- The document is a student research paper that numerically tests whether the negative time derivative of the cross-correlation of ambient noise between two points in a complex medium approximates the impulse response (Green's function) between those points. - It introduces the theoretical background of using noise cross-correlation to retrieve medium information without using a real source. It then describes testing this theory for a 2D lossy complex medium using numerical modeling and the Finite-Difference Time-Domain method. - The key variables tested are the number of time steps, random noise sources, and decay in the system to analyze their effects on how well the cross-correlation matches the impulse response.

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Project Title: Retrieval of the Green's function of a complex medium from the correlation of noise
Student Registration Number: 201045581
Date of submission: 14/01/2015
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2. Retrieval of the Green’s function of a complex medium from the correlation of noise
201045581∗
SUPA, Department of Physics, University of Strathclyde,
Glasgow G4 0NG, United Kingdom
(Dated: January 14, 2015)
An impulse applied at one point in a system with a response at another point (Green’s function)
has shown to be proportional to the cross-correlation at these two points in the presence of noise. A
real source is therefore not required as ambient noise can be used with the cross-correlation function
rather than a separate noisy source. This paper shows the testing of this comparison numerically for
a two-dimensional lossy complex medium. If there are a large number of time steps and a significant
number of random sources, the negative derivative of the cross-correlation between two points is
found to be approximately equal to the impulse response function between the same points.
The inevitable effects of noise and losses in a sys-
tem are frequently viewed as nuisances in many fields
of work, with many researchers trying to develop meth-
ods to suppress them. However, it has been proven
both theoretically and experimentally that the ambient
noise in a system can actually be used as an advantage
[1]. An impulse response is the response that the sys-
tem has when it has been excited in some way. It can
be accurately measured by the cross-correlation of the
ambient noise, which can provide information by either
probing the medium or through non-destructive testing
[2]. This paper looks at cross-correlating the ambient
noise between two points, and comparing it to the im-
pulse response function (Green’s function). This has
already been tried in a multitude of disciplines with the
added benefit of being more cost-effective and safe for
retrieving the medium information, as no real source is
needed to be injected into the system [1]. In the field of
ultrasonics, this method is used to analyse the acoustic
thermal fluctuations in a material [3, 4], whereas in seis-
mic interferometry the seismic coda waves can be used
to obtain different types of information [5–8]. In the
industry of structural engineering, they suggest a pos-
sibility of using this method to monitor the turbulence-
induced vibrations in real-time [9], and acoustics can
also utilise this method for the analysis of the acous-
tic quality in a room [10]. Ocean acoustics have also
benefitted greatly from this method. In a sensitive ma-
rine life environment, minimal disturbance is preferred
as studies claim that the marine life can be disturbed
if close to sonar equipment [11, 12]. Thus, by using the
cross-correlation between two points in the presence of
the ambient noise instead of the impulse response from
one point to another, the marine life does not need to
be disturbed [13, 14].
A few papers delved deeper into this theory by show-
ing that the cross-correlation and impulse response
functions could be related not only by a direct com-
parison, but by using the time-derivative of the cross-
correlation to compare to the impulse response [15–18].
This interesting method proved to be more accurate,
as these papers produced almost identical results of the
∗ Completed at LPMC, CNRS UMR 7336 Universit´e de Nice-
Sophia Antipolis, Parc Valrose 06108, Nice cedex 2
time-derivative of the correlation function and the im-
pulse response, which is what will be tested in this pa-
per. The other unavoidable real-life situation is that of
losses in the system. This will also be investigated to
see how it affects both the impulse response and cross-
correlation functions.
The Green’s function is an approach that can be used
in order to obtain the response of a continuous function,
which can be depicted as an infinite sum of Dirac delta
functions. It is called the impulse response, as in the
event of a Dirac delta function excitation in a linear sys-
tem, the response is the equivalent of the Green’s func-
tion [19]. This response will give details on the source
and receiver points’ displacement. In this paper the im-
pulse response from one arbitrary point to another is to
be compared to the cross-correlation between the same
two points in the presence of noise. This is tested un-
der a number of conditions to test the validity of the
following relation,
G (A, B; τ) ≈ −
∂
∂τ
⟨p(A, t)p(B, t + τ)⟩t (1)
where G (A, B; τ) is the impulse response between
points A and B, and the right term is the negative
derivative of the time averaged cross-correlation be-
tween the diffuse wave field p measured at points A
and B, respectively at time t and delayed time t + τ
[18, 20]. The continuous damped wave equation used
for the system is
∂2
p
∂t2
+ 2γ
∂p
∂t
− v2 ∂2
p
∂x2
= 0 (2)
This is a second order partial differential equation,
which is with respect to time t and space x. It describes
the evolution of a non-dispersive wave with velocity v
and a damping rate γ. Eq. (2) can be discretised in both
the time and space variables through a second order
approximation scheme, which allows for the system to
use the previous and current time steps to calculate the
wave field at the next time step by the Finite-Difference
Time-Domain (FDTD) method
pi,j,k+1 =
2pi,j,k − (1 − γ∆t)pi,j,k−1 + U
1 + γ∆t
(3)

3. 2
where
U = v2
∆t2
(pi−1,j,k − 2pi,j,k + pi+1,j,k
∆x2
+
pi,j−1,k − 2pi,j,k + pi,j+1,k
∆y2
)
where the subscripts i, j, k to the wave field p represent
the x and y grid positions and time, respectively. A sub-
script of the variable i, j, k with +1 represents a step for-
ward, whereas −1 represents a step back. Additionally,
∆t is the time step, ∆x and ∆y are the steps in space.
Using dimensionless variables, we chose ∆t = 1√
2
, v = 1,
and ∆x = ∆y = 1, in order to satisfy the Courant-
Friedrichs-Lewy (CFL) condition for a stable wave field
evolution [21].
To test the validity of Eq. (1), two scenarios are pre-
sented of two systems which begin at rest (p = 0). The
first scenario has a short impulse which occurs at point
A, and the response at point B is then recorded - this
is the impulse response (Green’s function). The type of
impulse that will be used at point A is a half period sine
squared wave with a width of nine steps. The second
is a system where there are a number of noise sources
with values calculated at random between −1
2 and 1
2
occurring for the length of the simulation at random
positions. The cross-correlation of the wave amplitudes
at points A and B is then calculated. For a nonlinear
system, only one noise source could occur per system,
i.e., for 50 noise sources, the system would have to be
processed 50 separate times. The cross-correlation for
the system would then be the average of these 50 in-
dividual cross-correlation signals. Since this is a linear
system, the random noise responses at A and B can be
from many sources in one system, thus 50 noise sources
in a system only needs to be processed once.
The cross-correlation of the signals at A and B can
be calculated by four main steps. First, the system’s
response to the random noise at A and B for all time
is recorded. Then the Fast Fourier Transform (FFT) of
these responses is multiplied by the FFT of the impulse
signal, which will allow for the high frequencies of the
random noise responses to be filtered out. The third
step is to conjugate the filtered FFT of response A and
then multiply it by the filtered FFT response of B. The
last step then involves taking the inverse FFT of the
product in the previous step in order to obtain the cross-
correlation of A and B. The resultant outputs of the
cross-correlation and the impulse response functions are
then normalised by their own energy by dividing the
function F by
√
⟨|F|2⟩ in order to compare them.
The area in which the system was tested was designed
to have an irregular shape, allowing for ergodic dynam-
ics to occur in a classical ray system. It was important
for the shape not to have any spatial symmetry or par-
allel lines as this would contribute to a ”bouncing ball
effect”, which allows for non-universal behaviours to oc-
cur in the system [22]. In order to achieve this, three
different variables were tested: the number of timesteps
Tn, the number of random sources Sn and the decay γ in
the system. Henceforth, these variables will be referred
to by their symbols, with their subscripts n being the
amount, i.e., T106 will represent 106
timesteps and S50
will be 50 randomly placed source points.
These variables were tested to see how the cross-
correlation function compares to the impulse response.
These variables relate to ergodicity, as T must be large
enough and be from a large S in order to have the waves
interacting with the boundaries. Thus, the waves will
be travelling to all parts of the system and have bounced
many times at the boundary, therefore the system will
be ergodic.
Most previous work has assumed γ = 0, i.e., a loss-
less system. This paper investigates the degree to which
Eq. (1) was satisfied in the presence of different types
of γ. Thus, a system was tested with γglobal, γlocal
and γnone, where the subscript represents the type of
γ, to investigate what type of effect that this had on
the comparison between the cross-correlation and im-
pulse response functions. The value for γglobal is cal-
culated through a proportionality factor of the system
area, which allows the value that is assigned for the
γlocal area to be applied to the whole grid area, thus
the two values would have the same average γ i.e.,
Arealocal × γlocal = Areaglobal × γglobal. The value for
γglobal is quoted throughout the paper, since from Eq.
(2), the typical decay time for the waves in the system
can be approximated as γ−1
global. Fig. 1 illustrates the
system under consideration, including the area for when
γlocal is being tested. The dark grey area in Fig. 1 acts
0 20 40 60 80 100 120
0
10
20
30
40
50
60
70
80
X
Y
Point A
R1
R
2
R3
γ
local
Point B
Example
source point
FIG. 1. Shape of the system with variable information la-
bels. R1 is a circle with a radius of 50 centred at (120,100),
R2 is an ellipse with a major radius of 119 and minor ra-
dius of 80, centred at (60,-69) and R3 is also an ellipse with
a major radius of 100 and minor radius of 90, centred at
(-92,38).
as the area in which the system will be tested. It has
properties which represent a system with v = 1, while
the white areas are set to v = 0 to produce boundaries
which act as a hard wall with perfect reflection. The
S positions shown on Fig. 1 are merely an example,
as the locations are calculated at random, but with a
validation procedure to ensure that the points do not
occur in any problematic areas. If this occurs, the lo-
cation is rejected and the process reiterated until no S
points occur at the receiver points A or B, extremely
close to either the v = 0 or γlocal areas or on the direct
line between A and B.

4. 3
A system with γglobal = 7.71 × 10−5
, S50 and T106 ,
provided an excellent comparison between the cross-
correlation and the impulse response functions, which
can be seen in Fig. 2. Although the system had T106 ,
0 50 100 150 200 250 300 350 400 450 500
−3
−2
−1
0
1
2
3
Time (∆t × T)
Amplitude
Impulse response Derived correlation
FIG. 2. Comparison of the differentiated cross-correlation
and the impulse response in a γglobal system with T106 and
S50, zoomed in for a time window of 0 to 500 (T707).
the time window which is shown in Fig. 2 is from
the time of 0 to 500, thus 707 timesteps, in order to
have a clear look at the two functions. Comparison be-
tween the cross-correlation and impulse response func-
tions does not start from the initial time, as it is recom-
mended in a paper by Weaver and Lobkis (2001) that
the early times of the signal should be discarded as they
can often be distorted by both noise and the waves not
being fully diffuse [3]. Thus all comparisons will be
taken from the time of 48, i.e., the 68th timestep, which
is shown by the dashed line in Fig. 2. This delay also
corresponds to the time it takes for the wave to get from
A to B.
To quantify the influence of these chosen variables,
a χ2
analysis was completed to determine the differ-
ence between the cross-correlation and impulse response
functions. The closer χ2
is to 0, the more accurate is the
resemblance between the two functions. The χ2
value
was calculated by
χ2
=
∑
( ˆFC − ˆFI)2
√
∑ ˆFC
2
×
√
∑ ˆFI
2
(4)
where the function ˆF followed by a subscript of C rep-
resents the normalised cross-correlation signal, and I
represents the normalised impulse response signal. The
variables which were tested in Fig. 3 with a χ2
analysis,
were T and S for each γ system.
It can be seen in Fig. 3 that with increasing S and
T, the value of χ2
can be reduced. A larger T gives
a greater accuracy than the lower T since the system
has time to gain ergodicity. Ergodicity is achieved at a
larger T as the waves cover the entire area and inter-
act many times with the boundaries of the system. In-
deed, Lobkis and Weaver touched upon the idea of wave
chaos theory when comparing the cross-correlation and
impulse response functions. They postulated that if the
system were not fully chaotic, i.e., if the system is not
ergodic, then the correlation function would be inaccu-
rate [15]. This could be due to the presence of a scar,
which is an area where localisation in the system has oc-
curred [22]. If a source or one of the receivers were to be
10
0
10
1
0
0.5
1
1.5
2
2.5
3
3.5
S (log scale)
χ
2
value
γglobal
T10
6
γglobal
T10
5
γglobal
T10
4
γ
local
T
10
6
γ
local
T
10
5
γ
local
T
10
4
γnone
T10
6
γ
none
T
10
5
γnone
T10
4
FIG. 3. χ2
value for each system in a time window of 48 to
500 (68th to the 707th timestep). Three different γ systems
of γglobal (red), γlocal (blue) and γnone (green), and change
in T of T106 (solid), T105 (dashed) and T104 (dashed dot).
Results are shown for S1, S2, S5 and S50 for the system on
a log scale.
present on the path of the scar, then it is not taking the
entire system into account since the wave may become
trapped in a periodic orbit of one area of the system.
This may produce inaccurate results. There was only a
marginal difference between the χ2
value for T105 and
T106 , thus it is impractical to increase the T past 106
as
the χ2
value is unlikely to be minimised further. In ad-
dition, it would greatly increase the computation time.
Fig. 3 not only eliminates the lower T but highlights
an inconsistency in the use of S1. Further investigation
into this showed that although the use of S1 could pro-
duce a good match between the cross-correlation and
the impulse response, it would only occur under certain
conditions e.g., if the source point was near the receivers
points of A or B. Otherwise a substantially inaccurate
cross-correlation function would be produced. As this is
a linear system, the use of a larger S is more reliable as
these inaccuracies would be self-averaged by the many
waves travelling in the system. S2 was found to have
similar inconsistencies to S1 since two source positions
could cancel one another out, thus removing crucial in-
formation. S5 provided satisfactory results with the χ2
value being relatively low, but S50 proved to be the most
accurate.
The value of γ in the system was also investigated
to see how this variable affected the comparison of the
cross-correlation and impulse response functions. This
χ2
analysis for different γ rates is presented in Fig. 4.
There are some inevitable fluctuations, which can be
seen by points around the red dashed line, but these
are not typically larger than ±0.1 for the γ values used.
Beyond the range of decay rates shown in Fig. 4, the χ2
value for both the γglobal and γlocal systems increased.
A test of a system with a γ value which was 2 orders
of magnitude larger than that chosen for Fig. 2 i.e.,
γglobal = 7.71 × 10−5
to γglobal = 7.71 × 10−3
, showed
an increase of the χ2
value from 0.22 to 0.52 and 0.02 to
0.27 for γlocal and γglobal systems, respectively. Hence,
as γ became larger, the accuracy of the comparison be-
tween the cross-correlation and impulse response func-
tions gradually decreased. Therefore, it is recommended
that the γglobal value is in the range 0.3 − 2.5 × 10−4
as

5. 4
otherwise the χ2
value increases rapidly. The χ2
value
for the system with γnone can also be viewed in Fig. 4
as the first point, i.e., where the γglobal value is 0.
0 0.5 1 1.5 2 2.5
x 10
−4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
γ
global
value
χ
2
value
γ
global
γ
local
Average of fluctuation
FIG. 4. χ2
analysis for a γlocal (blue) and γglobal (red) sys-
tem to see how γ affects the χ2
. The red dashed line rep-
resents the average of some additional values which were
included to show how χ2
can fluctuate. This was used on a
system of T106 and S50. γlocal is worked out proportionally.
An important aspect which affects the outcome of
not only the cross-correlation, but also the impulse re-
sponse function, is the placement of the receivers A and
B in the different γ systems. Through investigation of
these points and how their location affects the output,
it was found that if they were moved closer together,
the χ2
value would become larger. In a γglobal system,
this increase was only marginal, however in a γlocal sys-
tem, the increase was more significant. If the receivers
were moved further apart, the χ2
value decreased in
a γlocal system, thus obtaining a closer match of the
functions. However, in a γglobal system, the χ2
value
increased slightly. Hence, the location of A and B had
a direct impact on the outcome of the two functions,
and requires further investigation.
It has been demonstrated that a combination of a
large S and T were the most accurate variable values
since this allowed the system to achieve ergodicity. This
is because a large T allows for the signals from the S50
to travel to effectively all points in the system. This is
due to the area being designed as chaotic, as this allows
the signals travelling from the source points to cover
the area. Therefore for the most accurate comparison
of the cross-correlation and the impulse response, the
variables would have T106 and S50 in a γglobal system,
as can be seen by Fig. 2 as well as the χ2
value in Fig.
4.
This paper has shown that with a significant number
of random sources occurring in a chaotic area for a large
amount of time, Eq. (1) is satisfied. That is, an impulse
released at one point and received at another can be
approximated with great accuracy by the negative time
derivative of the cross-correlation between the same two
points in a two-dimensional lossy system. There are
many possibilities on which future work might focus.
As previously mentioned, the placement of the points A
and B had a direct impact on the comparison between
the cross-correlation and impulse response function
for a system with γglobal = 7.71 × 10−5
. For γglobal,
the fluctuations in the χ2
value were not significant.
They were only altered by a magnitude of 10−3
by
the difference in point locations. However, in the case
of a γlocal system, these χ2
values changed by 10−1
.
A future task could be to investigate the significance
between different γ rates and the distance between A
and B to see whether any sort of trend occurs. Another
potential future task would be to change the frequency
range used for both the impulse and random signals,
and determine the impact on the cross-correlation
signal and the impulse response function.
I would like to thank my supervisors O. Legrand,
U. Kuhl and F. Mortessagne, for their support and
guidance during my time with them and, along with
the CNRS, for allowing me to complete an internship
with them.
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