Tutorial on seismic interferometry:
Part 1 — Basic principles and applications
Kees Wapenaar
1
, Deyan Draganov
1
, Roel Snieder
2
, Xander Campman
3
, and Arie Verdel
3
ABSTRACT
Seismic interferometry involves the crosscorrelation of re-
sponses at different receivers to obtain the Green’s function be-
tween these receivers. For the simple situation of an impulsive
plane wave propagating along the x-axis, the crosscorrelation of
the responses at two receivers along the x-axis gives the Green’s
function of the direct wave between these receivers. When the
source function of the plane wave is a transient 共as in exploration
seismology兲 or a noise signal 共as in passive seismology兲, then the
crosscorrelation gives the Green’s function, convolved with the
autocorrelation of the source function. Direct-wave interferome-
try also holds for 2D and 3D situations, assuming the receivers
are surrounded by a uniform distribution of sources. In this case,
the main contributions to the retrieved direct wave between the
receivers come from sources in Fresnel zones around stationary
points. The main application of direct-wave interferometry is the
retrieval of seismic surface-wave responses from ambient noise
and the subsequent tomographic determination of the surface-
wave velocity distribution of the subsurface. Seismic interferom-
etry is not restricted to retrieving direct waves between receivers.
In a classic paper, Claerbout shows that the autocorrelation of the
transmission response of a layered medium gives the plane-wave
reflection response of that medium. This is essentially 1D reflect-
ed-wave interferometry. Similarly, the crosscorrelation of the
transmission responses, observed at two receivers, of an arbitrary
inhomogeneous medium gives the 3D reflection response of that
medium. One of the main applications of reflected-wave interfer-
ometry is retrieving the seismic reflection response from ambient
noise and imaging of the reflectors in the subsurface. A common
aspect of direct- and reflected-wave interferometry is that virtual
sources are created at positions where there are only receivers
without requiring knowledge of the subsurface medium parame-
ters or of the positions of the actual sources.
INTRODUCTION
In this two-part tutorial, we give an overview of the basic princi-
ples and the underlying theory of seismic interferometry and discuss
applications and new advances. The term seismic interferometry re-
fers to the principle of generating new seismic responses of virtual
sources
4
by crosscorrelating seismic observations at different re-
ceiver locations. One can distinguish between controlled-source and
passive seismic interferometry. Controlled-source seismic interfer-
ometry, pioneered by Schuster 共2001兲, Bakulin and Calvert 共2004兲,
and others, comprises a new processing methodology for seismic ex-
ploration data. Apart from crosscorrelation, controlled-source inter-
ferometry also involves summation of correlations over different
source positions. Passive seismic interferometry, on the other hand,
is a methodology for turning passive seismic measurements 共ambi-
ent seismic noise or microearthquake responses兲 into deterministic
seismic responses. Here, we further distinguish between retrieving
surface-wave transmission responses 共Campillo and Paul, 2003;
Shapiro and Campillo, 2004; Sabra, Gerstoft, et al., 2005a兲 and ex-
ploration reflection responses 共Claerbout, 1968; Scherbaum, 1987b;
Draganov et al., 2007, 2009兲. In passive interferometry of ambient
noise, no explicit summation of correlations over different source
positions is required because the correlated responses are a superpo-
sition of simultaneously acting uncorrelated sources.
In all cases, the response that is retrieved by crosscorrelating two
Manuscript received by the Editor 30 November 2009; published online 14 September 2010.
1
Delft University of Technology, Department of Geotechnology, Delft, The Netherlands. E-mail: c.p.a.wapenaar@tudelft.nl; d.s.draganov@tudelft.nl.
2
Colorado School of Mines, Center for Wave Phenomena, Golden, Colorado, U.S.A. E-mail: rsnieder@mines.edu.
3
Shell International Exploration and Production, Rijswijk, The Netherlands. E-mail: xander.campman@shell.com; arie.verdel@gmail.com.
© 2010 Society of Exploration Geophysicists.All rights reserved.
4
In the literature on seismic interferometry, the term virtual source often refers to the method of Bakulin and Calvert 共2004, 2006兲, which is discussed extensive-
ly in Part 2. However, creating a virtual source is the essence of nearly all seismic interferometry methods 共see e.g., Schuster 共2001兲, who already used this term兲.
In this paper 共Parts 1 and 2兲 we use the term virtual source whenever appropriate. When it refers to Bakulin and Calvert’s method, we will mention this explicitly.
GEOPHYSICS, VOL. 75, NO. 5 共SEPTEMBER-OCTOBER 2010兲; P. 75A195–75A209, 15 FIGS.
10.1190/1.3457445
75A195
Downloaded 28 Sep 2012 to 131.180.130.198. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/
receiver recordings 共and summing over different sources兲 can be in-
terpreted as the response that would be measured at one of the re-
ceiver locations as if there were a source at the other. Because such a
point-source response is equal to a Green’s function convolved with
a wavelet, seismic interferometry is also often called Green’s func-
tion retrieval. Both terms are used in this paper. The term interferom-
etry is borrowed from radio astronomy, where it refers to crosscorre-
lation methods applied to radio signals from distant objects 共Thomp-
son et al., 2001兲. The name Green’s function honors George Green
who, in a privately published essay, introduced the use of impulse re-
sponses in field representations 共Green, 1828兲. Challis and Sheard
共2003兲 give a brief history of Green’s life and theorem. Ramírez and
Weglein 共2009兲 review applications of Green’s theorem in seismic
processing.
Early successful results of Green’s function retrieval from noise
correlations were obtained in the field of ultrasonics 共Weaver and
Lobkis, 2001, 2002兲. The experiments were done with diffuse fields
in a closed system. Here diffuse means that the amplitudes of the nor-
mal modes are uncorrelated but have equal expected energies.
Hence, the crosscorrelation of the field at two receiver positions does
not contain cross-terms of unequal normal modes. The sum of the re-
maining terms is proportional to the modal representation of the
Green’s function of the closed system 共Lobkis and Weaver, 2001兲.
This means that the crosscorrelation of a diffuse field in a closed sys-
tem converges to its impulse response. Later, it was recognized 共e.g.,
Godin, 2007兲 that this theoretical explanation is akin to the fluctua-
tion-dissipation theorem 共Callen and Welton, 1951; Rytov, 1956;
Rytov et al., 1989; Le Bellac et al., 2004兲.
The earth is a closed system; but at the scale of global seismology,
the wavefield is far from diffuse.At the scale of exploration seismol-
ogy, an ambient-noise field may have a diffuse character, but the en-
compassing system is not closed. Hence, for seismic interferometry,
the normal-mode approach breaks down. Throughout this paper, we
consider seismic interferometry 共or Green’s function retrieval兲 in
open systems, including half-spaces below a free surface. Instead of
a treatment per field of application or a chronological discussion, we
have chosen a setup in which we explain the principles of seismic in-
terferometry step by step. In Part 1, we start with the basic principles
of 1D direct-wave interferometry and conclude with a discussion of
the principles of 3D reflected-wave interferometry. We present ap-
plications in controlled-source as well as passive interferometry and,
where appropriate, review the historical background. To stay fo-
cused on seismic applications, we refrain from a further discussion
of the normal-mode approach, nor do we address the many interest-
ing applications of Green’s function retrieval in underwater acous-
tics 共e.g., Roux and Fink, 2003; Sabra et al., 2005; Brooks and Ger-
stoft, 2007兲.
DIRECT-WAVE INTERFEROMETRY
1D analysis of direct-wave interferometry
We start our explanation of seismic interferometry by considering
an illustrative 1D analysis of direct-wave interferometry. Figure 1a
shows a plane wave, radiated by an impulsive unit source at x ⳱ x
S
and t ⳱ 0, propagating in the rightward direction along the x-axis.
We assume that the propagation velocity c is constant and the medi-
um is lossless. There are two receivers along the x-axis at x
A
and x
B
.
Figure 1b shows the response observed by the first receiver at x
A
.We
denote this response as G共x
A
,x
S
,t兲, where G stands for the Green’s
function. Throughout this paper, we use the common convention
that the first two arguments in G共x
A
,x
S
,t兲 denote the receiver and
source coordinates, respectively 共here, x
A
and x
S
兲, whereas the last
argument denotes time t or angular frequency
. In our example, this
Green’s function consists of an impulse at t
A
⳱ 共x
A
ⳮ x
S
兲/ c; there-
fore, G共x
A
,x
S
,t兲⳱
␦
共tⳮ t
A
兲, where
␦
共t兲 is the Dirac delta function.
Similarly, the response at x
B
is given by G共x
B
,x
S
,t兲⳱
␦
共tⳮ t
B
兲, with
t
B
⳱ 共x
B
ⳮ x
S
兲/ c 共Figure 1c兲.
Seismic interferometry involves the crosscorrelation of responses
at two receivers, in this case at x
A
and x
B
. Looking at Figure 1a,itap-
pears that the raypaths associated with G共x
A
,x
S
,t兲 and G共x
B
,x
S
,t兲
have the path from x
S
to x
A
in common. The traveltime along this
common path cancels in the crosscorrelation process, leaving the
traveltime along the remaining path from x
A
to x
B
, i.e., t
B
ⳮ t
A
⳱ 共x
B
ⳮ x
A
兲/ c. Hence, the crosscorrelation of the responses in Figure 1b
and c is an impulse at t
B
ⳮ t
A
共see Figure 1d兲. This impulse can be in-
terpreted as the response of a source at x
A
observed by a receiver at
x
B
, i.e., the Green’s function G共x
B
,x
A
,t兲. An interesting observation is
that the propagation velocity c and the position of the actual source
x
S
need not be known. The traveltimes along the common path from
x
S
to x
A
compensate each other, independent of the propagation ve-
locity and the length of this path. Similarly, if the source impulse
would occur at t⳱ t
S
instead of at t⳱ 0, the impulses observed at x
A
and x
B
would be shifted by the same amount of time t
S
, which would
be canceled in the crosscorrelation. Thus, the absolute time t
S
at
which the source emits its pulse need not be known.
Let us discuss this example a bit more precisely. We denote the
crosscorrelation of the impulse responses at x
A
and x
B
as
G共x
B
,x
S
,t兲ⴱ G共x
A
,x
S
,ⳮt兲. The asterisk denotes temporal convolu-
tion, but the time reversal of the second Green’s function turns the
convolution into a correlation, defined as G共x
B
,x
S
,t兲ⴱ G共x
A
,x
S
,ⳮ t兲
⳱ 兰G共x
B
,x
S
,t Ⳮ t
⬘
兲G共x
A
,x
S
,t
⬘
兲dt
⬘
. Substituting the delta functions
into the right-hand side gives 兰
␦
共t Ⳮ t
⬘
ⳮ t
B
兲
␦
共t
⬘
ⳮ t
A
兲dt
⬘
⳱
␦
共t
ⳮ 共t
B
ⳮ t
A
兲兲⳱
␦
共t ⳮ 共x
B
ⳮ x
A
兲/ c兲. This is indeed the Green’s func-
tion G共x
B
,x
A
,t兲, propagating from x
A
to x
B
. Because we started this
derivation with the crosscorrelation of the Green’s functions, we
have obtained the following 1D Green’s function representation:
Figure 1. A 1D example of direct-wave interferometry. 共a兲 A plane
wave traveling rightward along the x-axis, emitted by an impulsive
source at x⳱ x
S
and t⳱ 0. 共b兲 The response observed by a receiver
at x
A
. This is the Green’s function G共x
A
,x
S
,t兲. 共c兲 As in 共b兲 but for a re-
ceiver at x
B
. 共d兲 Crosscorrelation of the responses at x
A
and x
B
. This is
interpreted as the response of a source at x
A
, observed at x
B
, i.e.,
G共x
B
,x
A
,t兲.
75A196 Wapenaar et al.
Downloaded 28 Sep 2012 to 131.180.130.198. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/
G共x
B
,x
A
,t兲⳱ G共x
B
,x
S
,t兲ⴱ G共x
A
,x
S
,ⳮ t兲. 共1兲
This representation formulates the principle that the crosscorrelation
of observations at two receivers 共x
A
and x
B
兲 gives the response at one
of those receivers 共x
B
兲 as if there were a source at the other receiver
共x
A
兲. It also shows why seismic interferometry is often called
Green’s function retrieval.
Note that the source is not necessarily an impulse. If the source
function is defined by some wavelet s共t兲, then the responses at x
A
and
x
B
can be written as u共x
A
,x
S
,t兲⳱ G共x
A
,x
S
,t兲ⴱ s共t兲 and u共x
B
,x
S
,t兲
⳱ G共x
B
,x
S
,t兲ⴱ s共t兲, respectively. Let S
s
共t兲 be the autocorrelation of
the wavelet, i.e., S
s
共t兲 ⳱ s共t兲ⴱ s共ⳮ t兲. Then the crosscorrelation of
u共x
A
,x
S
,t兲 and u共x
B
,x
S
,t兲 gives the right-hand side of equation 1, con-
volved with S
s
共t兲. This is equal to the left-hand side of equation 1,
convolved with S
s
共t兲. Therefore,
G共x
B
,x
A
,t兲ⴱ S
s
共t兲⳱ u共x
B
,x
S
,t兲ⴱ u共x
A
,x
S
,ⳮ t兲. 共2兲
In words: If the source function is a wavelet instead of an impulse,
then the crosscorrelation of the responses at two receivers gives the
Green’s function between these receivers, convolved with the auto-
correlation of the source function.
This principle holds true for any source function, including noise.
Figure 2a and b shows the responses at x
A
and x
B
, respectively, of a
bandlimited noise source N共t兲 at x
S
共the central frequency of the
noise is 30 Hz; the figure shows only 4 s of a total of 160 s of noise兲.
In this numerical example, the distance between the receivers is
1200 m and the propagation velocity is 2000 m / s; hence, the travel-
time between these receivers is 0.6 s. As a consequence, the noise re-
sponse at x
B
in Figure 2b is 0.6 s delayed with respect to the response
at x
A
in Figure 2a 共similar to the impulse in Figure 1c delayed with re-
spect to the impulse in Figure 1b兲. Crosscorrelation of these noise re-
sponses gives, analogous to equation 2, the impulse response be-
tween x
A
and x
B
, convolved with S
N
共t兲, i.e., the autocorrelation of the
noise N共t兲. The correlation is shown in Figure 2c, which indeed re-
veals a bandlimited impulse centered at t ⳱ 0.6 s 共the traveltime
from x
A
to x
B
兲. Note that from registrations at two receivers of a noise
field from an unknown source in a medium with unknown propaga-
tion velocity, we have obtained a bandlimited version of the Green’s
function. By dividing the distance between the receivers 共1200 m兲
by the traveltime estimated from the bandlimited Green’s function
共0.6 s兲, we obtain an estimate of the propagation velocity between
the receivers 共2000 m / s兲. This illustrates that direct-wave interfer-
ometry can be used for tomographic inversion.
Until now, we considered a single plane wave propagating in the
positive x -direction. In Figure 3a, we consider the same configura-
tion as in Figure 1a, but now an impulsive unit source at x ⳱ x
S
⬘
radi-
ates a leftward-propagating plane wave. Figure 3b is the response at
x
A
, given by G共x
A
,x
S
⬘
,t兲⳱
␦
共t ⳮ t
A
⬘
兲, with t
A
⬘
⳱ 共x
S
⬘
ⳮ x
A
兲/ c. Similarly,
the response at x
B
is G共x
B
,x
S
⬘
,t兲⳱
␦
共t ⳮ t
B
⬘
兲, with t
B
⬘
⳱ 共x
S
⬘
ⳮ x
B
兲/ c
共Figure 3c兲. The crosscorrelation of these responses gives
␦
共t ⳮ 共t
B
⬘
ⳮ t
A
⬘
兲兲⳱
␦
(t Ⳮ 共x
B
ⳮ x
A
兲/ c), which is equal to the time-reversed
Green’s function G共x
B
,x
A
,ⳮ t兲. So, for the configuration of Figure
3a, we obtain the following Green’s function representation:
G共x
B
,x
A
,ⳮ t兲⳱ G 共x
B
,x
S
⬘
,t兲ⴱ G共x
A
,x
S
⬘
,ⳮt兲. 共3兲
We can combine equations 1 and 3 as follows:
G共x
B
,x
A
,t兲Ⳮ G共x
B
,x
A
,ⳮt兲⳱
兺
i⳱1
2
G共x
B
,x
S
共i兲
,t兲ⴱ G共x
A
,x
S
共i兲
,ⳮt兲,
共4兲
where x
S
共i兲
for i ⳱ 1,2 stands for x
S
and x
S
⬘
, respectively.
For the 1D situation, this combination may not seem very useful.
We analyze it here, however, because this representation better re-
sembles the 2D and 3D representations we encounter later. Note that
because G共x
B
,x
A
,t兲 is the causal response of an impulse at t⳱ 0
共meaning it is nonzero only for t ⬎ 0兲, it does not overlap with
G共x
B
,x
A
,ⳮ t兲 共which is nonzero only for t ⬍ 0兲. Hence, G共x
B
,x
A
,t兲
can be resolved from the left-hand side of equation 4 by extracting
the causal part. If the source function is a wavelet s共t兲 with autocor-
relation S
s
共t兲, we obtain, analogous to equation 2,
Figure 2. As in Figure 1 but this time for a noise source N共t兲 at x
S
. 共a兲
The response observed at x
A
, i.e., u共x
A
,x
S
,t兲⳱ G共x
A
,x
S
,t兲ⴱ N共t兲. 共b兲
As in 共a兲 but for a receiver at x
B
. 共c兲 The crosscorrelation, which is
equal to G共x
B
,x
A
,t兲ⴱ S
N
共t兲, with S
N
共t兲 the autocorrelation of the
noise.
Figure 3. As in Figure 1 but this time for a leftward-traveling impul-
sive plane wave. The crosscorrelation in 共d兲 is interpreted as the
time-reversed Green’s function G共x
B
,x
A
,ⳮt兲.
Tutorial on interferometry: Part 1 75A197
Downloaded 28 Sep 2012 to 131.180.130.198. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/
兵G共x
B
,x
A
,t兲Ⳮ G共x
B
,x
A
,ⳮ t兲其ⴱ S
s
共t兲
⳱
兺
i⳱1
2
u共x
B
,x
S
共i兲
,t兲ⴱ u共x
A
,x
S
共i兲
,ⳮ t兲. 共5兲
Here, G共x
B
,x
A
,t兲ⴱ S
s
共t兲 may have some overlap with G共x
B
,x
A
,
ⳮt兲 ⴱ S
s
共t兲 for small 兩t兩, depending on the length of the autocorrela-
tion function S
s
共t兲. Therefore, G共x
B
,x
A
,t兲ⴱ S
s
共t兲 can be extracted
from the left-hand side of equation 5, except for small distances 兩x
B
ⳮ x
A
兩.
The right-hand sides of equations 4 and 5 state that the crosscorre-
lation is applied to the responses of each source separately, after
which the summation over the sources is carried out. For impulsive
sources or transient wavelets s共t兲, these steps should not be inter-
changed. Let us see why. Suppose the sources at x
S
and x
S
⬘
act simul-
taneously, as illustrated in Figure 4a. Then the response at x
A
would
be given by u共x
A
,t兲⳱ 兺
i⳱1
2
G共x
A
,x
S
共i兲
,t兲ⴱ s共t兲 and the response at x
B
by
u共x
B
,t兲⳱ 兺
j⳱1
2
G共x
B
,x
S
共j兲
,t兲ⴱ s共t兲. These responses are shown in Fig-
ure 4b and c for an impulsive source 共s共t兲⳱
␦
共t兲兲. The crosscorrela-
tion of these responses, shown in Figure 4d, contains two cross-
terms at t
B
ⳮ t
A
⬘
and t
B
⬘
ⳮ t
A
that have no physical meaning. Hence, for
impulsive or transient sources, the order of crosscorrelation and
summation matters.
The situation is different for noise sources. Consider two simul-
taneously acting noise sources N
1
共t兲 and N
2
共t兲 at x
S
and x
S
⬘
, respec-
tively. The responses at x
A
and x
B
are given by u共x
A
,t兲
⳱ 兺
i⳱1
2
G共x
A
,x
S
共i兲
,t兲ⴱ N
i
共t兲 and u共x
B
,t兲⳱ 兺
j⳱1
2
G共x
B
,x
S
共j兲
,t兲ⴱ N
j
共t兲, re-
spectively 共see Figure 5a and b兲. Because each of these responses is
the superposition of a rightward- and a leftward-propagating wave,
the response in Figure 5b is not a shifted version of that in Figure 5a
共unlike the responses in Figure 2a and b兲. We assume that the noise
sources are uncorrelated; thus, 具N
j
共t兲 ⴱ N
i
共ⳮt兲典⳱
␦
ij
S
N
共t兲, where
␦
ij
is the Kronecker delta function and 具·典 denotes ensemble averag-
ing. In practice, the ensemble averaging is replaced by integrating
over sufficiently long time. In the numerical example the duration of
the noise signals is again 160 s 共only 4 s of noise is shown in Figure
5a and b兲. For the crosscorrelation of the responses at x
A
and x
B
,we
can now write
具u共x
B
,t兲ⴱ u共x
A
,ⳮ t兲典
⳱
冓
兺
j⳱1
2
兺
i⳱1
2
G共x
B
,x
S
共j兲
,t兲ⴱ N
j
共t兲
ⴱ G共x
A
,x
S
共i兲
,ⳮ t兲ⴱ N
i
共ⳮ t兲
冔
⳱
兺
i⳱1
2
G共x
B
,x
S
共i兲
,t兲ⴱ G共x
A
,x
S
共i兲
,ⳮ t兲ⴱ S
N
共t兲. 共6兲
Combining equation 6 with equation 4, we finally obtain
兵G共x
B
,x
A
,t兲Ⳮ G共x
B
,x
A
,ⳮ t兲其ⴱ S
N
共t兲⳱ 具u共x
B
,t兲ⴱ u共x
A
,ⳮt兲典.
共7兲
Expression 7 shows that the crosscorrelation of two observed fields
at x
A
and x
B
, each of which is the superposition of rightward- and left-
ward-propagating noise fields, gives the Green’s function between
x
A
and x
B
plus its time-reversed version, convolved with the autocor-
relation of the noise 共see Figure 5c兲. The cross-terms, unlike Figure
4d, do not contribute because the noise sources N
1
共t兲 and N
2
共t兲 are
uncorrelated.
Miyazawa et al. 共2008兲 apply equation 7 with x
A
and x
B
at different
depths along a borehole in the presence of industrial noise at Cold
Lake, Alberta, Canada. By choosing for u different components of
multicomponent sensors in the borehole, they retrieve separate
Green’s functions for P- and S-waves, the latter with different polar-
izations. From the arrival times in the Green’s functions, they derive
the different propagation velocities and accurately quantify shear-
wave splitting.
Despite the relative simplicity of our 1D analysis of direct-wave
interferometry, we can make several observations about seismic in-
terferometry that also hold true for more general situations. First, we
can distinguish between interferometry for impulsive or transient
Figure 4. As in Figures 1 and 3 but with simultaneously rightward-
and leftward-traveling impulsive plane waves. The crosscorrelation
in 共d兲 contains cross-terms that have no physical meaning.
Figure 5. As in Figure 4 but this time with simultaneously rightward-
and leftward-traveling uncorrelated noise fields. The crosscorrela-
tion in 共c兲 contains no cross-terms.
75A198 Wapenaar et al.
Downloaded 28 Sep 2012 to 131.180.130.198. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/
sources on the one hand 共equations 4 and 5兲 and interferometry for
noise sources on the other hand 共equation 7兲. In the case of impulsive
or transient sources, the responses of each source must be crosscor-
related separately, after which a summation over the sources takes
place. In the case of uncorrelated noise sources, a single crosscorre-
lation suffices.
Second, it appears that an isotropic illumination of the receivers is
required to obtain a time-symmetric response between the receivers
共of which the causal part is the actual response兲. In one dimension,
isotropic illumination means equal illumination by rightward- and
leftward-propagating waves. In two and three di-
mensions, it means equal illumination from all di-
rections 共discussed in the next section兲.
Finally, instead of the time-symmetric re-
sponse G共x
B
,x
A
,t兲Ⳮ G共x
B
,x
A
,ⳮt兲, in the litera-
ture we often encounter an antisymmetric re-
sponse G共x
B
,x
A
,t兲ⳮ G共x
B
,x
A
,ⳮt兲. This is merely
a result of differently defined Green’s functions.
Note that a simple time differentiation of the
Green’s functions would turn the symmetric re-
sponse into an antisymmetric one, and vice versa
共see Wapenaar and Fokkema 关2006兴 for a more
detailed discussion on this aspect兲.
2D and 3D analysis of direct-wave
interferometry
We extend our discussion of direct-wave inter-
ferometry to configurations with more dimen-
sions. In the following discussion, we mainly use
heuristic arguments, illustrated with a numerical
example. For a more precise derivation based on
stationary-phase analysis, we refer to Snieder
共2004兲.
Consider the 2D configuration shown in Figure
6a. The horizontal dashed line corresponds to the
1D configuration of Figure 1a, with two receivers
at x
A
and x
B
, 1200 m apart 共x denotes a Cartesian
coordinate vector兲. The propagation velocity c is
2000 m / s, and the medium is again assumed to
be lossless. Instead of plane-wave sources, we
have many point sources denoted by the small
black dots, distributed over a “pineapple slice,”
emitting transient signals with a central frequen-
cy of 30 Hz. In polar coordinates, the positions of
the sources are denoted by 共r
S
,
S
兲. The angle
S
is equidistantly sampled 共⌬
S
⳱ 0.25° 兲, whereas
the distance r
S
to the center of the slice is chosen
randomly between 2000 and 3000 m. The re-
sponses at the two receivers at x
A
and x
B
are
shown in Figure 6b and c, respectively, as a func-
tion of the 共polar兲 source coordinate
S
共for dis-
play purposes, only every sixteenth trace is
shown兲. These responses are crosscorrelated 共for
each source separately兲, and the crosscorrelations
are shown in Figure 6d, again as a function of
S
.
Such a gather is often called a correlation gather.
Note that the traveltimes in this correlation gather
vary smoothly with
S
, despite the randomness of
the traveltimes in Figure 6b and c. This is because in the crosscorre-
lation process only the time difference along the paths to x
A
and x
B
matters.
The source in Figure 6a with
S
⳱ 0° plays the same role as the
plane-wave source at x
S
in Figure 1a. For this source, the crosscorre-
lation gives a signal at 兩x
B
ⳮ x
A
兩/ c ⳱ 0.6 s, seen in the trace at
S
⳱ 0° in Figure 6d. Similarly, the source at
S
⳱ 180° plays the same
role as the plane-wave source at x
S
⬘
in Figure 3a and leads to the trace
at
S
⳱ 180° in Figure 6d with a signal at ⳮ0.6 s. Analogous to
equation 5, we sum the crosscorrelations of all sources, i.e., we sum
Figure 6. A 2D example of direct-wave interferometry. 共a兲 Distribution of point sources,
isotropically illuminating the receivers at x
A
and x
B
. The thick dashed lines indicate the
Fresnel zones. 共b兲 Responses at x
A
as a function of the 共polar兲 source coordinate
S
. 共c兲
Responses at x
B
. 共d兲 Crosscorrelation of the responses at x
A
and x
B
. The dashed lines indi-
cate the Fresnel zones. 共e兲 The sum of the correlations in 共d兲. This is interpreted as
兵G共x
B
,x
A
,t兲Ⳮ G共x
B
,x
A
,ⳮ t兲其ⴱ S
s
共t兲. The main contributions come from sources in the
Fresnel zones indicated in 共a兲 and 共d兲. 共f兲 Single crosscorrelation of the responses at
x
A
and x
B
of simultaneously acting uncorrelated noise sources. The duration of the noise
signals was 9600 s.
Tutorial on interferometry: Part 1 75A199
Downloaded 28 Sep 2012 to 131.180.130.198. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/