an

arbitrary inhomogeneous medium by cross correlation,” Phys. Rev. Lett.

93, 254301.

Wapenaar, K., and Fokkema, J. (2006). “Green’s function representations

for seismic interferometry,” Geophysics 71, SI33–SI46.

Wapenaar, K., Slob, E., and Snieder, R. (2010). “On seismic interferometry,

the generalized optical theorem, and the scattering matrix of a point

scatterer,” Geophysics 75(3), SA27–SA35.

Weaver, R. L., and Lobkis, O. I. (2001). “Ultrasonics without a source:

Thermal fluctuation correlations at MHz frequencies,” Phys. Rev. Lett.

87(13), 134301.



Documents

Green’s function retrieval through cross-correlations …hestia.lgs.jussieu.fr/~boschil/publications/colombi14.pdfcross-correlation are symmetric with respect to zero time (Fan and