COHERENT BACK-SCATTERING AND WEAK LOCALIZATION OF SEISMIC WAVES

ADVANCES IN GEOPHYSICS, VOL. 50, CHAPTER 1 COHERENT BACK-SCATTERING AND WEAK LOCALIZATION OF SEISMIC WAVES Ludovic Margerin Abstract I present a review of the weak localization effect in seismology. To understand this multiple scattering phenomenon, I begin with an intuitive approach illustrated by experiments performed in the laboratory. The importance of reciprocity and interference in scattering media is emphasized. I then consider the role of source mechanism, again starting with experimental evidence. Important theoretical results for elastic waves are summarized, that take into account the full vectorial character of elastic waves. Applications to the characterization of heterogeneous elastic media are discussed. Key Words: Multiple scattering, interference, reciprocity, elastic waves. ! 2008 Elsevier Inc. 1. Introduction In strongly scattering media, the propagation of multiply-scattered waves is best described by considering the transport of the energy. An elastic scattering medium is an inhomogeneous medium where the wavespeed and the density vary laterally. It can also contain embedded obstacles such as cracks or cavities. Upon propagation, an incoming plane wave with well-defined wavevector k will transfer energy to all possible space directions, a phenomenon known as scattering. The energy transport approach has been developed by astrophysicists at the beginning of the twentieth century and has given birth to the theory of radiative transfer (Chandrasekhar, 1960; Apresyan and Kravtov, 1996). Phenomenologically, the transfer equation for acoustic, electromagnetic, and elastic waves can be derived from a detailed local balance of energy that neglects the possible interference between wave packets (see, e.g., Sato, 1994; Sato and Fehler, 1998; and Margerin, 2005, for seismic applications). This important assumption is justified by the fact that the phase of the wave is randomized by the scattering events. Thus at a given point, the field can be written as a sum of random phasors and on average, intensities can be added, rather than amplitudes. This seemingly convincing argument can actually be shown to be wrong and one of the goals of this paper will be to put forward the role of interferences in scattering media in specific cases. To begin with, we present the results of an experiment of ultrasound propagation in a granular material, which can be defined as a material containing many individual solid particles with arbitrary sizes. An array of 128 acoustic transducers has been placed at the surface of a box with lateral dimensions 0.15 m ! 0.15 m ! 0.15 m containing commercial sand for aquariums. The sand does not have a well-defined granulometry but the typical size of a grain is "2 mm. The central transducer emits a short pulse in the 1#1.5 MHz frequency range and the waves are recorded 1 # 2008 Elsevier Inc. All rights reserved. ISSN: 0065-2687 DOI: 10.1016/S0065-2687(08)00001-0 2 MARGERIN by the whole array. The wavespeed of the dominant ballistic pulse is about 1000 m/s, which gives a dominant wavelength of "0.8 mm, which is of the same order as the size of one transducer of the array ("0.55 mm). The logarithm of the energy of the wavefield along the array after time averaging over four cycles is shown in Fig. 1, as a function of distance from the center of the array in millimeters (horizontal axis) and time in microseconds (vertical axis). In this figure, one can identify direct waves that propagate along the array and decay exponentially with distance because of the energy losses due to scattering. They are followed by a diffuse coda, which can be thought of as the result of the random walk of the energy p inffiffiffiffithe ffi scattering medium. As a rule of thumb, the multiple scattering halo grows like Dt, where D = ul*/3 is the diffusion constant of the waves in the medium and u is the wave velocity. The transport mean free path l* (see, e.g., Sheng, 1995, for a rigorous definition) represents the typical step length of the random walk of the energy in the scattering medium and is much larger than the wavelength. In the case of sand samples, the transport mean free path is roughly 10 times larger than the wavelength. According to diffusion theory (Akkermans and Montambaux, 2005), at fixed time t = t0, the energy distribution in the scattering medium is approximately proportional to 2 % e#3r =ð4ut0 l Þ , where r is the distance from the source. Therefore, at fixed time, the energy in the diffuse halo is expected to vary significantly on the scale of l*. Yet, at the center of Fig. 1, the reader will notice a clear, but highly localized increase of intensity. This is not 20 −4 30 −5 40 Time (microsec) Diffuse halo −3 −6 50 −7 60 70 −8 80 Weak localization 90 −9 −10 100 −11 Log (energy) Ballistic waves 10 110 −12 −60 −40 −20 0 20 Distance (mm) 40 60 FIG. 1. Energy of the wavefield recorded at the surface of a granular material as a function of time. A short pulse with a central frequency of 1.2 MHz is shot at the central transducer. Direct waves propagating along the array rapidly attenuate. They are followed by coda waves that form a diffuse halo in the medium. Note the sharp increase of intensity at the center of the array, where energy was initially released. Experiment performed at the Laboratoire Ondes et Acoustique, Paris, by R. Hennino and A. Derode. COHERENT BACK-SCATTERING AND WEAK LOCALIZATION 3 an artifact. In this particular experiment, the typical width of the zone of enhanced intensity is roughly equal to the size of one transducer. Other experiments, to be described later, have demonstrated that the zone of enhancement actually coincides with the central wavelength of the waves, which is the clear signature of an interference effect that takes place around the source in the multiple scattering medium: this is known as weak localization. In the next section, I provide a simple explanation for this observation. 2. Weak Localization Effect: A Heuristic View In what follows, I represent a scalar partial wave as a complex number c = Aeif, where A and f are real numbers denoting the amplitude and phase, respectively. Each partial wave follows an arbitrarily complicated scattering path from source to receiver in the medium. At a given point, the measured field u is a superposition of a large number of partial waves that have propagated along different scattering paths u¼ X Aj eifj ; ð1Þ j where Aj and fj are random and uncorrelated because of the multiple scattering events, and j can be understood as a “label” for the different paths. The representation (1) is strictly valid for point scatterers and will suffice for the present purposes. Typical examples of scattering paths are shown in Fig. 2. In Eq. (1), I now pair direct and reciprocal scattering paths as shown in Fig. 2. The direct and reciprocal paths are characterized by the fact that the same scatterers are visited, but the sequences of scattering events are opposite. To illustrate this definition, in Fig. 2, the sequence S, A, B, C, D, R (solid line) and S, D, C, B, A, R (dashed line) represent the direct and reciprocal paths, respectively. One obtains u¼ X" j0 # cdj0 þ crj0 ; ð2Þ where c denotes the complex partial waves, the superscripts d and r stand for “direct” and “reciprocal,” and a new label j0 has been introduced to emphasize the new representation of the field. The intensity I is proportional to |u|2 and reads I¼ X" j0 ;k0 cdj0 þ crj0 #$ % cdk0 þ crk0 ; ð3Þ where the overbar denotes complex conjugation. In Eq. (3), it is reasonable to assume that the waves visiting different scatterers will have random phase differences and after averaging over scatterer positions will have no contribution. Thus, we can restrict the summation to the case j0 ¼ k0 to obtain I¼ X'&& &&2 && &&2 ( X ðcdj0 crj0 þ cdj0 crj0 Þ: &cdj0 & þ &crj0 & þ j0 j0 ð4Þ 4 MARGERIN Configuration 1 S S R R D D A A C C B B Configuration 2 S R R S C C D D B B A A FIG. 2. Examples of multiple scattering paths from source S to receiver R. Scattering events are labeled with letters A, B, C, and D. Solid and dashed lines represent direct and reciprocal paths, respectively. The two configurations differ by the position of the scattering events. On the left: source and receiver coincide. On the right, source and receiver are typically a few wavelengths apart. Reprinted figure with permission from Margerin et al. (2001). Blackwell Publishing. The first sum in Eq. (4) represents the usual incoherent contribution to the measured intensity, which can be calculated with radiative transfer theory (see Wegler et al., 2006 for recent applications). The second sum can be interpreted as the interference term between the direct and reciprocal paths in the scattering medium. In a reciprocal medium, that is, a medium where the reciprocity principle is verified, the amplitude and phase of the direct and reciprocal wave paths are exactly the same, that is, Adj ¼ Arj and fdj ¼ frj , provided that source and receiver are located at the same place. Therefore, the total intensity which includes the interference term is exactly double of the classical incoherent term. This is the interference term which causes the intensity to be higher in the experiment shown in Fig. 1. Reciprocity is a general property of wave equations such as the acoustic and elastic wave equation. In a simple scalar case, it means that the response measured at r2 due to a source at r1 is the same as the response measured at r1 due to a source at r2. This remarkable property can be broken when an external field acts on the system. An interesting seismic example of broken reciprocity is the effect of the Coriolis force on the seismic wave motion at long period where the effect of the rotation of the Earth is important. A generalized reciprocity relation can still be given upon exchange of COHERENT BACK-SCATTERING AND WEAK LOCALIZATION 5 source and receiver, which involves the inversion of the instantaneous rotation vector of the Earth (see Dahlen and Tromp, 1998, for a thorough discussion). Although it had been theoretically predicted in several pioneering papers published around 1970 (Watson, 1969; de Wolf, 1971; Barabanenkov, 1973), it is only in the mideighties that the role of interference in multiple scattering has been appreciated with the discovery of the coherent backscattering of light (Kuga and Ishimaru, 1984; van Albada and Lagendijk, 1985; Wolf and Maret, 1985; Kaveh et al., 1986). Later the coherent backscattering effect has been predicted and observed for acoustic and elastic waves in both stationary and dynamic experiments (Bayer and Niederdrank, 1993; Sakai et al., 1997; Tourin et al., 1997; de Rosny et al., 2000). Today, coherent backscattering or weak localization is still a very active topic of research. The coherent backscattering for moving scatterers has been studied by Snieder (2006) and Lesaffre et al. (2006). Derode et al. (2005) have used the coherent backscattering effect to measure the heterogeneity of human bones. Aubry and Derode (2007) have devised an ingenious technique to measure lateral variations of the diffusion constant of strongly scattering media based upon the separation of the incoherent and coherent intensities. Note that the term “coherent backscattering” refers to the intensity enhancement observed in a small cone of direction in the far-field of a disordered sample for plane ^ Although the basic physics of coherent wave sources with fixed incident direction k. backscattering and weak localization are identical, the latter term indicates that the loops of interference occur inside the disordered sample, and should therefore be preferred to describe the seismic experiments. These interference loops result in a deviation from the diffusive behavior (Haney and Snieder, 2003). When the wavelength is of the same order as the mean free path, the interference effects can completely block the transport of energy away from the source, a phenomenon known as strong localization (see, e.g., Sheng, 1995; Akkermans and Montambaux, 2005). Weak localization is therefore a basic phenomenon to explain the transition from the diffuse to the localized propagation regime. Note that there exists a number of other mechanisms of intensity enhancement. One of the most famous is the “opposition effect” in astrophysics, which manifests itself as an increase of the reflectance of celestial bodies such as the Moon when the light of the Sun reflected from the regolith is observed close to the backscattering direction. Hapke et al. (1993) have shown that the opposition effect is partly explained by the coherent backscattering of light. I refer the interested reader to the paper by Barabanenkov et al. (1991) for an extensive review of backscattering enhancement phenomena in optics. In particular, these authors discuss the case of backscattering enhancement by several deterministic scatterers which can also be of interest in seismology. Let me finally point out that weak localization is only one manifestation of the role of the phase in the seismic coda (see Campillo, 2006, for a review). I have shown with a very simple argument that interference effects have to be incorporated in the usual transport theory, but for the moment, it has not been explained why the enhancement due to interference is so highly localized. In Fig. 2, I schematically represent the more usual case where source and detection are not collocated. In that case there is a phase shift between the two wavepaths which is acquired during the propagation from the source to the first scattering event and from the last scattering event to the receiver. Clearly, if the distance between source and detection is “large enough,” the phase shift will fluctuate randomly from one configuration of the scatterers to the other. Therefore, the interference term is expected to vanish upon averaging when source and receiver are sufficiently far apart. One of the goals of the paper is to demonstrate and 6 MARGERIN illustrate the fact that the enhancement zone is actually narrow and about the size of the wavelength. A final comment on the role of multiple scattering is in order. It is clear that the representation of the wavefield [Eq. (3)] makes sense only if one can pair the direct and reciprocal propagation paths. If there is a single scattering event, there is only one possible path from source to receiver and therefore no interference is possible. This basic observation proves that weak localization is indeed a genuine multiple scattering effect. In what follows, I pursue the experimental approach of weak localization by considering the role of the source mechanism. 3. The Role of Source Mechanism and Wavefield Polarization Because of their vectorial nature, the weak localization of elastic waves cannot be fully explained by the simple intuitive approach presented above. As will be shown shortly, the reciprocity principle of elastic waves in its full extent has to be obeyed in order to preserve the factor of 2 enhancement at the source position. This subtle effect is first examined through a laboratory experiment with ultrasound. 3.1. Effect of Source Mechanism The most common seismic sources, that is, explosions and earthquakes are combinations of dipoles and/or couples of forces. We must therefore consider more complex sources than simple isotropic point sources. In the case of earthquakes, the radiation is strongly anisotropic and the radiation pattern displays nodal planes with reversal of the polarity of first motions. de Rosny et al. (2001) have studied the weak localization of elastic waves propagating in a chaotic reverberant cavity. The nature of the disorder is different from the scattering medium, but until a time known as the Heisenberg time, the mechanisms of enhancement are similar and can be based on an intuitive ray description. Beyond the Heisenberg time, the eigenmodes of the system can be resolved and the statistical properties of the eigenfunctions lead to an enhancement of intensity by a factor 3 around the source, as demonstrated experimentally and theoretically by Weaver and Burkhardt (1994) and Weaver and Lobkis (2000). This result is valid in chaotic cavities only. Using an interferometer, de Rosny et al. (2001) have recorded the vertical motions of Lamb waves generated by vertical monopole and dipole sources in a thin (0.5 mm thickness) chaotic plate of total area 2335 mm2, with the shape of a quarter stadium. The dominant frequency of the signal is 1.0 MHz and the typical wavelength is "2.5 mm. After time averaging between lapse time t = 200 ms and t = 500 ms, they have measured the intensity patterns shown in Fig. 3. The beginning of the signal is excluded in order to avoid the first reflection on the boundary of the plate and the choice of the end of the time window is dictated by the signal to noise ratio. The distribution of energy is perfectly homogeneous in the plate, except in a small area centered around the source where, in the case of a monopole, it is the double of the background intensity. An important result of this study is the confirmation of the typical wavelength size of the zone of intensity enhancement, which shows that weak localization is a near-field effect. In the 3-D case, the increase of intensity would occur inside a sphere centered at the source. 1.5 1 7 2 2 Normalized intensity Normalized intensity COHERENT BACK-SCATTERING AND WEAK LOCALIZATION 1 0.5 0 -5 0 y(mm) 5 -5 0 x(mm) 1.5 1 0.5 0 5 2 1 y(mm) 0 -1 -2 -1 -2 1 0 x(mm) 2 FIG. 3. Enhanced backscattering of elastic waves in a 2-D chaotic cavity. The central frequency is 1 MHz and the dominant wavelength is 2.5 mm. The integrated intensity between lapse time t = 200 ms and t = 5 ms is represented as a function of position around the source. Left: Monopolar source. Right: Dipolar source. In the dipolar case, the enhancement disappears on the line going through the source and perpendicular to the dipole axis. Along the dipole, the intensity enhancement presents two maxima located about half a wavelength away from the source. Reprinted figure with permission from de Rosny et al. (2001). Copyright (2001) by the American Physical Society. In addition, Fig. 3 (right) highlights the importance of the source mechanism. In the dipolar case, there are actually two zones of enhancement with an enhancement factor of about 1.6, separated by a line of zero enhancement. To understand this puzzling observation, one must consider the full reciprocity principle for elastic waves. In the present experiment, there is a lack of symmetry between the dipolar source and the monopolar detection. To restore reciprocity one would like to measure not the field itself, but its directional derivative along the dipole axis. When this operation is performed and intensity is redefined as the square of the partial derivative, de Rosny et al. (2001) have demonstrated both experimentally and theoretically that the factor of 2 enhancement at the source is restored. Being motivated by this result, I now give an asymptotic but rigorous theory of weak localization for vector waves. 3.2. Review of Multiple Scattering Formalism To obtain a satisfactory theory of weak localization, one needs to develop a transport theory of the energy that keeps track of all polarization indices at both source and receiver. As shown by Weaver (1990), the necessary information is contained in the fourth rank coherence tensor Gij! kl of the elastic wavefield defined as ) * Gij!kl ðt; r1 ; r2 ! r3 ; r4 Þ ¼ Gaki ðt; r3 ; r1 ÞGljaðt; r4 ; r2 Þ ; ð5Þ where Gaki ðt; r3 ; r1 Þ is the element of the Green matrix corresponding to a point force applied at r1 in direction i, and measured displacements in direction k at r3. The superscript a is introduced to label the realizations of the random medium. To each a there corresponds exactly one medium of the statistical ensemble. t denotes the time elapsed since energy has been released by sources with a common origin time. Note that in the analysis that follows, the signal is assumed to be band-pass filtered in a narrow frequency band with central angular frequency o. In order to simplify notation, all tensor 8 MARGERIN quantities are assumed to depend implicitly on o. The tensor Gij!kl (t, r1, r2!r3, r4) describes the transfer of the displacement correlation function from source (displacement indices i, j and positions r1, r2) to receiver (displacement indices k, l and positions r3, r4). The brackets denote an average over a, that is, over an ensemble of random media with prescribed statistics. In what follows we assume that the property of statistical homogeneity holds, in which case the ensemble-averaged Green tensor depends on the difference of the position vectors of the source and detector only hGaki ðt; r3 ; r1 Þi ¼ Gki ðt; r3 # r1 Þ; ð6Þ where for notational simplicity Gki (without superscript) denotes the ensemble averaged Green tensor. The complete evolution of the tensor Gij!kl is described by the Bethe-Salpeter equation which contains all correlations among all possible scattering paths in the medium (Sheng, 1995). This is far too detailed for the present purposes, and one usually contents oneself with the approximate calculation of two terms: the classical contribution—also termed “diffuson”—and the interference term between reciprocal paths—also termed “cooperon”—(see Akkermans and Montambaux, 2005, for the origin of this terminology). In the radiative transfer equation, information on source is usually integrated out and the cooperon term is neglected (see Margerin, 2005, for details). Apresyan and Kravtov (1996) have suggested a modification of the radiative transfer equation, which includes contribution of the cooperon and thereby is able to describe coherent phenomena such as weak localization. The cooperon and diffuson contributions are conveniently represented by Feynman diagrams which are both computationally efficient and physically meaningful. Typical diagrams are shown in Fig. 4. In the ladder diagrams, the Green function (upper line) and its complex conjugate (lower line) visit the same scatterers in the same order. In the crossed diagrams, first introduced for multiple scattering of classical waves in the pioneering papers of Barabanenkov (1973, 1975), the upper line is unchanged but in the lower line the sequence of scattering is reversed. The ladder and crossed diagrams correspond to the classical (incoherent) and interference (coherent) contributions, respectively. To make the link with the elementary treatment given in Section I, the reader can think of the ladder diagrams alone, as the result of summing the intensities of the direct path (solid line in Fig. 2) and reciprocal path (dashed line in Fig. 2). The ladder and crossed diagrams altogether are the result of first summing and then squaring the fields of the direct and reciprocal paths. Below, I give a long-time asymptotic formula for the ladder term. To calculate the crossed term, one can make use of the following reciprocity argument: the field produced by a force in direction k at r2 and recorded in direction l at r4 after scattering at A, B, C, D, . . . is equal to the field produced by a force in direction l at r4 and recorded in direction k at r2 after scattering at . . ., D, C, B, A. This is equivalent to saying that every crossed diagram can be turned into a ladder diagram after suitable exchange of the polarization indices, and positions on the lower line. We now decompose the tensor Gij!kl into the fundamental diffuson and cooperon contributions and write Gij!kl ðt; r1 ; r2 ! r3 ; r4 Þ ¼ Lij!kl ðt; r1 ; r2 ! r3 ; r4 Þ þ Cij!kl ðt; r1 ; r2 ! r3 ; r4 Þ: ð7Þ The previous discussion implies the following fundamental reciprocity relation between Cij!kl and Lij!kl COHERENT BACK-SCATTERING AND WEAK LOCALIZATION rs + x/2 i Source j rD + y/2 x x x x x x x x k Detector l rs - x/2 rD - y/2 rs + x/2 rD + y/2 i Source j x x x x x x x x k Detector l rs - x/2 rD - y/2 p + q/2 p! + q/2 i Source j p - q/2 9 x x x x x x x x L C k Detector Lpp!(q) l p! - q/2 FIG. 4. Typical Feynman diagrams representing the classical (top) and interference (middle) contribution to the intensity pattern. Crosses connected by dashed lines represent the same scatterer. The crosses are also connected by solid lines which represent Green functions describing the propagation between different scatterers. Note that in the lower line, the Green functions are complex conjugated. The upper diagram L, often pictorially termed “ladder diagram,” gives the classical contribution to the measured intensity. The middle diagram termed “crossed diagram” represents the interference term between reciprocal wavepaths. Bottom: scattering diagram in the wavenumber domain. Reprinted figure with permission from van Tiggelen et al. (2001). Copyright (2001) by the American Institute of Physics. Cij!kl ðt; r1 ; r2 ! r3 ; r4 Þ ¼ Lil!kj ðt; r1 ; r4 ! r3 ; r2 Þ: ð8Þ As above, I draw the attention of the reader to the fact that relation (8) is true in multiple scattering only. Fortunately, in nonabsorbing media, the single scattering contribution vanishes exponentially (see, e.g., Sato and Fehler, 1998) while, at large lapse time, the ladder contribution can be shown to be the solution of a diffusion equation (Barabanenkov and Ozrin, 1991, 1995; Sheng, 1995; Akkermans and Montambaux, 2005) and therefore decays only algebraically. In the presence of absorption, both the single scattering and ladder contribution exhibit an algebro-exponential decay but the single scattering term still vanishes faster because it suffers from scattering losses. These facts have been illustrated in papers by Gusev and Abubakirov (1987), Hoshiba (1991), and Zeng et al. (1991), where solutions of the single scattering and full multiple scattering problem are presented. Therefore, relation (8) applies after one mean free time, which is the average time between two scattering events. In addition, from Eq. (8), one can easily infer that when source and detection coincide r1 ¼ r2 ¼ r3 ¼ r4, and the polarization of source and detection are identical i ¼ j ¼ k ¼ l, the interference term equals exactly the diffuson term. Our next task is to provide a general formula to predict the exact shape of the weak localization effect in the long time limit. This requires an asymptotic solution of the Bethe-Salpeter equation, which is presented in what follows. 10 MARGERIN 3.3. Theoretical Results for Acoustic and Elastic Waves The Bethe-Salpeter equation is an exact equation for the full coherence tensor of the wavefield. The complete solution of this equation seems out of reach in general, but asymptotic solutions for the diffuson contribution in the long time limit have been presented by Barabanenkov and Ozrin (1991, 1995), which can be applied to classical scalar and vectorial linear waves, independent of the underlying equation of motion. In the theoretical approach, one considers a narrowly band-passed signal with central period T which is much smaller than the typical duration of the coda. This is known in the literature as the slowly varying envelope approximation (Sheng, 1995). The results of Barabanenkov and Ozrin are valid in this limit and I refer the interested reader to the original publications for further technical details. Asymptotically t ! 1, the ladder or diffuson or classical contribution takes the form Lij!kl ðt; r1 ; r2 ! r3 ; r4 Þ ¼ e#t=ta ðDtÞ3=2 ImGij ðr1 # r2 ÞImGlk ðr4 # r3 Þ; ð9Þ where Im denotes the imaginary part of a complex number, D is the diffusion constant of the waves in the random medium, and ta denotes a phenomenological absorption term. Note that in the last equation, the tensor Gij stands for the spatial part of the ensembleaveraged elastic Green tensor at angular frequency o ¼ 2p/T. Because of the underlying assumption of statistical homogeneity, the tensor Gij depends on the separation vector between source and station only and is an implicit function of o. The tensor Gij!kl depends on both central frequency of the waves, and lapse time in the coda. Equation (9) is valid in the slowly varying envelope approximation. The reader is referred to Sheng (1995) and Apresyan and Kravtov (1996) for more mathematical details. According to Eq. (8), the cooperon term reads Cij!kl ðt; r1 ; r2 ! r3 ; r4 Þ ¼ e#t=ta ðDtÞ3=2 ImGil ðr1 # r4 ÞImGjk ðr2 # r3 Þ: ð10Þ Let us investigate some consequences of Eq. (10) for a given source station configuration: r1 ¼ r2, r3 ¼ r4. To illustrate the fact that interference effects are important only in a region of the size of one wavelength around the source, we consider the scalar case and introduce the enhancement factor defined as the total intensity normalized by the diffuson contribution: E ¼ 1 þ C/L. For scalar waves in 3-D, the enhancement profile E is given by E¼1þ ' ( sinðkr Þ 2 ; kr ð11Þ where k ¼ o/c is the central wavenumber of the scalar waves with velocity c and r is the source receiver distance. For scalar waves in 2-D, the corresponding formula is E ¼ 1 þ J0 ðkr Þ2 ð12Þ COHERENT BACK-SCATTERING AND WEAK LOCALIZATION 11 where J0 denotes the Bessel function of the first kind of order 0. These results are valid in the long lapse time limit and when the mean free path is much larger than the wavelength, which is the most common situation in practice. When scattering is extremely strong, the wavelength can be of the order of the mean free path, and strong localization can set in, a regime which is very difficult to reach (Akkermans and Montambaux, 2005). The cardinal sine and Bessel function in Eqs. (11) and (12) are proportional to the imaginary part of the Green function of the Helmoltz equation in 2-D and 3-D, respectively. Equation (11) has been verified by numerical simulations (Margerin et al., 2001), while Eq. (12) has been checked experimentally by de Rosny et al. (2000) (see also the left part of Fig. 3). It is important to note that the enhancement profile is independent from absorption. This is not surprising since absorption does not affect the reciprocity principle. This is often illustrated by the following sentence (van Tiggelen and Maynard, 1997): “If you can see me, I can see you!” that applies even in the fog. This independence of the weak localization effect on absorption is used later in the paper to give estimates of the scattering properties of heterogeneous media. I now explore theoretically the role of the source mechanism. As an example, I will give a simple formula that explains all the features of the experiments described in Fig. 3 with a dipole source. The radiation of the dipole is obtained by taking the directional partial derivative of the Green function with respect to source coordinates. For a dipole source along the x axis of the coordinate system, the coherence function G of the scalar field is given by Gðt; r1 ; r2 ! r3 ; r4 Þ ¼ h@x1 Ga ðt; r3 ; r1 Þ@x2 Gaðt; r4 ; r2 Þi: ð13Þ Because the operation of taking derivatives is linear, the @x1 and @x2 symbols can be taken outside of the ensemble average brackets. For the diffuson and cooperon contributions, one obtains respectively L¼ C¼ e#t=ta ðDtÞ3=2 e#t=ta ðDtÞ3=2 ImGð0Þ@x1 @x2 ImGðr2 # r1 Þjr2 ¼r1 ð14Þ @x1 ImGðr1 # r4 Þjr4 ¼r3 @x2 ImGðr3 # r2 Þjr2 ¼r1 : ð15Þ In the 2-D case, after some algebra, one obtains the following enhancement pattern for the dipole source E ¼ 1 þ ðJ1 ðkr ÞcosðyÞÞ2 ; ð16Þ where J1 denotes the Bessel function of the first kind of order 1. This theoretical prediction matches the observations of Fig. 3 (left) very closely. The calculation of the weak localization effect has thus been illustrated in simple cases. The calculations in the full elastic case with arbitrary moment tensor sources have been performed by van Tiggelen et al. (2001). The principle is the same as above but the calculations are much more tedious. To illustrate the effect of broken symmetry between source and receiver, I show in Fig. 5 the calculation of the weak localization profile for explosion and dislocation sources in an infinite elastic medium. As usual, an explosion Explosion and dislocation: Potential energy cone (b) 12 (a) Explosion: Kinetic cone 2.0 1.015 Enhancement Enhancement 1.8 1.6 1.4 1.010 1.005 1.2 −2 (c) 0 Distance [lambda S] 1.000 2 Dislocation: Kinetic cone between seismic axes −2 (d) 0 Distance [lambda S] 2 Dislocation: Kinetic cone along seismic axes 1.05 1.08 Enhancement Enhancement 1.04 1.06 1.04 1.02 1.03 1.02 1.01 1.00 −2 0 Distance [lambda S] 2 1.00 −2 0 Distance [lambda S] 2 MARGERIN 1.0 COHERENT BACK-SCATTERING AND WEAK LOCALIZATION 13 and a seismic dislocation will be represented by three mutually orthogonal dipole of forces, and by a double couple of forces, both with zero net applied linear and angular momentum, respectively. The enhancement profiles in Fig. 5 are calculated in the plane perpendicular to the axis of the double couple. In this plane, there exist two perpendicular directions, termed seismic axes, with respect to which the moment tensor is diagonal. For the simple dislocation source, the seismic axes define the direction of maximum radiation of P waves and make an angle 45) with the applied forces. Note that the direction of application of the forces also coincide with the maximum radiation of S waves. Figure 5 demonstrates that the backscattering enhancement can be totally destroyed if the operations carried out at the source and at the detection are different. For instance, the enhancement of kinetic energy due to a dislocation is typically less than 10% and vanishes exactly at the source position. The angular dependence of the enhancement profile in the plane containing the double couple is illustrated in Fig. 5c and d. It is seen that the enhancement is slightly higher along the direction of maximum radiation of S waves. Although I shall not prove it, it is always possible to recover the factor 2 enhancement by measuring the appropriate quantity. For the case of a dipole source, one needs to measure the derivative of the field along the dipole axis. This of course complicates the experimental setup but it can be done in practice (see, e.g., Hennino et al., 2001). For an explosion, one needs to evaluate the divergence of the wavefield (i.e., the compressional energy of the waves), which is even more demanding. Although this seems discouraging, I present below some experiments with seismic waves that demonstrate the potential usefulness of weak localization. 4. Geophysical Applications At this point, I would like to convince the reader that the weak localization effect is not only a theoretical or mathematical curiosity but a useful interference effect which can be used to probe the medium properties when multiple scattering hampers the detection of ballistic waves. I give two examples of applications of weak localization: the first one illustrates the measurement of elastic properties of a concrete slab (Larose et al., 2006); the second one reports the measurement of the mean free path in a volcano (Larose et al., 2004). 4.1. Measurement of the Dispersion Relation of Surface Waves Let me first consider an application at a scale which is intermediate between the laboratory and the field. An array of vertical accelerometers with typical bandwidth 0.1– 5 kHz has been set up on a steel reinforced concrete slab. Upon propagation, waves undergo strong multiple scattering and reflections. In the time domain, the ballistic signals are difficult to separate from the multiple reflections on the sides of the slab and FIG. 5 Enhancement profiles as a function of the distance from the source, in units of the shear wavelength, for an infinite Poissonian medium. (a) The enhancement of the potential energy for an explosion (solid) and a dislocation (dashed). (b) Enhancement profile for the kinetic energy near an explosion. (c) and (d) Enhancement profiles for the kinetic energy between (at an angle 45) to) and along the seismic axes of a dislocation source. The term “cone” shown on top of each plot refers to the zone of intensity enhancement around the source. Reprinted figure with permission from van Tiggelen et al. (2001). Copyright (2001) by the American Institute of Physics. 14 MARGERIN are also rapidly attenuating. In such a situation, an alternative to classical signal processing techniques is welcome. The elastic waves are generated by an approximately vertical hammer strike, thus ensuring that source and detection verify the general reciprocity condition previously discussed. In the frequency range of interest (from 500 to 1500 Hz), the elastic energy is transported through the slab by the fundamental antisymmetric Lamb mode, which is known to be strongly dispersive. The phase speed of the fundamental Lamb mode at 1 kHz is "1000 m/s. The geometry of the experiment can thus be considered quasi 2-D and the shape of the weak localization can be accurately predicted using the theoretical expression (12). I refer the reader to Larose et al. (2006) for further details. To illustrate the practical measurement of the weak localization effect, I show in Fig. 6 (top) a schematic view of the spatial distribution of the field intensity at a given instant of time. The upper plot in Fig. 6 is not the outcome of the experiment but serves as an Energy ratio S(Dr) 2 3 ms−10 ms 10 ms−100 ms Theoretical fit 1.5 1 [500 Hz−1500 Hz ] 0.5 −1.5 −1 −0.5 0 0.5 Distance Dr(m) 1 1.5 FIG. 6. (Top) Schematic snapshot of random spatial intensity fluctuations measured in the coda. One can imagine that the typical side length is "2 m. The weak localization effect is hidden in this speckle pattern. (Bottom) Reprinted figure with permission from Larose et al. (2006). Copyright (2006) by the American Physical Society: Typical profile of intensity enhancement around the source obtained after averaging over a large number of speckle patterns. COHERENT BACK-SCATTERING AND WEAK LOCALIZATION 15 illustration of the experimental process involved in the measurement of the weak localization effect. This complex interference pattern shows rapid spatial fluctuations and is known in optics as a speckle pattern. In the speckle, the maxima can be viewed as random constructive interferences between multiply-scattered waves that mask the weak localization spot (at the center of Fig. 6 (top)). At another time instant, the speckle pattern will have completely changed, except for the deterministic constructive interference between reciprocal waves around the source. By averaging speckle patterns over sufficiently long time windows, one suppresses unwanted fluctuations thus revealing the weak localization intensity pattern as shown in Fig. 6 (bottom). In the present example, the averaging is performed in a lapse time window ranging from 10 to 100 ms. This simple procedure makes the link between averaging in theory and in practice. The reader will notice that the enhancement factor is lower than 2 in this experiment. Presumably, this is caused by the application of a force with a significant horizontal component, thus breaking the symmetry between source and detection. Despite the imperfection of the reconstruction, it is still possible to determine with reasonable accuracy the width of the weak localization zone as a function of frequency using the theoretical relation (12), thus providing the dispersion of the fundamental antisymmetric Lamb mode. In Fig. 7, the results of the dispersion measurements are plotted together with a theoretical fit. Indeed, the phase velocity c of the fundamental Lamb mode obeys the following rather complicated dispersion relation (see, e.g., Royer and Dieulesaint, 2000) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffis u u b2 t 2p c ¼ pffiffi fhb 1 # 2 ; a 3 ð17Þ where f is the frequency, h is the slab thickness, and a and b denote the P and S wave speeds, respectively. In the present experiment, h is known and the values of the wavespeeds can be adjusted to fit the experimental data. This technique agrees with independent measurements to within a few percents, which is highly satisfactory. Phase velocity (m/s) 2000 1500 1000 From WL width Theory 500 0 0 500 1000 Frequency (Hz) 1500 2000 FIG. 7. Dispersion relation of the fundamental antisymmetric Lamb mode in a concrete slab. (Squares) Experimental data obtained from the width of the weak localization zone. The theoretical relation between the wavelength and the width of the weak localization zone is given by Eq. (12). (Solid line) Theoretical fit of the experimental data with Eq. (17). Reprinted figure with permission from Larose et al. (2006). Copyright (2006) by the American Physical Society. 16 MARGERIN 4.2. Measurement of the Diffusion Constant in Strongly Scattering Media Recent active experiments have shown the extreme character of the propagation in volcanic regions (see, e.g., Wegler and Lühr, 2001; Wegler, 2004). In the short period band, it is hardly possible and even sometimes impossible to extract the coherent ballistic waves from the complex recorded waveform. In such extreme cases, a meaningful measurement of the medium heterogeneity is the diffusion constant of the waves. However, in practice, its measurement can be affected by a number of factors, such as anelastic dissipation or local boundary conditions (see Friedrich and Wegler, 2005 for details). As demonstrated above, absorption does not influence the measurement of the weak localization effect. Unfortunately, the weak localization intensity profile does not depend on the scattering properties of the medium and therefore does not offer direct access to the scattering properties of the medium. However, we know that weak localization is a multiple scattering process which sets in only after the single scattering intensity has become negligible. Such a time dependence has been confirmed in a numerical study by Margerin et al. (2001), which concludes that after a time of the order of the mean free time, the weak localization pattern is measurable. This has also been verified experimentally by Larose et al. (2004). They measured the weak localization effect on a volcano in the 10–20 Hz frequency band and found that there exists a characteristic rise time of the intensity enhancement. Their result is shown in Fig. 8. In the case of the Puy des Goules, one can infer a mean free time of the order of 1 s which gives a mean free path of a few hundred meters. Note that in their experiment, the width of the weak localization depends in a nonlinear way on frequency. This is explained by the fact that according to the equipartition principle (Weaver, 1990; Hennino et al., 2001), the energy at the surface is largely dominated by the fundamental mode Rayleigh wave, whose dispersion is caused by the complex layering at the surface. Although the S(Dr) 2 1.7 s 0.7 s 0.3 s 1 0 −20 −10 0 Meters 10 20 FIG. 8. Emergence of the weak localization spot in the coda. The normalized intensity is represented as a function of distance around the source. The dominant frequency of the waves is 15 Hz and the dominant signal is the Rayleigh wave with a wavespeed "250 m/s. The wavelength is of the order of 15 m. The different curves correspond to the averaged intensity profiles obtained at different lapse time in the coda (as indicated in the inset). Reprinted figure with permission from Larose et al. (2004). Copyright (2004) by the American Physical Society. COHERENT BACK-SCATTERING AND WEAK LOCALIZATION 17 observation of weak localization on a volcano requires active sources, it is free of the effect of absorption and therefore offers direct access to the scattering properties of the medium. 5. Conclusion In this paper, I have given a brief introduction to the weak localization effect in seismology. I have summarized general formulas that enable the calculation of the weak localization effect for a wide range of practical cases. The fundamental role of reciprocity between source and detection has been emphasized and illustrated with experimental results. In practice, the control of the source mechanism is crucial. The simplest solution is to measure the vertical displacements generated by vertical forces. Applications to the characterization of scattering or bulk elastic properties have been presented. In particular, I have shown that the emergence time of weak localization yields an estimate of the scattering mean free path, independent of absorption effects. Thus, weak localization combined with other measurements such as time and space dependence of the coda could be used to discriminate anelastic and scattering attenuation. Acknowledgments I would like to thank E. Larose, A Derode, J. de Rosny, and R. Hennino for their invaluable help with the figures and for many discussions on the weak localization effect. I would like to thank M. Campillo and B. van Tiggelen for their constant input. Comments by H. Sato, Yu. Kravtsov, and an anonymous reviewer greatly helped to improve the presentation and the readability of this chapter. References Akkermans, E., Montambaux, G. (2005). 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