Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium
This paper investigates the reconstruction of elastic Green’s function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation, -the resolvent formula-, Green’s function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell -matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell -matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The -matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Green’s function reconstruction in a scattering medium is discussed.
Recently, there has been increasing interest in the use of coda and scattered waves to monitor temporal variations in dynamic structures such as volcanoes and faults. Such an approach is known as coda-wave interferometry in seismology and diffusive wave spectroscopy in acoustics (Snieder et al., 2002; Snieder and Page, 2007). While the early developments of the technique were based on the use of repeating small earthquakes termed “doublets” (Poupinet et al., 1984, 2008), the possibility to reconstruct Green’s function from ambient seismic noise has given rise to a new technique termed passive image interferometry (Sens-Schönfelder and Wegler, 2006; Wegler and Sens-Schoenfelder, 2007; Brenguier et al., 2008a, b). In coda-wave interferometry, temporal changes in the medium are detected by comparing the coda portions of the cross-correlation function of ambient noise, that were computed at different times. Therefore, understanding the basic physical processes that allow the reconstruction of scattered waves from the cross-correlation of random noise sources is of interest in seismology and acoustics.
The reconstruction of Green’s function from field cross-correlations has a rich and vast history as reviewed by Shapiro (2011). Green’s function retrieval in seismology and elastodynamics finds its roots in the fluctuation-dissipation theorem and in the theory of speckle correlations in optics. In recent years, the pioneering results described by Shapiro (2011) have been extended to different system configurations: open or closed, and to different types of excitations: deterministic or random sources located on a surface or distributed in a volume. For instance, general theorems on the reconstruction of Green’s function have been established for a variety of inhomogeneous, dissipative (Wapenaar et al., 2006; Snieder et al., 2007), and even non-reciprocal media (Wapenaar, 2006). An important ingredient of seismic interferometry is Green’s theorem or its elastodynamic generalization, i.e. the Rayleigh-Betti reciprocity theorem (Ramirez and Weglein, 2009). The importance of Green’s theorem and its generalizations for the reconstruction of Green’s function is well illustrated by Wapenaar and Fokkema (2006). Depending on the excitation mechanism -active or passive-, the acquisition geometry -surface or borehole measurements-, the nature of boundary conditions -radiation or reflection-, the type of Green’s function employed -analytical, numerical, empirical-, a great variety of interferometric methods can be derived from the application of Green’s theorem. A comprehensive review of applications to exploration seismology, emphasizing the strengths and limitations of each method is provided by Ramirez and Weglein (2009).
More specifically, the connection between the cross-correlation of wavefields and Green’s function between two points in an inhomogeneous elastic half-space has been established by Wapenaar (2004), based on Betti’s reciprocity theorem. The reconstruction of Green’s function of a homogeneous elastic medium has been discussed by Sánchez-Sesma and Campillo (2006) from a different perspective. These authors find that a set of uncorrelated plane and waves verifying the equipartition relation, i.e. carrying the same amount of energy, allow the elastic Green’s function to be reconstructed from the cross-correlation of noise wavefields. This result was extended to the case of one cylindrical inclusion in a 2-D elastic medium by Sánchez-Sesma et al. (2006). These authors demonstrated the complete reconstruction of Green’s function including scattered waves in the case of an illumination of the heterogeneous medium by an equipartition mixture of plane and waves.
The case of scattering media has been studied by Sato (2009, 2010). He demonstrated the reconstruction of singly-scattered coda waves from random noise sources in an acoustic medium composed of point-like velocity perturbations. He considered both volumetric and surface distribution of noise sources. An important connection between energy conservation in scattering and the reconstruction of scattered waves in Green’s function obtained from cross-correlations of random noise sources has recently been put forward by Snieder et al. (2008, 2009). Considering a scattering medium illuminated by noise sources distributed on an enclosing surface, Snieder et al. (2008) showed that the reconstruction of Green’s function of scalar waves is equivalent to a generalized optical theorem. Such theorems provide general symmetry relations that must be obeyed by the scattering amplitude, which describes the amplitude and phase relations between incident and scattered waves. Lu et al. (2011) extended the method of Snieder et al. (2008) to the case of vector elastic waves, for a scattering medium composed of a homogeneous and isotropic heterogeneity. Snieder and Fleury (2010) and Margerin and Sato (2011) developed a theory for the reconstruction of scattered arrivals in Green’s function obtained from the cross-correlation of noise signals in an acoustic medium, for surface and volumetric distributions of noise sources, respectively. These authors demonstrate the cancellation of spurious terms in the multiple scattering series expansion of Green’s function cross-correlations by repeated application of the optical theorem.
In this work, we analyze the case of a general -possibly anisotropic- heterogeneity embedded in an isotropic medium, illuminated by a volumetric distribution of random, uncorrelated forces. In the same spirit as Snieder et al. (2008), Lu et al. (2011), and Margerin and Sato (2011), we establish the equivalence between Green’s function reconstruction and some symmetry relations pertaining to the scattering of elastic waves which we refer to as “generalized optical theorems”. Compared to our previous work on scalar waves, we first use a general operator formalism to establish in a concise way Green’s function reconstruction theorem for a general -homogeneous or inhomogeneous- elastic medium. Next, we apply the theorem to Green’s function of a scattering medium to establish general symmetry relations in the scattering process. An analysis based on scattering theory finally shows that the symmetry relations are extensions of generalized forms of the optical theorem for elastic waves. The central result of this paper -Equation (61)- contains as special cases all forms of optical theorems found in the literature. In view of future applications to a collection of scatterers we develop a simple point-like scattering model for elastic waves. The formulas (80)-(81) present an extension of the point-scattering model originally developed for scalar waves to the case of vector elastic waves. The physical interpretation of the model is discussed in details.
The problem setting is depicted in Figure 1. We consider an arbitrary heterogeneity with bounded support embedded in an elastic medium illuminated by a randomly homogeneous distribution of forces oriented in all possible space directions. By considering the correlation of the displacements recorded at two points and , we will show the possibility to reconstruct the elastic Green’s tensor including the waves scattered by the heterogeneity. In a heterogeneous, slightly dissipative elastic medium, the wavefield generated by random forces satisfies the following equation:
where and denote the density and elasticity tensor of the isotropic matrix. The summation over repeated indices is assumed. The second term on the left-hand side introduces a characteristic absorption time . The deviations of the density and elastic properties from homogeneity are encapsulated in the terms and . No particular symmetry properties, except the usual ones, are imposed on . On the right-hand side of equation (1), we consider a randomly homogeneous and stationary distribution of forces with uniform random orientations. In addition, we assume that the spatial correlation of these forces is much shorter than any other scale length of the problem, which implies that they are well described by a white noise process in space. The basic observable in seismology is the cross-correlation of signals averaged over an ensemble of noise sources:
where the double brackets denote the ensemble average. If the noise sources are statistically stationary, the basic quantity to be evaluated is the integrand on the right-hand side of equation (2). Note that the definition of the cross-correlation adopted in Eq. (2) differs from that found in other references (see e.g. Wapenaar, 2004). For deterministic or stationary random signals, the change of variable allows us to recover the form of cross-correlation used in other references. The difference is therefore purely notational. Since calculations are much easier to perform in the frequency domain, we will develop a theory for the cross-spectral density:
In equation (3), the left-hand side does not depend on the variable for a stationary signal. Upon introducing the spectral representation of the random process :
we obtain the following basic relation:
The delta function condition which appears in equation (4) is characteristic of random processes that are stationary in time. We note that the usual Fourier transforms may not be well defined since we are interested in stationary random fields. A physically satisfying solution is to define the Fourier transforms for a finite observation time , and to take the limit after averaging over noise sources. Instead, we make use of the probabilistic interpretation of the spectral representation (4), where the coefficients of the exponential is itself a random process, and the equality is to be understood in the probabilistic (mean-squared) sense (Rytov et al., 1989; Yaglom, 2004). Since only the cross-spectral density of the field is needed in this work, and this quantity can be defined by traditional Fourier analysis, the formal representation (4) suffices. A link between the physical and mathematical approaches is provided by the periodogram:
where represents the Fourier transform of the original signal observed over a finite time window . As is well-known, Equation (6) expresses the fact that the periodogram is an asymptotically unbiased estimator of the cross-spectral density.
The paper is organized as follows. In section 2, we introduce some important notations and present a simple derivation of Green’s function reconstruction for a homogeneous medium illuminated by a set of randomly oriented and spatially uncorrelated forces. In section 3 we introduce the Dirac calculus which will be used to generalize our results to an inhomogeneous medium. The formalism is applied to demonstrate the reconstruction of Green’s function of a general -homogeneous or inhomogeneous- elastic medium illuminated by a homogeneous distribution of randomly oriented forces, or by a wavefield at equipartition. In section 4, we introduce the -operator for elastic waves and discuss its symmetry properties. In section 5, we explore the reconstruction of Green’s function for an arbitrary inhomogeneity embedded in a homogeneous medium and establish several forms of the generalized optical theorem for elastic waves. In section 6, we establish the generalized optical theorem from the governing equations of scattering in the limit of infinitesimal absorption. In section 7, we consider a simple configuration composed of a single point scatterer and demonstrate the Green’s function reconstruction including all the recurrent scattering loops. In section 8, our results are discussed in the framework of equipartition theory and compared to other works.
2 Green’s function reconstruction in a homogeneous medium
In this section, we introduce important notations and present in a simplified context our approach to Green’s function reconstruction. Although it is not the main theme of the paper, we give a short derivation of Green’s function reconstruction for elastic waves excited by random uncorrelated noise sources in a homogeneous medium. Our assumptions differ from the one adopted by Sánchez-Sesma and Campillo (2006) who considered a set of uncorrelated plane and waves at equipartition. The complete equivalence between the two models of noise source will be demonstrated later in the paper. Let us first rewrite the correlation function (3) in terms of Green’s function. The elastic wavefields recorded at two points and due to a distribution of forces can be expressed as:
where is Green’s function of a homogeneous, slightly dissipative elastic medium at angular frequency . Using the summation convention over repeated indices, is recognized as the fundamental solution to the equation of elastodynamics:
and has the following analytical form:
where Green’s function has been split into transverse and longitudinal parts. In equation (9) we have introduced the notation , the unit vector in the direction of , and the functions , and :
The following short-hand notations have been used: , and , where and are the and wave velocities in the background medium with Lamé parameters and . The definition of Green’s function adopted in equation (9) obeys an opposite sign convention to that usually found in the seismological literature. This choice will be clarified in section 3. In equation (9), we have explicitly separated the far-field terms and from the near field term .
Equation (7) allows us to express the cross-correlation of wavefields measured at and in the frequency domain as follows:
For spatially uncorrelated and randomly oriented forces, one further assumes that
where is the power spectral density of the forces. Therefore the key integral to be computed writes
In coordinate space, Green’s function has the complicated form (9). However, using Fourier transforms, it is found that Green’s function has the following exact expression in wavenumber space:
where we have introduced the following notations:
and the unit vector is in the direction of . In equation (14), Green’s function has been separated into longitudinal and transverse parts. Upon inverse Fourier transformation, both contribute to the near-field term in the spatial domain. In wavenumber space, the key integral is defined as
Inserting the spectral representation of Green’s function in equation (13), one obtains the following key integral in the wavenumber domain:
which establishes the reconstruction of Green’s function of a homogeneous, slightly dissipative elastic medium. More precisely, equations (3), (11) and (18) allow us to obtain the following closed form expression of the cross-correlation function of two wavefields in the space-time domain:
Hence, we retrieve the classical result that the temporal derivative of the cross-correlation of two wavefields is proportional to the difference of the retarded and advanced Green’s functions filtered by the source power spectral density. Strictly speaking, the limit and integral signs on the left-hand side of equation (19) are superfluous. However, introducing the temporal averaging process makes a closer link with the measurement procedure. It is worth noting that the result (19) is valid under the assumption of equal absorption times for and waves. We will come back to this point in section 8.
3 Green’s function reconstruction in an inhomogeneous medium
3.1 Formalism and notations
To perform the calculations required for the proof of the elastic Green’s function reconstruction in an inhomogeneous medium, it is convenient to introduce the Dirac formalism. This approach allows us to switch easily from coordinate to wavenumber space and is extremely efficient to perform compact formal derivations. In this paper, we work in a linear space of vector wavefunctions in which various representations can be introduced. It is convenient to think of this vector space as the tensor product of a 3-dimensional polarization space, and the usual space of scalar wavefunctions. In what follows, the symbol will be employed to denote the scalar product between two vectors. For instance, the space of vector wavefunctions is equipped with the following scalar product:
which is inherited from the tensor product structure. In what follows, we will switch among various representations by introducing generalized orthonormal bases which verify the continuum normalization. The three most important bases for this work are listed below.
(1) First, we define: , where the symbol denotes a tensor or direct product between vectors of the polarization and wavefunction space, respectively. The bra-ket notations and definitions adopted here follow the treatment of vector fields in quantum mechanics (see Messiah, 1999, p.549-551). The vectors form an orthonormal basis of the 3-D polarization space. The simplest choice is to take as unit vectors along the axes of a right-handed orthogonal reference frame. We adopt the convention that unit vectors are denoted by a hat. The symbol denotes generalized eigenvectors of the position operator. By forming the tensor product , one obtains a wavefunction which is perfectly localized in space at position and perfectly polarized along direction . The completeness of this set of orthonormal vectors is intuitively clear and the following normalization of the basis vectors is easily verified: Using this basis, we recover the usual definition of the recorded elastic wavefield in cartesian coordinates
(2) Next, we introduce the symbol = which denote a basis of eigenvectors of the vector Laplace operator with eigenvalue . The symbol represents a plane wave with wavenumber which obeys the normalization of the continuum. By forming the tensor product , one obtains a plane wave perfectly polarized along direction and propagating in direction . In the coordinate representation (1) introduced above, the eigenfunctions write
(3) Finally, we denote by the eigenvectors of the elastodynamic operator in free space , which is defined as
The notational conventions adopted in Eq. (23) are as follows: the operator acts on the vector wavefunction from the left to give a new wavefunction , and the right-hand side gives the usual expression of the wavefunction in coordinate space. The minus sign adopted in definition (23) makes a positive operator, i.e., such that . This property is easily proved since represents the deformation energy of the solid (see Ben-Menahem and Singh (1998) p.31, for details). In coordinate space, the eigenvectors of are expressed as:
where denotes the polarization vector and is the wavevector. Equation (24) introduces the complete orthonormal set of linearly polarized plane P and S waves. In what follows, latin subscripts refer to cartesian coordinates (representations 1 and 2), while greek letters will be used to distinguish the contribution of the 3 different polarizations in representation 3. The form a right-handed basis in polarization space which coincides with the usual spherical coordinate frame , where and point in the direction of increasing and , respectively. Our conventions for the definition of the polarizations are depicted in Figure 2. It is important to keep in mind that polarization vectors rotate with the incident wavevector . Note that with our notational conventions, there should be no risk of confusion between representations (2) and (3).
3.2 Resolvent formula and Green’s function retrieval
We now generalize the previous derivation to the case of an inhomogeneous elastic medium. We will consider simultaneously the case of a wavefield at equipartition, or a system driven by a white noise distribution of random forces. Our formalism allows us to treat the two cases on the same footing. Some basic facts about resolvents and their connection with Green’s functions are recalled hereafter. Let us first introduce the operator which pertains to elastic waves in inhomogeneous media:
where is the elastodynamic operator in free space introduced in equation (23) and is the scattering potential at the frequency :
where the usual summation convention over repeated index is used. Equation (26) defines the matrix elements of the scattering potential operator in the coordinate representation. This representation is most usually encountered in the Green’s function retrieval literature. For simplicity, we assume that the scattering potential vanishes outside a bounded region of . To demonstrate that is Hermitian, we let act on test functions and . Since the Hermitian symmetry of the term involving the density perturbation is easily verified, we only consider the term:
In the derivation of equation (27), we have used integration by parts twice and the usual symmetries of the elasticity tensor: . In the following, we assume that all Hermitian operators we have to deal with have unique self-adjoint extensions. Because the elastic tensor and density fields are real, relation (27) implies that the matrix elements of the scattering potential in the representation (1) obey the following symmetry relation:
Let us introduce the resolvent of a linear operator as follows:
where is the identity operator in the space of vector wavefunctions, and . Comparison of equations (29) and (9) reveals that is intimately related to , the resolvent of . By making the following substitutions , , in equation (29), we can formally define Green’s function at angular frequency with a small, finite absorption time . This justifies a posteriori the definition of Green’s function adopted in section 2. With this choice, is related to the resolvent of a positive operator with continuous spectrum on the positive real axis (see Equation (23) and the discussion that follows). Let us now introduce in a similar fashion the resolvent and Green’s function of the equation of elastodynamics in an inhomogeneous medium as follows:
An important point to be noted in the definition (31) is the fact that , the real part of the complex frequency, does not necessarily equal , the angular frequency at which the scattering potential is evaluated. For elastic waves, the only physically accessible Green’s function is obtained when we set in equation (31). Taking the matrix elements of equation (31) in representation (1), we recover the fact that is the solution of Equation (1) with the force density , as it should. In equation (31), everything happens as if the scattering potential has been “frozen” at the frequency . This mathematical trick allows us to use the standard tools of potential scattering theory and facilitates intermediate calculations. In elastodynamics, this idea has previously been introduced by Budreck and Rose (1991). Eventually, all key formulas will be obtained for and will involve physically accessible quantities only.
Let us consider the elastic wavefield at frequency excited by a distribution of forces and recorded at in an inhomogeneous medium:
In equation (32a), we have used the coordinate representation to express the field generated by the force distribution. But any other complete set of vectors can be inserted between and . As an example, in equation (32b), we make use of this freedom by introducing a decomposition of the identity operator using the complete orthonormal set (3), which results in a hybrid representation. Indeed, the symbol has its left foot in representation (1) and its right foot in representation (3). The quantity can be calculated from the usual coordinate representation of Green’s function by Fourier transformation over the variable and by expanding the standard basis over the rotating polarization basis . Clearly, the quantity gives the amplitude of the plane wave with polarization and wavenumber , which is excited by the force distribution. As previously noted, if the force distribution is statistically homogeneous, the wavenumber decomposition of the field has a formal character. Standard Fourier analysis can nevertheless be used after ensemble averaging (Rytov et al. (1989)). Using equations (32a)-(32b), the cross-correlation between the wavefields at and can be written as:
where denotes the Hermitian adjoint of . We now consider two possible physical assumptions regarding the excitations that drive the system. As in equation (12), we may assume that the forces are randomly oriented and spatially delta-correlated. Alternatively, we may consider that the medium is illuminated by a set of plane and waves whose amplitudes constitute a white-noise in modal space:
Assumption (34) may be interpreted as a requirement of equipartition: the system is illuminated by uncorrelated plane and waves coming from all possible directions with equal weights. This point will be further substantiated in section 8. Under which conditions such an equipartition state can be achieved in practice remains to be elucidated. Reporting Equations (12) and (34) in Equations (33a) and (33b) respectively, we find:
where in the last equation, we have used the completeness of the bases (1) and (3), respectively. In both cases -white noise distribution of forces, or wavefield at equipartition- we have reduced the computation of the cross-correlation of two random wavefields to the evaluation of the matrix elements of the operator . In fact, from the defining property of Green’s function as the resolvent of a self-adjoint operator -equation (31)-, the following identity follows:
with the resolvent operator introduced in Equation (30). The key point in the above derivation is the use of a fundamental operator equation known as the “first resolvent formula” in the second line of Equation (36). The resolvent formula is valid for a broad class of operators (closed linear operators) including the differential operators considered in this work (Richtmyer, 1978). The last step follows simply from the Hermitian character of the elastodynamic operator . Taking into account the power spectral density of the source and calculating the matrix elements of the right-hand side of equation (36) yields:
where the Hermitian symmetry of has been used. When the reciprocity theorem applies, the integrand in the right-hand side of equation (37) is recognized as the usual imaginary part of Green’s function. Thus, the derivative of the cross-correlation tensor of random signals yields a filtered version of the difference between advanced and retarded Green’s functions. Since can be substituted with the Green’s function of any self-adjoint operator, this implies the reconstruction of Green’s function for other systems governed by linear wave equations. Derivations of Green’s function reconstruction based on a different operator formalism have been further developped by Wapenaar et al. (2006) and Snieder et al. (2007). Our derivation highlights the fact that a wavefield at equipartition and a random distribution of white noise sources are indistinguishable from the point of view of cross-correlation functions. The complete equivalence between the two types of excitations will be demonstrated in section 8. The general result (36) does not constitute, however, the end of our investigations. We will show that interesting information on the scattering of elastic waves can be obtained by expressing the Green’s function of an inhomogeneous medium using the formalism of scattering theory. To do so, basic notions of scattering of elastic waves are recalled in the next section.
4 Summary of scattering of elastic waves
Our starting point is the observation that Equation (31) is formally identical to a Schrödinger equation, with a scattering potential . This allows us to use all the arsenal of scattering theory. In particular we can introduce the retarded Lippman-Schwinger eigenvectors of the operator which satisfy the following equation in the infinitesimal absorption limit (Economou (2006)):
with . It is worth noting that in equation (38), the potential is “frozen” at the frequency , and is scanned by plane and waves with all possible wavevectors with , and eigenfrequency . These eigenvectors do not depend on the sign of . For elastic waves, the physical wavefunction is obtained when the condition is satisfied. For later developments it is nevertheless convenient to introduce the more general Lippmann-Schwinger eigenvectors (38). All the scattering properties of an inhomogeneous elastic medium are conveniently encapsulated in an operator denoted by which is defined as follows:
In equation (39), we have carefully distinguished between the angular frequency of the target -- and of Green’s function --. Clearly, the physical result is obtained when equals and we introduce a special notation for this case:
For simplicity, we have dropped the frequency dependence in this definition. The superscript reminds us of the fact that . The definition of the -operator adopted in equation (39) is in keeping with our definition of Green’s function of an inhomogeneous medium (see equation (31)) and will facilitate intermediate calculations in section 6. Using the -operator, we can rewrite the Lippman-Schwinger equation as follows (, ):
which further implies:
Green’s function of the heterogeneous medium can also conveniently be rewritten in terms of the -operator and the free space Green’s function alone:
Since our approach is based on an analogy with quantum scattering, a word of comment about the sign of is in order at this point. The problem of negative frequencies is easily solved because we consider real fields in the space-time domain. This imposes the following Hermitian symmetry on the matrix elements of and in coordinate representation:
As a consequence, all results obtained for positive frequency are readily translated to negative frequency using the symmetry conditions (44).
In the far-field of the scatterer, i.e. in the limit with the observation point and the dimension of the scatterer (Figure 1), we may approximate the free space Green’s function as follows:
where denotes an arbitrary point of the scatterer. The position vectors are measured from an arbitrary origin located inside the scattering region and is a unit vector in the direction of . The asymptotic formula (45) allows us to express the Lippman-Schwinger wavefunction in the usual coordinate representation as follows:
valid in the limit , where . In equation (46), we have introduced the scattering amplitudes:
and the notations: . In the last line of equation (47), we have introduced “on-shell” matrix elements of the -operator. In general, the matrix elements are said to be “on-shell” when the condition is satisfied. Although this property does not depend on the fact that we are on the shell , we eventually specialize our results to this particular case. The scattering amplitudes contain information on the scattering pattern for all possible mode conversions. For instance, taking , , corresponds to the physical situation where a linearly polarized incident shear wave is scattered into a compressional wave. Note that the definition is valid for an arbitrary anisotropic inhomogeneity. From the symmetries of the scattering potential: and the reciprocity of the free space Green’s function , we deduce a similar reciprocity relation for the operator:
Using equation (48), our formalism allows us to recover well-known reciprocity relations for the scattering amplitudes: