Green’s function retrieval through cross-correlations...
Transcript of Green’s function retrieval through cross-correlations...
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:
1034 J. Acoust. Soc. Am. 135 (3), March 2014 0001-4966/2014/135(3)/1034/10/$30.00 VC 2014 Acoustical Society of America
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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
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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
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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.
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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.
1038 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Colombi et al.: Green’s function retrieval in complex media
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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.
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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.
1040 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Colombi et al.: Green’s function retrieval in complex media
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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.
J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Colombi et al.: Green’s function retrieval in complex media 1041
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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.
1042 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Colombi et al.: Green’s function retrieval in complex media
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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
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