Stationary-phase integrals in the cross-correlation of ambient1

noise2

Lapo Boschi1,2 and Cornelis Weemstra33

1Sorbonne Universites, UPMC Univ Paris 06, UMR 7193, Institut des Sciences de la4

Terre Paris (iSTeP), F-75005 Paris, France.5

1CNRS, UMR 7193, Institut des Sciences de la Terre Paris (iSTeP), F-75005 Paris,6

France.7

3Department of Geoscience and Engineering, Delft University of Technology,8

Stevinweg 1, 2628 CN Delft, The Netherlands.9

February 5, 201610

1

Abstract11

The cross-correlation of ambient signal allows seismologists to collect data even in the absence12

of seismic events. “Seismic interferometry” shows that the cross-correlation of simultaneous13

recordings of a random wave field made at two locations is formally related to the impulse14

response between those locations. This idea has found many applications in seismology, as15

a growing number of dense seismic networks become available: cross-correlating long seismic16

records, the Green’s function between instrument pairs is “reconstructed,” and used, just17

like the seismic recording of an explosion, in tomography, monitoring, etc. These applica-18

tions have been accompanied by theoretical investigations of the relationship between noise19

cross-correlation and the Green’s function; numerous formulations of “ambient-noise” theory20

have emerged, each based on different hypotheses and/or analytical approaches. The pur-21

pose of this study is to present most of those approaches together, providing a comprehensive22

overview of the theory. Understanding the specific hypotheses behind each Green’s function23

recipe is critical to its correct application. Hoping to guide non-specialists who approach24

ambient-noise theory for the first time, we treat the simplest formulation (the stationary-25

phase approximation applied to smooth unbounded media) in detail. We then move on to26

more general treatments, illustrating that the “stationary-phase” and “reciprocity-theorem”27

approaches lead to the same formulae when applied to the same scenario. We show that a28

formal cross-correlation/Green’s function relationship can be found in complex, bounded me-29

dia, and for nonuniform source distributions. We finally provide the bases for understanding30

how the Green’s function is reconstructed in the presence of scattering obstacles.31

1 Introduction32

In “seismic interferometry,” the Green’s function, or impulse response, of a medium can be33

determined empirically, based on the background signal recorded by two instruments over34

some time. The term “Green’s function”, ubiquitous in ambient-noise literature, indicates35

the response of a medium to an impulsive excitation: a point source [e.g., Morse and Ingard ,36

1986; Aki and Richards, 2002]. Measuring a Green’s function amounts to recording the ground37

oscillations that follow an explosion: exploiting the “ambient noise,” the same signal can be38

measured without setting off any explosive. This approach was foreshadowed by several39

early studies in ocean acoustics [e.g., Eckart , 1953; Cox , 1973] and small-scale seismology40

[Aki , 1957; Claerbout , 1968], and later applied successfully to helioseismology [Duvall et al.,41

2

1993; Woodard , 1997], ultrasound [Weaver and Lobkis, 2002; Malcolm et al., 2004], terrestrial42

seismology [Campillo and Paul , 2003; Shapiro et al., 2005], infrasound [Haney , 2009] and43

engineering [Snieder and Safak , 2006; Kohler et al., 2007] .44

Ambient-noise seismology on Earth takes advantage of the “ambient”, low-energy sig-45

nal that seems to be generated continuously by the coupling between oceans and solid Earth46

[e.g., Longuet-Higgins, 1950; Hasselmann, 1963; Stehly et al., 2006; Kedar et al., 2008; Hillers47

et al., 2012; Gualtieri et al., 2013; Traer and Gerstoft , 2014]. Its resolution is not limited48

by the nonuniform distribution of earthquakes, or by the difficulties inherent in setting up49

a man-made seismic source. Most ambient energy pertains to surface waves of period be-50

tween ∼ 5s and ∼ 30s, which are difficult to observe in the near field of an earthquake, and51

almost completely attenuated in the far field, where only longer-period surface waves can be52

measured. Ambient-noise-based surface-wave observations in this frequency range are thus53

complementary to traditional observations, and particularly useful to map the crust, litho-54

sphere and asthenosphere: the shorter the surface-wave period, the shallower the depth range.55

Some recent studies [Ruigrok et al., 2011; Ito and Shiomi , 2012; Poli et al., 2012a, b; Gorbatov56

et al., 2013; Nishida, 2013] show that not only surface-waves, but also body waves traveling57

over large distances and/or reflected by Earth discontinuities (the Moho, the upper-to-lower58

mantle boundary, etc.) can be extracted from ambient-signal cross-correlations.59

Ambient-noise correlation allows one to build a seismic data set even in the absence of60

earthquakes. In practice, the Green’s function is reconstructed only approximately and within61

a limited frequency band, but that is enough to estimate relevant parameters such as group62

and phase velocity of surface waves. This is valuable for a number of different applications,63

allowing passive imaging at the reservoir scale and in the context of hydrocarbon industry64

[De Ridder and Dellinger , 2011; Corciulo et al., 2012; Weemstra et al., 2013] as well as65

monitoring of time-dependent changes in material properties around an active fault [Wegler66

and Sens-Schonfelder , 2007], in the rigidity of a landslide-prone area [Mainsant et al., 2012],67

or in the shape and location of magma [Brenguier et al., 2011] and hydrocarbon [De Ridder68

et al., 2014] reservoirs.69

The last decade has seen the publication of a number of increasingly exhaustive math-70

ematical descriptions of the phenomenon of Green’s function reconstruction from ambient71

recordings [Lobkis and Weaver , 2001; Derode et al., 2003; Snieder , 2004; Wapenaar , 2004;72

Weaver and Lobkis, 2004; Roux et al., 2005; Wapenaar et al., 2005; Nakahara, 2006; Sanchez-73

Sesma and Campillo, 2006; Wapenaar et al., 2006; Tsai , 2009; Weaver et al., 2009]. In74

3

general, a mathematical relationship is found between the Green’s function associated with75

the locations of two receivers (i.e., the response, observed at one of the receivers, to a point76

source deployed at the other receiver) and the cross-correlation [e.g. Smith, 2011], computed77

over a long time interval, between the random ambient signal recorded by the receivers. (A78

detailed account of how ambient data are treated is given by Bensen et al. [2007].) Dif-79

ferent approaches to establishing this relationship have been followed, however, resulting in80

different formulations. Most authors first develop the theory for the simple case of acoustic81

(scalar) wave propagation in two- or three-dimensional media. Some formulations hold for82

heterogeneous and/or bounded media, while others are limited to infinite homogeneous me-83

dia. Defining ambient noise (a “diffuse field”) mathematically is a nontrivial problem per se:84

some authors describe it as the superposition of plane waves propagating in all directions;85

others prefer to superpose impulsive responses (Green’s functions), sometimes in two dimen-86

sions, sometimes in three; others yet define the noise field as one where all normal modes87

have the same probability of being excited. Finally, the expression “Green’s function” itself88

is ambiguous: in elastodynamics, it may refer to the response of a medium to an impulsive89

force, impulsive stress, or impulsive initial conditions in displacement, or velocity... While90

specialists are well aware of these differences and their implications, interferometry “users”91

(including the authors of this article) are sometimes confused as to the theoretical basis, and92

practical reliability, of the methods they apply.93

The goal of this study is to show in detail how several different derivations of “ambient-94

noise theory” lead to apparently different, but indeed perfectly coherent results. We first95

treat some particularly simple scenarios: ambient noise in a homogeneous lossless (i.e. non-96

attenuating) 2-D or 3-D medium, generated by azimuthally uniform, 2-D or 3-D distributions97

of point sources, or by plane waves propagating in all directions. The analogy between acous-98

tic waves in two dimensions and Rayleigh waves is discussed. Analytical expressions for the99

cross-correlation of ambient signals in all these physical settings are given in sec. 3 and im-100

plemented numerically in sec. 4. In all the simple cases we consider, the cross-correlation101

of ambient signal as derived in sec. 3 coincides with the integral, over the area occupied by102

sources, of an overall very oscillatory function, which becomes slowly varying only around a103

small set of so-called “stationary” points. We show in sec. 5 how such integrals are solved via104

the approximate “stationary-phase” method (appendix A); different analytical relationships105

between cross-correlation and Green’s function are thus found and discussed; they are later106

summarized in sec. 9. In sec. 8 we apply, again, the stationary-phase method to a more107

4

complicated medium including one scattering obstacle, which allows us to introduce the108

concept, often found in ambient-noise literature, of “spurious” arrival in cross-correlation.109

Finally, the so-called “reciprocity-theorem” approach provides an analytical relationship be-110

tween Green’s function and cross-correlation that is valid in arbitrarily complex, attenuating111

media. We describe it in detail in sec. 6, together with a few other “alternative” approaches112

to ambient-noise theory; the consistency between reciprocity-theorem and stationary-phase113

results is verified. To avoid ambiguity as much as possible, we provide an overview of the114

underlying theory for both acoustics and elastodynamics in sec. 2, and a derivation of acous-115

tic Green’s functions in appendix E. By collecting all this previously scattered material in116

a single review, we hope to provide a useful tool for graduate students and non-specialists117

approaching the theory of acoustic and seismic ambient noise for the first time. Most of the118

results presented here are limited to non-scattering, non-dissipative, homogeneous acoustic119

media, and to surface-wave (and not body-wave) propagation in elastic media. These sim-120

plifications have the advantage of allowing a self-contained, relatively uncluttered derivation.121

We explore more realistic setups (non-uniform source distributions; scattering) in secs. 7 and122

8. The thorough understanding of ambient-noise cross-correlation in simple media will serve123

as a solid platform for more advanced investigations. Readers that are mostly interested in124

applications to realistic environments are referred to the differently minded reviews of Curtis125

et al. [2006], Larose et al. [2006], Gouedard et al. [2008], Wapenaar et al. [2010a, b], Snieder126

and Larose [2013], Ritzwoller [2014] and Campillo and Roux [2014].127

2 Governing equations128

In this study, analytical relationships between Green’s functions and the cross-correlation129

of ambient signal are derived in a number of different scenarios: spherical acoustic waves130

in two and three dimensions, Rayleigh waves, plane waves. We first summarize the theory131

underlying each of these cases.132

2.1 Acoustic waves from a point source in free space133

Pressure p in homogeneous, inviscid fluids occupying a three-dimensional space obeys the134

linear, lossless wave equation135

∇2p− 1

c2

∂2p

∂t2=∂q

∂t, (1)

5

where ∇ is the gradient operator, c2 the ratio of adiabatic bulk modulus to density, t denotes136

time and the forcing term q is the (apparent) mass production per unit volume per unit time,137

representing e.g. an explosion or a loudspeaker [e.g., Kinsler et al., 1999, chapter 5].138

If ∂q/∂t = δ(r− r0)δ(t− t0), with δ the Dirac distribution, r the position vector and r0,139

t0 the location and time of an impulsive sound source, respectively, eq. (1) is referred to as140

“Green’s problem”, and its solution for p as “Green’s function”. Once the Green’s problem141

is solved, the response of the system to however complicated a source is found by convolving142

the Green’s and source functions [e.g., Morse and Ingard , 1986; Aki and Richards, 2002].143

The Green’s function G3D associated with (1) is derived in appendix E here, eqs. (200) and144

(201), working in spherical coordinates and choosing (without loss of generality) t0 = 0 and145

r0 = 0.146

2.2 Membrane waves from a point source147

Eq. (1) also describes the motion of an elastic membrane. In this case, p can be interpreted as148

displacement in the vertical direction (for an horizontal membrane), ∇ is the surface gradient149

and c the ratio of the membrane tension per unit length to its surface density [e.g., Kinsler150

et al., 1999, section 4.2]. The associated Green’s function G2D is determined analytically in151

appendix E, eqs. (194) and (195).152

2.3 Rayleigh waves from a point source153

The “potential” representation of surface-wave propagation, adopted e.g. by Tanimoto [1990],154

consists of writing Rayleigh-wave displacement155

uR = U(z) zΨR(x, y; t) + V (z)∇1ΨR(x, y; t), (2)

where x, y, z are Cartesian coordinates (z denotes depth), the “vertical eigenfunctions” U156

and V depend on z only and the Rayleigh wave “potential” ΨR varies only laterally (but157

varies with time). x, y, z are unit vectors in the directions of the corresponding coordinates,158

and the operator ∇1 = x ∂∂x + y ∂

∂y . Love-wave displacement is accordingly written159

uL = −W (z)z×∇1ΨL(x, y; t), (3)

with × denoting a vector product.160

Substituting the Ansatze (2) and (3) into the 3-D equation of motion, one finds that both161

6

potentials ΨR and ΨL satisfy162

∇21ΨR,L −

1

c2R,L

∂2ΨR,L

∂t2= TR,L, (4)

i.e. eq. (1), with cR,L denoting phase velocity, and TR, TL scalar forcing terms [e.g. Tanimoto,163

1990; Tromp and Dahlen, 1993; Udıas, 1999].164

The Green’s problem associated with (4) coincides with the 2-D, membrane-wave problem165

of sec. 2.2 and appendix E, and the Green’s function corresponding to ΨR,L is, again, G2D166

of eqs. (194) and (195). The surface-wave Green’s function can be obtained from the scalar167

Green’s function G2D via eq. (2) [Tromp and Dahlen, 1993, sec. 5].168

Importantly, according to (2) the vertical component of Rayleigh-wave displacement uR169

is simply U(z)ΨR(x, y; t); U(z) is a function of z that does not change with time, so that170

the phase of the vertical component of uR coincides with that of the potential ΨR(x, y; t).171

The analysis of 2-D ambient noise that we conduct here holds therefore for Rayleigh waves172

propagating in a 3-D medium (in the Earth) and measured on the vertical component of173

seismograms, as far as the phase is concerned; amplitude needs to be scaled.174

The Rayleigh-wave derivation of Snieder [2004] is also based on this idea, although in that175

study the separation between x,y,t-dependence and z-dependence follows from a normal-mode176

formulation [e.g., Snieder , 1986]. Halliday and Curtis [2008] explore the effects of sub-surface177

sources, or lack thereof, on Green’s function reconstruction; their results are consistent with178

ours and with those of Snieder [2004] whenever their setup is the same as ours, i.e. uniform179

source distribution over the Earth’s (laterally invariant) surface.180

2.4 Plane waves181

Some authors [e.g., Aki , 1957; Sanchez-Sesma and Campillo, 2006; Tsai , 2009; Boschi et al.,182

2013] have studied diffuse wave fields with cylindrical symmetry, which they described as the183

superposition of plane waves (rather than spherical or circular as seen so far) traveling along184

many different azimuths. If its right-hand side ∂q∂t=0, eq. (1) is satisfied by a monochromatic185

plane wave186

p(r, t) = S(ω) cos(ωt+ k · r), (5)

where the vector k is constant and defines the direction of propagation, and the amplitude187

function S(ω) is arbitrary. A plane wave approximates well the response of a medium to a188

real source, at receivers that are sufficiently far from the source.189

7

ϕ = 0

θ = 0

θ = π2 , ϕ = π

2

PR1 R2

ϕ

θ

Figure 1: Spherical reference frame used throughout this study. The origin is placed at thelocation of “receiver 1” (R1), while “receiver 2” (R2) lies on the ϕ = 0, θ = π/2 axis. Thedistance between the origin and a point P is denoted r.

3 Cross-correlation in a diffuse wave field190

We are interested in the cross-correlation between recordings of a diffuse wave field made191

at a pair of receivers. In acoustics, a diffuse wave field is such that the energy associated192

with propagating waves is the same at all azimuths of propagation [e.g. Kinsler et al., 1999,193

sec. 12.1]. (The expression “equipartitioned field”, often intended as a synonym of “diffuse”,194

is used in the literature with slightly different meanings depending on the context [Snieder195

et al., 2010], and we chose not to employ it here to avoid ambiguity.) While ambient noise196

recorded on Earth is not strictly diffuse [Mulargia, 2012], diffuse-field theory successfully197

describes many seismic observations, and as such it is at least a useful first approximation of198

real ambient noise.199

We simulate an approximately diffuse wave field by averaging (or, in seismology jargon,200

“stacking”) cross-correlations associated with sources distributed uniformly over a circle or201

a sphere surrounding the receivers. This is equivalent to the source azimuth being random,202

i.e., over time, all azimuths are sampled with equal frequency/probability.203

We center our spherical reference frame at the location of “receiver 1” (R1), and orient204

it so that “receiver 2” (R2) lies on the ϕ = 0, θ = π/2 axis (Fig. 1). We choose sources205

to be separated in time, that is to say, a receiver never records signal from more than one206

8

source at the same time: while diffuse wave fields in the real world might result from multiple,207

simultaneous sources, our simplification is justified by the mathematical finding that the so-208

called “cross-terms”, i.e. the receiver-receiver cross-correlation of signal generated by different209

sources, are negligible when “ensemble-averaged”. A proof is given e.g. by Weemstra et al.210

[2014] and is summarized here in appendix D.211

Let us call S a source location, and riS the distance between S and the i-th receiver. If212

the source at S is an impulsive point source, those signals coincide with the Green’s function213

associated with the source location S, evaluated at R1 and R2. In 3-D, the Green’s function214

is given by eq. (201), and in the frequency domain the cross-correlation reads215

G3D(r1S , ω)G∗3D(r2S , ω) =1

(4πc)2

1

2π

eiωc

(r2S−r1S)

r1S r2S(6)

(here and in the following the superscript ∗ denotes complex conjugation). Since the216

dimension of G3D in the frequency domain is time over squared distance (app. E), that of217

eq. (6) is squared time over distance to the power of four, or time over distance to the power218

of four in the time domain.219

In 2-D (membrane waves, Rayleigh waves), G2D is given by (195), and the cross-correlation220

G2D(r1S , ω)G∗2D(r2S , ω) =−i

32πc4H

(2)0

(ωr1S

c

) [−iH

(2)0

(ωr2S

c

)]∗. (7)

The cumulative effect of multiple sources S is obtained by rewriting expressions (6) or221

(7) for each S, with r1S and r2S varying as functions of the distance r and azimuth ϕ (and,222

in 3-D, inclination θ) of S with respect to the origin, and summing.223

We shall consider several different scenarios:224

(i) In 3-D space, for a continuous distribution of sources along a circle in the θ = π/2 plane225

(centered, for the sake of simplicity, at R1) summing the cross-correlations over S leads226

to an integral over ϕ,227

IC(ω) =1

(4πc)2

1

4π2

∫ π

−πdϕnC(ϕ)

eiωc

(r2S(ϕ)−r1S(ϕ))

r1S(ϕ) r2S(ϕ), (8)

with nC the number of sources per unit azimuth, or source density.228

(ii) If sources are distributed on a sphere centered at R1, the double integral is over the229

9

surface of the sphere (4π solid radians), and230

IS(ω) =1

4π

∫ π

0dθ sin θ

∫ π

−πdϕnS(θ, ϕ)G3D(r1S , ω)G∗3D(r2S , ω)

=1

4π

1

(4πc)2

1

2π

∫ π

0dθ sin θ

∫ π

−πdϕnS(θ, ϕ)

eiωc

(r2S(θ,ϕ)−r1S(θ,ϕ))

r1S(θ, ϕ) r2S(θ, ϕ),

(9)

where nS denotes the number of sources per unit of solid angle on the sphere.231

(iii) Switching to 2-D, i.e. to Rayleigh waves or elastic waves propagating on a membrane,232

we need to integrate expression (7) over the position occupied by S. Assuming a source233

distribution analogous to that of case (i), with sources along a circle, and source density234

nM a function of ϕ only, we find after some algebra that cross correlation is described235

by236

IMW (ω) =1

32πc4

1

2π

∫ π

−πdϕnM (ϕ) H

(2)0

(ωr1S(ϕ)

c

)H

(2)0

∗(ωr2S(ϕ)

c

)≈ 1

16π2c3

1

2π

∫ π

−πdϕnM (ϕ)

eiωc

(r1S(ϕ)−r2S(ϕ))

ω√r1S(ϕ)r2S(ϕ)

,

(10)

where we have replaced the Hankel function H(2)0 with its asymptotic (high-frequency237

and/or far-field) approximation, eq. (9.2.4) of Abramowitz and Stegun [1964]; this238

approximation is necessary to later solve the integral in (10) via the stationary-phase239

method. Notice that we do not yet require the wave field to be isotropic: at this point240

nC , nS , nM are arbitrary functions of θ, ϕ. We will show in sec. 5, however, that they241

need to be smooth or constant for the stationary-phase approximation to be applicable.242

(iv) In the plane-wave approach of sec. 2.4 no source location is specified, but one combines243

plane waves (5) traveling along all azimuths ϕ. At R1, the monochromatic plane wave244

of frequency ω0 traveling in the direction ϕ is245

p(R1, t) = S(ω0) cos(ω0t); (11)

at R2 the same signal is recorded at a different time, and246

p(R2, t) = S(ω0) cos

[ω0

(t+

∆ cosϕ

c

)]. (12)

Based on eqs. (11) and (12), Boschi et al. [2013] show that the source-averaged, two-247

10

station cross-correlation of monochromatic plane waves at frequency ω0 can be written248

IPW (ω0, t) =q(ω0)

2π

∫ π

0dϕ cos

[ω0

(t+

∆ cosϕ

c

)], (13)

obtained from eqs. (18) and (21) of Boschi et al. [2013] through the change of variable249

td = ∆ cos(ϕ)/c, and assuming source density and the amplitude term q(ω) (reflecting250

the frequency spectrum of the source) to be constant with respect to ϕ. The Fourier251

transform of (13) is252

IPW (ω0, ω) =q(ω0)√

8π

∫ π

0dϕ [δ(ω + ω0) + δ(ω − ω0)] eiω∆ cosϕ

c . (14)

Eq. (14) is valid for any ω0; using the properties of the Dirac δ function we generalize253

it to254

IPW (ω) =q(ω)√

2π

∫ π

0dϕ cos

(ω

∆ cosϕ

c

). (15)

We integrate eqs. (8)-(10) and (15) numerically in sec. 4; this exercise will serve to255

validate the results, illustrated in sec. 5, of integrating the same equations analytically via256

the stationary-phase approximation.257

The case of a spatially (and not just azimuthally) uniform distribution of sources over a258

plane or the 3-D space is not treated here in the interest of brevity. It essentially requires259

that expressions (8), (9) or (10) be additionally integrated over source location; in practice,260

one needs to integrate over the distance between source and origin, along the azimuths of261

stationary points only (one-dimensional integrals) [Snieder , 2004; Sato, 2010].262

4 Numerical integration263

Before deriving approximate analytical solutions for the integrals in eqs. (8)-(10) and (15),264

we implement these equations numerically. In practice, we evaluate eqs. (195) and (201) at265

R1, R2, to determine of G2D or G3D numerically for discrete sets of sources and values of ω.266

For each source, we cross-correlate the Green’s function calculated at R1 with that calculated267

at R2. We stack the resulting cross-correlations in the time-domain, which is equivalent to268

calculating the ϕ- and θ-integrals in the equations above. The stacks can be compared with269

predictions based on the analytical formulae that we shall illustrate below.270

11

4.1 Ring of sources271

The setup associated with eq. (8) is implemented numerically, placing 720 equally-spaced272

sources along the planar ring centered at R1. R1 and R2 are 20km away from one another and273

wave speed is 2km/s. We implement eq. (8) directly at frequencies between ±10Hz, with274

sampling rate 20Hz; we taper the highest and lowest frequencies to avoid ringing artifacts, and275

inverse-Fourier-transform via numerical FFT [Cooley and Tukey , 1965]. The corresponding276

cross-correlations are shown in the “gather” plot of Fig. 2a. Sources aligned with the two277

receivers (azimuth 0 or 180) result in the longest delay time between arrival of the impulse278

at R1 and R2: cross-correlation is nonzero near ±10s, which corresponds to the propagation279

time between R1 and R2. Sources at azimuth around ∼90 and ∼270 are approximately280

equidistant from R1 and R2, resulting in the impulse hitting R1 and R2 simultaneously, and281

cross-correlation in Fig. 2a being nonzero at ∼ 0s for those source azimuths.282

The result of averaging over all sources is shown in Fig. 2b. It is clear that the imaginary283

part of the time-domain signal is 0, as it should be. As for the real part, cross-correlations284

associated with sources at azimuth 0 or 180 add up constructively; the cumulative contribu-285

tion of other sources is smaller, but non negligible. The source-averaged cross-correlation is286

accordingly dominated by two peaks corresponding to energy traveling in a straight path from287

R2 to R1 (generated by sources at azimuth 0) and from R1 to R2 (sources at 180). The two288

peaks are often labeled “causal” and (not quite appropriately) “anticausal” (or “acausal”),289

respectively. In this particular case, the causal and anticausal peaks have different amplitude.290

This asymmetry reflects the asymmetry in the locations of R1 and R2 with respect to the291

source distribution: R2 is closer to sources at ϕ = 0 than R1 is to sources at ϕ = 180,292

and waves hitting R2 have accordingly larger amplitude. The description of seismic/acoustic293

interferometry in terms of a stationary-phase integral (sec. 5.1 and appendix A) explains294

mathematically why sources at 0 or 180 are most relevant, and provides an analytical re-295

lationship between the source-averaged cross-correlation in Fig. 2b and the Green’s function296

of the medium.297

4.2 Sources on a sphere298

We next distribute ∼103 approximately equally-spaced sources over the surface of a sphere299

of radius R = 100km, centered at R1. R2 lies 20km away from R1. In analogy with sec. 4.1,300

the signal recorded at R1 and that recorded at R2 (eq. (201)) are cross-correlated for each301

source, and cross-correlations are stacked, implementing eq. (9). The source-averaged302

12

a

b

Figure 2: Cross-correlations associated with a circular, planar distribution of sources sur-rounding the two receivers. R1 is at the center of a circle of sources with radius R=100km.R2 lies on the plane defined by R1 and the sources, ∆=20km away from R1. Phase velocityc=2km/s. Source density nC(ϕ) = 2 sources per degree, independent of ϕ. G3D(r1S , ω)and G3D(r2S , ω) are evaluated numerically and cross-correlated for a ring of equally-spacedsources. The result is inverse-Fourier-transformed, and averaged over all sources. (a) Time-domain single-source cross-correlations for all azimuths. (b) Real (solid line) and imaginary(dashed) parts of the source-averaged, time-domain cross-correlation, resulting from stackingthe traces in (a). The dimension is that of G3D(r, ω), squared and integrated over frequency(inverse-Fourier tranformed).

13

Figure 3: Real (solid line) and imaginary (dashed) parts of the average time-domain cross-correlation, resulting from stacking all cross-correlations associated with a uniformly densedistribution of sources over a sphere. R1 is at the center of a sphere of sources with radiusR=100km. R2 lies ∆=20km away from R1. Phase velocity c=2km/s. The calculation isconducted numerically (sec. 4.2), by evaluating G3D(r1S , ω) and G3D(r2S , ω) for each sourcevia eq. (201), and subsequently applying cross-correlation, inverse-Fourier-transformation,and averaging over all sources.

frequency-domain cross-correlation is computed between ±10Hz, with sampling rate 20Hz,303

and tapered at high frequency to avoid artifacts. The time-domain result is shown in Fig. 3;304

in this case, no “gather” plot was made because of the difficulty of visualization when sources305

are distributed in three dimensions. The source-averaged cross-correlation is real, as it should306

be; it is different from that of Fig. 2b, in that cross-correlations associated with sources at307

all azimuths add up constructively, resulting in a boxcar function. Again, the maximum308

delay time between arrival of the impulse at R1 and R2 (10s in our setup) corresponds to309

sources at 0, 180, and no energy is recorded at t > 10s; cross-correlations are accordingly310

0 at t < −10s and t > 10s. Despite the asymmetry in receiver location with respect to311

the sources, the stacked cross-correlation is now symmetric. All these results are reproduced312

analytically in sec. 5.2.313

4.3 Membrane waves/Rayleigh-wave potential314

The setup associated with eq. (10) is also reproduced numerically, placing 720 sources at 0.5315

azimuth ϕ intervals along the ring centered at R1. Frequency varies between ±10Hz with316

sampling rate=20Hz, high frequencies are tapered as above. As we shall show analytically317

in sec. 5.3, eq. (10) leads to a non-convergent ω-integral when a time-domain expression318

14

a

b

Figure 4: Cross-correlations, differentiated with respect to time, associated with a circular,planar distribution of sources surrounding the two receivers in two dimensions, i.e. on amembrane. The setup is the same as in Fig. 2. G2D(r1S , ω) and G2D(r2S , ω) are evaluatednumerically and cross-correlated for a ring of 720 equally-spaced sources. The result is inverse-Fourier-transformed, and averaged over all sources. (a) Time-domain single-source cross-correlations for all azimuths. (b) Real (solid line) and imaginary (dashed) parts of thestacked time-derivative of the cross-correlation, resulting from stacking the traces in (a).

15

for IMW is sought; we therefore compute the time-derivative of IMW (multiplication by iω319

in the frequency domain) before inverse-Fourier transforming. In the interest of speed, we320

implement the asymptotic approximation of H(2)0 (i.e., the second line of eq. (10)) rather321

than H(2)0 itself. The results are shown in Fig. 4. The “gather” (Fig. 4a) is qualitatively322

similar to that of Fig. 2a, but Fig. 4b shows that cross-correlations associated with sources323

away from azimuth 0 and 180 cancel out when stacking. After stacking (Fig. 4b) we verify324

that the time-derivative of IMW is real. We also find that it is antisymmetric with respect325

to time, indicating that IMW is symmetric, in agreement with the results of sec. 4.2 but not326

with sec. 4.1.327

5 Analytical formulae for 2-receiver cross-correlations of a dif-328

fuse wave field329

As pointed out by Snieder [2004], integration with respect to θ and ϕ in eqs. (8)-(10) and330

(15) can also be conducted analytically by means of the stationary-phase approximation, the331

details of which are given in appendix A. The numerical results of sec. 4 will serve as a332

reference to validate the approximate results presented below.333

5.1 Ring of sources in free space334

The integrand at the right-hand side of eq. (8) coincides with that in (150), after replac-335

ing λ=ω, x=ϕ and defining f(ϕ)=nC(ϕ)/[4π2 r1S(ϕ) r2S(ϕ)

]and ψ(ϕ)=[r1S(ϕ)− r2S(ϕ)] /c.336

We shall first identify the values of ϕ such that ψ′(ϕ)=0 (stationary points), then use eq. (151)337

to evaluate the contribution of each stationary point to the integral, and finally combine them.338

Importantly, the stationary phase approximation is valid at high frequencies ω −→ ∞ (cor-339

responding to λ −→ ∞ in sec. A.1). Choosing the reference frame as described in sec. 3340

(Fig. 1), by definition r1S is constant (we shall call it R) and, based on some simple geomet-341

rical considerations,342

r2S =[(R cosϕ−∆)2 +R2 sin2 ϕ

] 12 =

(R2 + ∆2 − 2∆R cosϕ

) 12 , (16)

where ∆ is interstation distance and ∆ < R in our setup. It follows that343

f(ϕ) =1

(4πc)2

nC(ϕ)

4π2R (R2 + ∆2 − 2∆R cosϕ)12

, (17)

16

ψ(ϕ) =1

c(r1S − r2S) =

1

c

[(R2 + ∆2 − 2∆R cosϕ

) 12 −R

], (18)

and upon differentiating ψ with respect to ϕ,344

ψ′(ϕ) =∆R sinϕ

c (R2 + ∆2 − 2∆R cosϕ)12

, (19)

345

ψ′′(ϕ) =∆R cosϕ

c (R2 + ∆2 − 2∆R cosϕ)12

− ∆2R2 sin2 ϕ

c (R2 + ∆2 − 2∆R cosϕ)32

. (20)

We infer from (19) that the stationary points of ψ(ϕ) within the domain of integration are346

ϕ = −π, 0, π (corresponding to sinϕ = 0). Following Bender and Orszag [1978], we rewrite347

eq. (8) as a sum of integrals limited to the vicinity of stationary points, and with a stationary348

point as one of the integration limits:349

IC(ω) ≈∫ −π+ε

−πf(ϕ)eiωψ(ϕ)dϕ+

∫ 0

−εf(ϕ)eiωψ(ϕ)dϕ+

∫ ε

0f(ϕ)eiωψ(ϕ)dϕ+

∫ π

π−εf(ϕ)eiωψ(ϕ)dϕ,

(21)

which is valid for ω −→ ∞ and arbitrarily small ε. Eq. (151) can now be used directly to350

integrate each of the terms at the right-hand side of (21), and after noticing that both f(ϕ)351

and ψ(ϕ) are symmetric with respect to ϕ = 0 we find352

IC(ω) ≈ 2f(π)ei(ωψ(π)±π4 )√

π

2ω|ψ′′(π)|+ 2f(0)ei(ωψ(0)±π

4 )√

π

2ω|ψ′′(0)|. (22)

We now need to evaluate f , ψ and ψ′′ at 0 and π. It follows from the definition of f ,353

from eqs. (18) and (20), and from the fact that R > ∆ (and hence |R−∆| = R−∆) that354

f(0) =1

(4πc)2

nC(0)

4π2R(R−∆), (23)

355

f(±π) =1

(4πc)2

nC(±π)

4π2R(R+ ∆), (24)

356

ψ(0) = −∆

c, (25)

357

ψ(±π) =∆

c, (26)

358

ψ′′(0) =∆R

c(R−∆), (27)

359

ψ′′(±π) = − ∆R

c(R+ ∆). (28)

To obtain eqs. (25) through (28), one must also recall that√R2 + ∆2 −∆R cosϕ = r2S is,360

17

Figure 5: Source-averaged cross-correlations associated with a circular, planar distribution ofsources surrounding the two receivers, obtained (green line) implementing eq. (32), comparedwith (black) the real part of that resulting from the numerical approach of sec. 4.1 (shownalready in Fig. 2). The source/station set-up is the same as for Fig. 2.

physically, a positive distance: when ϕ = 0,±π, it follows that√R2 + ∆2 − 2∆R = R−∆.361

Substituting expressions (23) through (28) into eq. (22), we find after some algebra that362

IC(ω) ≈ 1

(4πc)2

1

2π

√c

2π∆R3

[nC(π)

ei(ω∆c−π

4 )√(R+ ∆)ω

+ nC(0)e−i(ω∆

c−π

4 )√(R−∆)ω

], (29)

where the sign of π/4 in the argument of the exponential function was selected based on the363

sign of ψ′′ as explained in sec. A.1. Comparing eq. (29) with (196), it is apparent that both364

terms at the right-hand side of (29) are proportional to the high-frequency/far-field form of365

the Green’s function G2D(∆, ω) (and, interestingly, not G3D).366

Equation (21) and the subsequent expressions for IC(ω) are only valid for large and367

positive ω, so that IC(ω) remains undefined for ω < 0. We know, however, that a sum of368

cross-correlations of real-valued functions of time should be real-valued in the time domain,369

requiring that IC(ω) = I∗C(−ω), and370

F−1[IC(ω)] =1√2π

∫ +∞

−∞dω IC(ω)eiωt =

1√2π

(∫ +∞

0dω IC(ω)eiωt +

∫ +∞

0dω I∗C(ω)e−iωt

)(30)

After substituting (29) into (30), the integration over ω can be conducted analytically, making371

18

use of the identity372 ∫ ∞0

dxeiax

√x

=

1√a

√π2 (1 + i) if a > 0,

1√−a√

π2 (1− i) if a < 0,

(31)

and integrating separately for the three cases t < −∆c , −∆

c < t < ∆c and t > ∆

c . After a373

considerable amount of algebra we find374

IC(t) =1

(4πc)2

1

2π

√c

π∆R3×

nC(0)√R−∆

1√∆c−t

if t < −∆c ,

nC(π)√R+∆

1√∆c

+t+ nC(0)√

R−∆1√∆c−t

if − ∆c < t < ∆

c ,

nC(π)√R+∆

1√∆c

+tif t > ∆

c ,

(32)

which, as required, has zero imaginary part. We infer from eq. (32) that, in the current setup,375

there is no explicit relationship between the source-averaged cross-correlation IC(t) and the376

Green’s functions G2D or G3D (App. E). Fig. 5 shows that IC(t) as obtained from eq. (32)377

is consistent with the numerical result of Fig. 2. The discussion of sec. 4.1 remains valid. A378

slight discrepancy between the “analytical” and “numerical” results in Fig. 5 is explained by379

the stationary-phase approximation being strictly valid only at high frequency (ω −→∞).380

5.2 Sources over a spherical surface381

The integral in eq. (9) can be solved analytically with the help of eq. (161). Equation (9) is382

indeed a particular case of (152), with x=θ, y=ϕ, λ=ω, f(θ, ϕ) = nS sin θ/[8π2(r2S(θ, ϕ) −383

r1S(θ, ϕ))] and ψ(θ, ϕ) = [r2S(θ, ϕ)− r1S(θ, ϕ)] /c. Again, the reference frame is centered on384

R1, so that r1S(θ, ϕ) = R for all values of θ, ϕ. After some algebra, we find385

r2S =(R2 + ∆2 − 2∆R sin θ cosϕ

) 12 , (33)

and consequently386

f(θ, ϕ) =1

(4πc)2

nS(θ, ϕ) sin θ

8π2R (R2 + ∆2 − 2∆R sin θ cosϕ)12

, (34)

387

ψ(ϕ) =1

c(r2S − r1S) =

1

c

[(R2 + ∆2 − 2∆R sin θ cosϕ

) 12 −R

]. (35)

19

Differentiating (35) with respect to θ and ϕ we find388

ψθ = − ∆R cos θ cosϕ

c (R2 + ∆2 − 2∆R sin θ cosϕ)12

, (36)

where the compact ψθ stands for ∂ψ∂θ . Likewise,389

ψϕ =∆R sin θ sinϕ

c (R2 + ∆2 − 2∆R sin θ cosϕ)12

. (37)

If we continue differentiating,390

ψθθ = − ∆R cos2 θ cos2 ϕ

c (R2 + ∆2 − 2∆R sin θ cosϕ)32

+∆R sin θ cosϕ

c (R2 + ∆2 − 2∆R sin θ cosϕ)12

, (38)

391

ψϕϕ = − ∆R sin2 θ sin2 ϕ

c (R2 + ∆2 − 2∆R sin θ cosϕ)32

+∆R sin θ cosϕ

c (R2 + ∆2 − 2∆R sin θ cosϕ)12

, (39)

392

ψθϕ =∆R sin θ sinϕ cos θ cosϕ

c (R2 + ∆2 − 2∆R sin θ cosϕ)32

+∆R cos θ sinϕ

c (R2 + ∆2 − 2∆R sin θ cosϕ)12

. (40)

Equations (36) and (37) allow us to identify the stationary points (θ,ϕ such that ψθ, ψϕ =393

0, 0) of the integrand at the right hand side of (9): namely (θ=0, ϕ=±π/2), (π,±π/2),394

(π/2, 0), (π/2, π). It is sufficient to evaluate (161) at these points, and sum the results,395

to find an analytical expression for the integral (9). We notice first of all that f(θ, ϕ)=0 if396

θ = 0,±π: the corresponding stationary points give no contribution to the integral and will397

be neglected in the following. We are left with the stationary points at θ = π/2 and ϕ=0,π.398

Let us evaluate f , ψ, ψθθ, ψϕϕ, ψθϕ at those points.399

f(π

2, 0)

=1

(4πc)2

nS(π2 , 0)

8π2R (R−∆), (41)

400

f(π

2, π)

=1

(4πc)2

nS(π2 , π

)8π2R (R+ ∆)

, (42)

401

ψ(π

2, 0)

= −∆

c, (43)

402

ψ(π

2, π)

=∆

c, (44)

403

ψθθ

(π2, 0)

=∆R

c(R−∆), (45)

404

ψθθ

(π2, π)

= − ∆R

c(R+ ∆), (46)

20

405

ψϕϕ

(π2, 0)

=∆R

c(R−∆), (47)

406

ψϕϕ

(π2, π)

= − ∆R

c(R+ ∆), (48)

and it follows immediately from (40) that ψθϕ=0 at all stationary points.407

The above expressions can be substituted into (161) to find408

IS(ω) ≈ 1

(4πc)2

i c

4πR2∆

[nS

(π2, 0) e−iω∆

c

ω− nS

(π2, π) eiω∆

c

ω

], (49)

which satisfies IS(−ω) = −I∗S(ω). Notice that, substituting eq. (201) into (49),409

IS(ω) ≈√

2π

(4π R)2

i

ω

[nS

(π2, 0)G3D(∆, ω)− nS

(π2, π)G∗3D(∆, ω)

], (50)

and if nS(π2 , 0)

= nS(π2 , π

), eq. (50) can be further simplified to410

IS(ω) ≈ −√

2π

8π2R2nS

(π2, 0) 1

ω= [G3D(∆, ω)] . (51)

We next inverse-Fourier transform eq. (49) to the time domain. In our convention,411

F−1

(1

ω

)= i

√π

2sgn(t) (52)

(Dirichlet integral), where the sign function sgn is +1 or -1 for positive and negative values412

of the argument, respectively. In the time domain, IS is then413

IS(t) ≈ 1

(4πc)2

c

4√

2π R2 ∆

[nS

(π2, π)

sgn

(t+

∆

c

)− nS

(π2, 0)

sgn

(t− ∆

c

)], (53)

illustrated in Fig. 6. This result is consistent with eq. (15) of Roux et al. [2005] and eq. (27)414

of Nakahara [2006], who treated the same physical problem in different ways, and with Fig.415

1a of Harmon et al. [2008]. By comparison with eq. (200), it is apparent that the terms at the416

right-hand side of eq. (53) are time-integrals of the time-domain Green’s function G3D(∆, t).417

Equations (49) and (53) mean that the Green’s function G3D corresponding to propaga-418

tion from receiver R1 to R2 and vice-versa can be found by (i) recording at both receivers the419

signal emitted by a dense distribution of sources covering all azimuths in three dimensions;420

(ii) cross-correlating, for each source, the signal recorded at R1 with that recorded at R2; (iii)421

integrating the cross-correlation over the source location r.422

21

Figure 6: Stacked cross-correlation resulting from a spherical distribution of sources surround-ing the two receivers, predicted (green line) by the approximate analytical formula (53), and(black) computed numerically as described in sec. 4.2 and already illustrated in Fig. 3 (onlythe real part is shown here). The source/station setup is the same as for Fig. 3.

Our eq. (49) coincides with eq. (11) of Snieder [2004], except that here the differ-423

ence between the right-hand-side terms is taken, while in Snieder [2004] they are summed.424

Inverse-Fourier transforming the formula of Snieder [2004] would lead to a non-physical cross-425

correlation, nonzero at times where cross-correlation is necessarily zero at all ϕ. We infer426

that eqs. (11) and (12) in Snieder [2004] are wrong. However, this error did not affect the427

subsequent, Rayleigh-wave derivation of Snieder [2004].428

5.3 Membrane (or Rayleigh) waves from 2-D distribution of point sources429

5.3.1 Source-averaged cross-correlation430

We treat the integral in eq. (10) with the stationary-phase method as illustrated in previous431

sections. The phase term ψ(ϕ) in (10) is the additive inverse of that in (8), and the two432

integrands share the same stationary points ϕ=0, ϕ=±π. At those points,433

f(0) =nM (0)

32π3c3ω√R(R−∆)

, (54)

434

f(±π) =nM (π)

32π3c3ω√R(R+ ∆)

, (55)

435

ψ(0) =∆

c, (56)

436

ψ(±π) = −∆

c, (57)

22

ψ′′(0) = − ∆R

c(R−∆), (58)

437

ψ′′(±π) =∆R

c(R+ ∆). (59)

Substituting into eq. (151) (which is multiplied by 2 for each stationary point) and438

summing over our set of two stationary points,439

IMW (ω) ≈ 1

16π3c3R

√πc

2∆ω−

32

[nM (0)ei(ω∆

c−π

4 ) + nM (π)e−i(ω∆c−π

4 )], (60)

coherently with eqs. (23) and (24) of Snieder [2004], which however involve summation over440

normal modes. Comparing eq. (60) with (196), we can also write441

IMW (ω) ≈ i

4π2ωcR

√π

2[nM (π)G2D(∆, ω)− nM (0)G∗2D(∆, ω)] , (61)

valid in the asymptotic (high-frequency/far-field) approximation.442

Eqs. (60) and (61) are valid in the stationary-phase approximation, i.e. for large and443

positive ω only. We know, however, that IMW (t) must be real: as in sec. 5.1, IMW at ω < 0444

is thus defined by IMW (ω) = I∗MW (−ω), and eq. (30) remains valid after replacing IC with445

IMW , i.e.446

F−1[IMW (ω)] =1√2π

(∫ +∞

0dω IMW (ω)eiωt +

∫ +∞

0dω I∗MW (ω)e−iωt

). (62)

After substituting the expression (60) for IMW into (62), it becomes apparent that finding a447

time-domain expression for IMW requires the solution of a non-convergent integral, namely448 ∫∞0 dxx−

32 cos(x).449

5.3.2 Derivative of the source-averaged cross-correlation450

One can still use the present theoretical formulation to interpret 2-D data in the time domain,451

by simply taking the time-derivative of both the observed source-averaged cross-correlation452

and its analytical expression IMW (ω). Based on the properties of Fourier transforms, the453

latter is achieved by multiplying IMW (ω) by iω,454

I ′MW (ω) ≈ i

16π3c3R

√πc

2∆

[nM (0)

ei(ω∆c−π

4 )√ω

+ nM (π)e−i(ω∆

c−π

4 )√ω

]. (63)

23

Figure 7: Time-derivative I ′MW (t) of the stacked cross-correlation associated with a circulardistribution of sources surrounding the two receivers on a membrane, predicted (green line) bythe approximate analytical formula (64), and (black line) computed numerically as describedin sec. 4.3 (Fig. 4). Only the real parts are shown here since we have verified that imaginaryparts are null as required by physics. For comparison, the Green’s function -G2D from eq.(194) with x = ∆ is also shown (red line); it is normalized so that its amplitude coincideswith that of the numerical stack. The phase velocity and source/station setup are the sameas for Fig. 4.

We infer from eq. (63) that for acoustic waves in 2-D, and within the stationary-phase455

approximation, the derivative of the source-averaged cross-correlation is proportional to the456

sum of causal and anticausal Green’s functions G2D(ω) given by eq. (196) [Snieder , 2004].457

Like (60), eq. (63) is only valid for large positive ω, but the fact that I ′MW (t) is real458

can be used to define I ′MW (ω) at ω < 0 via eq. (30). The inverse Fourier transform of the459

resulting expression for I ′MW (ω) involves a convergent integral (eq. (31)) and can be found460

analytically. The procedure is similar to that of sec. 5.1, eqs. (30) through (32). The result461

is quite different, as can be expected since the imaginary unit multiplies the right-hand side462

of eq. (63). Namely,463

I ′MW (t) ≈ 1

16π3c3R

√πc

∆×

nM (0)√−t−∆

c

if t < −∆c ,

0 if − ∆c < t < ∆

c ,

− nM (π)√t−∆

c

if t > ∆c .

(64)

Fig. 7 shows that eq. (64) is validated by the numerical results of sec. 4.3. No explicit464

mathematical relationship between I ′MW and G2D can be inferred from eq. (64), although465

Fig. 7 shows that the behaviour of the two functions is qualitatively similar.466

24

5.3.3 Symmetric source distribution467

Inverse Fourier transformation turns out to be simpler when nM (π)=nM (0): setting them468

both to 1 for simplicity, eq. (61) simplifies to469

IMW (ω) ≈ i

4π2ωcR

√π

2[G2D(∆, ω)−G∗2D(∆, ω)]

≈ − 1

2π2ωcR

√π

2= [G2D(∆, ω)] ,

(65)

and470

IMW (t) ≈ − 1

2π2cR

√π

2F−1

= [G2D(∆, ω)]

ω

. (66)

Let us denoteGo2D the odd functionGo2D(t)=12G2D(t)− 1

2G2D(−t). We know from appendix B,471

eq. (170), that = [G2D(∆, ω)]=−iGo2D(∆, ω), so that472

IMW (t) ≈ − 1

2π2cR

√π

2F−1

[−iGo2D(∆, ω)

ω

]. (67)

It also follows from appendix B, eq. (165), that473

F−1

[−iGo2D(∆, ω)

ω

]=

∫ t

−∞Go2D(τ)dτ , (68)

and by definition of Go2D,474

IMW (t) ≈ − 1

4π2cR

√π

2

∫ t

−∞[G2D(τ)−G2D(−τ)] dτ

. (69)

Differentiating with respect to time,475

I ′MW (t) ≈ 1

4π2cR

√π

2[G2D(−t)−G2D(t)] , (70)

consistent with Fig. 7. The apparent discrepancy between eqs. (64) and (70) is explained by476

the fact that the derivation of (70) involved identifying G2D with its asymptotic approxima-477

tion (196), which was not the case for the derivation of eq. (64).478

25

5.4 Plane waves479

5.4.1 Exact integration480

An elegant way of reducing eq. (15) to a simple and useful identity is to compare it with the481

integral form of the 0th-order Bessel function of the first kind,482

J0(z) =1

π

∫ π

0dϕ cos(z cosϕ) (71)

[Abramowitz and Stegun, 1964, eq. (9.1.18)]. Substituting (71) into (15),483

IPW (ω) =

√π

2q(ω)J0

(ω∆

c

), (72)

an exact equality that does not require the stationary-phase approximation. Importantly,484

eq. (72) was originally obtained in the early study of Aki [1957], providing the basis for485

much of the later work in ambient-noise seismology. Comparing (72) with (195), we infer486

that the source-averaged cross correlation IPW (ω) is proportional to the imaginary part of487

the membrane-wave Green’s function G2D

(ωrc

)if r = ∆.488

5.4.2 Approximate integration489

In analogy with previous sections, the integral in eq. (15) can also be solved by means of the490

stationary phase approximation. Let us rewrite it491

IPW (ω) =q(ω)√

2π<[∫ π

0dϕ eiω∆ cosϕ

c

]. (73)

The integral at the right-hand side of eq. (73) coincides with that in (150) after replacing492

a = 0, b = π, x = ϕ, λ = ω and ψ(ϕ) = ∆ cosϕc . Taking the ϕ-derivative of ψ,493

ψϕ(ϕ) = −∆

csinϕ, (74)

and we see immediately that there are two stationary points within the integration domain,494

at the integration limits ϕ=0,π. Differentiating again, we find495

ψϕϕ(ϕ) = −∆

ccosϕ. (75)

26

At the stationary point ϕ = 0 we have ψ(0) = ∆c and ψϕϕ(0) = −∆

c . At ϕ = π, ψ(0) = −∆c496

and ψϕϕ(0) = ∆c . For each stationary point, we substitute the corresponding values into497

(151), choosing, as usual, the sign of π/4 in the argument of the exponential based on that498

of ψϕϕ. We next sum the contributions of both stationary points, finding499

∫ π

0dϕ eiω∆ cosϕ

c ≈√

2π c

ω∆cos

(ω∆

c− π

4

). (76)

Substituting, in turn, (76) into (73),500

IPW (ω) ≈ q(ω)

√c

ω∆cos

(ω∆

c− π

4

)≈√π

2q(ω) J0

(ω∆

c

), (77)

where we have used the asymptotic approximation of J0 [Abramowitz and Stegun, 1964, eq.501

(9.2.1)], valid in the high-frequency (and/or far-field) limit, i.e. in the range of validity of the502

stationary-phase approximation. Eq. (77) shows that the stationary-phase approximation503

leads to an estimate of source-averaged cross-correlation IPW (ω) consistent, at large ω, with504

the result (72).505

The relationship (72) between observed stacked cross-correlations and the Bessel’s func-506

tion J0 has been applied e.g. by Ekstrom et al. [2009] and Ekstrom [2014] to analyze ambient-507

noise surface-wave data in the frequency domain, and measure their velocity. The inverse-508

Fourier transform of (72) is obtained and discussed by Nakahara [2006], who also shows509

that the Hilbert transform of the stacked cross-correlation coincides with the (causal minus510

anticausal) G2D [Nakahara, 2006, eq. (19)].511

5.4.3 Monochromatic plane waves in the time domain512

The treatments of Sanchez-Sesma and Campillo [2006], Tsai [2009] and Boschi et al. [2013]513

are slightly different from the plane-wave formulation presented here, in that they work with514

monochromatic plane waves in the time domain. Following Tsai [2009], Boschi et al. [2013]515

make use of the properties of the Bessel and Struve functions to solve the integral in eq.516

(13), and are eventually able to write the source-averaged cross-correlation of plane waves of517

frequency ω0 traveling along all azimuths as518

ITD(ω0, t) = q(ω0)

√c

8πω0∆

cos

[ω0

(t+

∆

c

)− π

4

]+ cos

[ω0

(t− ∆

c

)+π

4

](78)

27

[Boschi et al., 2013, eqs. (35) and (41)], valid in the far-field (large ∆) and/or high-frequency519

(large ω0) approximations.520

To verify that our eq. (77) is consistent with (78), let us take the Fourier transform of the521

latter. From the equality F cos[ω0(t+ k)] =√

π2 [δ(ω + ω0) + δ(ω − ω0)] e−iωk it follows522

that523

ITD(ω0, ω) =q(ω0)

2

√c

ω0∆[δ(ω + ω0) + δ(ω − ω0)] cos

[ω

(π

4ω0− ∆

c

)]. (79)

Like (14), eq. (79) is valid for any ω0. Implicitly repeating the monochromatic-wave, time-524

domain analysis at all ω0’s,525

ITD(ω) = q(ω)

√c

ω∆cos

(ω∆

c− π

4

), (80)

which coincides, as expected, with our expression (77) for IPW (ω).526

6 Other derivations527

6.1 Time-domain approach528

We loosely follow the treatment of Roux et al. [2005] (sec. II), with acoustic sources uniformly529

distributed throughout an unbounded, infinite 3-D medium. This setup is similar to sec. 5.2530

here. Although in sec. 5.2 sources are all lying on the surface of a sphere (azimuthally uniform531

distribution), we show in the following that the formula we find for the source-averaged cross-532

correlation is proportional to that derived by Roux et al. [2005].533

In free 3-D space, the time-domain cross-correlation between impulsive signals emitted at534

S and recorded at R1 and R2 (i.e. the time-domain version of eq. (6)) reads535

1

T

∫ +T2

−T2

dτ G3D(r1S , τ)G3D(r2S , t+ τ) =1

(4πc)2

1

T

∫ +T2

−T2

dτδ(τ − r1S

c

)δ(τ + t− r2S

c

)r1S r2S

=1

(4πc)2

1

T

δ(t+ r1S

c −r2Sc

)r1S r2S

,

(81)

where we have used expression (200) for G3D, and we have limited cross-correlation to a finite536

time interval (−T/2, T/2), whose length T is related, in practice, to interstation distance and537

wave speed.538

As discussed in sec. 3, if “cross terms” are neglected (app. D), the cross-correlation of a539

diffuse wave field recorded at R1 and R2 is estimated by summing expression (81) over many540

uniformly distributed sources S. For sources densely distributed throughout the entire 3-D541

28

space R3,542

I3D(t) =1

(4πc)2

1

T

∫R3

d3rδ(t+ r1S

c −r2Sc

)r1S r2S

, (82)

where r is the source location, and, said r1 and r2, respectively, the locations of R1 and R2,543

r1S = |r− r1|, r2S = |r− r2|. Eq. (82) is equivalent to eq. (9) of Roux et al. [2005].544

As noted by Roux et al. [2005], eq. (82) shows that a wiggle in the cross-correlation545

I3D at time t is necessarily associated to one or more sources whose location r satisfies546

ct = |r− r2| − |r− r1|. In two dimensions, the latter is the equation of a hyperbola with foci547

at R1 at R2; in three dimensions, it is the equation of the single-sheet hyperboloid obtained548

by rotation of said hyperbola around the vertical axis. It follows that the integral in (82)549

can be reduced to a surface integral over the hyperboloid. After this simplification, Roux550

et al. [2005] are able to solve the integral in (82) analytically: they find I3D(t) to be a boxcar551

function between t = ±∆/c, equivalent to eq. (49) in sec. 5.2 here.552

6.2 Reciprocity-theorem approach553

All descriptions of seismic and acoustic interferometry that we discussed so far rest on eq.554

(1) and on the Green’s function formulae of appendix E, which are strictly valid only for g,555

infinite media. More general formulations have been developed by Wapenaar [2004], Weaver556

and Lobkis [2004], van Manen et al. [2005], Wapenaar and Fokkema [2006], Snieder [2007]557

and others, based on the reciprocity, or Betti’s theorem [e.g., Aki and Richards, 2002, sec.558

2.3.2].559

6.2.1 acoustic waves560

The propagation of acoustic waves in a nonhomogeneous, lossless stagnant gas of density ρ561

and compressibility κ is described by eq. (6.2.7) of Morse and Ingard [1986], plus the forcing562

term q corresponding to mass injection. In the frequency domain,563

∇p+ iωρv = 0, (83)

564

∇ · v + iωκp− q = 0. (84)

In the more general case of a sound-absorbing medium, κ is complex and its imaginary part565

is proportional to the rate of energy loss (attenuation) [Kinsler et al., 1999, chapter 8]. Eqs.566

(83) and (84) are equivalent to eqs. (2) and (3) of Wapenaar and Fokkema [2006], or eqs. (1)567

29

and (2) of Snieder [2007] where a different Fourier-transform convention applies. Eq. (83)568

implies v = 1iωρ∇p, and this expression for v can be substituted into (84), to obtain569

∇ ·(

1

ρ∇p)

+ ω2κp = −iωq. (85)

For homogeneous or smooth media, where the spatial derivatives of ρ and κ are zero or570

approximately zero, (85) further simplifies to the Fourier transform of eq. (1),571

1

ρ∇2p+ ω2κp = −iωq. (86)

Following Wapenaar and Fokkema [2006], let us call G the solution of (86) when q(x, t) =572

δ(x)δ(t):573

1

ρ∇2G + ω2κG = −iωδ(x). (87)

G can be interpreted as the Green’s function associated with eq. (84) plus the condition (83).574

The relationship between G and G3D as defined in app. E can be determined if one575

considers that eq. (87) is the usual scalar wave equation (whose Green’s function is G3D)576

with forcing term −iωδ(x); based on eq. (213), the time-domain solution to (87) is then the577

convolution of G3D with the inverse-Fourier transform of the forcing term −iωδ(x); in the578

frequency domain the convolution reduces to the product of the functions in question, and579

G = −iωG3D. (88)

Wapenaar and Fokkema [2006] also introduce a “modified Green’s function” (see their eq.580

(32)) which coincides with our G3D except for the sign.581

Let us next consider a volume V bounded by a surface ∂V . (∂V is just an arbitrary582

closed surface within a medium, and generally does not represent a physical boundary.) Let583

qA(r, ω), pA(r, ω) and vA(r, ω) denote a possible combination of forcing, pressure and velocity,584

respectively, co-existing at r in V and ∂V . A different forcing qB would give rise, through585

eqs. (83) and (84), to a different “state” B, defined by pB(r, ω) and vB(r, ω).586

A useful relationship between the states A and B, known as “reciprocity theorem”, is587

obtained by combining eqs. (83) and (84) as follows,588

∫V

d3r [(83)A · v∗B + (83)∗B · vA + (84)A p∗B + (84)∗B pA] = 0 (89)

30

[e.g., Wapenaar and Fokkema, 2006; Snieder , 2007], where (83)A is short for the expression589

one obtains after substituting p = pA(r, ω) and v = vA(r, ω) into the left-hand side of eq.590

(83), etc. Namely,591

(83)A · v∗B = ∇pA · v∗B + iωρvA · v∗B (90)592

(83)∗B · vA = ∇p∗B · vA − iωρv∗B · vA (91)593

(84)A p∗B = ∇ · vA p∗B + iωκpAp

∗B − qAp∗B (92)

594

(84)∗B pA = ∇ · v∗B pA − iωκ∗p∗BpA − q∗BpA. (93)

Since (83)A=(83)B=(84)A=(84)B=0 by virtue of eq. (83) and (84), it follows that the ex-595

pression at the left-hand side of (89) equals 0 as anticipated. After substituting (90)-(93)596

into (89),597

∫V

d3r (∇pA · v∗B +∇ · v∗B pA) +

∫V

d3r (∇p∗B · vA +∇ · vA p∗B)

−∫V

d3r iω(κ∗ − κ)pAp∗B −

∫V

d3r (qAp∗B + q∗BpA) = 0.

(94)

It is convenient to apply the divergence theorem [e.g. Hildebrand , 1976] to the first two598

integrals at the left hand side of eq. (94), which then simplifies to599

∫∂V

d2r (pAv∗B + p∗BvA) · n + 2iω

∫V

d3r=(κ)pAp∗B −

∫V

d3r (q∗BpA + qAp∗B) = 0, (95)

where n is the unit vector normal to ∂V , and =(κ) denotes the imaginary part of κ. Eq. (95)600

is equivalent to the “reciprocity theorem of the convolution type”, eq. (5) of Wapenaar and601

Fokkema [2006] or eq. (4) of Snieder [2007].602

An equation relating the cross-correlation of ambient signal to Green’s functions can be603

found from (95) by considering the case qA,B(r) = δ(r− rA,B), with rA,B arbitrary locations604

in V . It follows that pA,B=G (r, rA,B). Substituting these expressions for p and q into (95)605

and using (83) to eliminate the velocity,606

G (rB, rA) + G ∗(rA, rB) =

=1

iω

∫∂V

d2r1

ρ[G ∗(r, rB)∇G (r, rA)− G (r, rA)∇G ∗(r, rB)] · n

+ 2iω

∫V

d3r=(κ)G (r, rA)G ∗(r, rB)

(96)

[e.g. Wapenaar and Fokkema, 2006; Snieder , 2007; Campillo and Roux , 2014].607

31

Notice that the treatment that lead from eq. (89) to (96) remains valid for heterogeneous608

κ and ρ; eq. (96) holds for a heterogeneous, attenuating medium that could be bounded or609

unbounded. it is thus more general than similar eqs. (32), (53), (63), (72), which are only610

strictly valid if the propagation medium is homogeneous.611

Provided that the medium be smooth or homogeneous at and near ∂V , G still coincides612

with −iωG3D on ∂V , with G3D the homogeneous-medium Green’s function. ∇G = −iω∇G3D613

can then be computed through eq. (201), which implies614

∇G3D(r, ω) =r

r

[2

rG3D(r, ω)− iω

cG3D(r, ω)

]. (97)

If we make the further assumption that all sources are far from ∂V (r is always large), it615

follows that 2rG3D can be neglected in eq. (97), while r

r · n ≈ 1, so that616

∇G3D(r, ω) · n ≈ − iω

cG3D(r, ω), (98)

and617

∇G (r, ω) · n ≈ − iω

cG (r, ω) (99)

[e.g. Wapenaar and Fokkema, 2006; Snieder , 2007; Campillo and Roux , 2014]. The surface618

integral in eq. (96) is accordingly simplified,619

G (rB, rA) + G ∗(rA, rB) =

≈ − 2

ρc

∫∂V

d2r [G ∗(r, rB)G (r, rA)]

+ 2iω

∫V

d3r=(κ)G (r, rA)G ∗(r, rB).

(100)

Our treatment in sec. (5.2) was limited to non-attenuating media, where =(κ)=0. In this620

case eq. (100) reduces to621

G (rB, rA) + G ∗(rA, rB) ≈ − 2

ρc

∫∂V

d2r [G ∗(r, rB)G (r, rA)] . (101)

Applying eq. (88), i.e. replacing G = −iωG3D,622

iρc

2ω[G∗3D(rA, rB)−G3D(rB, rA)] = −

∫∂V

d2r [G∗3D(r, rB)G3D(r, rA)] , (102)

32

and by virtue of the reciprocity G3D(rB, rA) = G3D(rA, rB), etc.,623

ρc

ω=[G3D(rA, rB)] = −

∫∂V

d2r [G∗3D(r, rB)G3D(r, rA)] . (103)

The right-hand side of eq. (103) is simply IS(ω) as defined by eq. (9); since nS in eq. (51)624

is arbitrary, eqs. (103) and (51) are equivalent. In a smooth, lossless medium illuminated625

from all azimuths, the stationary-phase and reciprocity-theorem approaches lead to the same626

relationship between Green’s function and cross-correlation, establishing, in practice, that the627

Green’s function between rA and rB can be reconstructed from observations as long as the628

medium is illuminated by a dense distribution of sources covering its boundary ∂V (sec. 5.2).629

The more general, reciprocity-theorem-based results (96) and/or (100) apply to attenu-630

ating media. =(κ)6=0 implies that the volume-integrals at their right-hand sides cannot be631

neglected; reconstruction of an attenuating medium’s Green’s function from the data requires632

that the medium be illuminated by sources within V [e.g. Campillo and Roux , 2014].633

6.2.2 seismic waves634

In a series of articles, Kees Wapenaar and co-workers have applied the above ideas to the case635

of an elastic medium, where both compressional and shear deformation exist [e.g., Wapenaar ,636

2004; Wapenaar and Fokkema, 2006; Wapenaar et al., 2006]. Their procedure, based on637

applying the reciprocity theorem to a pair of states both excited by impulsive point sources, is638

qualitatively similar to the acoustic-wave formulation of Snieder [2007], illustrated here in sec.639

6.2.1. The most complete description of the reciprocity-theorem approach is that of Wapenaar640

et al. [2006], who allow for medium inhomogeneity and attenuation. Wapenaar et al. [2006]641

show that, in analogy with eqs. (100)–(103) for the acoustic case, the Green’s function can642

be reconstructed from noise cross-correlation provided that the medium is illuminated by643

noise sources densely distributed throughout a volume V , where receivers are immersed.644

If the medium is lossless, sources within V are unnecessary, but illumination from sources645

distributed throughout the surface that bounds V is still needed.646

6.3 Normal-mode approach647

Lobkis and Weaver [2001] use a normal-mode approach to find an analytical expression for648

diffuse-field cross-correlation in a bounded medium. Following Snieder et al. [2010], we briefly649

repeat their treatment for the simpler case of a lossless bounded medium. Normal modes are650

33

defined as the real functions pn(r) such that pn(r) cos(ωnt), pn(r) sin(ωnt) (n=1,2,3,. . . ,∞),651

with eigenfrequencies ωn, form a complete set of solutions to the homogenous version of the652

scalar wave equation (1). Any wave field p(r, t) propagating in the medium under consider-653

ation can be written as a linear combination of modes with coefficients an, bn654

p(r, t) =∑n

[anpn(r) cos(ωnt) + bnpn(r) sin(ωnt)] , (104)

where∑

n denotes summation over the integer values of n from 1 to infinity.655

6.3.1 Green’s function as a linear combination of modes656

Let us define the Green’s function GM as the solution of the homogeneous version (∂F∂t = 0)657

of eq. (1), with initial conditions p = 0 and ∂p∂t = δ(r − s) (point source at s). GM (r, t) can658

be written as a linear combination of modes,659

GM (r, t) =∑k

[αk cos(ωkt) + βk sin(ωkt)] pk(r). (105)

Substituting into (105) the initial condition on p we find660

∑k

αkpk(r) = 0; (106)

after differentiating (105) with respect to time, the initial condition on ∂p∂t gives661

∑k

ωkβkpk(r) = δ(r− s). (107)

We multiply both sides of eqs. (106) and (107) by the eigenfunction pn(r) and integrate over662

r. From eq. (107) we find663

∑k

ωkβk

∫R3

d3r pk(r)pn(r) =

∫R3

d3r δ(r− s)pn(r), (108)

which, after taking advantage of the orthonormality of the modes (left-hand side) and apply-664

ing the properties of the δ function (right-hand side), collapses to665

βnωn = pn(s). (109)

34

Eq. (106) likewise reduces to αk = 0, and eq. (105) becomes666

GM (r, t) =∑n

pn(s)pn(r)sin(ωnt)

ωn, (110)

valid for t > 0, consistent with equation (4) of Lobkis and Weaver [2001] and equation (1) of667

Snieder et al. [2010].668

6.3.2 Ensemble-averaged cross-correlation as a linear combination of modes669

Let us write the ambient signals recorded at R1 and R2 as linear combinations of modes with670

random coefficients an, bn. Their cross-correlation then reads671

∫ +T/2

−T/2dτ p∗(r1, τ)p(r2, t+ τ) =

∑n,k

akanpk(r1)pn(r2)

∫ +T/2

−T/2dτ cos(ωkτ) cos[ωn(t+ τ)]

+ akbnpk(r1)pn(r2)

∫ +T/2

−T/2dτ cos(ωkτ) sin[ωn(t+ τ)]

+ bkanpk(r1)pn(r2)

∫ +T/2

−T/2dτ sin(ωkτ) cos[ωn(t+ τ)]

+ bkbnpk(r1)pn(r2)

∫ +T/2

−T/2dτ sin(ωkτ) sin[ωn(t+ τ)]

.(111)

Lobkis and Weaver [2001] make the assumption that “modal amplitudes are uncorrelated672

random variables”, which is equivalent to noise sources being spatially and temporally un-673

correlated so that “cross-terms” can be neglected (App. D). This means in practice that674

if one repeats the cross-correlation (111) at many different times, the normal modes of the675

medium stay the same (medium properties do not change), but the coefficients an, bn change676

in a random fashion at each realization. When the average of all realizations is taken, the677

products akan and bkbn both average to δknM(ωn), with the function M indicating how678

strongly different eigenfrequencies are excited on average, and δkn = 1 if k=n, 0 otherwise;679

akbn, on the other hand, averages to 0 for all values of k, n. Using 〈•〉 to denote the averaging680

35

procedure, it follows from (111) that681

⟨∫ +T/2

−T/2dτ p∗(r1, τ)p(r2, t+ τ)

⟩=

∑n,k

〈akan〉 pk(r1)pn(r2)

∫ +T/2

−T/2dτ cos(ωkτ) cos[ωn(t+ τ)]

+ 〈akbn〉 pk(r1)pn(r2)

∫ +T/2

−T/2dτ cos(ωkτ) sin[ωn(t+ τ)]

+ 〈bkan〉 pk(r1)pn(r2)

∫ +T/2

−T/2dτ sin(ωkτ) cos[ωn(t+ τ)]

+ 〈bkbn〉 pk(r1)pn(r2)

∫ +T/2

−T/2dτ sin(ωkτ) sin[ωn(t+ τ)]

=∑n

M(ωn)pn(r1)pn(r2)

∫ +T/2

−T/2dτ cos(ωnt)

=∑n

M(ωn)pn(r1)pn(r2)T cos(ωnt), (112)

where the trigonometric identity cos(ωnτ) cos[ωn(t+τ)]+sin(ωnτ) sin[ωn(t+τ)] = cos(ωnt) is682

used [Snieder et al., 2010]. Eq. (111) is equivalent to eqs. (9) or (10) of Snieder et al. [2010],683

except that we have chosen not to normalize the coefficents an, bn by the corresponding684

eigenfrequency ωn.685

Provided that M is constant with respect to ω (all modes are equally excited), the time-686

derivative of the right-hand side of eq. (112) is proportional to the right-hand side of (110):687

in other words, the ensemble-averaged cross-correlation of a diffuse field recorded at r1 and688

r2 is proportional to the time-derivative of the Green’s function GM (eq. (110) in sec. 6.3.1),689

for a source located at r1 and a receiver located at r2. This is equivalent to eqs. (50) and690

(103), valid for non-steady-state 3-D acoustic media.691

6.4 Analogy between diffuse field and time-reversal mirror692

Derode et al. [2003] explain the relationship between diffuse-field cross-correlation and Green’s693

function via the concept of time-reversal mirror [e.g., Fink , 1999, 2006]. A time-reversal694

mirror can be thought of as an array of transducers that record sound, reverse it with respect695

to time, and emit the time-reversed acoustic signal; if the array is sufficiently large and dense,696

it will time-reverse the entire propagating wave field, focusing time-reversed waves back to697

the origin of the initial signal.698

Following Stehly [2007], we next summarize the reasoning of Derode et al. [2003] in three699

simple steps. First of all (sec. 6.4.1), cross-correlating two signals is equivalent to time-700

reversing the first signal, then convolving it with the second. We then show (sec. 6.4.2)701

that the convolution of two impulsive signals emitted from r and recorded at the locations702

36

r1 and r2 coincides with the signal recorded at r2 after being emitted by a source at r1703

and then time-reversed and re-emitted by a transducer at r. It follows (sec. 6.4.3) that, if704

instead of a single transducer at r an entire array of transducers forming a time-reversal705

mirror are present, the mentioned convolution coincides with the Green’s function: through706

the relationship between convolution and cross-correlation, an equation connecting Green’s707

function and cross-correlation is thus determined.708

6.4.1 Convolution, cross-correlation and time reversal709

Let us first recall the definition of the convolution f ⊗ g of two real-valued functions f(t),710

g(t):711

f ⊗ g(t) =

∫ +∞

−∞f(τ)g(t− τ)dτ. (113)

It can be shown that the convolution operator is commutative,712

f ⊗ g = g ⊗ f, (114)

and associative,713

(f ⊗ g)⊗ h = f ⊗ (g ⊗ h). (115)

Convolution and cross-correlation are closely related: if one denotes f−(t) ≡ f(−t) the714

function found by systematically changing the sign of the argument of f , that is to say, by715

time-reversing f , it follows that716

f− ⊗ g(t) =

∫ +∞

−∞f(−τ)g(t− τ)dτ

= −∫ −∞

+∞f(τ)g(t+ τ)dτ

=

∫ +∞

−∞f(τ)g(t+ τ)dτ, (116)

i.e., the convolution of f− with g coincides with the cross-correlation of f with g [e.g. Smith,717

2011].718

Now consider a source at the location r emitting the (real-valued) signal e(t), and two719

receivers at the locations r1 and r2. By definition of Green’s function G, the receiver at r1720

records a signal e ⊗ G(r1, r)(t), while that at r2 records e ⊗ G(r2, r)(t) (Fig. 8). Let us721

cross-correlate the signal recorded at r1 with that recorded at r2, and denote C12 the cross722

correlation. Making use of equation (116) and of the fact that convolution is commutative723

37

Figure 8: Setup of sec. 6.4.1, where a signal e(t) emitted by a source at r (star) is recordedby receivers at r1 and r2 (triangles).

and associative,724

C12 =[e− ⊗G−(r1, r)(t)

]⊗[e⊗G(r2, r)(t)] =

[G−(r1, r)⊗G(r2, r)(t)

]⊗[e− ⊗ e(t)

]. (117)

6.4.2 Propagation from one source to a transducer and from the transducer to725

a receiver726

Let us now place at the location r1 a sound source that emits an impulsive signal, and at r a727

transducer that records sound, time-reverses it and emits it back (Fig. 8). A receiver is still728

placed at r2. The signal recorded at r coincides with the Green’s function G(r, r1, t). The729

signal p(r2, t) recorded at r2 at a time t is the convolution of the signal emitted by r, that is,730

the time-reversed Green’s function G(r, r1,−t), with the Green’s function G(r2, r, t),731

p(r2, t) = G−(r, r1)⊗G(r2, r)(t). (118)

(This is valid in the assumption that the emitted wavelet is short enough, and/or r is far732

enough from both r1 and r2 for the the time-reversed wave packet p(r2, t) to be easily isolated733

from the “direct” arrival at r2.) By the reciprocity of G, G−(r, r1, t) = G−(r1, r, t). We can734

thus substitute eq. (118) into (117), and735

C12 = p(r2, t)⊗[e− ⊗ e(t)

], (119)

38

Figure 9: Setup of sec. 6.4.2, where an impulse emitted by a source at r1 (triangle) is recordedby a transducer at r (star), which then time-reverses it and emits it back. A receiver at r2

(triangle) would then record the convolution of the time-reversed initial impulse with theGreen’s function corresponding to the locations r and r2.

i.e., the cross-correlation of the recordings, made at r1 and r2, of the signal emitted by a736

source at r coincides with the recording made at r2 of the same signal, emitted at r1 and737

recorded and time-reversed at r. For impulsive signals e(t) = δ(t), eq. (119) collapses to738

C12 = p(r2, t). (120)

6.4.3 Multiple sources and time-reversal mirror739

We next consider a dense, uniform distribution of transducers surrounding the locations740

r1 and r2. Such a set of transducers forms a time-reversal mirror [e.g., Fink , 1999, 2006].741

Equations (117) and (118) remain valid, after replacing G(r1, r, t), G(r2, r, t) with the sum742

of Green’s functions associated with all locations r where transducers are now placed; at the743

limit of a continuous source distribution,744

C12 =

∫Ω

dr[G−(r1, r)⊗G(r2, r)(t)

]⊗[e− ⊗ e(t)

], (121)

where Ω is the entire solid angle.745

In this setup (Fig. 10), an impulsive signal emitted at r1 is first recorded at r2 as G(r2, r1);746

it then hits the transducer array, which, by definition of time-reversal mirror, sends the signal747

back in such a way that the same wave field propagates backwards in time: the receiver at748

39

(a) (b)

Figure 10: Setups of sec. 6.4.3. (a) Signals emitted by a circle of sources (stars) are recordedby a receiver pair at r1 and r2 (triangles); this is the same setup as in Fig. 8 (sec. 6.4.1), butnow there is more than one source. (b) An impulse emitted by a source at r1 is recorded by aset of transducers which time-reverse it and emit it back. This is the same setup at in Fig. 9(sec. 6.4.2), but with more than one transducer. The transducers occupy the same locations(stars) as the sources in (a) and thus form a time-reversal mirror (in two dimensions). Derodeet al. [2003] and Stehly et al. [2006] use the properties of acoustic time reversal to prove that,in a setup such as (a), the cross-correlation between the signals recorded at r1 and r2 can beassociated with the Green’s function between the same two locations.

r2 records the time-reversed Green’s function G(r2, r1,−t), and the back-propagated signal749

eventually focuses back on r1 in the form of an impulse at time −t = 0.750

This means, in practice, that if the individual transducer at r (Fig. 8) is replaced by751

a time-reversal mirror, then p(r2, t) in eq. (120) can be replaced by the Green’s function752

G(r2, r1,−t),753

C12 = G(r2, r1,−t). (122)

Now recall from sec. 6.4.1 that C12 is also the cross-correlation of the recordings made754

at r1 and r2 of signal generated at r; the individual source at r must now be replaced by a755

set of sources occupying the transducer locations, so that cross-correlations associated with756

individual sources are summed. Derode et al. [2003] correctly infer that, if a signal is generated757

by a distribution of sources with the geometry of an effective time-reversal mirror (i.e. energy758

propagating along all azimuths, diffuse field), the Green’s function between any two points759

r1 and r2 can be found by cross-correlating the recordings of said signal made at r1 and r2.760

This statement is, again, equivalent to eqs. (50) and (103).761

40

Figure 11: Distribution of sources (stars) and receivers (triangles) in the setup of sec. 7.Only 10% of the simulated sources are shown. In some of the simulations discussed in thefollowing, the effects of uneven source distributions including only the sources denoted bygreen vs. yellow stars are modeled.

7 Uneven source distributions and “spurious arrivals”762

Our derivation so far is based on the hypothesis that the geographic distribution of noise763

sources be close to uniform with respect to source-receiver azimuth. The stationary-phase764

formulae of appendix A only hold if f is a smooth function of x in eq. (150), and of both x765

and y in eq. (152); the source distributions nC , nM , nS must accordingly be smooth with766

respect to ϕ and/or θ for the treatment of sec. 5 to be valid. The integral in eq. (103)767

likewise extends to the whole boundary of the volume V containing the receivers: if ∂V is768

not covered densely and uniformly by sources, noise cross-correlation does not coincide with769

the right-hand side of eq. (103), and G is not properly reconstructed.770

Noise sources are generally not uniformly distributed in practical applications, and we771

know e.g. from Mulargia [2012] that seismic ambient noise on Earth is not strictly diffuse.772

We illustrate the consequences of significant inhomogeneities in source distribution with a773

simple model. As in secs. 4.3 and 5.3, receivers R1 and R2, lying 20km from one another774

on a membrane of infinite extension, are surrounded by a circle of sources whose center is775

R1 and whose radius is 100km (Fig. 11). We numerically convolve a Ricker wavelet (central776

frequency of 1Hz) with the Green’s function G2D for each of the sources in question. Using777

a wavelet rather than an impulse allows to better visualize the effects we are interested in.778

41

For each location of the source, we cross-correlate the corresponding signals at R1 and R2779

and plot the cross-correlations in Fig. 12a. The result of stacking the cross-correlations,780

shown in Fig. 12b, is consistent with the results of sec. 5.3, after modulating the Green’s781

function with the Ricker wavelet (we shall speak of “Ricker response” instead of Green’s782

function). We next average only the cross-correlations associated with sources denoted in783

green in Fig. 11, and, finally, only those associated with the “yellow” sources of Fig. 11.784

Two inferences can be made from Fig. 12c, where both averages are shown: (i) if only785

the “yellow” sources are “on”, and energy only travels in the direction R2−→R1, only the786

anticausal Ricker’s response between R1 and R2 emerges from averaging; likewise, only the787

causal part shows up if only sources to the left of R1 are active. (ii) While both causal and788

anti-causal arrivals in Fig. 12b approximately coincide with those of Fig. 12c, the curves in789

Fig. 12c contain two additional arrivals, corresponding to the two azimuths where both source790

distributions in Fig. 11 abruptly end. These arrivals, usually referred to as “spurious”, have791

no relation to the Ricker response; they are artifacts caused by strong inhomogeneities in the792

source distribution. Spurious arrivals are likely to affect field data, and can be identified in793

laboratory (physical acoustics) data.794

8 Ambient-signal cross-correlation in the presence of a scat-795

terer: a stationary-phase derivation796

The stationary-phase derivations carried out above have established mathematical relation-797

ships between two-station ambient-signal cross-correlation and a medium’s Green’s function,798

in the simple case of homogeneous, unbounded media. The same approach can also usefully be799

applied to a homogeneous medium including a limited number of point scatterers. Following800

Snieder et al. [2008], we shall treat in some detail the case of a homogeneous, 3-D acoustic801

medium containing a single point scatterer: extension to more scatterers [Fleury et al., 2010]802

is then straightforward, albeit cumbersome.803

It is convenient to place the origin of the coordinate system at the location of the scatterer,804

and to choose the x- and z-axes so that the plane they identify contains the locations r1, r2805

of receivers R1 and R2. In this setup, the Green’s function GS3D is the sum of G3D from eq.806

(201) plus an additional, “scattered” term involving propagation from the source (located at807

42

Figure 12: Cross-correlations associated with a circular, planar distribution of sources sur-rounding the two receivers as sketched in Fig. 13. Each source generates a Ricker wavelet(central frequency=1Hz); the wavelet is convolved with the Green’s function (phase veloc-ity c=2Km/s) to evaluate the signals observed at the two receivers, which are then cross-correlated. (a) Single-source cross-correlations for all source azimuths, the dashed lines markthe azimuths separating the two (yellow vs. green) subsets of sources as defined in Fig. 11.(b) Stacked cross-correlation resulting from (a). (c) Stacked cross-correlations that one wouldobtain if signal was generated only at the locations identified by green stars in Fig. 11 (greenline) vs. the yellow stars (yellow line).

43

a point s) to the scatterer and from the scatterer to the receiver,808

GS3D(r1,2, s, ω) = G3D(r1,2S , ω) +G3D(r1,2, ω)G3D(s, ω)h(r1,2, s), (123)

where r1,2 and s denote the moduli of r1,2 and s, respectively, and r1,2, s are unit-vectors par-809

allel to r1,2 and s. The scattering function (or “matrix”) h(r1,2, s) accounts for the azimuth-810

dependence of the amount of scattered energy [e.g. Snieder , 1986; Groenenboom and Snieder ,811

1995; Marston, 2001].812

The frequency-domain cross-correlation of the signals recorded at R1 and R2 reads813

GS3D(r1, s, ω)GS∗3D(r2, s, ω) = G3D(r1S , ω)G∗3D(r2S , ω)

+G3D(r1S , ω)G∗3D(r2, ω)G∗3D(s, ω)h∗(r2, s)

+G3D(r1, ω)G3D(s, ω)G∗3D(r2S , ω)h(r1, s)

+G3D(r1, ω)G3D(s, ω)G∗3D(r2, ω)G∗3D(s, ω)h(r1, s)h∗(r2, s).

(124)

We next assume sources to be distributed over a spherical surface of radius s=R sur-814

rounding the receivers, as in sec. 3, scenario (ii); for the sake of simplicity, we limit ourselves815

to uniform source density nS(θ, ϕ) = 1. Neglecting cross-terms (appendix D), the cumulative816

effect of such a source distribution is obtained by integrating eq. (124) in s over the whole817

solid angle. The integral IS(ω) of the first term at the right-hand side of (124) has been818

treated in detail in sec. 5.2 and its analytical form is given, e.g., by eq. (50); substituting819

nS(θ, ϕ) = 1,820

IS(ω) ≈√

2π

(4π R)2

i

ω[G3D(∆, ω)−G∗3D(∆, ω)] , (125)

where ∆ denotes, as usual, the distance between R1 and R2. Let us call I2(ω), I3(ω) and821

I4(ω) the source-averages of the remaining three terms at the right-hand side of (124). The822

first of these integrals,823

I2(ω) =1

4π

∫ π

0dθ sin θ

∫ π

−πdϕG3D(r1S(θ, ϕ), ω)G∗3D(r2, ω)G∗3D(s(θ, ϕ), ω)h∗(r2, s(θ, ϕ))

=1

(4π)4

1(c√

2π)3 ∫ π

0dθ sin θ

∫ π

−πdϕ

eiωc

(R+r2−r1S(θ,ϕ))

Rr2 r1S(θ, ϕ)h∗(r2, s(θ, ϕ)),

(126)

can be simplified by making the hypothesis that the source be very far from both receivers824

44

and from the scatterer [Snieder et al., 2008], i.e. R r1,2, which implies that r1,2S can be825

replaced with R at the denominator of (126). At the exponent of the numerator care must be826

taken to evaluate the difference R−r1S which is of the same order as r2; in the 3-D Cartesian827

reference frame828

s− r1 = (R sin θ cosϕ− r1 sin θ1, R sin θ sinϕ,R cos θ − r1 cos θ1) , (127)

and consequently829

r1S = |s− r1| =[R2 + r2

1 − 2Rr1(sin θ cosϕ sin θ1 + cos θ cos θ1)] 1

2 . (128)

Eq. (128) implies that830

r1S ≈ R− r1(sin θ cosϕ sin θ1 + cos θ cos θ1) (129)

to first order in r1/R [Snieder et al., 2008]. Substituting into eq. (126),831

I2(ω) =1

(4π)4

1(c√

2π)3R2r2

eiωcr2

∫ π

0dθ sin θ

∫ π

−πdϕ eiω

cr1(sin θ cosϕ sin θ1+cos θ cos θ1)h∗(r2, s(θ, ϕ)).

(130)

The integral at the right-hand side of eq. (130) coincides with that in (152), after replacing832

λ=ω, x=θ, y=ϕ , f(θ, ϕ)=h∗(r2, s(θ, ϕ))sin θ and ψ(θ, ϕ)=r1(sin θ cosϕ sin θ1+cos θ cos θ1)/c;833

the stationary-phase formula (151) can then be applied, provided that h be a smooth function834

of θ, ϕ.835

In analogy with the procedure of sec. 5.2, we differentiate ψ(θ, ϕ) with respect to θ and836

ϕ to find the stationary points of (130),837

ψθ =r1

c(cos θ cosϕ sin θ1 − sin θ cos θ1), (131)

838

ψϕ = −r1

csin θ sinϕ sin θ1. (132)

Equation (132) establishes that stationary points can be found at either ϕ=0 or ϕ=π, i.e. on839

the plane where the scatterer and both receivers are. Substituting cosφ=±1 into eq. (131)840

we further identify the two stationary points (θ=θ1,ϕ=0) and (θ=π-θ1,ϕ=π). If we continue841

differentiating,842

ψθθ = −r1

c(sin θ cosϕ sin θ1 + cos θ cos θ1), (133)

45

ψϕϕ = −r1

csin θ cosϕ sin θ1, (134)

843

ψθϕ =r1

csin θ sinϕ sin θ1. (135)

At the stationary points,844

ψ(θ1, 0) =r1

c, (136)

845

ψ(π − θ1, π) = −r1

c, (137)

846

ψθθ(θ1, 0) = −r1

c, (138)

847

ψθθ(π − θ1, π) = +r1

c, (139)

848

ψϕϕ(θ1, 0) = −r1

csin2 θ1, (140)

849

ψϕϕ(π − θ1, π) =r1

csin2 θ1, (141)

850

ψθϕ(θ1, 0) = ψθϕ(π − θ1, π) = 0. (142)

After substituting (136)-(142) into the stationary-phase formula (161),851

I2(ω) ≈ 1

(4π)4

1√2πc2R2

eiωcr2

r2

i

ω

[h∗(r2, s(π − θ1, π))

e−iωcr1

r1− h∗(r2, s(θ1, 0))

eiωcr1

r1

]. (143)

The integral in I3(ω) is similar to that in I2(ω), and the stationary-phase approximation852

leads to853

I3(ω) ≈ 1

(4π)4

1√2πc2R2

e−iωcr1

r1

i

ω

[h(r1, s(θ2, 0))

e−iωcr2

r2− h(r1, s(π − θ2, π))

eiωcr2

r2

]. (144)

The integral in I4(ω),854

I4(ω) =1

(4π)5

1

(2π)2 c4

eiωc

(r2−r1)

R2 r2 r1

∫ π

0dθ sin θ

∫ π

−πdϕh(r1, s(θ, ϕ))h∗(r2, s(θ, ϕ)), (145)

is not a stationary-phase integral in general.855

Notice that, by eq. (201), and since s(θ1, 0)=r2, the term856

eiωcr2

r2

eiωcr1

r1h∗(r1, s(θ1, 0)) = 25π3c2G∗3D(r2, ω)G∗3D(r1, ω)h∗(r1, r2), (146)

appearing in (143), describes (the complex conjugate of) an impulse propagating from R2 to857

46

R1 via the scatterer. Likewise, in eq. (144),858

e−iωcr1

r1

e−iωcr2

r2h(r2, s(θ2, 0)) = 25π3c2G3D(r2, ω)G3D(r1, ω)h(r2, r1) (147)

is an impulse propagating from R1 to R2 via the scatterer. The remaining terms in (143),859

(144) and (145) do not have an immediate physical explanation. The source-averaged cross-860

correlation, coinciding with the integral of (124), takes the form861

IS(ω) + I2(ω) + I3(ω) + I4(ω) ≈√

2π

(4π R)2

i

ω[G3D(∆, ω)−G∗3D(∆, ω)]

−√

2π

(4πR)2

i

ωG∗3D(r2, ω)G∗3D(r1, ω)h∗(r1, r2)

+

√2π

(4πR)2

i

ωG3D(r2, ω)G3D(r1, ω)h(r2, r1)

+1

(4π)4

1√2πc2

i

ω

eiωc

(r2−r1)

R2 r2 r1

×

[h∗(r2,−r1)− h(r1,−r2)]− 1

4πc2

ω

(2π)32

∫ π

0dθ sin θ

∫ π

−πdϕh(r1, s(θ, ϕ))h∗(r2, s(θ, ϕ))

(148)

where we have used eq. (49) with nS = 1, and the identities s(π − θ1, π)=−r1, s(π −862

θ2, π)=−r2.863

Equation (148) was first derived, using a slightly different convention/notation, by Snieder864

et al. [2008], who observed that the term in . . . is zero as a consequence of the generalized865

optical theorem. We are left with866

IS(ω) + I2(ω) + I3(ω) + I4(ω) ≈√

2π

(4π R)2

i

ω

[GS3D(∆, ω)−GS∗3D(∆, ω)

]≈ −

√2π

8(π R)2

1

ω=[GS3D(∆, ω)

],

(149)

similar to eqs. (51) and (103). The procedure of sec. 5.3.3 could be applied to show that,867

in the time domain, the source-averaged cross-correlation (149) is proportional to the sum of868

GS3D(t) (causal part) and -GS3D(−t) (anticausal).869

This result confirms that, as first pointed out by Weaver and Lobkis [2004], diffuse-field870

cross-correlation in heterogeneous (rather than just homogeneous) media allows in principle871

to reconstruct the full Green’s function of the medium, with all reflections and scatterings872

and propagation modes. This is implicit in the reciprocity-theorem formulation of sec. 6.2,873

and has been verified experimentally and numerically by, e.g., Larose et al. [2006], Mikesell874

47

Figure 13: Distribution of sources (stars), receivers (triangles) and a point scatterer (redcircle) in the setup of sec. 8.1. As in Fig. 11, only 10% of the simulated sources are shown.

et al. [2012] and Colombi et al. [2014].875

8.1 “Spurious” arrivals and their cancellation876

The result (149) might be surprising if one considers the scatterer as a (“secondary”) source;877

no matter where the actual (“primary”) source is, the scatterer is always at the same location878

relative to R1 and R2, so that the delay between the arrival of the scattered signal at R1879

and its arrival at R2 is always the same. This would give rise to a peak in the stacked cross-880

correlation, that does not correspond with either of the two arrivals in GS3D as defined by eq.881

(123).882

This is illustrated in Figs. 13 and 14. Following Snieder et al. [2008], we convolve a Ricker883

wavelet (central frequency of 1Hz) with the Green’s function associated with a uniformly884

dense, circular source distribution (Fig. 13) plus a single scatterer. For simplicity and in885

analogy with Snieder et al. [2008], we work in two dimensions (membrane waves from point886

sources) and assume isotropic scattering. We implement the expression for h derived by887

Groenenboom and Snieder [1995] via the optical theorem [Newton, 2002]. For each location888

of the source, we cross-correlate the corresponding signals at R1 and R2, which we choose to889

be equidistant from the scatterer, and plot the cross-correlations in Fig. 14a. As anticipated,890

whatever the source azimuth, a peak in the cross-correlation appears at t=0, corresponding891

to the scattered signal hitting simultaneously R1 and R2. This peak, or “arrival”, at t=0892

48

Figure 14: Cross-correlations associated with a circular, planar distribution of sources sur-rounding the two receivers, and one isotropic point scatterer, as sketched in Fig. 13. Eachsource generates a Ricker wavelet (central frequency=1Hz); the wavelet is convolved with theGreen’s function (phase velocity c=2Km/s), including the contribution of scattering, to evalu-ate the signals observed at the two receivers, which are then cross-correlated. (a) Single-sourcecross-correlations for all source azimuths; the color scale saturates at 20% of the maximumamplitude to emphasize the spurious arrivals; the dashed lines mark the azimuths where the“yellow”/“green” source distribution is discontinuous (Fig. 13). (b) Source-averaged cross-correlation resulting from (a). (c) Source-average cross-correlations that one would obtain ifsignal was generated only at the locations identified by green stars in Fig. 13 (green line) vs.the yellow stars (yellow line).

49

clearly does not exist in GS3D, which would seem to contradict eq. (149). However, when893

stacking all single-azimuth cross-correlations, the t = 0 peak cancels out with the “knees”894

of some other cross-correlation peaks (Fig. 14b), and eq. (149) is indeed confirmed. The895

four stationary points identified in sec. 8 appear as knees of the cross-correlation peaks in896

Fig. 14a, corresponding in turn to the two causal and two anticausal peaks of the system’s897

Ricker response (Fig. 14b).898

It is critical that source illumination be uniform, for the spurious term associated with899

the scatterer to disappear. To emphasize this point, we show in Fig. 14c the stacks obtained900

by considering only half of the available sources [Snieder et al., 2008], on either side of the901

receiver array. Depending on which side, only the causal or anticausal part of the Ricker902

response is reconstructed, as in Fig. 12. Furthermore, three artifacts emerge; the one at903

t=0 clearly results from the spurious wavelet in Fig. 14a, which does not cancel out since904

the illumination is (strongly) nonuniform. The other two are simply the spurious arrivals of905

Fig. 12, which result from the sharp inhomogeneities in source distribution, and would be906

found even in the absence of scatterers.907

Scattering thus has a complex effect on Green’s function reconstruction. Each scatterer908

further complicates the Green’s function, introducing an additional, physical term to be909

reconstructed. It also hinders its retrieval, generating a spurious term which will only cancel910

out if the wave field is sufficiently diffuse. On the other hand, scatterers themselves contribute911

to the wave field’s diffusivity and azimuthal uniformity of illumination. Recent and current912

work [e.g. Fleury et al., 2010; Mikesell et al., 2012; Ravasi and Curtis, 2013; Colombi et al.,913

2014] aims at disentagling the specific role of scattered ambient signal in reconstructing the914

main peaks of the Green’s function as well as its “coda”.915

9 Summary916

In a diffuse ambient wave field, i.e. a random wave field where energy propagates with equal917

probability in all directions, cross-correlating the signal recorded by a pair of receivers is a918

way to measure the impulse response (Green’s function) between the receivers. In practice,919

long recordings of seismic ambient noise often include waves traveling along approximately920

all azimuths: the combination of their cross-correlations approximates that of a diffuse field.921

The relationship between cross-correlation and Green’s function changes depending on some922

properties of the medium and of the wave field:923

50

(i) in a homogeneous, lossless, unbounded membrane where circular waves are generated by924

point sources, the cross-correlation of diffuse noise is generally not explicitly related to925

the corresponding Green’s function G2D (sec. 5.3). If, however, the source distribution is926

symmetric along the receiver-receiver axis, eq. (70) stipulates that G2D is proportional927

to the time-derivative of the cross-correlation. The membrane setup is relevant to928

seismology, because it corresponds in practice to the propagation of Rayleigh waves929

(sec. 2.3) on Earth.930

(ii) In homogeneous, lossless, unbounded 3-D acoustic media, the Green’s function can be931

exactly reconstruced from diffuse noise generated at point sources that are smoothly dis-932

tributed over a sphere surrounding the receivers. More precisely, the cross-correlation933

coincides with the time-derivative of the Green’s function (sec. 5.2). This relation-934

ship holds for both causal and anticausal contributions to the Green’s function/cross-935

correlation, and remains valid if the source distribution is asymmetric (provided that it936

be smooth).937

(iii) The latter result does not hold if noise in free space is generated by point sources938

uniformly distributed along a circle surrounding the receivers (sec. 5.1). Still, even in939

this case, the time of maximum cross-correlation clearly coincides with that of G2D’s940

peak (Fig. 5).941

(iv) If noise is made up of plane waves propagating in all directions (sec. 5.4), the same942

formulation holds for 2-D and 3-D unbounded, lossless media. In this case, the source-943

averaged cross-correlation is proportional, in the frequency domain, to the real part944

of the 2-D Green’s function G2D, which in turn is proportional to the Bessel function945

J0(ωr/c), with r the inter-receiver distance. Hence eq. (72). The corresponding time-946

domain relationship was derived by Nakahara [2006], who showed that the Hilbert947

transform of diffuse-field cross-correlation coincides with the (causal minus anticausal)948

G2D.949

(v) In bounded, heterogeneous, attenuating media, a general relationship between source-950

averaged cross-correlation and Green’s function is found via the reciprocity theorem951

(sec. 6.2). Essentially, source-averaged cross-correlation coincides with the sum of a952

volume (V ) integral, and an integral over the boundary ∂V of such volume. It follows953

that densely distributed noise sources throughout a volume are needed for the Green’s954

51

function to be reconstructed via cross-correlation; sources within V are unnecessary if955

the medium is lossless, provided that all azimuths are illuminated.956

The results (i) and (iv) are relevant for most seismology applications seen so far, since,957

provided that attenuation can be neglected, they apply to surface waves in the “membrane”958

approximation [Tanimoto, 1990] (sec. 2.1). E.g. Ekstrom et al. [2009] have shown explicitly959

that the phase of the Green’s function is in agreement with the data, thus validating the960

“lossless-medium” approximation for phase/group-velocity-based seismic imaging, while re-961

producing the Green’s function’s amplitude remains problematic [e.g., Weemstra et al., 2014].962

Acknowledgments963

This study greatly benefitted from our discussions with M. Campillo, L. Campos-Venuti, A.964

Colombi, A. Curtis, J. de Rosny, J. P. Montagner, P. Poli, R. Snieder, P. Roux, J. Verbeke,965

K. Wapenaar, and four anonymous reviewers. Data and software used to produce the results966

of this paper can be obtained by E-mailing the first author: lapo.boschi@upmc.fr.967

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A The stationary phase approximation1191

A.1 1-D integrals1192

The stationary phase approximation [e.g., Bender and Orszag , 1978] applies to integrals of1193

the form1194

I(λ) =

∫ b

af(x)eiλψ(x)dx, (150)

where a and b are arbitrary constants, and λ is large enough for eiλψ(x) to be a rapidly1195

oscillating function of x, with respect to the smooth functions f(x), ψ(x). It is based on the1196

finding [e.g., Bender and Orszag , 1978, sec. 6.5] that the leading contribution to I(λ) comes1197

from a small interval surrounding the “stationary points” of ψ(x), i.e. the locations x such1198

that ψ′(x) = 0 (where the “prime” denotes differentiation with respect to x). Elsewhere, the1199

integrand oscillates quickly and its average contribution to the integral is negligible. Bender1200

and Orszag [1978] demonstrate that, for a single stationary point at a,1201

I(λ) ≈ f(a)ei(λψ(a)±π4 )√

π

2λ|ψ′′(a)|, (151)

valid for λ −→∞ and in the assumption that ψ′′(a) 6= 0. The sign of π/4 at the exponent of1202

e is positive if ψ′′(a) > 0 and negative otherwise. Since any integral of the form (150) can be1203

written as a sum of integrals with one of the integration limits coinciding with a stationary1204

60

point, eq. (151) is sufficient to solve all 1-D stationary phase integrals like (150), regardless1205

of the number and location of stationary points [Bender and Orszag , 1978].1206

A.2 Extension to 2-D integrals1207

The result of section A.1 can be used to find a general approximate formula for integrals of1208

the form1209

I(λ) =

∫ b

adx

∫ d

cdyf(x, y)eiλψ(x,y). (152)

If we, again, only consider the limit λ −→ ∞, the integrand in (152) turns out to be very1210

strongly oscillatory, and the only nonnegligible contribution to the integral comes from the1211

vicinity of the stationary points (xi, yi), defined by1212

(∂ψ

∂x(xi, yi),

∂ψ

∂y(xi, yi)

)= (0, 0) (153)

[e.g., Wong , 1986]. Equation (152) can thus be approximated by1213

I(λ) ≈∑i

f(xi, yi)

∫ xi+ε

xi−εdx

∫ yi+δ

yi−δdyeiλψ(x,y), (154)

where the sum is extended to all stationary points i, and ε and δ are small. f(x, y) is1214

approximated by f(xi, yi) since f varies much more slowly than eiλψ when λ is large. We1215

next conduct a Taylor series expansion of ψ(x, y) around (xi, yi),1216

ψ(x, y) ≈ ψ(xi, yi) +1

2

[ψxxi(x− xi)2 + ψyyi(y − yi)2 + 2ψxyi(x− xi)(y − yi)

], (155)

where ψxxi stands for ∂2ψ∂x2 (xi, yi), and so on. Substituting (155) into (154) and after the1217

changes of variable ui = x− xi and vi = y − yi, we find1218

I(λ) ≈∑i

f(xi, yi)eiλψ(xi,yi)

∫ +ε

−εdui

∫ +δ

−δdvie

iλ2 (ψxxu2

i+ψyyv2i+2ψxyuivi). (156)

The integral in (156) can be solved analytically by first rewriting1219

ψxxiu2i + ψyyiv

2i + 2ψxyiuivi = ψxxi

(ui +

ψxyiψxxi

vi

)2

+ v2i

(ψyyi −

ψ2xyi

ψxxi

), (157)

61

and substituting into (156),1220

I(λ) ≈∑i

f(xi, yi)eiλψ(xi,yi)

∫ +δ

−δdvi e

iλ2v2i (ψyyi−

ψ2xyi

ψxxi)∫ +ε

−εdui e

iλ2ψxxi

(ui+

ψxyiψxxi

vi

)2

. (158)

Since λ −→ ∞, both integrals at the right-hand side are 1-D stationary phase integrals as1221

seen in sec. A.1. The only stationary point in the vi-integration domain is at vi = 0; the phase1222

term in the ui-integral is likewise stationary at ui = 0 (provided that vi ≈ 0). Application of1223

eq. (151) to the ui-integral in (158) then gives1224

∫ +ε

−εdui e

iλ2ψxxi

(ui+

ψxyiψxxi

vi

)2

= e±iπ4

√2π

λ|ψxxi|, (159)

where we have assumed vi ≈ 0 since this integral is to be evaluated near the stationary point1225

of the vi-integral, and we have implicitly multiplied by 2 since (151) is valid for a single1226

stationary point located at one of the integration limits. Similarly,1227

∫ +δ

−δdvi e

iλ2v2i (ψyyi−

ψ2xyi

ψxxi)

= e±iπ4

√√√√ 2π

λ|ψyyi −ψ2xyi

ψxxi|. (160)

Substituting eqs. (159) and (160) into (158), we are left with the final formula1228

I(λ) ≈ 2π

λ

∑i

f(xi, yi)ei[λψ(xi,yi)±π4±

π4 ]

√√√√ 1

|ψxxi||ψyyi −ψ2xyi

ψxxi|, (161)

valid for λ −→ ∞ and in the assumption that ψxxi 6= 0, ψyyi −ψ2xyi

ψxxi6= 0. The signs of the1229

π/4 terms in (161) depend on those of ψxxi and ψyyi − ψ2xyi/ψxxi (see sec. A.1) [Bender and1230

Orszag , 1978].1231

B Fourier transform convention1232

Inverse and forward Fourier transformations are applied frequently throughout this study.1233

We denote F and F−1 the forward and inverse Fourier transform operators, respectively. F1234

and F−1 can be defined in various ways. We adopt here the following convention: for any1235

function f ,1236

f(ω) = F [f(t)] =1√2π

∫ +∞

−∞dt f(t) e−iωt, (162)

62

and consequently1237

f(t) = F−1[f(ω)] =1√2π

∫ +∞

−∞dω f(ω) eiωt, (163)

where we have chosen to simply specify the argument (time t or frequency ω, respectively)1238

to distinguish a time-domain function f(t) from its Fourier transform f(ω).1239

Some properties of the Fourier transform are particularly useful in the context of ambient-1240

noise cross-correlation. First of all, it follows from (162) that the Fourier transform of the1241

derivative of f with respect to t is1242

F

[df(t)

dt

]= iωf(ω). (164)

The Fourier transform of the integral of f is1243

F

[∫ t

−∞f(τ)dτ

]= − i

ωf(ω), (165)

provided that f(t)−→0 when t−→−∞.1244

Other useful properties of the Fourier transform concern even and odd functions. The1245

Green’s functions G(t) we work with throughout this study (App. E) are real (in the time1246

domain) and have the property G(t) = 0 if t < 0. Let us define the real, even function1247

Ge(t) =1

2G(t) +

1

2G(−t), (166)

and the real, odd function1248

Go(t) =1

2G(t)− 1

2G(−t). (167)

The definitions (166) and (167) imply1249

G(t) = Ge(t) +Go(t). (168)

It follows from (162) that the Fourier transform of a real even function is even and purely1250

real, while the Fourier transform of a real odd function is odd and purely imaginary. Then,1251

< [G(ω)] = Ge(ω) (169)

and1252

= [G(ω)] = −iGo(ω), (170)

63

where < [G(ω)] and = [G(ω)] denote the real and imaginary part of G(ω), respectively.1253

C Bessel functions1254

Bessel functions emerge frequently in noise literature, starting with the early works of, e.g.,1255

Eckart [1953], Aki [1957], Cox [1973]. Their mathematical properties are described in detail1256

by Abramowitz and Stegun [1964]. In sesimic interferometry we are in practice only interested1257

in 0th-order Bessel functions of the first and second kind, denoted J0 (•), Y0 (•), respectively,1258

which together with the Hankel functions H(1)0 (•) = J0 (•) + iY0 (•) and H

(2)0 (•) = J0 (•)−1259

iY0 (•) can be defined as the solutions of the 0th-order Bessel equation1260

d2f(x)

dx2+

1

x

df(x)

dx+ f(x) = 0 (171)

[Abramowitz and Stegun, 1964, eq. (9.1.1)].1261

In our implementation we employ the far-field (large r) and/or high-frequency (large ω)1262

approximations for J0 (x), Y0 (x) and H(1)0 (x), H

(2)0 (x), namely1263

J0 (x) ≈√

2

πxcos(x− π

4

), (172)

1264

Y0 (x) ≈√

2

πxsin(x− π

4

), (173)

1265

H(1)0 (x) ≈

√2

πxei(x−π/4), (174)

1266

H(2)0 (x) ≈

√2

πxe−i(x−π/4) (175)

[Abramowitz and Stegun, 1964, eqs. (9.2.1)-(9.2.4)]. In practice, J0 and Y0 can roughly be1267

thought of as “damped” sinusoidal functions, whose amplitude decays exponentially as their1268

argument tends to infinity; at large x, J0 approximates a cosine with a π/4 phase shift, and1269

Y0 approximates a sine with a π/4 phase shift.1270

D Cancellation of “cross terms”1271

Throughout this study, simple analytical formulae for the cross-correlation of an ambient1272

wave field are obtained neglecting the contribution of the so-called “cross-terms”, i.e. the1273

receiver-receiver cross-correlation of signal generated by a couple of different sources. We1274

64

propose in the following a simple proof, similar to that of Weemstra et al. [2014], of the1275

validity of this assumption.1276

The pressure due to an impulse emitted by source j at a random time tj is given by1277

equation (200), after replacing t−x/c with t− tj −x/c as argument of the Dirac function; in1278

the frequency domain, this is equivalent to adding a phase anomaly −iωtj to the argument of1279

the exponential in eq. (201). If NS such sources are active during the time interval over which1280

a cross-correlation is conducted, the pressure pi recorded at receiver i is a linear combination1281

of the signals originating from these sources, i.e., in 3-D,1282

pi(ω) =

NS∑j=1

e−i(ωrijc

+ωtj

)√

2π4πcrij, (176)

where rij is the distance between receiver i and source j.1283

Cross-correlating the signal recorded at R1 with that recorded at R2, we find1284

p1(ω)p∗2(ω) =

NS∑j=1

NS∑k=1

e−iωc

(r1j−r2k)

2π(4πc)2r1jr2ke−iω(tj−tk), (177)

and the cross-term contribution can be isolated by separating j 6=k terms (cross terms) from1285

j=k ones:1286

p1(ω)p∗2(ω) =

NS∑j=1

e−iωc

(r1j−r2j)

r1jr2j+

NS∑j=1

∑j 6=k

e−iωc

(r1j−r2k)

r1jr2ke−iω(tj−tk). (178)

The first term at the right-hand side of eq. (178) is a sum of single-source cross-correlations1287

in 3-D (eq. (6)). (The algebra is similar in 2-D (eq. (7)), if the far field approximation is1288

applied.) Its average is given e.g. by equations eqs. (8)-(10) above, which are obtained from1289

(6) or (7) by neglecting the cross terms j 6=k and replacing the sum over sources j with an1290

integral over the area or volume occupied by the sources.1291

While j 6=k terms in (178) are non-negligible, we next show that their contribution to the1292

average of (178) over a large set of sources is negligible. We make the assumptions that (i)1293

at each realization, the values tj change randomly, and (ii) they are uniformly distributed1294

between 0 and Ta, i.e., 0 < tj < Ta, where Ta = 2π/ω. Let us introduce the phase φj = ωtj ,1295

which is randomly distributed between 0 and 2π. The exponent −iω(tj − tk) is accordingly1296

replaced by −i(φj−φk). In the process of averaging, impulses will be generated multiple times1297

at each source location j, resulting in random phases φj . This is equivalent to requiring, as it1298

65

is usually done in the literature, that noise sources be spatially and temporally uncorrelated1299

[e.g., Snieder , 2004; Roux et al., 2005]. To take into account the effects of random variations1300

in phase, the j 6=k term must consequently involve (besides the usual sum or integral over1301

sources) an integral over all possible values (0 to 2π) of each source phase,1302

Av

NS∑j=1

∑j 6=k

eiωc

(r1j−r2k)

2π r1jr2kei(φj−φk)

=1

(2π)NS

∫ ∫...

∫ 2π

0

NS∑j=1

∑j 6=k

eiωc

(r1j−r2k)

2π r1jr2kei(φj−φk)dφ1dφ2...dφNS .

(179)

All Integrals evaluate to zero because the integrands traverse a circle in the complex plane1303

from 0 to 2π. Consequently, if cross-correlations are averaged over sufficient realizations, the1304

cross-correlation in eq. (178) reduces to a sum of single-source cross-correlations (non-cross-1305

terms).1306

The averaging procedure we just described is often referred to as “ensemble-averaging”1307

in ambient-noise theory. This expression is borrowed from statistical mechanics: an “ensem-1308

ble” is a set of states of a system, each described by the same set of microscopic forces and1309

sharing some common macroscopic property. The ensemble concept then states that macro-1310

scopic observables can be calculated by performing averages over the states in the ensemble1311

[Tuckerman, 1987]. In our case, one state consists of the same acoustic or elastic medium (its1312

response to a given impulse is always the same) being illuminated by one or more randomly1313

located sources with random phases: our ensemble average is the average over all possible1314

combination of source locations and phases. Cross-correlating recordings associated with a1315

single state (a unique combination of sources) yields many cross-terms; we have just shown,1316

however, that ensemble-averaging cross-correlations over different states implies that these1317

cross-terms stack incoherently and hence become negligible. In the real world, the location1318

and phase of noise sources (e.g., ocean microseisms) are not well known; the assumption is1319

made that, over time, a sufficiently diverse range of sources is sampled, so that averaging1320

the cross-correlation over time (i.e., computing the cross-correlation over a very long time1321

window) is practically equivalent to ensemble-averaging.1322

66

E Green’s functions of the scalar wave equation (homoge-1323

neous lossless media)1324

E.1 Green’s problem as homogeneous equation1325

Following e.g. Roux et al. [2005], Sanchez-Sesma and Campillo [2006], Harmon et al. [2008],1326

we call Green’s function G = G(x,xS , t) the solution of1327

∇2G− 1

c2

∂2G

∂t2= 0 (180)

with initial conditions1328

G(x,xS , 0) = 0, (181)1329

∂G

∂t(x,xS , 0) = δ(x− xS), (182)

i.e. an impulsive source at xS . We are only interested in causal Green’s functions, satisfying1330

the radiation condition, i.e. vanishing at t < 0.1331

Once G is known, it can be used to solve rapidly more general initial-value problems1332

associated with (180). Consider for example1333

∇2f − 1

c2

∂2f

∂t2= 0 (183)

with the more general initial conditions1334

f(x, 0) = 0, (184)

1335

∂f

∂t(x, 0) = h(x). (185)

It can be proved by direct substitution that if G solves (180)-(182) then1336

f(x, t) =

∫Rd

ddx′G(x,x′, t)h(x′) (186)

solves (183)-(185). d in eq. (186) denotes the number of dimensions: 2 or 3 in our case.1337

For the sake of simplicity, we shall set xS=0 in the following. G(x,xS , t) can be recovered1338

from G(x, 0, t) by translation of the reference frame.1339

67

E.2 Solution in the spatial-Fourier-transform domain1340

Equation (180) can be solved via a spatial Fourier transform, i.e. by the Ansatz1341

G(x, t) =1

(2π)d

∫Rd

ddkG(k, t)eik·x, (187)

which, substituted into (180), gives1342

1

c2

∂2G

∂t2(k, t) + k2G(k, t) = 0, (188)

with k = |k|. Eq. (188) is solved by1343

G(k, t) = A(k) cos(kct) +B(k) sin(kct), (189)

where A(k) and B(k) must be determined by the initial conditions. Transforming (181) and1344

(182) to k-space, with1345

δ(x) =1

(2π)d

∫Rd

ddk eik·x, (190)

and replacing G in the resulting equations with its expression (189), we find A(k) = 0,1346

B(k) = 1kc , and1347

G(k, t) =1

kcsin(kct). (191)

Eq. (187) can now be used to determine G(x, t) from its spatial Fourier transform G(k, t),1348

and the result differs importantly depending on d.1349

E.3 Inverse transform of the solution to 2-D space1350

Substituting (191) into (187) in the d = 2 case,1351

G2D(x, t) =1

4π2

∫ +∞

−∞dk1

∫ +∞

−∞dk2

sin(kct)

kceik·x. (192)

We call ξ the angle between k and x. It follows that k · x = kx cos ξ, with x = |x|. For each1352

x the integration can be conducted over k and ξ, using dk1dk2=kdkdξ,1353

G2D(x, t) =1

4π2c

∫ +∞

0dk sin(kct)

∫ 2π

0dξ eikx cos ξ

=1

2π2c

∫ +∞

0dk sin(kct)

∫ π

0dξ cos(kx cos ξ)

=1

2πc

∫ +∞

0dk sin(kct)J0(kx),

(193)

68

where we have used the symmetry properties of sine and cosine, and the integral form of the1354

0th-order Bessel function of the first kind J0 [Abramowitz and Stegun, 1964, eq. (9.1.18), and1355

appendix C here]. The remaining integral in (193) is solved via eq. (11.4.38) of Abramowitz1356

and Stegun [1964], resulting in1357

G2D(x, t) =1

2πc2

H(t− x

c

)√t2 − x2

c2

(194)

[Sanchez-Sesma and Campillo, 2006; Harmon et al., 2008], where H denotes the Heaviside1358

function. Eq. (194), as well as other time-domain formulae for the Green’s function, is1359

only physically meaningful for t > 0. Dimensional analysis of (194) shows that G2D in this1360

formulation has units of time over squared distance. The Fourier transform of G2D(x, t) is1361

inferred from eq. (9.1.24) of Abramowitz and Stegun [1964], after applying our definition1362

(162) to eq. (194):1363

G2D(x, ω) =1√

8π3c2

∫ ∞xc

dte−iωt√t2 − x2

c2

=1√

8π3c2

∫ ∞1

dt′e−iωx

ct′

√t′2 − 1

= − 1

4√

2πc2

[Y0

(ωxc

)+ iJ0

(ωxc

)]=

1

4i√

2πc2H

(2)0

(ωxc

)(195)

[Sanchez-Sesma and Campillo, 2006; Harmon et al., 2008], where Y0 and H(2)0 denote the1364

0th-order Bessel function of the second kind and the 0th-order Hankel function of the second1365

kind [Abramowitz and Stegun, 1964, and appendix C here]. Based upon the far-field/high-1366

frequency asymptotic form (175) of H(2)0 , we can also write1367

G2D(x, ω) ≈ 1

4iπc3/2

e−i(ωxc −π4 )

√ωx

, (196)

which is analogous to eq. (14) of Snieder [2004].1368

Eq. (9.1.24) of Abramowitz and Stegun [1964] is only valid for positive frequency ω > 0,1369

and so is, as a consequence, our relation (195). We know, however, that G2D(x, t) is a real-1370

valued function: the relationship G2D(x, ω) = G∗2D(x,−ω) then holds, and allows us to define1371

G2D in the entire frequency domain.1372

69

E.4 Inverse transform of the solution to 3-D space1373

We now substitute (191) into the 3-D version of (187), and find1374

G3D(x, t) =1

8π3

∫ +∞

−∞dk1

∫ +∞

−∞dk2

∫ +∞

−∞dk3

sin(kct)

kceik·x. (197)

The integral in (197) is simplified by switching from k1, k2, k3 to spherical coordinates k, ξ, χ,1375

with the ξ=0-direction coinciding with that of x. Then dk1dk2dk3=−k2dkdχd(cos ξ), k ·1376

x=kx cos ξ, and1377

G3D(x, t) = − 1

8π3

∫ +∞

0dk

k sin(kct)

c

∫ 2π

0dχ

∫ −1

1d(cos ξ) eikx cos ξ

=1

2π2cx

∫ +∞

0dk sin(kct) sin(kx).

(198)

The k-integral in (198) is solved via the equality sinα sinβ = 12 [cos(α− β)− cos(α+ β)],1378

which gives1379

G3D(x, t) =1

4π2cx

∫ +∞

0dk cos[k(ct− x)]− cos[k(ct+ x)]

=1

4π2cxlimz→∞

[sin[k(ct− x)]

ct− x

]k=z

k=0

−[

sin[k(ct+ x)]

ct+ x

]k=z

k=0

=1

4πcx

[δ(t− x

c

)− δ

(t+

x

c

)],

(199)

where we have used the property of the Dirac δ function that δ(x) = 1π limz→∞

sin(zx)x [e.g.,1380

Weisstein, 1999-2013]. In our formulation both t and x are positive, so that δ(t+ x

c

)= 01381

and we are left with1382

G3D(x, t) =1

4πc

δ(t− x

c

)x

(200)

[e.g., Aki and Richards, 2002, chapter 4]. Notice that, since the dimension of the Dirac δ is1383

the inverse of that of its argument, that of G3D is one over squared distance. According to1384

(162), F [δ(t− t0)] = 1√2π

e−iωt0 , and consequently1385

G3D(x, ω) =1√2π

1

4πc

e−iωxc

x. (201)

70

E.5 Green’s problem as inhomogeneous equation1386

The Green’s problem is also often written as eq. (1) plus an impulsive forcing term, i.e.1387

∇2g − 1

c2

∂2g

∂t2= δ(x)δ(t), (202)

with initial conditions1388

g(x, 0) = 0, (203)1389

∂g

∂t(x, 0) = 0. (204)

The solution g to this problem is related to the solution G of (180)-(182); namely1390

g(x, t) =

∫ t

0dsG(x, t− s). (205)

We demonstrate that (205) solves (202)-(204) via Duhamel’s principle [e.g. Hildebrand ,1391

1976; Strauss, 2008]: let us consider the more general case1392

∇2u− 1

c2

∂2u

∂t2= h(x, t), (206)

1393

u(x, 0) = 0, (207)1394

∂u

∂t(x, 0) = 0, (208)

with h an arbitrary forcing term. Suppose that a solution v(x, t) to the following homogeneous1395

problem, closely related to (206)-(208), can be found:1396

∇2v − 1

c2

∂2v

∂t2= 0, (209)

1397

v(x, 0; s) = 0, (210)1398

∂v

∂t(x, 0; s) = h(x, s). (211)

(In practice, a solution of eq. (209) that satisfies the initial condition (211) must be deter-1399

mined for any possible value of the parameter s in (211); s replaces t in the expression of h1400

first encountered in (206).) Then, the following relation between u and v holds:1401

u(x, t) =

∫ t

0ds v(x, t− s; s). (212)

71

One can verify that (212) solves (209)-(211) by direct substitution, applying Leibniz’s rule1402

for differentiating under the integral sign. Equation (205) is a particular case of (212).1403

Result (212) can also be combined with (186) to write the solution of the general inho-1404

mogeneous problem (206) in terms of the Green’s function G (e.g., G2D or G3D). Replacing1405

v in eq. (212) with expression (186),1406

u(x, t) =

∫ t

0ds

∫Rd

ddx′G(x,x′, t− s; s)h(x′, s) (d = 2, 3). (213)

1407

72