Attenuation, transport and diffusion of scalar waves in textured random media

2006) 229–244www.elsevier.com/locate/tecto

Tectonophysics 416 (

Attenuation, transport and diffusion of scalar waves intextured random media

L. Margerin ⁎

Laboratoire de Géophysique Interne et Tectonophysique, Université Joseph Fourier/CNRS, BP 53, 38041, Grenoble, France

Accepted 28 November 2005Available online 13 February 2006

Abstract

Most theoretical investigations of seismic wave scattering rely on the assumption that the underlying medium is statisticallyisotropic. However, deep seismic soundings of the crust as well as geological observations often reveal the existence of elongatedor preferentially oriented scattering structures. In this paper, we develop mean field and radiative transfer theories to describe theattenuation and multiple scattering of a scalar wavefield in an anisotropic random medium. The scattering mean free path is foundto depend strongly on the propagation direction. We derive a radiative transfer equation for statistically anisotropic random mediafrom the Bethe–Salpeter formalism and propose a Monte Carlo method to solve this equation numerically. At longer times, theenergy density is shown to obey a tensorial diffusion equation. The components of the diffusion tensor are obtained in closed formand excellent agreement is found between Monte Carlo simulations and analytical solutions of the diffusion equation. The theoryhas important potential implications for lithospheric models where scatterers are for example flat structures preferentially alignedalong the surface. In this simple geometry, analytical expressions of the Coda Q parameter can be given explicitly in the diffusiveregime. Our results suggest that pulse broadening and coda decay are controlled by different parameters, related to the eigenvaluesof the diffusion tensor. These eigenvalues can differ by more than one order of magnitude. This theory could be applied to probethe anisotropy of length scales in the lithosphere.© 2005 Elsevier B.V. All rights reserved.

Keywords: Multiple scattering; Radiative transfer; Random media; Anisotropy; Coda

1. Introduction

The propagation of seismic waves in heterogeneousmedia is a topic of continued interest for seismologists.Among the many approaches of the subject, stochastictheories have undergone a vigorous development in thelast 20 years. The traditional range of applicationincludes the modeling of the amplitude and phase ofcoherent arrivals, as well as the transport of the scattered

⁎ Tel.: +33 4 76 82 80 25; fax: +33 4 76 82 81 01.E-mail address: [email protected].

0040-1951/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.tecto.2005.11.011

energy. Important advances in the understanding ofdirect seismic wave attenuation due to scattering havebeen made by Sato (1982), Wu (1982), Shapiro andKneib (1993), who developed new theories with specificapplication to seismic experiments. As an example, thetravel-time corrected mean wave formalism of Sato(1982) has nicely reconciled known discrepanciesbetween seismic measurements of attenuation andstandard mean field theories. The study of seismicwave travel times in random media has also led tounexpected results. It has for example been shown thatthe first arrivals can propagate at velocities that are

230 L. Margerin / Tectonophysics 416 (2006) 229–244

significantly higher than the volume-averaged velocity.This phenomenon has been termed ‘velocity shift’ andnumerous extensive studies can be found in theliterature (see, e.g. Roth et al., 1993; Shapiro et al.,1996).

Beyond a length scale known as the mean free path alarge part of the energy of the coherent waves has beentransferred to scattered waves which eventually form thecoda of short period seismic events. The scattered wavestransport the energy at distances larger than the meanfree path and are responsible for the rapid fieldfluctuations, which are averaged out in the mean fieldapproach. Thus, the information pertaining to the energytransport in a scattering medium is contained in thesecond statistical moment of the field. Field–fieldcorrelation theory is an extremely powerful tool thatestablishes the link between wave and radiative transferequations and provides a theoretical background toexplain many striking wave phenomena, such aselectron's localization in condensed matter (Vollhardtand Wölfle, 1982), or coherent backscattering of opticalwaves (Akkermans et al., 1988). Correlation theory ofrandom fields has also known some remarkableachievements in seismology with the development ofcoda wave interferometry (Snieder et al., 2002) or theGreen function reconstruction from coda waves (Cam-pillo and Paul, 2003). These methods make use of thephase information contained in multiply scattered wavesto gain information about the medium. Such approachesgo beyond the scope of this paper where attention willmostly be given to the energy density.

In seismology, the acoustic transport equation hasbeen introduced by Wu (1985). Later, the theory hasbeen extended to time-dependent cases by Zeng et al.(1991), Sato (1995), and to elastic waves includingmode conversions by Zeng (1993) and Sato (1994).Data analysis tools such as the multiple-lapse-time-window analysis have been developed and applied innumerous regions of the world (e.g. Fehler et al., 1992;Hoshiba et al., 2001). Recent years have seen theemergence of even more realistic and challengingmodelings including the possible depth dependence ofbackground velocities and scattering properties (Mar-gerin et al., 1998; Hoshiba et al., 2001; Lacombe et al.,2003), the interpretation of short period scattered wavesat the global scale (Margerin and Nolet, 2003; Shearerand Earle, 2004) and the scattering of long periodRayleigh waves (Sato and Nishino, 2002). In most ofthese works, radiative transfer is introduced as aphenomenological theory, leaving a gap with theunderlying wave equation. However, in condensedmatter physics, radiative transfer emerges as a rigorous

consequence of correlation theory, and this is theapproach that will be adopted throughout the paper.

Although anisotropic media play a fundamental rolein explaining many seismic observations such asanomalous splitting of core-sensitive normal modes orshear wave birefringence, most studies on multiplescattering of seismic waves assume implicitly that theunderlying medium is statistically isotropic (see how-ever the works of Iooss, 1998; Samuelides and Mukerji,1998; Müller and Shapiro, 2003; Kravtsov et al., 2003).Yet, deep seismic soundings and geological maps oftenreveal elongated or laminated structures (see, e.g.Thybo, 2002). Thus, a scattering theory of anisotropicrandom media would be a useful tool to quantify thepotential impact of textured geological structures on thewavefield. Important efforts have been invested in thestudy of wave propagation through anisotropic randommedia, mostly in optics and condensed matter physics.Furutsu (1980) proposed a phenomenological acousticradiative transfer equation with broken rotationalsymmetry, and developed a diffusion approximationbased on a diagonalization of the collision operator byperturbation theory. Anisotropic random media havealso been studied by Wölfle and Bhatt (1984), inconnection with the problem of electron localization. Astationary electromagnetic transport equation has beenderived by Mischenko (2002) for arbitrarily shapeddiscrete scatterers using the Lax–Tversky's theory. Aradiative transfer theory has also been proposed forelectromagnetic waves propagating through nematicliquid crystals (van Tiggelen et al., 1996; Stark andLubensky, 1997), and some experiments have beenconducted (Wiersma et al., 2000; Johnson et al., 2002)that reveal an anisotropic transport of light.

The primary goal of the present work will be toinvestigate through a simple model, where modeconversions are neglected, how preferentially orientedand shaped inhomogeneities influence the transport ofenergy. We will employ the Dyson and Bethe–Salpeterformalism developed in condensed matter physics toquantify the attenuation of the mean field and derivefrom first principles a radiative transfer equation forscalar waves in anisotropic random media. We shall usethe term ‘anisomeric’ random media since this refers toan isotropy of scale lengths. At longer times, the diffuseintensity will be shown to obey a tensorial diffusionequation that can be solved analytically. We will presentnumerical solutions of the anisotropic transfer equationand comparisons with diffusion theory. Finally, we willapply the diffusion theory to an anisomeric waveguide(for example the earth's crust) to show that broadeningof seismogram envelopes and Coda Q may be used to

231L. Margerin / Tectonophysics 416 (2006) 229–244

infer the anisotropy of scale lengths in the medium. Ananalytical expression of Coda Q including the leakageeffect (Margerin et al., 1998, 1999; Wegler, 2004) willalso be derived.

2. Presentation of the model

2.1. Examples of anisotropic random media

Anisotropy is a phenomenon that has several possibleorigins. Crystals such as olivine possess the mostfundamental form of anisotropy, which is related to thespatial arrangements of atoms. Yet, a piece of rockwhich is made of randomly oriented olivine grains willbehave as an isotropic elastic body. Hence, anisotropyrequires a preferential alignment of the grains in thewhole sample. But anisotropy can also be caused by thepresence of laminated isotropic structures as is well-known in the theory of 1-D randomly layered media(e.g. Shapiro and Hubral, 1998). The anisotropy that weconsider in this paper is of statistical nature and is oftenreferred to as anisomery (Rytov et al., 1989). Weimagine that the medium contains scatterers that looklike lentils or cigars that are preferentially aligned. Atthe microscopic scale, described by the wave equation,the medium is perfectly isotropic. But at the macro-scopic scale, that is beyond the mean free path, therandom medium will display anisotropic attenuation andtransport properties.

2.2. Quantitative description of statistical anisotropy

We consider scalar waves propagating in a mediumwhere the velocity fluctuates around a mean value c0. Ina given realization of the statistical ensemble of randommedia, the field u at frequency ω satisfies the followingHelmoltz equation:

Duþ k20ð1þ eðrÞÞu ¼ 0; ð1Þ

where possible source terms have been omitted. Thefield describes fluctuations in inverse slowness squared.In most cases, one assumes that the power spectrum Фof the fluctuations is a function of the norm of thewavenumber k only, which is equivalent to saying thatthe field is isotropic. For example, the power spectrumof Gaussian random media can be expressed as:

UðkÞ ¼ ð2kÞ3=2a3he2ie−k2a2=2; ð2Þ

where ⟨ϵ2⟩ is the total variance of the squared slownessfluctuations and a is the correlation length. This

spectrum can be seen as a special case of the moregeneral expression:

UðkÞ ¼ ð2kÞ3=2abche2ie−12ðk2x a2þk2y b

2þk2z c2Þ; ð3Þ

where different correlation lengths have been introducedalong the axis (x, ŷ, z) of a Cartesian coordinate system.Eq. (3) can be rewritten as:

UðkÞ ¼ ð2kÞ3=2a3he2ie−k2lðkÞ2=2; ð4Þwhere l can be understood as a correlation length thatdepends on the space direction k. The hat symbol willbe used throughout the paper to denote a unitary vector.Depending on the choice of a, b, c, the scatterers takedifferent forms. Let us clarify this on a simple examplewhere we introduce cylindrical symmetry by lettinga=b. Although it is by no means a restriction of thetheory, in numerical applications, this cylindricalsymmetry will be adopted in the rest of the paper.When c≫a, the inhomogeneities look like elongatedcigars, while in the opposite case they look likesquashed-flat objects. This interpretation is valid in astatistical sense only and it has to be emphasized thatmedia with very different spatial distributions ofheterogeneity can have exactly the same correlationfunction. The procedure that has been adopted abovecan be applied to other correlation functions. As is well-known the power spectrum of isotropic exponentialmedia reads:

UðkÞ ¼ 8kabche2ið1þ k2a2Þ2 ; ð5Þ

and can be generalized to the anisotropic case asfollows:

UðkÞ ¼ 8kabche2ið1þ k2x a

2 þ k2y b2 þ k2z c

2Þ2 : ð6Þ

For the moment, we have only considered thespectral representation of inhomogeneities. Yet, anothercustomary method of description is the correlationfunction, i.e., the inverse Fourier transform of the powerspectrum. For Gaussian media, the result is trivialbecause of the well-known properties of the Fouriertransform. As regards the power spectrum (6), thecorrelation function can be expressed as:

CðrÞ ¼ ache2ie−r=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2sin2hþc2cos2h

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2cos2hþ c2sin2hÞða2sin2hþ c2cos2hÞ

q ;

ð7Þ


where the additional symmetry a=b has been intro-duced for simplicity, and θ denotes the angle betweenthe position vector, r, and the axis of symmetry of thesystem, z. Formula (7) shows that the correlation in realspace is exponential with a correlation length thatdepends on the space direction. This agrees with theterminology “exponential power spectrum” used above,and justifies the intuitive procedure used to introducestatistical anisotropy. For a more general introduction tocorrelation functions in seismology, we refer to Klimeš(2002).

3. Mean free path and phase velocity

We now introduce the multiple scattering formalismin order to determine the phase velocity and spatialdecay rate of a coherent plane wave propagating indirection p, in an anisomeric random medium. Thiscoherent wave is defined as the ensemble averageresponse of the medium. Like in an isotropic medium,one expects both an attenuation of the coherent wavedue to scattering, and a slight shift of the propagationvelocity. At a given frequency ω the coherent fieldexcited by a point sourceG – the mean Green function –obeys the following Dyson equation in wavenumberspace (Economou, 1990):

Gðx; pÞ ¼ ð2kÞ3k20−p2−ð2kÞ−3Rðx; pÞ

; ð8Þ

where k0 is the wavenumber in the reference medium,and Σ is a new object known as the self-energy or massoperator. Σ can be considered as a unit-cell from whichall the possible scattering paths in the random mediumcan be constructed. Eq. (8) is exact but cannot be solvedunless an approximate form of Σ is introduced. Tolowest order of the perturbations ⟨ϵ2⟩, the mass operatoris given by (Frisch, 1968):

Rðx; pÞ ¼ k40ð2kÞ3

Zℝ3

G0ðx; qÞUðp−qÞd3q: ð9Þ

Eq. (9) is known as the Bourret or first-ordersmoothing approximation. Physically, it implies thatthe waves can visit an arbitrary number of inhomoge-neities, provided they are different. Thus, recurrentscattering is not taken into account, which is accurate forsufficiently small perturbations. In Eq. (9), G0 denotesthe Green function of the reference medium:

G0ðx; pÞ ¼ ð2kÞ3k20−p2 þ ig

; ð10Þ

where η is a positive and infinitesimal imaginary partwhich ensures that the retarded solution is selected. Asusual, the location of the poles of the mean Greenfunction provides the dispersion relation in the randommedium (Sheng, 1995). This is still a complicated taskbecause the poles are solutions of an integral equation. Itis customary to further simplify the problem byassuming that the wavenumber inside any inhomoge-neity differs very little from k0, which is also known asBorn approximation (Weaver, 1990). One then obtainsthe following explicit expression of the effectivewavenumber ke in the random medium:

keðx; pÞ ¼ k0−1

2ð2kÞ3k0RfRðx; k0pÞg

−i1

2ð2kÞ3k0IfRðx; k0pÞg; ð11Þ

where symbolsR, I denote the real and imaginary part,respectively. The product of the inverse real part of theeffective wavenumber with the frequency defines thephase velocity v. Eq. (11) also shows that any planewave propagating in the random medium has a finiteimaginary part. As is well-known, this has nothing to dowith absorption but reflects the amplitude decay of thecoherent wave caused by the random scattering events.In scattering theory, 1=ð2IfkegÞ is defined as the meanfree path l, which is the length scale beyond which asignificant amount of energy has been scattered awayfrom the incident direction. At this point it is useful tointroduce the following functional limit (η→0+):

G0ðx; pÞ ¼ ð2kÞ3 P:V:1

k20−p2−ikdðk20−p2Þ

� �; ð12Þ

which enables to derive simple expressions for v and l:

xvðpÞ ¼ k0 þ k30

16k3P:V:

Zℝ3

1

q2−k20Uðk0p−qÞd3q; ð13Þ

1

lðpÞ ¼k40

16k2

Z4kUðk0p−k0qÞd2q: ð14Þ

In Eqs. (12) and (13), the symbol P.V. denotes theCauchy principal value, and the formula assumes thatthe parameter ⟨ϵ2⟩ of the power spectrum Ф representsthe total variance of the squared-slowness (which is 4times larger than the variance of velocity). These resultsput forward the angular dependence of both ν and l. TheCauchy principal value can be evaluated numerically byregularizing the integrand, a procedure which isfrequently encountered in the literature in connectionwith Kramers–Krönig relations (see, e.g. Byron andFuller, 1992, p. 346). The angular integration in Eq. (14)

Fig. 2. Top: angular dependence of the mean free path in anexponential anisotropic medium with a=0.5 and c=1, 2, 3, 4, 5 andrms velocity fluctuations 5%. For each curve, the maximum of themean free path has been normalized to 1. Bottom: angular dependenceof the perturbation of phase velocity (same medium). The correlationlengths are indicated next to each curve. The angle θ between thewavevector and the symmetry axis varies between 0° and 90° onlybecause the medium is invariant by reflection with respect to a planeperpendicular to this axis.


poses no special problem. Eqs. (13) and (14) are verygeneral and will now be applied to Gaussian andexponential media possessing a rotational symmetryaxis (a=b in Eqs. (6) and (3)), as depicted in Fig. 1. Inthis geometry, l and ν only depend on the polar angle θbetween the wavevector and the axis of symmetry. Thiscase is very relevant to seismological applications and isrelatively easier to investigate.

In Fig. 2, we illustrate the angular dependence of themean free path and phase velocity in an exponentialmedium with 5% rms velocity fluctuations, fixedcorrelation length a=0.5 and varying correlation lengthc=1, 2, 3, 4, 5. The wavenumber in the referencemedium is normalized to one. With increasing c, thescatterers tend to look more and more like elongatedcigars. Broader range of parameters have been investi-gated but it was found that the effects of anisomerybecome important only in the regime max(k0a, k0c)≥1 /2, that is when the wavelength and the size of the objectsbecome comparable. The upper bound of the correlationlengths is imposed by the condition of validity of theBourret approximation max(k20a

2⟨ϵ2⟩, k20c2⟨ϵ2⟩)≪1

(Rytov et al., 1989). Physically this means that awave propagating through a scatterer acquires anegligible phase delay. In Fig. 2, it is noticeable thatthe difference between the phase velocity in thereference medium and in the random medium neverexceeds 0.5%. Thus, the slowness surface can hardlybe distinguished from a spherical surface with radius1=v ¼ 1=4k

R4k 1=vðpÞd2p. In certain directions the

phase velocity can exceed the reference value c0,which is never the case in an isotropic medium (Rytovet al., 1989). This does not constitute a violation ofcausality because our approximation automaticallyverifies the Kramers–Krönig relations. It is also tobe noted that the phase velocity averaged over allpropagation directions is always smaller than c0. These

Fig. 1. Schematic view of the geometry of the random medium. Therandom medium is assumed to have rotational symmetry with respectto a fixed axis. The correlation length equals c along the symmetryaxis, and a along any perpendicular direction.

features are further illustrated in Fig. 3, which issimilar to Fig. 2 but with a=4.5 and c=1, 2, 3, 4, 5.With decreasing c, the scatterers tend to look more andmore squashed flat. In Fig. 4, we also show the resultsof calculations with a=2.5 and c=1, 2, 3, 4, 5. Ourcalculations imply that the anisotropy of scale lengthscannot explain the strong velocity anisotropy some-times observed in the Earth. Even by boosting theperturbations up to 10%, the deviation from isotropywould still be less than 1%, but the theory may breakdown for such large perturbations. Finally, wecomment on the limits of validity of the theory forthe case of extremely flat scatterers (c→0, a→∞, seeFig. 1). In that case, the medium can be considered as




locally layered. For such finely layered media,transport theory is known to break down because ofinterference effects between multiply reflected/trans-mitted waves, and localization theory should then beapplied (Shapiro and Hubral, 1998).

It is noticeable that the scattering mean free pathdepends very strongly on the propagation direction. Thisdependence is rich and complex and it is apparent thatthe extrema of the mean free path do not necessarilycoincide with the principal axes of the medium. In arandom medium composed of very flat objects withanisotropy parameter c /a=9, (see Fig. 1) the ratiobetween the max and min mean free path can be as largeas 2, which would certainly be detectable if the mean

field could accurately be measured. Note that in mostcases, seismologists do not have access to the mean fieldbut rather to its travel-time corrected version. In thiscase, the measured attenuation length can differsignificantly from the mean free path as has beenexplained by Sato (1982) and Wu (1982). These authorsalso developed theoretical approaches such as the travel-time corrected mean wave formalism, which are inbetter agreement with the seismological practice. We donot discuss these theories in this paper, but it isanticipated that the attenuation length should stilldepend strongly on the propagation direction.


4. Transport theory

4.1. General definitions and approximations

Beyond one mean free path, most of the energy hasbeen transferred from the coherent wave to diffusescattered waves which will form the coda of the seismicsignal. As will be shown below, the space–timedistribution of energy in the coda is governed by aradiative transfer equation. We present a concisederivation of this equation for an anisomeric randommedium. We first introduce the energy density of thescalar field u for a source at r′ and detection at r as:

Eðt; r; r VÞc 12

��c−10 A

Atuðt; r; r VÞ

��2

þ Ep; ð15Þ

where c0 is the velocity in the reference medium andEp denotes the potential energy. Note that Eq. (15)assumes that the velocity departs only slightly from c0.Following Weaver (1990), we assume that the virialtheorem applies and focus our attention on the firstterm, i.e. the kinetic energy. The field can be decom-posed over different frequency components by Fouriertransformation:

uðt; r; r VÞ ¼ 12k

Zℝuðx; r; r VÞe−ixtdx: ð16Þ

which yields the following expression for the intensity:

EðtÞ¼ 1

ð2kÞ2Z Z

ℝ2

x1x2uðx1Þu1ðx2Þe−ix1tþix2tdx1dx2;

ð17Þwhere the star denotes complex conjugate and constantprefactors have been dropped. For notational simplic-ity, the dependence on the spatial variables r and r′ isomitted. Upon introducing the barycentric variablesω=(ω1+ω2) / 2 and ωd=ω1−ω2, the intensity can inturn be written as a Fourier integral:

EðXÞ¼ 1

ð2kÞ2ZZZ

ℝ3

ðxþ xd=2Þðx−xd=2Þ

� uðxþ xd=2Þu1ðx−xd=2Þe−ixd tþiXtdxdxddt:

ð18ÞIntegration over time yields a 2πδ(Ω−ωd) factor and

further integration over ωd gives:

EðXÞ ¼ 1ð2kÞ

Zℝðxþ X=2Þðx−X=2Þ

� uðxþ X=2Þu1ðx−X=2Þdx: ð19Þ

We now assume that the field u is narrowly band-passed which means that E(Ω) will be non-zero forΩ≪ω only. Because Ω is the conjugate variable ofthe time t, this implies that the intensity evolution isslow compared to the typical oscillation period of thefield. This separation of time scales is known as theslowly varying envelope approximation. In agreementwith signal theory terminology, ω and Ω aresometimes referred to as the carrier (or internal) fre-quency, and the modulation (or external) frequency,respectively. Following Weaver (1990), we invokean ergodic hypothesis and assume that integratingover the internal frequency is equivalent to an en-semble average. Taking into account the band-passednature of u, one finds the following approximateexpression:

EðX; r; r VÞcx2Dx

2kc20huðxþX=2; r; r VÞu1ðx−X=2; r; r VÞi;

ð20Þwhere Δω is the typical bandwidth of the signal, andbrackets denote an ensemble average. From thepreceding argument, it becomes clear that the densityof radiation per unit frequency E ¼ 2pE=Dx can beobtained from the two-point, two-frequency correlationfunction defined as:

huðxþ X=2;R þ r=2Þu1ðx−X=2;R−r=2Þi¼ 1

ð2kÞ6Z Z

ℝ3

Cx;pðX;#Þeiðpdrþ#dRÞd3pd3D; ð21Þ

where r denotes the separation between the twomeasurement points, and R is the position with respectto a point-like source. Since ensemble averagingrestores the translational symmetry, the field–fieldcorrelation function does not depend explicitly on thesource position, which explains the simplified nota-tions (dependence on source position has beendropped). In Eq. (21), Γ is recognized as the spatialFourier transform of the two-point correlation functionof the wavefield. As for temporal variables, we willassume that the average energy distribution is smoothat the wavelength scale. It implies that the energy iscarried in space by wavepackets that have fast internaloscillations modulated by a smooth envelope. Byidentifying p and Δ as the central and modulationwavenumbers of the wavepackets, respectively, werequire that p≪Δ (Sheng, 1995). In the definition ofΓ, the internal variables appear as subscripts, aconvention that will be adopted where possible in thefollowing discussion.


4.2. Radiative transfer approximation of the Bethe–Salpeter equation

The correlation theory of random wavefields showsthat the two-point correlation function of the wavefieldobeys a Bethe–Salpeter equation. This equation hasbeen derived in details in several papers and textbooksand we refer the interested reader to Frisch (1968),Rytov et al. (1989) or van Rossum and Nieuwenhuizen(1999). The Bethe–Salpeter is an exact (perturbative)equation that retains all the possible correlations amongthe various scattering paths in the medium. In theFourier domain, this equation writes (Lagendijk and vanTiggelen, 1996):

−Ak20Ax

Xþ 2pd#þ DRx;pðX;#Þð2kÞ3

" #Cx;pðX;#Þ

¼ ð2kÞ3DGx;pðX;#Þ

� SðxÞð2kÞ6þ

1

ð2kÞ12Zℝ3

Ux;p;sðX;#ÞCx;sðX;#Þd3s�;"

ð22Þwhere the following notations have been introduced:

DGx;pðX;#Þ¼ G xþ X

2; pþ#

2

� �−G1 x−

X2; p−

#

2

� �; ð23Þ

with a similar definition for ΔΣ; S is a source term thatdepends on the energy released by the point source in anarrow frequency band around ω. Eq. (22) introducesthe new object U known as the Vertex function, whichcan be interpreted as a unit cell from which all thepossible multiple scattering paths involving two fieldscan be described. For more details about the definitionand construction of the Vertex function, the reader isreferred to Frisch (1968). As it stands, the Bethe–Salpeter equation is intractable and approximations ofU, ΔΣ and ΔG are required. The conservation of energyis an important constraint that must be fulfilled by theapproximate expressions. In a stationary situation, arelatively simple equation known as Ward identity, orgeneralized optical theorem guarantees the conservationof energy (Lagendijk and van Tiggelen, 1996):

ð2kÞ6DRx;pðX;#Þ¼Zℝ3

DGx;p VðX;#ÞUx;p V;pðX;#Þd3p V;ð24Þ

with Ω=0. When calculating the mean free path, wehave already approximated Σ to lowest order of the

perturbations. The corresponding approximation of theVertex function is provided by Frisch (1968):

Uðp; p V; q; q VÞ ¼ k40ð2kÞ3Up−qþ p V−q V

2

� �: ð25Þ

Eq. (25) is often termed the ‘ladder’ approximation, aterm deriving from the diagrammatic representation ofUoften employed in condensed matter physics. Thisapproximation implies that waves can visit an arbitrarynumber of inhomogeneities, but neglects the occurrenceof recurrent scattering between two (or more) scatterers.It can be shown that the ladder diagrams represent thedominant part of the multiple-scattered wavefield exceptin a sphere of diameter a wavelength around the sourcewhere crossed diagrams representing interference be-tween reciprocal wavepaths contribute an equal amountof energy, giving rise to backscattering enhancement andweak localization (Akkermans et al., 1988; Larose et al.,2004). We nowmake use of the slowly varying envelopeapproximation, and note that for sufficiently small Δ, thetwo terms defining ΔG and ΔΣ nearly become complexconjugates of each other, which yields: DGc−2iIfGg,with a similar approximation for Σ. The final stepconsists in replacing IfGg by IfG0g~−ikdðk2−k20Þ,which is equivalent to neglecting the renormalization ofthe phase velocity in the random medium. This is not asevere approximation since we have shown that for rmsfluctuations of order 5%, the phase velocity differs by atmost 0.5% from the reference value. This is completelyequivalent to the Born approximation that has beenapplied to evaluate the mean free path and phasevelocity, where the slight perturbation of velocity insidethe scatterer was neglected. Actually, it can be seen thatEq. (25) contains as a special case the definition of themean free path. The Bethe–Salpeter equation nowsimplifies to:

−ik0Ak0Ax

Xþ ipd#−IfRðx; pÞg

ð2kÞ3" #

Cx;pðX;#Þ

¼kdðk20−p2Þ SðxÞþk40

ð2kÞ3Zℝ3

Uðp−sÞCx;sðX;#Þd3s�:"

ð26ÞThe presence of the delta function imposes that Γ be

sharply peaked around k0 and therefore suggests asolution of the form:

Cx;pðX;#Þ ¼ 16k3c20x3

dðk20−p2ÞIx;pðX;DÞ; ð27Þ

where I denotes the specific intensity (written in theFourier domain). In Eq. (27), the prefactor has been


chosen in such a way that the specific intensity in thespace–time domain, obtained after inverse Fouriertransformation over Ω and Δ obeys the customaryrelation (Chandrasekhar, 1960):

Eðt;RÞ ¼ 1

c0

Z4kIx;pðt;RÞd2p ð28Þ

where ε denotes the energy density. According to Eq.(21), the specific intensity can be rigorously defined asthe angular spectrum of the field correlation function.It can also be understood as an angularly resolvedenergy flux as illustrated in Eq. (28). This is the usualphenomenological definition encountered in textbooks(e.g. Chandrasekhar, 1960). Noting the followingFourier transform equivalence A

AtX−iX;jrX iD, and using

the definition of the mean free path (14), one obtainsthe radiative transfer equation in anisomeric randommedia:

AIx;pðt;RÞc0At

þ pdjIx;pðt;RÞ

¼−Ix;pðt;RÞlðpÞ þ

Z4krðp; p VÞIx;p V ðt;RÞd2p VþSx;pðt;RÞ;

ð29Þwhere the differential scattering cross-section isdefined as:

rðp; p VÞ ¼ k4016k2

Uðk0p−k0p VÞ; ð30Þ

and the energy release by (seismic) sources isencapsulated in S. This equation bears a closeresemblance with the transfer equation in an isotropicmedium. The term on the left-hand side describes thechange of intensity of an energy beam propagating indirection p. The first and second terms on the right-hand side represent losses and gains caused byrandom scattering events, respectively. This equationrepresents a local detailed energy balance and can beshown to obey energy conservation as it was statedbefore. In a weak scattering regime and for lapse timest is sufficiently small compared to the man free timeτ= l /c0, the transport equation is equivalent to theusual Born approximation. In addition, the transportequation provides a rigorous description of the energypropagation in the multiple scattering regime t≥τ.The notable difference between Eq. (29) and the usualtransport equation is the explicit dependence of thescattering mean free path on space direction, as wellas the explicit dependence of the differential scatteringcross-section σ on both the incoming and outgoingpropagation directions. σ describes the angular

dependence of the scattering pattern and is completelydetermined by the power spectrum of the fluctuations.It obeys the general reciprocity relation:

rðp; p VÞ ¼ rð−p V;−pÞ: ð31ÞTo gain insight into the physics contained in Eq. (29),

we will now solve it by two customary methods. Firstwe will investigate asymptotic solutions of the transportequation in the limit (t→∞). We will find that theenergy density obeys a generalized tensorial diffusionequation which can be solved analytically in simplecases. These asymptotic solutions will then be comparedto ‘exact’ numerical solutions of the transport equationby the Monte Carlo method.

5. Diffusive regime

Multiple scattering processes tend to uniformize theangular dependence of intensity because each scatteringevent distributes energy randomly in phase space. Aftera sufficiently large number of scattering events, we thusexpect the intensity to differ only slightly from isotropy.The physical idea of the diffusion approximation can beexpressed mathematically by an expansion of thespecific intensity in a linear combination of its firsttwo angular moments. The first moment is simplyrelated to the local energy density:

c0Eðt;RÞ ¼Z4kI pðt;RÞd2p; ð32Þ

and the second moment defines the energy currentvector J:

Jðt;RÞ ¼Z4kI pðt;RÞpd2p: ð33Þ

The current vector gives the local direction ofmaximum energy flow. Note that the ω dependence ofall quantities has been dropped for notational conve-nience. Combining Eqs. (32) and (33), the followingexpansion of the specific intensity is obtained:

I pðt;RÞ ¼ c04k

Eðt;RÞ þ 34k

Jðt;RÞd p þ : : :; ð34Þ

where the dots stand for higher order multipoles that areneglected and the 3 / 4π prefactor guarantees theconsistency of Eqs. (33) and (34). After integration ofthe transport equation over the whole solid angle, oneobtains the following exact continuity equation:

AEðt;RÞAt

þjdJðt;RÞ ¼ 0; ð35Þ


where the source term has been omitted for simplicity.Eq. (35) expresses the conservation of energy in therandom medium. The next step is to establish a relationbetween the spatial gradient of energy density and thecurrent vector. Following the standard procedure, weinsert expansion (34) in the transport equation andcalculate its second moment. After some algebra, thisyields:

Jðt;RÞ ¼ −DjEðt;RÞ; ð36Þwhere D is a symmetric diffusion tensor. This diffusiontensor can be written in the following suggestive formby analogy with the isotropic case:

D ¼ c0L1

3; ð37Þ

where L★ denotes the transport mean free path tensor.In the isotropic case, the transport mean free path tensoris diagonal with elements:

l1 ¼ l

1−hpd p Vi ; ð38Þ

where l is the usual (direction independent) scatteringmean free path, p and p′ are the incoming and outgoingpropagation direction. The brackets denote a weightedaverage over the whole solid angle, where the weightingfunction is the differential cross-section, which in theisotropic case depends solely on the scalar product por p′. In an anisomeric random medium, the expressionof the transport mean free path tensor is far morecomplex:

L1 ¼ ðK1−K2Þ−1; ð39Þ

where the tensors K1 andK2 are fully determined by thepower spectrum of heterogeneities:

K1 ¼ 34k

Z Z4krðp; p VÞp V� p Vd2pd2p V; ð40Þ

K2 ¼ 34k

Z Z4krðp; p VÞp � p Vd2pd2p V; ð41Þ

where the symbol⊗ denotes the usual tensor product. Inthe course of the derivation use is made of thereciprocity relation (31), and of the symmetry of thephase function to an inversion: p→− p , p′→− p′. Thesesymmetries should hold for quite a general class ofrandom media and do not represent a restriction of thetheory. Contrary to what happens in an isotropicmedium, the direction of maximum energy flow doesnot necessarily coincide with the gradient of the energy

density, which is intuitively appealing. Thus the textureof the material tends to “guide” the energy alongpreferential directions. After inserting relation (36) intothe continuity Eq. (35) one obtains the followingtensorial diffusion equation for the energy density:

AEðt;RÞAt

−jdDjEðt;RÞ ¼ Sðt;RÞ; ð42Þ

where S denotes a source term. In the next section, wewill investigate both numerical and approximateanalytical solutions of the transport equation foranisotropic random media.

6. Comparison of Monte Carlo simulations anddiffusion approximation

We consider anisotropic random media with asymmetry axis directed along the z (vertical) directionof a Cartesian system (see Fig. 1). By symmetry, thediffusion tensor must be diagonal in the set of axis (x, ŷ,z), where x and ŷ denote orthogonal vectors in thehorizontal plane. The eigenvalues of the diffusion tensorperpendicular and parallel to the z axis will be denotedby D|| and D⊥, respectively. The choice for this notationwill become clear in the next part of the paper. Theanalytical solution of the diffusion equation with sourceterm S(t, R)=δ(t)δ(R) writes (Carslaw and Jaeger,1959):

Eðt;RÞ ¼ ð4ktÞ−3=2D−1jj D

−1=28 e−½z

2=4D8tþðx2þy2Þ=4Djjt�:

ð43ÞSolving analytically the transport equation is a very

difficult task and in most cases, one must resort tonumerical methods. Considering the large dimensional-ity of our problem and the complexity introduced by theanisomery of the medium, Monte Carlo simulationsconstitute a convenient and flexible tool. This powerfulmethod has already been applied to complex seismo-logical situations and we refer the interested reader tothe literature for an introduction to the topic (Hoshiba,1991; Margerin et al., 1998). The basis of the methodcan better be understood by using an analogy with thekinetic theory of gases where the radiative transferequation (known in this field as the Boltzmannequation) is used to model the flow of particles. In thispicture, the transport of seismic energy in a scatteringmedium is analogous to the random walk of manyparticles that collide and get randomly deflected. Themean free path can therefore be interpreted as theaverage distance between two collisions. The

Fig. 5. Energy density as a function time (in seconds) in a Gaussiananisomeric random medium with k0a=0.6 and k0c=2. The pointsource has unit energy. The location of the receivers is indicated next toeach curve. The heavy lines correspond to analytical solutions of thediffusion equation, while the wiggly lines show the results of theMonte Carlo simulations. Top: weak scattering regime (rms velocityfluctuations 3.5%). Bottom: strong scattering regime (rms velocityfluctuations 7%).


normalized differential scattering cross-section σ givesthe probability distribution for a transition from state pto state p′. From a more technical point of view, twodifficulties have to be overcome in the case of ananisomeric random medium. First, we note that themean free path depends on the propagation direction.This is easily handled by introducing an angulardependent average distance between two collisions.Thus the probability law to determine the free pathlength L of a particle propagating in direction p is givenby F=− l(p) ln(u), where u is a uniformly distributedrandom number in the interval ]0, 1[. Second, thedifferential cross-section depends explicitly on theincoming direction p′ and the two angles defining theoutgoing direction do not decouple. After normaliza-tion, we interpret the differential cross-section as theconditional probability for outgoing state p knowingincoming state p′. The outgoing scattering direction issubsequently chosen by applying a rejection algorithm(Press et al., 1992). Note that this technique does notrequire the introduction of an intermediate localreference frame and provides the new propagationdirection directly in the global frame. This advantage issomewhat balanced by the relatively low efficiency ofthe method when the scattering pattern is highly peakedin certain directions.

We have calculated the point source solutions of theradiative transfer and diffusion equations in Gaussianrandom media, for both weak and strong scatteringmedia with rms velocity fluctuations equal to 3.5% and7%, respectively. Two detectors are located 100 kmaway in the x and z directions, from an isotropic,instantaneous, and point-like source. The centralwavenumber of the waves is k0=10 which correspondsto high-frequency (6.4 Hz) waves in a medium withvelocity c0= 4 km/s. In Fig. 5, we show the results of thecalculations for elongated objects (k0a=0.6, k0c=2.0).The diffusion constants have been calculated by meansof the theory developed above. We find D|| =20.4 km

2/s,D⊥=66.4 km2/s for 7% rms velocity fluctuations, andD|| =81.6 km2/s, D⊥=265.5 km2/s for 3.5% rms, whichshows that the transport is more efficient in the directionof elongation of the objects. In the strong scatteringmedium, the Monte Carlo solutions show irregular andunphysical glitches. These fluctuations of the numericalsolution are characteristic of the statistical method andcould be damped by increasing the number of realiza-tions. In weak scattering media, the diffusion solutionapproximates poorly the behavior of the full numericalsolution at early times. The Monte Carlo solutionconfirms that the transport is facilitated along thedirection of stretching of the scatterers. At late times,

both solutions agree very well as expected. The twomethods are thus complementary: the diffusion equationyields a simple and accurate approximation at late times,where convergence of the Monte Carlo method is slowerand numerical accuracy generally less. In the strongscattering regime it is striking to see how well thediffusion approximation can capture the complex time-dependence of the energy density. The largest differenceof order 20% occurs at early times in the direction wherethe diffusion constant is larger. This is a remarkablygood agreement considering the tremendous simplifica-tion of the physics introduced in the diffusionapproximation. Thus, the diffusion solution can beused as an excellent first guess in solutions of multiplescattering problems. For comparison, we show in Fig. 6

Fig. 6. Energy density as a function time (in seconds) in a Gaussiananisomeric random medium with k0a=2.0 and k0c=0.6. The pointsource has unit energy. The location of the receivers is indicated next toeach curve. The heavy lines correspond to analytical solutions of thediffusion equation, while the wiggly lines show the results of theMonte Carlo simulations. Top: weak scattering regime (rms velocityfluctuations 3.5%). Bottom: strong scattering regime (rms velocityfluctuations 7%).


the results obtained for flat object with k0a=2 andk0c=0.6, with all other parameters unchanged. In thiscase, one obtains the following diffusion constants:D|| =37.3 km2/s, D⊥=10.9 km2/s with 7% rms, andD|| =149.1, D⊥=43.6 km2/s with 3.5% rms. This showsthat the energy will be preferentially transported alongthe direction of flattening of the scatterers. Apart fromthis difference, the essential features are similar toFig. 5.

7. Application to a waveguide

For the moment, we have considered the propagationof energy in infinite space. This is not very satisfying

since there are major velocity discontinuities inside theearth that cause the reflection of the seismic energy.Below we develop a simple model for the propagation inthe lithosphere, as illustrated in Fig. 7. We assume thatthe lithosphere is composed of a random heterogeneousand anisotropic crust overlying a homogeneous mantle.Such a model has already been shown by Margerin et al.(1999), Hoshiba et al. (2001), Lacombe et al. (2003) tosuccessfully predict the coda decay of regional earth-quakes. We further assume that the free surface isperpendicular to the symmetry axis of the anisotropicscatterers as depicted in Fig. 7, i.e. the scatterers areeither stretched perpendicular to the surface or flattenedparallel to the surface. Taking the z axis perpendicular tothe surface, and an arbitrary set (x , ŷ) of orthogonalvectors parallel to the surface diagonalizes the diffusiontensor. We call D|| and D⊥ the eigenvalues of thediffusion tensor parallel and perpendicular to the freesurface, respectively. Our model also includes a stepincrease of wavespeed at the Moho. We assume that theeigenvalues of the transport mean free path tensor aresmaller than the crustal thickness. In this regime, thediffusion approximation should apply (Margerin et al.,1998) but must be supplemented with boundaryconditions. They can be obtained by writing down adetailed balance of energy on an infinitesimal portion ofinterface. The presence of statistical anisotropy does notyield any new difficulties, and we refer the reader to theliterature (Zhu et al., 1991; Margerin et al., 1998) forfurther details of this procedure. Writing down anenergy balance at the free surface and at the Moho, oneobtains the following boundary conditions, respectively:

AEðt;RÞAz

¼ 0 at z ¼ 0 ð44ÞEðt;RÞ þ 2c−10 gD8

AEðt;RÞAz

¼ 0; at z ¼ H ð45Þ

where γ is a function of the reflection coefficient at theMoho at depth H. The zero flux condition at the surfaceexpresses the total reflection of energy, while theboundary condition at the Moho describes the partialtrapping of energy in the crust. The factor γ is related tothe usual energy reflection coefficient R at the Moho asfollows (Zhu et al., 1991; Margerin et al., 1998):

g ¼1þ 3

Z 1

0RðlÞl2dl

1−2Z 1

0RðlÞldl

; ð46Þ

where μ denotes the cosine of the incidence angle. Sincethe scattering mean free path in the mantle is assumed to

Fig. 7. Schematic view of the wave propagation in a waveguide with statistically anisotropic scatterers, preferentially aligned along the free surface.Note that in a single realization of the randommedium, the scatterers are expected to look very different from the cartoon above.H denotes the crustalthickness. As the waves leave the source they are multiply scattered by velocity fluctuations and multiply reflected by the Moho and the free surface.It will be assumed that the mantle is weakly heterogeneous, and does not backscatter a significant amount of energy.

Fig. 8. Contour plot of ln(D|| /D⊥) as a function of horizontal (a) andvertical (c) correlation lengths. Labels on the curve indicate the valueof the ratio D|| /D⊥. The wavenumber is normalized (k0=1). Top:anisotropic Gaussian medium. Bottom: anisotropic exponentialmedium.


be much larger than in the crust, Eq. (45) neglects thebackscattering from the mantle. The system of Eqs. (43−45) can be solved using the results of Margerin et al.(1998), after the following change of variables:x→D⊥x /D||,y→D⊥y /D||, which maps the anisotropicdiffusion equation onto an isotropic one. For a simplecase of a source located near the surface, one obtains:

Eðt;RÞft−1eðR2=4Djjtþxt=QcÞ: ð47Þ

This expression is formally identical to that proposedby Aki and Chouet (1975) to describe the decay of thecoda. The Coda Q parameter Qc is obtained explicitly asfollows:

Qc ¼ xH2

D8n2 ; ð48Þ

where H is the crustal thickness and ξ is the smallestsolution of the equation ξ tan ξ=H /γ, usually of order 1.Following Margerin et al. (1999), it is also possible todefine the residence time of the diffuse waves in thewaveguide as:

sr ¼ Qc

x: ð49Þ

The residence time characterizes the rate of decay ofthe coda. From Eq. (47), it is possible to derive acharacteristic time for the broadening of the seismicpulse τb defined as the time to reach the maximumenergy density. For sufficiently large source receiverdistance R, τb can be approximated as:

sbcHR

2nffiffiffiffiffiffiffiffiffiffiffiffiDjjD8

p : ð50Þ

Formulas (49) and (50) are instructive because theyshow that in a simple but realistic geometry the

broadening of the initial pulse and the coda decay arecontrolled by independent parameters.

ffiffiffiffiffiffiffiffiffiffiffiffiDjjD8

pgoverns

the lateral spreading of the pulse while D⊥ controls the


leakage rate of the diffuse waves. In order to evaluate thepotential impact of statistical anisotropy on the transportproperties, we have studied systematically the ratio ofparallel to transverse components of the diffusion tensorin exponential and Gaussian media. The results areshown in Fig. 8 where we plot the natural logarithm ofthe ratio D|| /D⊥ as a function of the vertical andhorizontal correlation lengths a and c. Note that weassume that the medium has rotational symmetry aroundthe vertical axis. The results show that the ratio D|| /D⊥can indeed become very large for sufficiently aniso-tropic media. This in turn implies that values of thetransport mean free path inferred from pulse broadeningand coda decay can be significantly different.

8. Perspectives and conclusion

In the context of wave propagation in the crust andlithosphere at local and regional distances, the presenttheory could be applied to infer the anisotropy of scalelengths from the joint analysis of direct and coda waveenergy envelops following the work of Abubakirov andGusev (1990), who measured the scattering properties ofthe lithosphere under Kamchatka. It is striking that theirestimates of the transport mean free path deduced frompulse broadening and coda wave analysis can differ by afactor as large as 2 (see table 1 of Abubakirov and Gusev,1990). This discrepancy could be reconciled byintroducing statistical anisotropy in the earth. Indeed,we have shown that in anisomeric media the broadeningof the pulse and the decay rate of the coda are controlledby two different transport mean free paths, in sharpcontrast with the isotropic case. The theory of multiplescattering in a waveguide could also be applied to inferthe scattering properties of the crust by analyzing thespatio-temporal distribution of energy in the coda ofhigh-frequency (N1Hz) Lgwaves, following the work ofLacombe et al. (2003). Recent modeling of scattering byupper mantle structures (Nielsen et al., 2003) have putforward the role of elongated objects to explain thecharacteristics of the wavefield at epicentral distancestypically smaller than 40 degrees. Transport theory couldbe used to further investigate the proposed models ofscattering by comparing modeled envelopes of multiply-scattered waves with observed short period codaenvelopes. At the global scale, transport theory ofanisomeric media could be applied to the coda ofteleseismic wavefields (Shearer and Earle, 2004) orprecursors of some phases such as PKP (Hedlin et al.,1997) and PKKP (Earle and Shearer, 1997). In particular,the energy envelopes of PKP precursors have beenextensively studied in recent years (Hedlin and Shearer,

2000; Margerin and Nolet, 2003) in order to determinethe depth and size distribution of small-scale hetero-geneities in the lower mantle. Cormier (1999) hasproposed that anisomeric structures may be superposedonto statistically isotropic fluctuations in the deepestparts of the lower mantle. Our transport equation couldbe used to model the multiple scattering of P waves insuch media and give better constraints on the presence ofstatistical anisotropy in D″. In conclusion, we havedeveloped a transport theory for acoustic and anisomericrandom media, as well as methods of solution. Thebiggest restriction of the present work is the absence ofenergy exchange between P and Smodes as would be thecase for elastic media. Nevertheless, it is hoped that thepresent results can be used to reveal the signature ofanisotropy by analyzing attenuation and transportproperties of short period seismic waves.

Acknowledgments

The topic of this paper was motivated by discussionswith M. Campillo. I thank my office mate E. Chaljub fornumerous advices that greatly helped me in the course ofthis work. B. van Tiggelen guided me in the multiplescattering formalism and corrected some of mymistakes. Reviewers are acknowledged for suggestingimprovements to the original manuscript.

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