Diffusion approximation with polarization and resonance effects for the ...

Geophysical Journal International Geophys. J. Int. (2013) 192, 326–345

doi: 10.1093/gji/ggs022

Diffusion approximation with polarization and resonance effects for the modelling of seismic waves in strongly scattering small-scale media Ludovic Margerin Institut de Recherche en Astrophysique et Planétologie, Observatoire Midi-Pyrénées, C.N.R.S., Université Paul Sabatier, 14 Avenue Edouard Belin, Toulouse, France. E-mail: [email protected]

Accepted 2012 October 12. Received 2012 October 11; in original form 2012 May 9

GJI Seismology

SUMMARY This paper presents an analytical study of the multiple scattering of seismic waves by a collection of randomly distributed point scatterers. The theory assumes that the energy envelopes are smooth, but does not require perturbations to be small, thereby allowing the modelling of strong, resonant scattering. The correlation tensor of seismic coda waves recorded at a threecomponent sensor is decomposed into a sum of eigenmodes of the elastodynamic multiple scattering (Bethe-Salpeter) equation. For a general moment tensor excitation, a total number of four modes is necessary to describe the transport of seismic waves polarization. Their spatiotemporal dependence is given in closed analytical form. Two additional modes transporting exclusively shear polarizations may be excited by antisymmetric moment tensor sources only. The general solution converges towards an equipartition mixture of diffusing P and S waves which allows the retrieval of the local Green’s function from coda waves. The equipartition time is obtained analytically and the impact of absorption on Green’s function reconstruction is discussed. The process of depolarization of multiply scattered waves and the resulting loss of information is illustrated for various seismic sources. It is shown that coda waves may be used to characterize the source mechanism up to lapse times of the order of a few mean free times only. In the case of resonant scatterers, a formula for the diffusivity of seismic waves incorporating the effect of energy entrapment inside the scatterers is obtained. Application of the theory to high-contrast media demonstrates that coda waves are more sensitive to slow rather than fast velocity anomalies by several orders of magnitude. Resonant scattering appears as an attractive physical phenomenon to explain the small values of the diffusion constant of seismic waves reported in volcanic areas. Key words: Volcano seismology; Theoretical seismology; Wave scattering and diffraction.

1 I N T RO D U C T I O N Since the tail or ‘coda’ of seismograms has been correctly interpreted by Aki (1969) as scattered waves from lithospheric heterogeneities, coda waves have been used in a variety of applications such as source studies, deep Earth structure, and monitoring of temporal variations (see Sato et al. 2008, for a review and recent references). Aki & Chouet (1975) noted the dominance of shear waves in the seismic coda and introduced the first quantitative models of energy transport based on the single-scattering and diffusion model. Following the breakthrough of Aki, many studies of coda waves focused on the modelling of the energy envelopes of seismograms (see Sato & Fehler 1998, for more details). In recent years, the field of seismic multiple-scattering has been enriched with the observation of interference phenomena such as the weak localization effect (Larose et al. 2004), and the retrieval of Green’s function from the cross-correlation of coda waves (Campillo & Paul 2003; Paul et al. 2005). In connection with Green’s function retrieval, the crucial role played by the equipartition principle was recognized (Hennino et al. 2001; Lobkis & Weaver 2001; Van Tiggelen 2003; Malcolm et al. 2004; Sánchez-Sesma & Campillo 2006). For scalar waves in infinite space, it demands that the wavefield be composed of uncorrelated plane waves coming from all possible directions with equal weights. In the case of vector elastic waves, equipartition demands in addition that far from the boundaries, the shear and compressional energy are partitioned in the ratio 2c3p /cs3 , with cp and cs the longitudinal and transverse wave speeds (Weaver 1982). Besides Green’s function retrieval from seismic coda waves (Paul et al. 2005), equipartition of P and S energy in multiply scattered elastic waves has a number of other applications. The typical time to establish equipartition may be exploited to characterize the nature of heterogeneities in polycrystals (Turner

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Diffusion approximation for seismic waves

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& Weaver 1995), Earth’s lithosphere (Margerin et al. 2000), and volcanoes (Yamamoto & Sato 2010). Recent applications of equipartition also include the assessment of site effects from coda waves (Margerin et al. 2009; Nakahara & Margerin 2011). The goal of this paper is to develop a generic model of multiple-scattering of elastic waves by a collection of small, possibly resonant scatterers. Particular attention will be paid to the depolarization of the emitted wavefield and the convergence towards equipartition. One approach to the problem of seismic wave multiple-scattering is provided by the theory of radiative transfer for elastic waves (Weaver 1990; Zeng 1993; Sato 1994; Turner & Weaver 1994; Ryzhik et al. 1996). Radiative transfer governs the temporal and spatial evolution of angularly resolved energy fluxes, the specific intensities. While this powerful method allows accurate modelling of scattering anisotropy and mode conversions, it does have has certain limitations and drawbacks. First, solutions of the time-dependent transfer equation are only accessible through numerical simulations. Discrete ordinates (Turner & Weaver 1994), Monte Carlo simulations (Margerin et al. 2000; Przybilla & Korn 2008) and fast Fourier transform methods (Yamamoto & Sato 2010) are the main tools that have been developed so far. Second, the applicability of radiative transfer is limited by an assumption of weak heterogeneity (Weaver 1990; Ryzhik et al. 1996), which excludes resonant scattering. Finally, the specific intensities which contain information on the correlation properties of the wavefield are not easily accessible in seismology. Their measurement requires taking the Fourier transform of the wavefield sampled at an array with subwavelength interstation distance. A more generic approach is provided by the diffusion approximation which models the transport of the seismic energy by a scalar diffusion equation (Weaver 1990; Papanicolaou et al. 1996; Turner 1998). A sophisticated version of the diffusion equation which considers the temporal evolution of coupled P and S energy densities has also been derived from the radiative transfer equation by Trégourès & van Tiggelen (2002). The diffusive regime sets in when (1) energy fluxes depart sufficiently little from isotropy and (2) the energy partition between shear and longitudinal waves obeys the equipartition prescription. It has been noted that the first requirement is much more stringent than the second (Paul et al. 2005). The diffusion approximation has proved useful to model the seismic coda in highly heterogeneous regions such as volcanoes (Wegler & Lühr 2001; Wegler 2004) and in heterogeneous materials such as concrete (Anugonda et al. 2001). However, the diffusion approach developed in previous works focuses on energy densities and does not include precise information on the polarization of the wavefield. The approach followed in this study uses as a starting point the elastodynamic version of the Bethe-Salpeter equation for the so-called structure factor (Weaver 1990). The structure factor transports the field-field correlations between two scatterers through a sequence of scattering events (Van Rossum & Nieuwenhuizen 1999; Akkermans & Montambaux 2007). At the outset, we restrict ourselves to a random collection of ‘point scatterers’, and assume that the disorder is sufficiently weak so that the independent scattering approximation holds (Lagendijk & van Tiggelen 1996). While largely developed and studied in the electromagnetic literature (see de Vries et al. 1998, for a review), the point scattering model for seismic waves has been introduced fairly recently only (Margerin 2011; Margerin & Sato 2011). It represents a simple but realistic model of scattering as long as the probing wavelength is larger than the typical size of the inclusions. In the usual limit of slow temporal and spatial evolutions, the structure factor will be decomposed into a sum of eigenmodes of the Bethe-Salpeter equation using standard perturbation theory. From the structure factor, we obtain an asymptotic but complete expression of the correlation tensor of seismic waves excited by moment tensor sources. Note that contrary to previous approaches, the perturbations are not limited by a smallness assumption and the scatterers are not assumed to be weak. In particular, resonant interaction between the incident waves and the inclusions is allowed. This gives rise to a revision of the usual formula for the diffusivity of seismic waves, which incorporates the time delay caused by the energy entrapment inside the scatterers, as first put forward for electromagnetic waves (van Tiggelen et al. 1992; Lagendijk & van Tiggelen 1996). The present developments may find applications in the characterization of strongly scattering solids. Multiple scattering of ultrasound in concrete (Anugonda et al. 2001) or diffusion of seismic waves in volcanoes (Wegler & Lühr 2001; Larose et al. 2004; Wegler 2004; Yamamoto & Sato 2010) constitute acoustical and seismological examples of potential interests, respectively. Further methodological developments may also be envisaged in connection with the diagnostic of damages in concrete and the monitoring of volcanic eruptions. In particular, the calculation of the sensitivity kernels of diffuse waves to local changes (Rossetto et al. 2011) may be generalized to vector elastic waves.

2 M U LT I P L E S C AT T E R I N G M O D E L 2.1 The propagation regime We consider the propagation of elastic (seismic) P and S waves in a collection of point-like scatterers embedded in a homogeneous matrix. We make no a priori assumption on the smallness of the contrast between the scattering objects and their environment. The density of scatterers is nevertheless assumed to be small enough for the weak disorder condition k p,s l p,s 1

(1)

to apply, where kp, s and lp, s denote the P (resp. S) central wavenumber and the P (resp. S) scattering mean free path, respectively. This implies that the mean distance between two scattering events is on average much larger than the wavelength and authorizes the use of the far-field form of Green’s function to propagate the waves between two scatterers. In addition, any scattering process involving the recurrent visit of a scatterer will be neglected: this is the usual independent scattering approximation (ISA). Slight departures of the matrix from elasticity may

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a be modelled phenomenologically by introducing the absorption time τ p,s of P (resp. S) waves. Together with the weak disorder condition, the propagation regime considered in this work is summarized in the following equation: a , τ p,s ≤ t τ p,s

(2)

where τ p, s denotes the mean free time between two scattering events for P or S waves. The first inequality ensures that multiple scattering processes set in, while the second guarantees that the detection of signals is not hampered by anelastic attenuation. In addition, the perturbation approach employed below requires that the spatial and temporal evolution of the energy carried by the waves be slow compared to the central frequency and wavenumber of the signal. This has been termed the slowly varying envelope approximation in the literature (Lagendijk & van Tiggelen 1996). 2.2 Point scattering model Point-scattering is a popular model in condensed matter physics and has received much attention in the past. In seismology, point-scattering has surfaced only recently in the literature (Margerin 2011; Margerin & Sato 2011). It may be particularly well adapted to treat wave propagation in media containing high-contrast inclusions. For instance in volcanic regions, the overall propagation characteristics seem on the whole well modelled by isotropic scattering, thereby suggesting that the scatterers are small compared to the wavelength (Yamamoto & Sato 2010). In addition, the diffusivity measured in such heterogeneous regions is anomalously low, which has been ascribed to small values of the mean free path (Wegler 2004). However, the possibility that the low diffusivity results from an accumulation of time delays in the scattering process has never been examined. In elastodynamics, a point-scatterer located at x0 may be defined by regularizing the multiple-scattering series for a singular potential of the form V (x, x ) = γ (ω)δ(x − x0 )δ(x0 − x )I.

(3)

In eq. (3), I denotes the identity operator in the 3-D polarization space, and γ (ω) is a frequency dependent coupling parameter between the incident waves and the scatterer. Although the regularization procedure is non-unique, the T-matrix of the point scatterer assumes the following form in the wavenumber domain (Margerin 2011; Margerin & Sato 2011)

T 0 (p, p ) =

eix0 · (p −p) t(ω)I , (2π )3

(4)

where t(ω) is a scalar t-matrix whose general form follows: t(ω) =

γ (ω) . 1 + γ (ω)(α + itrImG 0 (x0 , x0 ; ω)/3)

(5)

In eq. (5), tr denotes the trace of a matrix (the sum of diagonal elements), and Im G 0 (x0 , x0 ; ω) represents the imaginary part of the retarded return Green’s tensor of the embedding medium. The quantity α > 0 in the denominator depends on the physical size a of the scatterer and on the adopted regularization procedure. It may be modified arbitrarily without violating the following optical theorem: Imt(ω)I = |t(ω)|2 ImG 0 (x0 , x0 ; ω),

(6)

which guarantees the conservation of energy in the scattering process (Margerin & Sato 2011). The point scatterer offers a simple mathematical model of elastic wave scattering by high-contrast objects which are small compared to the wavelength, that is, kp, s a ≤ 1 . The key point is that the t-matrix (5) results from a summation of the full multiple scattering series for a point scatterer, thereby ensuring a coupling between the waves and the heterogeneity at all orders of the perturbation. The range of validity of the model is discussed in details in Margerin (2011). It is important to remark that contrary to its scalar counterpart, point scattering of elastic waves is anisotropic, the scattering pattern being simply obtained by taking the dot product between the polarization vectors of the incident and scattered waves. It is as the same time non-preferential because it follows easily from the definition of the scattering pattern that the mean cosine of the scattering angle is 0. This implies in particular that the same amount of energy is scattered in the forward and backward directions. In Section 5, I will provide an explicit expression of the t-matrix applicable to perfectly correlated P and S velocity anomalies. 2.3 Mean-field propagator As long as the propagation distance between source and detectors is smaller than the mean free path, the recorded signal is dominated by the coherent part of the wavefield, governed by the following Dyson equation (Frisch 1968; Weaver 1990): G = G 0 + G 0 G,

(7)

In eq. (7), denotes the so-called ‘mass-operator’ and G is the ensemble averaged retarded displacement response to an applied force. In the ISA, the mass-operator is expressed as: Ti , = (8) i


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where the double brackets indicate an ensemble average, and Ti denotes the T-matrix of a scatterer located at xi . In the wavenumber domain, the matrix elements of Ti are readily obtained by the substitution x0 → xi in eq. (4). In a translationally invariant medium all operators take a diagonal form in Fourier space and we shall employ the following special notations: G 0 (p, p ) = g 0 (p)δ(p − p ), (p, p ) = σ (p)δ(p − p ), etc., in the following. In addition, in view of the rotational invariance of the medium, the second rank tensors g0 and σ may be decomposed into transverse and longitudinal parts as follows: g 0 (p; ω) =

pˆ pˆ I − pˆ pˆ + 2 ρ0 ω+2 − c2p p2 ρ0 ω+ − cs2 p2

ˆ σ (p; ω) = nt(ω)pˆ pˆ + nt(ω)(I − pˆ p),

(9a) (9b)

where n is the number density of scatterers, cp , cs denote the P and S wave speeds in the matrix, and the symbol ω+ indicates that the frequency possesses an infinitesimal imaginary part to select the retarded solution. Eq. (9b) follows from (8) after averaging over the scatterer positions in a finite volume V, and subsequently letting the volume tend to infinity at fixed scatterer concentration (the thermodynamic limit). Substituting expressions (9a)–(9b) into eq. (7), yields the following expression of the mean Green’s function ˆ g(p; ω) = g p ( p; ω)pˆ pˆ + gs ( p; ω) (I − pˆ p)

(10a)

1 . ρ0 (ω2 − c2p,s p2 − nt(ω)/ρ0 )

(10b)

g p,s ( p; ω) =

The location of the poles of the mean Green’s function in the complex p plane provides the dispersion relation in the disordered medium. In the point scattering model, the attenuation of the coherent wave may be characterized by a single mean free time τ for P and S waves Imnt(ω)ce Imnt(ω) 1 =− =− + O(n 2 ). τ ρ0 ωc ρ0 ω

(11)

In the effective medium the phase speed of the longitudinal and transverse waves cep,s are renormalized in the same proportion cep,s ce 1 nRet(ω) = =1+ + O(n 2 ). = nRet(ω) c c p,s 2ρ0 ω2 1 − ρ ω2

(12)

0

As indicated by the symbol O(n2 ), equalities (11) and (12) are correct to first order in the scatterer density n. In what follows, the mean free path will be defined as the product between the mean free time and the matrix wave speed. Some authors would rather connect the mean free path to the effective phase speed, but the difference between the two definitions is O(n2 ) only as shown by the second equality in eq. (11). An obvious consequence is the following relation between the P and S mean free paths: cs lp = cp ls , implying that the S mean free path is the shortest attenuation length. Yet another remarkable property of the point-scattering model is expressed by the relation 2c3p 1 1 1 1 (13) + = 1 + = , τ sp τ ps τ sp cs3 τ where τ ps and τ sp denote the P to S and S to P scattering mean free time. The first equality is a consequence of reciprocity and is valid for an arbitrary scatterer, but the second one is specific to point scatterers.

2.4 Transport of correlations: the Bethe-Salpeter equation For propagation times larger than the scattering mean free time, the coherent wave starts to be dominated by the fluctuating part of the wavefield, apparent in the coda of the seismic signals (Aki 1969; Aki & Chouet 1975). In this regime, the recorded signals are composed of multiply scattered waves, and the relevant statistical information is contained in the second moment of the wavefield. The necessary mathematical apparatus is provided by the theory of field correlations in random media which has been exposed in a number of books and treatises (Frisch 1968; Sheng 2006; Akkermans & Montambaux 2007). I briefly review the most important material below. The central quantity of our multiple-scattering approach is the structure factor which connects two arbitrary scatterers of the medium through a sequence of scattering events. The structure factor satisfies the following Bethe-Salpeter equation (Sheng 2006, eq. 5.18), which governs the transport of wavefield correlations through a sequence of multiple scattering = U + U : G ⊗ G ∗ : .

(14)

In eq. (14), G∗ denotes the advanced (complex conjugated) mean Green’s function, the symbol ⊗ denotes a tensor product between retarded and advanced wavefields, and the dots imply a contraction of indices as explained below. Following the language of diagrammatic theory, the structure factor is also a reducible kernel because it is built from an elementary brick, the irreducible kernel U, from which any multiplescattering path involving two wavefields may be constructed (Frisch 1968). In this work, I adopt the independent scattering approximation (ISA) which stipulates (1) that the two wavefields visit exactly the same scatterers in the same order, (2) that the scatterers are spatially uncorrelated and (3) that each scatterer may be visited only once in a given sequence of scattering events. The complete expression of U

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Figure 1. Top: schematic view of the multiple scattering process. Scattering events are represented by open circles. The solid and dashed lines represent the mean Green’s function and its complex conjugate, respectively. The waves launched at the seismic sources (x3,4 ) propagate to the first scattering points (r3,4 ) and are transported through a sequence of independent scattering events by the structure factor . After exiting the last scattering points (r1,2 ), the waves are propagated to the detection points (x1,2 ) with the aid of the mean Green’s function G and G∗ . For seismological applications, the source is described by a unique moment tensor and we further require x3 = x4 . In seismic interferometry, the detection points may not necessarily coincide. Bottom: diagrammatic representation of eq. (14) showing the input (right-hand side) and output (left-hand side) channels of the tensor in wavenumber space (left-hand side). Each channel is equipped with a polarization index. On the right-hand side, the dotted line indicates that the upper and lower fields visit the same scatterer. By iteration of the Bethe-Salpeter equation, the multiple scattering series is generated.

corresponding to the ISA will be given below. ISA is a reasonable approximation at sufficiently low scatterer concentration and we will discuss its limitations in Section 2.5. A typical multiple scattering sequence is depicted in Fig. 1 (top). In the case of vector elastic waves the structure factor is a tensor operator with two input channels on the right-hand side and two output channels on the left-hand side, each channel being equipped with a polarization index. The input and output channels must subsequently be connected to the source and detection points, respectively. A diagrammatic representation of illustrating the concept of channels is shown in Fig. 1 (bottom). To clarify the interpretation of eq. (14), we now detail the expression of each term in the wavenumber domain, in component form, and discuss its physical meaning. Exploiting the translational invariance of the medium, and introducing centre-of-mass coordinates, the Fourier space representation of takes the form i, p+q/2 k, p +q /2 j, p−q/2 l, p −q /2

= γ (q, ; p, p , ω)i j;kl δ(q − q ),

(15)

where i, j, k, l represent polarization indices (see Fig. 1). In coordinate space, depends in general on four variables ri , i = 1, . . . , 4, as shown in Fig. 1. The variables p and p are conjugate of the variables r1 − r2 and r3 − r4 , respectively. Similarly, q and q are conjugate of the variables (r1 + r2 )/2 and (r3 + r4 )/2. The delta function in eq. (15) follows from translational invariance. On the one hand, the wavenumbers p, p (and the frequency ω) govern the rapid spatial (and temporal) oscillations of the wavefield. On the other hand, the momentum q (q p) and the frequency ( ω) serve to describe the transport of correlations on large spatial (and temporal) scales. The scalings (q p, ω) define the regime of slow spatial and temporal variations of the energy envelopes, and are consistent with the separation of space and timescales observed in the seismic coda. At typical frequencies of 10 Hz, the decay time of the seismic coda typically exceeds several tens of seconds. In the ISA adopted throughout this study, the kernel U may be expressed as (Frisch 1968) ∗ Ti ⊗ Ti . (16) U≈ i

Like the structure factor, the kernel U has two input channels on the right-hand side and two output channels on the left-hand side. In the case of point-scatterers, the representation of U is particularly simple, since it does not depend on the wavenumbers p, p i, p+q/2 k, p +q /2 j, p−q/2 U l, p −q /2

=

nt(ω + /2)t(ω − /2)∗ δik δ jl δ(q − q ). (2π )3

(17)


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The form of the kernel U (17) in turn implies that the structure factor depends solely on the modulation wavenumber q, which describes the transport of correlations on large length scales. Fig. 1 (bottom) illustrates diagrammatically the multiple scattering series generated by iteration of the Bethe-Salpeter eq. (14) in the ISA. Combining eqs (14), (15) and (17), one obtains the seismic Bethe-Salpeter equation for point-scatterers in the wavenumber–frequency domain γ (q, ; ω)i j;kl =

nt(ω + /2)t(ω − /2)∗ δik δ jl + Q(q, ; ω)i j;i j γ (q, ; ω)i j ;kl , (2π )3

(18)

where the summation convention over repeated indices is understood, and we have introduced the operator Q with matrix elements

nt(ω + /2)t(ω − /2)∗ gik (p + q/2, ω + /2)g jl (p − q/2, ω − /2)∗ d 3 p. Q(q, ; ω)i j;kl = (2π )3 R3

(19)

Formally, the solution of eq. (18) can be obtained straightforwardly by matrix inversion γ (q, ; ω) =

nt(ω + /2)t(ω − /2)∗ [I − Q(q, ; ω)]−1 , (2π )3

(20)

where I denotes the identity operator in the (9-D) polarization space, which is the tensor product of two vector spaces of three components vector fields. Before constructing an explicit solution of eq. (20), it is worthwhile to underline the physical content and limitations of the present theory.

2.5 Physical content of the independent scattering approximation While ISA is a general framework to analyse the multiple scattering of waves it does have a number of limitations which we now discuss. First, ISA neglects the possible correlations between scatterer positions. Therefore, it cannot describe the scattering by clusters of particles and does not impose that particles do not overlap. Since the support of our T-matrix reduces to a point, this last issue is not fundamental in the present work. ISA also imposes strong restrictions on the scattering paths, since it does not allow the same scatterer to be visited even twice. In dense systems, this approximation is certainly doomed to fail since one may expect the waves to bounce several times between neighboring objects. In addition, scatterers may screen each other so that the coherent wave attenuation does not increase linearly with the number density, contrary to what is asserted in eq. (11). Recurrent scattering has been detected experimentally by Wiersma et al. (1995) with optical waves propagating in very strongly scattering samples. According to their results, one should be very cautious with the outcome of ISA if the disorder parameter kl becomes typically lower than 5. When discussing applications of the present theory to seismic data in section 5, I will make sure that kl (as predicted by ISA) is of the order or larger than 10. Despite these warnings, it is worthwhile to underline that ISA does not require the perturbations of the elastic parameters to be small. It is only when the scattering process is simplified by introducing the Born approximation that the strength of perturbations is limited. The T matrix used in this paper stems from a full summation of the Born series and is therefore adapted to model high-contrast materials.

3 P E RT U B AT I V E S O L U T I O N I N F O U R I E R S PA C E 3.1 The static problem To compute the inverse matrix in eq. (20), we proceed by diagonalization of the matrix Q. In the limit → 0, q → 0, this goal may be achieved by perturbation. In a first step, we determine the eigenvalues and eigenprojectors of the matrix Q at q = 0, = 0 ⎡

∞

∞

2 n|t(ω)|2 ⎣ 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ p p p p p p p p δ δ d d |g ( p; ω)| p d p + − p − p |gs ( p; ω)|2 p2 d p Q(0, 0; ω)i j;kl = i j k l p ik i k jl j l (2π )3 4π 4π 0

∞

pˆ i pˆ k δ jl − pˆ j pˆ l d pˆ

+ 4π

0

∗

∞

g p ( p; ω)gs ( p; ω) p d p +

2

0

pˆ j pˆ l (δik − pˆ i pˆ k )d pˆ

2

2

4π

⎤ gs ( p; ω)g p ( p; ω) p d p⎦ ∗

2

(21)

0

In eq. (21) and in the analysis that follows, we repeatedly need to evaluate integrals of products of Green’s functions gs, p . Some key formulas which are useful for this purpose are summarized in Appendix A. The last two terms of eq. (21) involve cross-products of the longitudinal and transverse parts of the mean Green’s function and vanish in the limit kp, s lp, s → ∞, as discussed in Appendix . Terms of order (kp, s lp, s )−1 or higher will be consistently neglected throughout this work. This approximation forms the basis of current multiple scattering theories in the regime of weak disorder encapsulated in eq. (1) (Van Rossum & Nieuwenhuizen 1999; Akkermans & Montambaux 2007). This leaves us with c3p c3 δi j δkl + 6δik δ jl + δil δ jk . Q(0, 0; ω)i j;kl = 3 s 3 δi j δkl + δik δ jl + δil δ jk + 3 5 cs + 2c p 5 cs + 2c3p

(22)

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The spectral decomposition of the matrix Q(0, 0; ω) writes λn Pn , Q(0, 0; ω) =

(23)

i=0,1,2

where λn and Pn denote the eigenvalues and the corresponding mutually orthogonal projectors, respectively λ0 = 1, P0 i j;kl =

1 δi j δkl 3

(24a)

c3p 1 , P1 i j;kl = δik δ jl − δil δ jk λ1 = 3 3 2 2c p + cs λ2 =

7c3p + 2cs3 10c3p + 5cs3

, P2 i j;kl =

(24b)

1 1 δik δ jl − δil δ jk − δi j δkl . 2 3

(24c)

The dimensions of the associated eigenspaces are 1, 3 and 5, respectively. Such degeneracies are characteristic of the rotational symmetry of the underlying system.

3.2 Calculation of the perturbation matrix The second step of our program is the computation of the perturbation matrix δ Q = Q(q, ; ω) − Q(0, 0; ω) to order O( , q2 ) in the weak disorder limit introduced in eq. (1). It is convenient to disentangle the role played by the different parameters and to express δQ as follows: δ Q = δ Qq + δ Q + δ Qa ,

(25)

where the notation used for the first two terms is self-explanatory, and the last term incorporates a small amount of dissipation due to the a (or equivalently anelasticity of the matrix. On a phenomenological level, anelasticity may be introduced in the form of absorption times τ p,s a absorption lengths l p,s ) which are assumed to be much larger than the characteristic mean free time (mean free path) of the waves. In addition, it is assumed that the absorption properties do not vary too strongly in the frequency band interest. It is relatively straightforward to see that the perturbation of the tensorial part of the matrix Q does not contribute to the matrix δQ. Indeed, the perturbation of the polarization vectors caused by the substitution p → p + q/2 is proportional to q/p. Since q and p are typically of order 1/lp, s and kp, s , respectively, the weak disorder limit guarantees that the corresponding terms are negligible. This qualitative argument is confirmed by direct calculations, based on the fundamental integral formulas developed in Appendix A. To complete the calculation of the perturbation matrices δQ we expand the scalar part of Green’s function as follows: a = g ap,s p + q/2; ω + /2; τ p,s

ρ0 [(ω +

≈ g p,s

/2)2

−

c2p,s (p

1 a ] + q/2)2 − nt(ω + /2)/ρ0 + iω/τ p,s

c2p,s q 2 iω ndt(ω) − ω + c2p,s p · q + − a 1 + ρ0 g p,s + ρ02 c4p,s g 2p,s (p · q)2 2ρ0 dω 4 τ p,s

(26)

a where the short-hand notation g p,s = g ap,s (p; ω; ∞), and the absorption times τ p,s , have been introduced. The superscript a serves to remind the reader that a small amount of absorption has been introduced. After similar Taylor expansion of the product of t-matrices in eq. (19), evaluation of wavenumber integrals based on the formulas of Appendix A and application of the weak disorder condition (1), the following results are obtained:

1 1 −3τ 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ × p p p p p p p × p p p p , (27a) δ δ p q q d − p − p q q d δ Q q (q)i j;kl = i j k l m n m n ik i k jl j l m n m n −3 4π (c−3 c p 4π cs 4π p + 2cs )

δ Q ( )i j;kl

n dt(ω) c dt(ω) ce cs3 δi j δkl + δik δ jl + δil δ jk 1− Re + e Im = i τ 3 3 c 2ρ0 ω dω cτ ωdω 5 cs + 2c p c3p

δi j δkl + 6δik δ jl + δil δ jk + 3 5 cs + 2c3p δ Q iaj;kl = −

τ c3p τ c3 s δi j δkl + δik δ jl + δil δ jk − δi j δkl + 6δik δ jl + δil δ jk . a 3 3 a 3 3 5τ p cs + 2c p 5τs cs + 2c p

Eq. (27b) introduces the timescale τ ∗ n dt(ω) c dt(ω) ce τ 1− Re + e Im τ∗ = c 2ρ0 ω dω c τ t(ω)dω

(27b)

(27c)

(28)


333

a Table 1. Decomposition of the eigenvalues λm n (q, ; τ p,s ) of the matrix Q in terms of different type a . Note that the dependence of the λm on of perturbations, correct to lowest-order in , q and τ/τ p,s n the circular frequency ω is implicit.

Mode

Unperturbed

q perturbation

perturbation

n=0

1

−2ws q 2 ls2 /3

i τ ∗

Anelastic perturbation −

wpτ 2ws τ − a τsa τp

−w p q 2 l 2p /3 i τ ∗ ws

n = 1, m = 0

ws

−3ws q 2 ls2 /5

n = 1, m = 1

ws

−ws q 2 ls2 /5

i τ ∗ ws

−ws τ/τsa

n = 2, m = 0

(7ws + 2w p ) 5

−29q 2 ls2 ws /105

i τ ∗ (7ws + 2w p ) 5

−

2w p τ 7ws τ − 5τsa 5τ pa

i τ ∗ (7ws + 2w p ) 5

−


i τ ∗ (7ws + 2w p ) 5

−


−ws τ/τsa

−22q 2 l 2p w p /105 n = 2, m = 1

(7ws + 2w p ) 5

−13q 2 ls2 ws /35 −6q 2 l 2p w p /35

n = 2, m = 2

(7ws + 2w p ) 5

−23q 2 ls2 ws /35 −2q 2 l 2p w p /35

which plays a crucial role for the transport properties of the random medium at a resonance. Its physical interpretation is deferred until Section 5. For the sake of conciseness, it is convenient to introduce the notation ws, p =

c3p,s 2c3p

+ cs3

,

(29)

which will be used repeatedly below. Note the interchange of indices s and p in the R.H.S. and L.H.S. of eq. (29). The matrices δQa and δQ inherit of the rotational symmetry of the unperturbed matrix Q(0, 0; ω) and may therefore be decomposed in terms of the projectors Pn defined in eqs (24a)–(24c). The perturbation matrix δQq depends quadratically on the components of q and requires a special treatment which is briefly discussed in the next paragraph. 3.3 Dynamic solution Following standard degenerate perturbation theory, we proceed by diagonalizing the projection of the matrix δQq in each of the subspaces spanned by Pn . Note that P0 is 1-D and the associated eigenvalue problem is therefore non-degenerate. A total of 6 mutually orthogonal ˆ n ∈ {0, 1, 2}, 0 ≤ m ≤ n is necessary to decompose the operator Q(q, ; ω). Their matrix elements are listed in Appendix eigenprojectors Pnm (q), B and are the same as those found by Müller & Miniatura (2002) in the case of light scattering by atoms. The lower index of the projectors ˆ which Pnm refers to the corresponding eigenspace of the matrix Q(0, 0; ω). In particular, we note the important relation Pn = 0≤m≤n Pnm (q), n=2,m=n m ˆ The eigenvalue λmn corresponding to in turn implies the following decomposition of the identity in polarization space: I = n=0,m=0 Pn (q). the eigenprojector Pnm is decomposed in terms of each type of perturbation in Table 1. The spectral decomposition of the perturbation matrix a ) forms the basis of the diffusion approximation. Armed with the projectors δQ obtained by perturbation theory up to order O( , q 2 , τ/τ p,s m Pn , the matrix inversion of eq. (20) is immediate and provides the structure factor γ in the Fourier domain, in the form γ (q, ; ω)i j;kl =

n=2 m=n ˆ i j;kl Pnm (q) n|t(ω)|2 + O( , q 2 ). a ) (2π )3 n=0 m=0 1 − λmn (q, ; τ p,s

(30)

Examination of the components of the projectors Pnm reveals that the tensor γ is left invariant by an interchange of the right- and left-hand indices (i, j) ↔ (k, l). This is a manifestation of the reciprocity principle at the level of the diffusion approximation. Eq. (30) is the central result of this paper and its seismological implications will be developed in the sections that follow. 4 G R E E N ’ s F U N C T I O N R E T R I E VA L A N D E Q U I PA RT I T I O N 4.1 Separation of source, transport and detection processes We begin by examining the passage from the Fourier domain solution (30) to the usual coordinate space representation. More precisely, we want to calculate the correlation tensor of multiply scattered waves excited by a seismic source (to be specified) and detected at x1,2

T 1 Ci j = lim u i (x1 , ω + /2)u j (x2 , ω − /2)∗ e−i t dt. (31) T →∞ 2π −T

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We recall that γ models the transport of correlation from one scatterer to another through a sequence of scattering events. Therefore the calculation of Cij requires that we connect the input and output channels of the tensor γ to a realistic source and to the detection points with the aid of mean Green’s functions, as illustrated in Fig. 1. For geophysical applications, we consider a moment tensor source of the form Mij s(ω), where s(ω) accounts for the source spectrum and Mij is a symmetric tensor which depends on the source mechanism. Let us write the expression to be evaluated in full, using centre-of-mass coordinates Ci j (R − R , t; r, ω) =

Mkα Mlβ ∂x3α ∂x4β (2π )7

gii (p + q/2; ω + /2)g j j (p − q/2; ω − /2)∗ gk k (p + q/2; ω + /2)gl l (p − q/2; ω − /2)∗

R3 ×R3 ×R3 ×R

× γ (q, ; ω)i j ;k l s(ω + /2)s(ω − /2)∗ eipr−ip r +iq(R−R )−i t d 3pd 3 pd 3qd |r =x3 −x4 =0 .

(32)

In eq. (32), x3 and x4 are source positions which are eventually made to coincide, and the following notations are introduced: r = x1 − x2 , R = (x1 + x2 )/2, R = (x3 + x4 )/2 (see Fig. 1 for details). The evaluation of the integral above is only possible through a series of approximations. Consistent with the diffusion approximation, we neglect the dependence of the mean Green’s functions on the wavenumber q and on the modulation frequency . This approximation is physically adequate because the structure factor γ has spatial and temporal variations which are slow compared to the mean Green’s function. As will be illustrated below, γ varies significantly on the scale of the mean free path, while the mean Green’s function oscillates at the wavelength scale. Using a similar argument, we may neglect the action of the partial derivatives on the slow variable R . Finally, we assume that the source spectrum s(ω) does not vary too rapidly in the bandwidth of interest. These approximations allow us to disentangle the detection, the source excitation and the propagation. Detection may be viewed as an operator R that contracts with the tensor γ from the left-hand side

1 gii (p; ω)g j j (p; ω)∗ eipr d 3p. (33) Ri j;i j (r, ω) = (2π )3 R3 The propagation from the first to the last scattering point is provided by the inverse Fourier transform of γ and may be decomposed as a sum of six eigenmodes n=2,m=n

ˆ i j;kl Pnm (q) n|t(ω)|2 (34) γ (R − R , t; ω)i j;kl = e−i t+iq(R−R ) d 3 qd . a ) (2π )4 n=0,m=0 R3 ×R 1 − λmn (q, ; τ p,s Finally, the source process may be viewed as an operator that contracts with the propagator γ from the right-hand side

|s(ω)|2 M M ∂ ∂ gk k (p ; ω)gl l (p ; ω)∗ e−ip r d 3p |r =0 . Sk l (m, ω) = − kα lβ xα xβ (2π )3 R3

(35)

The evaluation of the source term S requires special care and will be examined in detail in Section 6. In the next subsection, the asymptotic form of the correlation tensor of seismic waves is established. 4.2 Correlation tensor of seismic waves at long lapse time At long lapse time the dominant contribution to the correlation tensor Cij comes from the 0-mode, as may be seen by noting the singular behaviour of this mode in the limit q → 0, → 0 in eq. (30). As will be demonstrated in Section 7, all other modes have a finite lifetime, that is, they vanish exponentially fast, independent of the presence of absorption. The asymptotic form of the correlation tensor, denoted by C0 , is therefore given by

gii (p; ω)g j j (p; ω)∗ δi j δk l Sk l (m, ω) ipr+iq(R−R )−i t 3 3 n|t(ω)|2 e d pd qd C0i j (R − R , t; r, ω) = 7 a )) (2π ) 3(1 − λ0 (q, ; τ p,s R3 ×R3 ×R (R−R )2

=

n|t(ω)|2 trS(M, ω)e− 4D∗ t −t/τ 3τ ∗ (4π D ∗ t)3/2

−Img p (p; ω) pˆ i pˆ j 1 −Imgs (p; ω)(δi j − pˆ i pˆ j ) ipr 3 + × e d p. (2π )3 R3 ρ0 ω 1/τ + 1/τ pa ρ0 ω 1/τ + 1/τsa a

(36)

The overall transport of the energy is governed by the classical diffusion solution (R−R )2

e− 4D∗ t l0 (R − R , t; D ) = , (4π D ∗ t)3/2

∗

(37)

where the diffusivity of seismic waves is given by an equipartition mixture of the P and S diffusion constants D∗ =

w p v ep l p 3

+

2ws vse ls . 3

(38)

Diffusion approximation for seismic waves Anelasticity entails exponential damping of the diffusing wave and is characterized by the absorption time τ a −1 2ws τ wpτ a ∗ τ =τ + a , τsa τp

335

(39)

which is proportional to the equipartition average of the P and S absorption times. While formula (38) is formally identical to the result of Weaver (1990), it is important to remark that it introduces the energy velocities v ep,s which may differ quite significantly from the velocity in the matrix, or from the phase velocity. Section 5 is devoted to a detailed analysis of these new velocities. Note also that because point-scattering is non-preferential, that is, the amount of forward and backward scattered energy is the same, the transport mean free path reduces to the usual mean free path in the diffusivity formula. The last line of eq. (36) shows that the local correlation function of coda waves—the r dependent part—is proportional to a wavenumber integral of the longitudinal and transverse spectral functions, that is, the imaginary part of Green’s function in the (ω, p) domain. These spectral functions count the number of P and S modes per unit volume (Sheng 2006). In the case of weak absorption, eq. (36) therefore implies that the energy of S and P waves partitions in the ratio c3p (1 + τ/τ pa ) Es , =2 3 Ep cs (1 + τ/τsa )

(40)

where the slight renormalization of the wave speed in the random medium has been neglected. Eq. (40) is in exact agreement with previous results of Trégourès & van Tiggelen (2002) based on radiative transfer theory. When the absorption times of P and S waves are large compared to the mean free times as assumed in this work, the pre-factors which show up in the last line of eq. (36) are approximately equal, which leaves us with the following important formula: n|t(ω)|2 τ trS(M, ω)l0 (R − R , t; D ∗ )e−t/τ ImG i j (r, ω). 3ρ0 ωτ ∗ a

C0i j (R − R , t; r, ω) ≈ −

(41)

Eq. (41) expresses the relation between the cross-correlation of random wavefields and the Green’s function, which has seen fascinating developments recently. The above derivation shows that elastic Green’s function retrieval follows from rigorous multiple-scattering theory and allows quantitative estimates of the effect of anelasticity. In this respect, this work extends the analysis of Van Tiggelen (2003) to polarized elastic waves. As announced earlier, eq. (41) shows that the correlation tensor of seismic waves contains two different scale lengths. The diffusive part varies spatially at the scale of the mean free path, while the local correlation function oscillates at the scale of the wavelength. When the absorption times of the P and S waves are either equal or infinitely large, eq. (40) implies that S and P energies partition exactly in the ratio 2c3p /cs3 and eq. (41) becomes exact. This illustrates the equivalence between Green’s function retrieval and equipartition for multiply scattered waves as put forward by Malcolm et al. (2004) and Paul et al. (2005). In this theory, the mean Green’s function is recovered in an ensemble average sense and exhibits attenuation due to both scattering and absorption. At r = 0, formula (41) further simplifies to trS(M, ω)l0 (R − R , t; D ∗ )e−t/τ δi j , (42) 3τ ∗ where the optical theorem (6) has been used. The correlation tensor (42) is diagonal which shows that the diffusive solution is depolarized, each direction of motion being equally represented, independent of the source process. Information on the polarization of the wavefield is carried by the non-diffusive eigenmodes of the Bethe-Salpeter equation. The convergence of the correlation tensor Cij towards the depolarized state depends on the respective lifetimes of the modes and on their excitation by seismic sources. These points will be examined in Section 6. The next section pursues the investigation of the diffusive mode. a

C0i j (R − R , t; 0, ω) ≈

5 D I F F U S I O N O F E L A S T I C WAV E S : S L O W V E R S U S FA S T A N O M A L I E S In this section we evaluate the diffusivity of seismic waves which has been introduced in eq. (38). Two quantities enter in the definition of the diffusion constant. (1) The mean free paths may be evaluated from eq. (11) and the relation lp, s = cp, s τ . (2) The energy velocities may be expressed as −1 c2p,s τ dt(ω) c p,s τ dt(ω) nτ e Re + e Im , (43) v p,s = c p,s ∗ = e τ− τ cp 2ρ0 ω dω c p,s t(ω)dω where terms of order O(n2 ) have been neglected. Following the usual terminology, the sum of the last two terms inside the parenthesis of eq. (43) defines the dwell time of the waves (Van Rossum & Nieuwenhuizen 1999). There exists an interesting connection between the energy stored inside the scatterer and the dwell time. We refer the reader to Lagendijk & van Tiggelen (1996) for a detailed discussion of the electromagnetic case. Hence, the timescale τ ∗ is essentially equal to the sum of the mean free time and of the dwell time. There are two contributions to the dwell time. (1) Using the polar representation of the t-matrix, t(ω) = |t(ω)|eiφ(ω) , the last term of eq. (43) may be identified (up to a pre-factor c/ce ) as the frequency derivative of the scattering phase shift φ(ω), and is known in the literature as the Wigner delay time. (2) To elucidate the role of the middle term inside the parenthesis of eq. (43), it is profitable to differentiate the dispersion relation (k p,s )2 = ω2 /c2p,s − nRe t(ω)/ρ0 c2p,s with respect to ω. The outcome of the calculation demonstrates that the derivative of the real part of the

336

L. Margerin

Figure 2. Scattering properties of a medium with slow velocity inclusions with radius a ≈ 21 m, volume fraction 0.3 per cent, and shear (longitudinal) wave speed 350 m s−1 (700 m s−1 ) embedded in a homogeneous matrix with P and S wave speeds cp = 3000 ms−1 and cs = 1500 ms−1 , respectively. The horizontal axis of each sub-plot displays the a-dimensional frequency ka = ωa/cs . Top left-hand panel: scattering cross-section of shear waves normalized by the geometric cross-section as a function of a-dimensional frequency. A resonance is identified at the frequency f = 5 Hz (ωr = 10π ). Top right-hand panel: mean free time in seconds as a function of a-dimensional frequency. At the resonance frequency, the mean free time is of the order of 0.3 s, implying a mean free path ranging between 500 and 1000 m. Bottom left-hand panel: delay times normalized by the mean free time as a function of a-dimensional frequency. Solid line: Wigner delay time; dashed line: group delay time. At the resonance frequency, the Wigner delay time becomes large and positive. Bottom right-hand panel: velocities in the scattering medium normalized by the velocity in the matrix as a function of a-dimensional frequency. Solid line: energy velocity, dashed line: group velocity, dotted line: phase velocity. At the resonance frequency, the energy velocity is significantly reduced (≈30 per cent) compared to the matrix velocity.

t-matrix accounts for the group delay of the coherent wave. In other words, when the Wigner delay time is negligible, the energy velocity is simply given by v ep,s ≈ v gp,s ,

(44)

where v gp,s denotes the group velocity of the P, S waves in the random medium. This result is intuitively reasonable, as the group velocity is often interpreted as the speed at which energy propagates in a dispersive medium. As shown below, the group delay will be largely counterbalanced by the Wigner delay at resonance. In order to illustrate the seismological implications of the theory, we calculate the scattering properties of a medium containing either slow or fast small-scale inclusions with radius a. Based on the calculations of Margerin (2011), such a medium may be modelled by a t-matrix of the form −ρ0 cs2 /cs21 − 1 ω2 Vs , (45) t(ω) = 1 − cs2 /cs21 − 1 [2 ω2 a 2 /c2p + 2ω2 a 2 /cs2 /15 + i ω3 a 3 /c3p + 2ω3 a 3 /cs3 /9] where cp, s denote the P, S wave speeds in the matrix, cs1 is the shear wave speed inside the inclusions and Vs is the volume of the scatterer. It is further assumed that the perturbation of P and S wave speeds are perfectly correlated. For slow velocity anomalies cs1 < cs , the real part of the denominator of t(ω) vanishes at the resonance frequency ωr . A single low-frequency resonance may be incorporated in this simple model. In Fig. 2, we show the scattering cross-section, mean free time, time delays and wave velocities as a function of a-dimensional frequency ka = ωa/cs , for a medium with low-velocity inclusions with cs1 ≈ cs /4. The inclusions radius is approximately 21 m and their volume fraction is 0.3 per cent. As a consequence of the large and negative velocity contrast, a resonance shows up at a-dimensional frequency ka ≈ 0.5 (ωr = 10π ), where the scattering cross-section exceeds 25 times the geometric cross-section. The scattering cross-section at resonance takes the physically appealing form σrs =

3γ 3 λr2 , π (1 + 2γ 3 )

(46)


337

where γ is the P-to-S wave speed ratio of the matrix, and r is a subscript which stands for ‘resonance’. The large peak of σ r still satisfies the conservation of energy because the t-matrix (45) obeys the optical theorem (6). In high-contrast materials, the shear wavelength at resonance λr ≈ π acs /cs1 is much larger than the physical size of the scatterers. The relation (46) in turn allows one to understand why the scattering is so strong even at low scatterer concentrations. Consider a beam of S waves of width λr2 propagating during one period. Within the sampled volume (λr3 ), there will be on average only one obstacle, but this single object redistributes approximately half of the beam energy. Accordingly, the mean free time (Fig. 2 top right-hand panel) reaches a minimum at the resonance frequency where it equals approximately 0.25 s. This rather low value yields a mean free path of the order of 400 m for shear waves and 800 m for longitudinal waves, which is the right order of magnitude in volcanic regions (Wegler & Lühr 2001; Larose et al. 2004; Yamamoto & Sato 2010). The normalized time delays associated with the multiple scattering process are shown in Fig. 2 (bottom left-hand panel). Note that for point-scatterers, the time delays are identical for P and S waves. At low frequency, the group delay is small and positive, and becomes large and negative at the resonance frequency. The Wigner delay time is zero at low and high-frequency and becomes large and positive at the resonance frequency, where it is roughly equal to the mean free time. Hence, the waves may spend a significant amount of time inside the scatterers, which comes in addition to the typical mean free time τ spent between two scattering events (van Tiggelen et al. 1992; Lagendijk & van Tiggelen 1996). In the multiple-scattering regime, the time delays induced by scatterer resonances accumulate and may entail drastic deviations of the group and energy velocities from the velocity of the matrix. This phenomenon is illustrated in Fig. 2 (bottom right-hand panel), where the relevant velocities are plotted. The phase velocity is weakly frequency dependent and is close to the matrix velocity. The energy and group velocity are approximately equal at low-frequency, while at the resonance frequency, the group velocity exceeds the matrix velocity by a factor 2 while the energy velocity of the diffusing waves is reduced by about 30 per cent compared to the matrix value. These results are independent of the wave type. Note that the large group velocity is not necessarily unphysical but pertains to the coherent part of the wavefield only. It is important to underline that the novel aspect of the present work is not the calculation of the phase and group velocities by the mean-field approach, for which a number of other methods have been devised (see e.g. Beltzer 1988). The key result is the reduction of the diffusion velocity caused by the energy entrapment inside the scatterers, an effect which has not been considered so far in seismology. This property pertains to the mean intensity, not the mean field. Within the framework of the point-scattering model, it is found that the reduction of the diffusivity—of the order of 30 per cent—is moderate but not negligible. It implies in particular that the mean free path deduced from measurements of the diffusion constant in volcanic areas might be underestimated. It is worthwhile to underline that energy entrapment of electromagnetic waves and its subsequent impact on the speed of diffusing waves have been observed experimentally by Van Albada et al. (1991). In addition, these authors show that the low density ISA predicts remarkably well the energy velocity measured experimentally. These facts provide strong support for the theoretical analysis proposed in this paper. To illustrate the important role played by scatterer resonances, we explore the scattering properties of fast inclusions embedded in a homogeneous matrix. We consider objects with velocity cs1 = 3000 ms−1 , volume fraction 10 per cent, the other parameters being identical to the case of slow inclusions discussed above. Such fast inclusions do not give rise to resonance phenomena because the real part of the denominator of the t-matrix (45) never vanishes. Note that the velocity contrast (cs1 = 2cs ) is not as large as in the case of slow inclusions. The reason for this choice is that faster anomalies would be quite geophysically unrealistic. To compensate for the weaker contrast, I increase the volume fraction of scatterers to 10 per cent which is 30 times larger than in the slow inclusion case. The results of the calculations are shown in Fig. 3. It is noticeable that the scattering cross-section is roughly 200 times smaller for fast inclusions than for slow inclusions. As a consequence, the calculated scattering mean free time is at least one order of magnitude larger than the values reported in the seismological literature for volcanic regions. The impact of fast heterogeneities on the various velocities of the disordered medium is characterized by an increase which is typically less than 5 per cent compared to the matrix wave speed. As illustrated in Fig. 3 (bottom left-hand panel), the modest enhancement of the diffusivity is mostly caused by an increase of the group velocity in the random medium. Although the present analysis has been limited to point-like objects characterized by perfectly correlated P and S wave speeds, I expect that the dominant role of slow versus fast anomalies is not likely to be affected by a more sophisticated analysis of the scattering process based on exact solutions. The key phenomenon at work is the scattering resonance which may only be significant if slow anomalies are present. Fluid resonances, which may give rise to particularly sharp resonance peaks would deserve accrued attention. Such effects go beyond the capabilities of the simple t-matrix (45) and will not be discussed in this paper. To conclude the discussion of resonance effects, we remark that the timescale τ ∗ which enters into the definition of the diffusion constant also appears in the absorption time formula (39). In the vicinity of a resonance, the factor τ ∗ tends to increase the absorption time compared to the standard equipartition averaged values of the matrix absorption times obtained by Trégourès & van Tiggelen (2002). This result makes sense physically because we have assumed the heterogeneities to be free from absorption. Hence the time spent by the waves inside the scatterers does not contribute to the dissipation of energy. Away from resonances, we recover their result exactly.

6 R A D I AT I O N O F A M O M E N T T E N S O R S O U R C E In this section, we calculate the excitation of the eigenmodes of the Bethe-Salpeter equation by seismic sources, represented by a symmetric moment tensor. We begin by noting that if we evaluate the integral (35), and subsequently compute the partial derivatives with respect to x , the result diverges at x = 0. This is an issue of physical nature. It turns out that the mean free path dependence of the mean Green’s function in the vicinity of the source is responsible for the divergence. Such dependence, however, is not physical as the mean free path

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L. Margerin

Figure 3. Scattering properties of a medium with high-velocity inclusions with radius a ≈ 21 m, volume fraction 10 per cent, and shear (longitudinal) wave speed 3000 ms−1 (6000 ms−1 ), embedded in a homogeneous matrix with P and S wave speeds cp = 3000 ms−1 and cs = 1500 ms−1 , respectively. The horizontal axis of each subplot displays the a-dimensional frequency ka = ωa/cs . Top left-hand panel: scattering cross-section of shear waves normalized by the geometric cross-section as a function of a-dimensional frequency. Top right-hand panel: mean free time in seconds as a function of a-dimensional frequency. Bottom left-hand panel: delay times normalized by the mean free time as a function of a-dimensional frequency. Solid line: Wigner delay time; dashed line: group delay time. Bottom right-hand panel: velocities in the scattering medium normalized by the velocity in the matrix as a function of a-dimensional frequency. Solid line: energy velocity, dashed line: group velocity (eq. 44), dotted line: phase velocity.

acquires a meaning for source propagation distances that are at least larger than the wavelength. In the limit of infinitely large mean free path, a physically meaningful result should nevertheless be obtained. This offers an attractive alternative to the brute force calculation of partial derivatives outlined above. We begin by rewriting the source term as the following integral:

|s(ω)|2 Mkα Mlβ gk k (p ; ω)gl l (p ; ω)∗ pα pβ d 3p (47) Sk l (M, ω) = (2π )3 R3 which is obtained by formal differentiation under the integral sign. Evidently, the integral (47) is also divergent but it is amenable to evaluation by regularization. We propose two methods to circumvent the large p divergence in eq. (47). The first method consists in noting that in the weak scattering regime, the product of Green’s function |gp, s (p, ω)|2 is sharply peaked around p = ω/cep,s , and may be approximated with the aid of the Dirac delta function cep,s τ π ω2 2 2 δ p − e 2 . |g p,s ( p, ω)| ≈ 2 3 (48) c p,s ρ0 c p,s |ω| When the so-called ‘shell’ approximation (48) is applied, the behavior of the integrand at infinity becomes irrelevant and the wavenumber integral (35) may be evaluated in the sense of distributions. A more physical way of making sense of the integral (47) is to remark that any physical process such as detection or source excitation can never take place at a point. There is always a small and finite scale length 1/ < λp, s lp, s , at which the source operates. This implies that details smaller than 1/ are not physically relevant. Therefore, we may regularize the integral by multiplying the integrand by a factor f (, p ) =

2

2 . + p2

(49)

The function f does not modify the physics for scale lengths larger than and guarantees the convergence of the integral (47) at large p . For sufficiently large lp, s , more precisely in the limit 1 (kp, s −1 )(kp, s lp, s ), the result turns out to be independent of and identical to the


339

outcome of the shell approximation outlined above. This leaves us with the following angular integrals:

τ |s(ω)|2 ω2 Mkα Mlβ pˆ k pˆ k pˆ l pˆ l pˆ α pˆ β d 2 pˆ Sk l (M, ω) = (4π )2 c3p cep 2 4π +

τ |s(ω)|2 ω2 Mkα Mlβ (4π )2 cs3 cse 2

(δk k − pˆ k pˆ k )(δl l − pˆ l pˆ l ) pˆ α pˆ β d 2 pˆ ,

(50)

4π

which, after contraction with the moment tensor M yield the final expression of the source term Sk l (M, ω) =

τ |s(ω)|2 ω2 2tr(M 2 ) + (trM)2 δk l + 8Mk2 l + 4(trM)Mk l 2 2 3 e 420π ρ0 c p c p +

τ |s(ω)|2 ω2 2tr(M 2 ) + (trM)2 δk l + 15Mk2 l − 10(trM)Mk l . 2 2 3 e 420π ρ0 cs cs

(51)

The second term is related to the excitation of shear waves and vanishes exactly when the moment tensor is diagonal, that is, in the case of an explosion. Using formula (51), we examine in the next section how the polarization information is transported from the source to the receiver.

7 D E P O L A R I Z AT I O N O F S E I S M I C WAV E S 7.1 Evaluation of the correlation tensor Cij We now calculate the contribution of the non-diffusive modes to the correlation tensor of seismic waves (Cij ) detected at a single threecomponent sensor (r = 0). Evaluation of the integral (33) yields the following result for the detection term: 0 1 1 1 1 0 7 τ 1 1 2 P P P + , (52) + + + P + + P + P R(0, ω) = 0 1 2 2 2 6π ρ0 2 2c p 3 cs 3 2cs 3 1 5c p 3 10cs 3 j

where the expression of the projectors Pi may be found in Appendix B. By lumping together the propagation and detection terms, we obtain the correlation tensor of a given eigenmode Cnm in the form

ˆ i j;k l e−i t+iq(R−R ) 3 Sk l (M, ω)Pnm (q) λn Cnm i j (R − R , t; 0, ω) = (53) d qd , a ) (2π )4 1 − λmn (q, ; τ p,s R3 ×R

where λn is the unperturbed eigenvalue of mode n as defined in eq. (24a)–(24c). The diagonal elements of the matrix C are proportional to the kinetic energy of the wavefield along the three components of the sensor. Remarkably, the trace of the matrices Cnm (n > 0) are equal to zero and therefore do not modify the total kinetic energy. This property is easily proven by examining the components of the projectors Pnm given in Appendix B. The matrix C is also symmetric and may therefore always be diagonalized by a rotation of the sensor axes. Hence, all information on the polarization of the wavefield is contained in the three eigenvalues of the matrix C and in the orientation of the sensor with respect to a global coordinate system. The preceding discussion demonstrates that the role of the n > 0 modes is to transport information on the polarization of the wavefield from the source to the receivers. Using the symmetry of the matrix Skl and the formulas of Appendix B, it is easily seen that the modes that belong to the subspace n = 1 are projected out by moment tensor sources. If the matrix Skl were antisymmetric as a result of the application of simple torques, these modes would participate to the correlation properties of the wavefield. Clearly, the (n = 1)-modes transport polarization information for shear waves exclusively and have a finite lifetime t1 given by t1 =

c3p /cs3 τ ∗ λ1 τ ∗ = 1 − λ1 1 + c3p /cs3

(54)

which is slightly smaller than the mean free time. We emphasize that the n = 1 modes are not absorbed but disappear because of the mixing of polarizations caused by scattering. We are left with the evaluation of the contribution of the (n = 2)-modes. These modes have a lifetime 2 + 7c3p /cs3 τ ∗ λ2 τ ∗ , = (55) t2 = 1 − λ2 3 1 + c3p /cs3 which is typically of the order of two mean free times. These modes suffer from an additional decay due to dissipation with a characteristic time −1 2w p τ 7ws τ + . (56) τ2a = τ ∗ 5τsa 5τ pa From eq. (56), one may deduce that the n = 2 modes transport polarization information for shear and longitudinal waves in the ratio 7ws /2wp , which is of the order of 25. Their spatio-temporal dependence may be calculated, based on the integral formulas given in Appendix C and the

340

L. Margerin

expression of the projectors P2m given in Appendix . Introducing the constants d2m 3 2 3 2 3 2 29c p cs + 22cs3 c2p τ 2 13c p cs + 6cs3 c2p τ 2 23c p cs + 2cs3 c2p τ 2 0 1 2 d2 = d2 = d2 = , 21 7c3p + 2cs3 τ ∗ 7 7c3p + 2cs3 τ ∗ 7 7c3p + 2cs3 τ ∗

(57)

and denoting by r the source station vector R − R , the contribution of each (n = 2)-mode may be expressed as a

C20 i j (R − R , t; 0, ω) =

e−(1/τ2 +1/t2 )t Skl (M, ω) 6τ ∗ × δi j δkl (l0 − 6l4 + 9l1 )(r, t; d20 ) + 9(δik δ jl + δil δ jk )l1 (r, t; d20 ) + (δi j rˆk rˆl + δkl rî rˆ j )(9l2 − 3l5 )(r, t; d20 ) + 9(δil rˆ j rˆk + δ jl rî rˆk + δik rˆ j rˆl + δ jk rî rˆl )l2 (r, t; d20 ) + 9ˆri rˆ j rˆk rˆl l3 (r, t; d20 ) ,

(58)

a

C21 i j (R − R , t; 0, ω) =

e−(1/τ2 +1/t2 )t Skl (M, ω) 2τ ∗ × (δik δ jl + δil δ jk )(2l4 − 4l1 )(r, t; d21 ) − 4δi j δkl l1 (r, t; d21 ) + (δik rˆ j rˆl + δ jl rî rˆk + δil rˆ j rˆk + δ jk rî rˆl )(l5 − 4l2 )(r, t; d21 ) −4(δi j rˆk rˆl + δkl rî rˆ j )l2 (r, t; d21 ) −4ˆri rˆ j rˆk rˆl l3 (r, t; d21 )

(59)

a

C22 i j (R − R , t; 0, ω) =

e−(1/τ2 +1/t2 )t Skl (M, ω) 2τ ∗ × (δik δ jl + δil δ jk )(l0 − 2l4 + l1 )(r, t; d22 ) + δi j δkl (2l4 − l0 + l1 )(r, t; d22 ) + (δik rˆ j rˆl + δil rˆ j rˆk + δ jk rî rˆl + δ jl rî rˆk )(l2 − l5 )(r, t; d22 ) +(δi j rˆk rˆl + δkl rî rˆ j )(l2 + l5 )(r, t; d22 ) +ˆri rˆ j rˆk rˆl l3 (r, t; d22 ) .

(60)

The functions li , i = {1, 2, 3, 4, 5} are defined in Appendix and l0 is defined in eq. (37). These functions have been introduced for notational simplicity and are not linearly independent. It is important to note that while the constants d2m have units of diffusivity, the functions li do not describe diffusion processes but rather the relaxation towards the depolarized state. The typical depolarization time is rather short, of the order of two mean free times only.

7.2 Application to seismic sources Eqs (58)–(60) show that the excitation of each mode depends on the source type and on the orientation of the source-station vector R − R with respect to the source axes. Armed with these formulas we may examine the imprint left by the source process on the polarization of multiply scattered waves, within the limits of our diffusion approximation. It must first be noted that in the diffusive regime, the sensitivity to the source process is considerably reduced compared to the ballistic regime. It is for instance striking that two moment tensors such as ⎛ ⎞ ⎛ ⎞ 0 1 0 −1 0 0 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ M =⎜ (61) 1 0⎟ ⎝ 1 0 0 ⎠, and M = ⎝ 0 ⎠ 0

0

0

0

0

0

are indistinguishable because they yield the same source term Skl . In addition, because the trace of the projectors Pnm (n > 1) over the rightor left-hand indices is exactly 0, the diffusion approximation will not carry any polarization information for source terms Skl which are proportional to the identity matrix. This includes in particular explosive sources, which can only excite the diffusive mode (n = 0). Obviously, depolarization and equipartition of the wavefield is not instantaneous for such sources but it cannot be studied within the framework of our diffusion approximation. In view of this limitation, we restrict the present discussion to the case of traceless moment tensor sources. Two representative examples are illustrated in Figs 4 and 5, where the strike-slip source (61) and the compensated linear vector dipole ⎛ ⎞ −1 0 0 ⎟ 1⎜ (62) M= ⎜ 0 −1 0 ⎟ ⎝ ⎠ 2 0 0 2


341

Figure 4. Illustration of the depolarization process in the seismic coda. The kinetic energies have been calculated at a receiver located four mean free paths away from the seismic source along the x-axis. Dotted line: x component; dashed line: y component; solid line: z component. Note that the x and y components are superposed. The horizontal axis shows the lapse time in mean free time units. Left-hand panel: strike-slip fault along the x-axis. Right-hand panel: tensile crack opening in the z direction.

Figure 5. Illustration of the depolarization process in the seismic coda. The kinetic energies have been calculated at a receiver located four mean free paths away from the seismic source along the z-axis. Dotted line: x component; dashed line: y component; solid line: z component. The horizontal axis shows the lapse time in mean free time units. Left-hand panel: strike-slip fault along the x-axis. Right-hand panel: tensile crack opening in the z direction.

are compared. The three diagonal elements of the complete correlation tensor Cij are plotted at a distance of four shear mean free paths from the source as a function of time (in mean free time units). The beginning of the time window coincides with the arrival of the ballistic shear waves (not shown). Note that the standard diffusion result is obtained by taking the trace of the tensor Cij . Figs 4 and 5 demonstrate that the energy partitioning in the coda depends critically on the source mechanism. The distribution of the kinetic energy on the three components of the motion may differ from the depolarized state by about 20 per cent, a signature that persists up to lapse times of the order of 6 mean free times in the coda. In particular, it is noticeable that the shear (clvd) source generates considerably more horizontal (vertical) motions than vertical (horizontal) motions. This property may facilitate the identification of the source mechanism in volcanic areas where the ballistic waves may be severely attenuated by scattering and anelasticity. Alternatively, the observation by Rautian & Khalturin (1978) that the coda envelopes do not exhibit signatures of the radiation pattern at lapse times of the order of 3ts (the ballistic time of shear waves) can be used to give some bounds on the mean free time in the lithosphere. Considering typical source station distance of the order of 200 km, this implies a mean free time of the order of 30 s or less. A word of caution is nevertheless necessary: polarization information may be preserved on much longer timescales if the scattering is strongly anisotropic. In the case of crustal propagation, scattering anisotropy is important to explain envelope broadening with distance. Since the present theory is valid for point scatterers and not for scatterers of arbitrary size, more work is needed to give precise estimates of the mean free time from polarization analysis.

8 C O N C LU S I O N A N D O U T LO O K In this work, a solution of the Bethe-Salpeter equation for point-scatterers has been developed in the regime of slow spatial and temporal variations. The transport of polarization information from source to receivers has been obtained in analytical form for arbitrary moment tensor sources. A formula for the diffusivity of seismic waves incorporating the effect of scattering delays has been established, which highlights the crucial role played by resonant scattering in high-contrast elastic media. A comparative study of slow versus fast anomalies suggests that the very small mean free paths values reported in the volcano-seismological literature are likely due to the presence of low-velocity inclusions, possibly related to the presence of fluid-filled cavities in volcanic edifices. More generally, the sensitivity of coda waves leans strongly towards low-velocity heterogeneities in high-contrast materials.

342

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From the perspective of Green’s function retrieval, we have shown that point scatterers play a role similar to a distribution of homogeneously distributed and randomly oriented uncorrelated forces in the correlation theory of noise sources. There is however an important distinction between the clean and scattering media. In a clean, absorbing medium, Green’s function retrieval from noise sources is valid only when the absorption times of P and S waves are exactly equal or if the noise sources compensate exactly for the local dissipation rates of P and S waves (Wapenaar et al. 2006). The presence of scattering alleviates the impact of dissipation, so that the amplitude of the P and S waves in the coda are always approximately the same as in the ‘true’ Green’s function. This contrasts with the case of the high-frequency noise wavefield, which is in many cases excited by sources that are located at close distance from the receivers. In such a situation, the local amount of propagating P and S waves is controlled by their absorption rates and by the excitation mechanism of the noise sources, and may differ very significantly from what is expected from equipartition theory. This work represents a first effort to develop a multiple scattering theory for small scatterers beyond the weak perturbation regime. This theory is still not quite satisfactory as it does not yield information on the depolarization rate of explosion sources, which are commonly used in seismological experiments. Modelling such sources requires to go beyond the regime of slow spatio-temporal variations adopted in this work. This problem may be tackled by considering better approximations of the transport kernel γ by either perturbative or numerical approaches. AC K N OW L E D G M E N T S The author would like to thank an anonymous reviewer and Yingcai Zheng for their careful comments which helped to improve the quality of the manuscript. REFERENCES Aki, K., 1969. Analysis of the seismic coda of local earthquakes as scattered waves, J. geophys. Res., 74, 615–631. Aki, K. & Chouet, B., 1975. Origin of coda waves: source, attenuation, and scattering effects, J. geophys. Res., 80, 3322–3342. Akkermans, E. & Montambaux, G., 2007. Mesoscopic Physics of Electrons and Photons, Cambridge University Press, Cambridge. Anugonda, P., Wiehn, J.S. & Turner, J.A., 2001. Diffusion of ultrasound in concrete, Ultrasonics, 39, 429–435. Beltzer, A., 1988. Dispersion of seismic waves by a causal approach, Pure appl. Geophys., 128, 147–156. Campillo, M. & Paul, A., 2003. Long-range correlations in the diffuse seismic coda, Science, 299, 547–549. de Vries, P., van Coevorden, D.V. & Lagendijk, A., 1998. Point scatterers for classical waves, Rev. Mod. Phys., 70, 447–466. Frisch, U., 1968. Wave propagation in random media, in Probabilistic Methods in Applied Mathematics, pp. 75–198, ed. Bharucha-Reid, A.T., Academic Press, New York. Hennino, R., Trégourès, N., Shapiro, N.M., Margerin, L., Campillo, M., van Tiggelen, B.A. & Weaver, R.L., 2001. Observation of equipartition of seismic waves, Phys. Rev. Lett., 86, 3447–3450. Lagendijk, A. & van Tiggelen, B.A., 1996. Resonant multiple scattering of light, Phys. Rep., 270, 143–215. Larose, E., Margerin, L., van Tiggelen, B. & Campillo, M., 2004. Weak localization of seismic waves, Phys. Rev. Lett., 93, 48501. Lobkis, O.I. & Weaver, R.L., 2001. On the emergence of the Green’s function in the correlations of a diffuse field, J. acoust. Soc. Am., 110, 3011–3017. Malcolm, A.E., Scales, J. & van Tiggelen, B.A., 2004. Extracting the Green function from diffuse, equipartitioned waves, Phys. Rev. E, 70, 015601, doi:10.1103/PhysRevE.70.015601. Margerin, L., 2011. Mean-field T-matrix approach to elastic wave scattering by small and point-like objects, Waves in Random and Complex Media, 21, 628–644. Margerin, L. & Sato, H., 2011. Generalized optical theorems for the reconstruction of Green’s function of an inhomogeneous elastic medium, J. acoust. Soc. Am., 130, 3674–3690. Margerin, L., Campillo, M. & van Tiggelen, B., 2000. Monte carlo simulation of multiple scattering of elastic waves, J. geophys. Res., 105, 7873–7892. Margerin, L., Campillo, M., van Tiggelen, B.A. & Hennino, R., 2009. Energy partition of seismic coda waves in layered media: theory and application to Pinyon Flats Observatory, Geophys. J. Int., 177, 571–585. Müller, C. & Miniatura, C., 2002. Multiple scattering of light by atoms

with internal degeneracy, J. Phys. A, 35, 10163, doi:10.1088/03054470/35/47/314. Nakahara, H. & Margerin, L., 2011. Testing Equipartition for S-Wave Coda Using Borehole Records of Local Earthquakes, Bull. seism. Soc. Am., 101, 2243–2251. Papanicolaou, G., Ryzhik, L. & Keller, J., 1996. Stability of the P-to-S energy ratio in the diffusive regime, Bull. seism. Soc. Am., 86, 1107– 1115. Paul, A., Campillo, M., Margerin, L., Larose, E. & Derode, A., 2005. Empirical synthesis of time-asymmetrical Green functions from the correlation of coda waves, J. geophys. Res., 110, 08302, doi:10.1029/2004JB003521. Przybilla, J. & Korn, M., 2008. Monte-Carlo simulation of radiative energy transfer in continuous elastic random media-three-component envelopes and numerical validation, Geophys. J. Int., 173, 566–576. Rautian, T. & Khalturin, V., 1978. The use of the coda for determination of the earthquake source spectrum, Bull. seism. Soc. Am., 68, 923–948. Rossetto, V., Margerin, L., Planès, T. & Larose, E., 2011. Locating a weak change using diffuse waves: theoretical approach and inversion procedure, J. appl. Phys., 109, 034903, doi:10.1063/1.3544503. Ryzhik, L., Papanicolaou, G. & Keller, J., 1996. Transport equations for elastic and other waves in random media, Wave Motion, 24, 327– 370. Sánchez-Sesma, F.J. & Campillo, M., 2006. Retrieval of the Green’s function from cross correlation: the canonical elastic problem, Bull. seism. Soc. Am., 96, 1182–1191. Sato, H., 1994. Multiple isotropic scattering model including P-S conversions for the seismogram envelope formation, Geophys. J. Int., 117, 487– 494. Sato, H. & Fehler, M., 1998. Seismic Wave Propagation and Scattering in the Heterogeneous Earth, AIP Press/Springer Verlag, New York. Sato, H., Fehler, M. & Dmowska, R., Eds., 2008. Advances in Geophysics, Vol. 50: Earth Heterogeneity and Scattering Effects on Seismic Waves, Elsevier Science & Technology, Amsterdam, The Netherlands. Sheng, P., 2006. Introduction to Wave Scattering Localization and Mesoscopic Phenomena, Springer, Berlin. Trégourès, N.P. & van Tiggelen, B.A., 2002. Generalized diffusion equation for multiple scattered elastic waves, Waves Random Media, 12, 21–38. Turner, J.A., 1998. Scattering and diffusion of seismic waves, Bull. seism. Soc. Am., 88, 276–283. Turner, J.A. & Weaver, R.L., 1994. Radiative transfer of ultrasound, J. acoust. Soc. Am., 96, 3654–3674. Turner, J.A. & Weaver, R.L., 1995. Time dependence of multiply scattered diffuse ultrasound in polycrystalline media, J. acoust. Soc. Am., 97, 2639– 2644.

Diffusion approximation for seismic waves Van Albada, M.P., Van Tiggelen, B.A., Lagendijk, A. & Tip, A., 1991. Speed of propagation of classical waves in strongly scattering media, Phys. Rev. Lett., 66, 3132–3135. Van Rossum, M. & Nieuwenhuizen, T., 1999. Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion, Rev. Mod. Phys., 71, 313–371. Van Tiggelen, B.A., 2003. Green function retrieval and time reversal in a disordered world, Phys. Rev. Lett., 91, 243904. Van Tiggelen, B.A., Lagendijk, A., van Albada, M.P. & Tip, A., 1992. Speed of light in random media, Phys. Rev. B, 45, 12 233– 12 243. Wapenaar, K., Slob, E. & Snieder, R., 2006. Unified Green’s function retrieval by cross correlation, Phys. Rev. Lett., 97, 234301, doi:10.1103/PhyRevLett.97.234301. Weaver, R.L., 1982. On diffuse waves in solid media, J. acoust. Soc. Am., 71, 1608–1609.

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Weaver, R.L., 1990. Diffusivity of ultrasound in polycrystals, J. Mech. Phys. Sol., 38, 55–86. Wegler, U., 2004. Diffusion of seismic waves in a thick layer: Theory and application to Vesuvius volcano, J. geophys. Res., 109, B07303, doi:10.1029/2004JB003048. Wegler, U. & Lühr, B.-G., 2001. Scattering behaviour at Merapi volcano (Java) revealed from an active seismic experiment, Geophys. J. Int., 145, 579–592. Wiersma, D., Van Albada, M., Van Tiggelen, B. & Lagendijk, A., 1995. Experimental evidence for recurrent multiple scattering events of light in disordered media, Phys. Rev. Lett., 74, 4193–4196. Yamamoto, M. & Sato, H., 2010. Multiple scattering and mode conversion revealed by an active seismic experiment at Asama volcano, Japan, J. geophys. Res., 115, B07304, doi:10.1029/2009JB007109. Zeng, Y., 1993. Theory of scattered P-and S-wave energy in a random isotropic scattering medium, Bull. seism. Soc. Am., 83, 1264–1276.

APPENDIX A: SOME USEFUL INTEGRAL FORMULAS In this appendix, we give some key formulas which facilitate the evaluation of integrals of products of Green’s functions. Following Van Rossum & Nieuwenhuizen (1999) let us define

+∞ Ik,l (m 1 , m 2 ) =

p2 d p ( p2

0

+∞ I˜k,l (m 1 , m 2 ) = 0

−

m 21 )k ( p2

(A1)

− m 22 )l

p4 d p . ( p2 − m 21 )k ( p2 − m 22 )l

(A2)

Using the following recurrence relations: Ik+1,l (m 1 , m 2 ) =

1 ∂ Ik,l (m 1 , m 2 ) 2km 1 ∂m 1

(A3)

Ik,l+1 (m 1 , m 2 ) =

1 ∂ Ik,l (m 1 , m 2 ) 2lm 2 ∂m 2

(A4)

m 1 ∂ Ik,l (m 1 , m 2 ) ˜ m 2 ∂ Ik,l (m 1 , m 2 ) I˜k+1,l (m 1 , m 2 ) = Ik,l (m 1 , m 2 ) + Ik,l+1 (m 1 , m 2 ) = Ik,l (m 1 , m 2 ) + 2k ∂m 1 2l ∂m 2

I1,1 =

iπ , 2(m 1 − m 2 )

(A5)

(A6)

the integrals of products of Green’s function containing an arbitrary number of factors may be evaluated. Armed with these results, let us show that the cross-terms involving the products of the P and S parts of Green’s function are negligible. Let us consider the integral

∞ I ps =

g p ( p; ω)gs ( p; ω)∗ p2 d p.

(A7)

0

Identifying m1 with ω/cep + i/2l p and m2 with ω/cse − i/2ls , Ips may be rewritten as I ps = =

I1,1 (ω/cep + i/2l p , ω/cep − i/2ls ) ρ02 c2p cs2

2ρ02 c2p cs2

iπ 1 ω cep − c1se + i l1p +

1 ls

!.

Because of the finite separation between the P and S propagation speeds, Ips vanishes in the limit kp, s lp, s → ∞.

(A8)

344

L. Margerin

A P P E N D I X B : E I G E N P R O J E C T O R S O F T H E M AT R I X Q(q, ; ω) In this appendix, we provide the matrix elements of the eigenprojectors which serve to diagonalize (approximately) the operator Q. It is ˆ convenient to introduce the following short-hand notations for the (q-dependent) longitudinal and transverse projectors L i j = qˆ i qˆ j

(B1)

Ti j = δi j − qˆ i qˆ j .

(B2)

The projector P0 which gives the diffusive behavior is independent of qˆ P0i j;kl =

1 δi j δkl . 3

(B3)

ˆ P11 (q), ˆ with associated eigenspaces of dimension 2 and 1, The eigenprojector P1 may be written as the sum of the projectors P10 (q), respectively: ˆ i j;kl = P10 (q)

1 Tik T jl − Til T jk 2

(B4)

ˆ i j;kl = P11 (q)

1 Tik L jl + L ik T jl − Til L jk − L il T jk . 2

(B5)

The eigenprojector P2 may similarly be written as a sum of three projectors P20 , P21 , P22 which span subspaces of dimension 1, 2 and 2, respectively ˆ i j;kl = P20 (q)

1 Ti j − 2L i j (Tkl − 2L kl ) 6

(B6)

ˆ i j;kl = P21 (q)

1 Tik L jl + L ik T jl + Til L jk + L il T jk 2

(B7)

ˆ i j;kl = P22 (q)

1 1 Tik T jl + Til T jk − Ti j Tkl . 2 2

(B8)

It may readily be checked that these projectors are mutually orthogonal, and that their sum equals the identity matrix in polarization space.

A P P E N D I X C : I N T E G R A L F O R M U L A S F O R T H E C A L C U L AT I O N O F T H E C O R R E L AT I O N T E N S O R C To calculate the double inverse Fourier transforms in eq. (53), we make repeated use of some integral formulas which are briefly derived below. Define

1 2 e−dq t+iqr qˆ i qˆ j qˆ k qˆ l d 3 q (C1) I a (r, t; d)i jkl = (2π )3 R3 and I b (r, t; d)i j =

1 (2π )3

e−dq

2 t+iqr

R3

qˆ i qˆ j d 3 q.

(C2)

Performing the angular integrals leaves us with

(δi j δkl + 2 perm.) +∞ −dq 2 t e j2 (qr )dq I a (r, t; d)i jkl = 2π 2 r 2 0

(δi j rˆk rˆl + 5 perm.) +∞ −dq 2 t e j3 (qr )qdq − 2π 2 r 0

rî rˆ j rˆk rˆl +∞ −dq 2 t e j4 (qr )q 2 dq + 2π 2 0 and: I b (r, t; d)i j =

δi j 2π 2 r

+∞ 0

e−dq t j1 (qr )qdq − 2

rî rˆ j 2π 2

0

+∞

(C3)

e−dq t j2 (qr )q 2 dq. 2

(C4)


345

In eq. (C3), the symbol ‘x perm.’ implies that the parenthesis contains x additional terms obtained by permutations and the notation jn refers to the usual spherical Bessel functions. After performing the wavenumber integrals, one arrives at I a (r, t; d)i jkl = l1 (r, t; d)(δi j δkl + 2 perm.) + l2 (r, t; d)(δi j rˆk rˆl + 5 perm.) + l3 (r, t; d)ˆri rˆ j rˆk rˆl

(C5)

I b (r, t; d)i j = l4 (r, t; d)δi j + l5 (r, t; d)ˆri rˆ j ,

(C6)

where the following functions have been introduced √ √ 2 3e−r /4dt dt (r 2 − 6dt)erf (r/2 dt) + l1 (r, t; d) = 4π 3/2 r 4 8πr 5

l2 (r, t; d) =

√ 2 3(10dt − r 2 )erf (r/2 dt) (r 2 + 15dt)e−r /4dt − √ 8πr 5 4π 3/2 r 4 dt

l3 (r, t; d) =

(r 4 + 20r 2 dt + 210d 2 t 2 )e−r (4π dt)3/2 r 4

l4 (r, t; d) =

√ 2 erf (r/2 dt) e−r /4dt − √ 4πr 3 4π 3/2 r 2 dt

l5 (r, t; d) =

e−r

2 /4dt

−

√ 15(14dt − r 2 )erf (r/2 dt) 8πr 5

√ (r 2 + 6dt) 3erf (r/2 dt) − . (4π dt)3/2 r 2 4πr 3

(C7)

(C8)

(C9)

(C10)

2 /4dt

(C11)

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