The effect of Rayleigh–Love coupling in an anisotropic medium

SUMMARY

For a weakly anisotropic medium, Rayleigh and Love wave phase speeds at angular frequency |$\omega$| and propagation azimuth |$\psi$| are given approximately by |$V(\omega ,\psi ) = A_0 + A_{2c} \cos 2 \psi + A_{2s} \sin 2 \psi + A_{4c} \cos 4 \psi + A_{4s} \sin 4 \psi$|⁠. Earlier theories of the propagation of surface waves in anisotropic media based on non-degenerate perturbation theory predict that the dominant components are expected to be |$2\psi$| for Rayleigh waves and |$4\psi$| for Love waves. This paper is motivated by recent observations of the the 2|$\psi$| component for Love waves and 4|$\psi$| for Rayleigh waves, referred to here as ‘unexpected anisotropy’. To explain these observations, we present a quasi-degenerate theory of Rayleigh–Love coupling in a weakly anisotropic medium by applying Rayleigh-Ritz variational method based on Hamilton’s Principle in Cartesian coordinates, benchmarking this theory with numerical results based on a three-dimensional spectral-element solver (SPECFEM3D). We show that unexpected anisotropy is expected to be present when Rayleigh–Love coupling is strong and recent observations of Rayleigh and Love wave 2|$\psi$| and 4|$\psi$| anisotropy can be fit successfully with physically plausible models of a depth-dependent tilted transversely isotropic (TTI) medium. In addition, when observations of the 2|$\psi$| and 4|$\psi$| components of Rayleigh and Love anisotropy are used in the inversion, the ellipticity parameter |$\eta _X$|⁠, introduced here, is better constrained, we can constrain the absolute dip direction based on polarization measurements, and we provide evidence that the mantle should be modelled as a tilted orthorhombic medium rather than a TTI medium. Ignoring observations of unexpected anisotropy may bias the estimated seismic model significantly. We also provide information about the polarization of the quasi-Love waves and coupling between fundamental mode Love and overtone Rayleigh waves in both continental and oceanic settings. The theory of SV-SH coupling for horizontally propagating body waves is presented for comparison with the surface wave theory, with emphasis on results for a TTI medium.

Body waves, Seismic anisotropy, Surface waves and free oscillations, Theoretical seismology

1 INTRODUCTION

Based on non-degenerate perturbation theory, Smith & Dahlen (1973) showed that the azimuthal variation of Rayleigh and Love wave phase and group speeds at angular frequency |$\omega$| in a slightly anisotropic medium is of the well-known form

$$\begin{eqnarray} V(\psi ) &=& A_0 + A_{2c} \cos 2 \psi + A_{2s} \sin 2 \psi\\ && + A_{4c} \cos 4 \psi + A_{4s} \sin 4 \psi \end{eqnarray}$$

(1)

where |$\psi$| is the azimuth of propagation. They also provided expressions for the sensitivity of each of the coefficients in this expansion to the depth dependence of 13 independent elastic parameters. They argued that the azimuthal dependence of Rayleigh wave speeds will be dominated by the 2|$\psi$| terms in eq. (1), whereas the Love wave phase speeds will be dominated by the 4|$\psi$| terms. In first-order non-degenerate perturbation theory, mode coupling will perturb the eigenvectors to first order and the eigenfrequencies only to second order, as discussed by (Tanimoto 2004, eqs 16 and 17). Thus, the inherent assumption has been that Rayleigh and Love waves propagate largely independently and couple at most very weakly. Following (Smith & Dahlen 1973; Montagner & Nataf 1986) presented straightforward integral expressions for each of the coefficients in eq. (1) to be used to invert observational estimates of the coefficients as a function of frequency for the depth-dependent components of the elastic tensor.

The aforementioned studies have strongly influenced the subsequent observation and interpretation of surface wave anisotropy. In particular, focus has been placed on observing and interpreting the 2|$\psi$| component of Rayleigh wave anisotropy and to a lesser extent the 4|$\psi$| component of Love wave anisotropy. Many studies have presented and interpreted the 2|$\psi$| component of Rayleigh wave anisotropy observed with earthquake waves, dating back to the mid-1970s (e.g. Forsyth 1975; Tanimoto & Anderson 1985; Montagner & Jobert 1988; Nishimura & Forsyth 1988; Lévěque et al. 1998; Yuan & Romanowicz 2010). More recently, these observations have been expanded to include ambient noise observations (e.g. Yao et al. 2010; Lin et al. 2011). Observations of the 4|$\psi$| component of Love wave anisotropy are much more rare (e.g. Montagner & Tanimoto 1990; Trampert & Woodhouse 2003; Ekström 2011; Russell et al. 2019). Much less effort has been devoted to observing the 2|$\psi$| component of Love wave anisotropy or the 4|$\psi$| component of Rayleigh wave anisotropy. We refer to the 2|$\psi$| component for Rayleigh waves and the 4|$\psi$| component for Love waves as ‘expected’ anisotropy, according to non-degenerate perturbation theory. Similarly, the 4|$\psi$| component for Rayleigh waves and the 2|$\psi$| component for Love waves are referred to here as ‘unexpected’.

Based on ambient noise data, a recent study in an oceanic setting presented strong evidence for the observation of unexpected anisotropy (Russell et al. 2019). They show that the 2|$\psi$| component of Love wave anisotropy is observed and its amplitude is commensurate with the 4|$\psi$| component of Love wave anisotropy and the 2|$\psi$| component of Rayleigh wave anisotropy, at least at short periods. Broader band ambient noise methods are now being employed in a continental setting based on eikonal tomography (Lin et al. 2009) to observe unexpected anisotropy. Fig. 1 presents an example for a point in western Alaska (Liu et al. “Observations of Rayleigh and Love wave anisotropy across Alaska”, manuscript in preparation, 2024). Strong 2|$\psi$| Love wave anisotropy is observed at 20 s period as well as the weaker 4|$\psi$| component of Rayleigh wave anisotropy. As expected, the 2|$\psi$| component of the Rayleigh wave and the 4|$\psi$| component of the Love wave anisotropy are also observed at this point.

$Observations of azimuthal anisotropy for the 20 s Rayleigh (left column) and Love (right column) waves based on ambient noise observations in western Alaska (64$^\circ$N, 159$^\circ$W, Data Source 4). Total azimuthal variation is shown in the top row and 2$\psi$ and 4$\psi$ variations are shown in the middle and bottom rows, respectively. The series $V(\psi ) = A_0 + A_2 \cos (\psi -\psi _2) + A_4\cos (\psi -\psi _4)$ is fit to the total variation, and fit values with uncertainties are presented at the top of each column. Errors bars are 1$\sigma$ variations in each of the 36 azimuthal bins.$

Figure 1.

Observations of azimuthal anisotropy for the 20 s Rayleigh (left column) and Love (right column) waves based on ambient noise observations in western Alaska (64|$^\circ$|N, 159|$^\circ$|W, Data Source 4). Total azimuthal variation is shown in the top row and 2|$\psi$| and 4|$\psi$| variations are shown in the middle and bottom rows, respectively. The series |$V(\psi ) = A_0 + A_2 \cos (\psi -\psi _2) + A_4\cos (\psi -\psi _4)$| is fit to the total variation, and fit values with uncertainties are presented at the top of each column. Errors bars are 1|$\sigma$| variations in each of the 36 azimuthal bins.

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Such strong Love wave 2|$\psi$| and Rayleigh wave 4|$\psi$| anisotropy cannot be explained by the non-degenerate perturbation theory applied by Smith & Dahlen (1973). Fig. 2 illustrates this by presenting predictions from non-degenerate perturbation theory based on the model of the depth-varying elastic tensor estimated by Liu & Ritzwoller (2024). Liu and Ritzwoller inverted these observations of the Rayleigh wave 2|$\psi$| component of anisotropy along with the isotropic components of both Rayleigh and Love waves for a tilted transversely isotropic (TTI) model of the crust and uppermost mantle. As expected, this model and theory predict the 2|$\psi$| component of Rayleigh wave anisotropy well but strongly underpredict the observed amplitude of the 2|$\psi$| component of Love wave anisotropy.

$Comparison of observations of the amplitude of the 2$\psi$ and 4$\psi$ components of Rayleigh and Love wave anisotropy (black 1$\sigma$ error bars) from 8 to 50 s period at location (64$^\circ$N, 159$^\circ$W) in western Alaska with predictions using the elastic tensor model of the crust and uppermost mantle of Liu $\&$ Ritzwoller (Liu & Ritzwoller 2024), Data Source 2. Predictions (blue dashed lines) are computed using non-degenerate perturbation theory (Smith & Dahlen 1973; Montagner & Nataf 1986), which does not include Rayleigh–Love coupling. The amplitudes of the Love wave 2$\psi$ observations are too large to be fit with non-degenerate perturbation theory.$

Figure 2.

Comparison of observations of the amplitude of the 2|$\psi$| and 4|$\psi$| components of Rayleigh and Love wave anisotropy (black 1|$\sigma$| error bars) from 8 to 50 s period at location (64|$^\circ$|N, 159|$^\circ$|W) in western Alaska with predictions using the elastic tensor model of the crust and uppermost mantle of Liu |$\&$| Ritzwoller (Liu & Ritzwoller 2024), Data Source 2. Predictions (blue dashed lines) are computed using non-degenerate perturbation theory (Smith & Dahlen 1973; Montagner & Nataf 1986), which does not include Rayleigh–Love coupling. The amplitudes of the Love wave 2|$\psi$| observations are too large to be fit with non-degenerate perturbation theory.

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We argue in this paper that the unexpected signals arise from Rayleigh–Love coupling. Tanimoto (2004) presented an update to the theory of Smith & Dahlen (1973) based on a quasi-degeneracy condition that introduces Rayleigh–Love coupling. Formally, Tanimoto does not apply quasi-degenerate perturbation theory but, consistent with Maupin (1989), applies Hamilton’s Principle valid for weak anisotropy based on the quasi-degeneracy condition that coupling Love and Rayleigh waves have the same wavenumber but slightly different frequencies. The polarizations of the resulting quasi-Rayleigh and quasi-Love waves in an anisotropic medium are then superpositions of the polarizations in the reference medium (⁠|$\hat{\bf a}_R$|⁠, |$\hat{\bf a}_L$|⁠):

$$\begin{eqnarray} \tilde{\bf a} = a_\mathit{ R} \hat{\bf a}_\mathit{ R} + a_\mathit{ L} \hat{\bf a}_\mathit{ L} , \end{eqnarray}$$

(2)

where |$a_\mathit{ L}$| and |$a_\mathit{ R}$| are coupling coefficients following the notation of Tanimoto (2004). Tanimoto (2004) set the coupling coefficients to be real and argued that the strength of coupling for realistic anisotropy in the Earth will be small. Therefore, his quasi-degenerate theory also is unable to explain observations of strong 2|$\psi$| Love wave or 4|$\psi$| Rayleigh wave anisotropy and types of anisotropy remained unexpected.

In this paper, we present a revised quasi-degenerate theory that does explain observations of strong 2|$\psi$| Love wave and 4|$\psi$| Rayleigh wave anisotropy. When Rayleigh–Love coupling is strong enough, significant 2|$\psi$| Love wave anisotropy is expected although the 4|$\psi$| Rayleigh wave anisotropy is typically weaker than the other components. We follow the methods of Tanimoto (2004), with the principal revision that the coupling coefficients are allowed to be complex in accordance with Maupin (1989) because the polarization vectors are complex for surface waves and because, as we shall see, the vertical derivatives of the eigenfunctions add further complexity. We show that this greatly enhances Rayleigh–Love coupling and allows observations, such as those presented in Fig. 1, to be fit with physically plausible models of the depth-variation of the elastic tensor.

The data sources we use for examples and computations are described in Section 2. Because of their similarity, the theoretical preliminaries for both body waves and surface waves are presented together in Section 3. Like Smith & Dahlen (1973), for purposes of comparison and to provide guidance about interpreting the surface wave results, we reproduce results for horizontally propagating body waves in an infinite, homogeneous anisotropic medium. To further tighten the comparison between the body wave and surface wave treatments, in Section 4 we apply Hamilton’s Principle based on a quasi-degeneracy condition to derive the body wave formalism, which models SV–SH coupling. We believe that this is the first time this approach has been taken, but the results are identical to those produced by the degenerate perturbation theory of Jech & Pšenčík (1989) (also Chapman 2004; Chen & Tromp 2007; Červenỳ & Pšenčík 2020). In Section 4, we then present expressions for the phase speeds and polarizations of coupled Rayleigh and Love waves and we benchmark our theory against numerical results obtained with a three-dimensional spectral-element solver (SPECFEM3D, Komatitsch & Tromp 1999). We then use the theory in Section 5 to show that the simultaneous observation of expected and unexpected anisotropy in Alaska can be fit with physically plausible models of the depth-dependent elastic tensor. We also highlight new information that results from using Love wave 2|$\psi$| and 4|$\psi$| and Rayleigh wave 4|$\psi$| observations in the inversion and discuss several other issues in Section 5. These include evidence that a tilted orthorhombic elastic tensor in the mantle should be used in place of the TTI elastic tensor, differences in the nature of Rayleigh–Love coupling in oceanic and continental settings with focus on the role of overtones, and the utility of polarization measurements for quasi-Love waves to constrain anisotropy, which was a point emphasized by Park & Yu (1993), Tanimoto (2004) and Maupin & Park (2015). Principal derivations are presented in the Supporting Information.

For clarification, we note that in the results we present the reference medium for body waves does not matter but the reference medium for surface waves is the effective transversely isotropic part of the 21 component elastic tensor (Appendix B), which matters because we use the eigenfunctions from the reference medium (for detail, see Section 4.1 for surface wave theory). One could also use an isotropic medium as the reference medium with similar results. The method we use sometimes is called the Rayleigh–Ritz variational principle (e.g. Aki & Richards 2002; Dahlen & Tromp 2020). Thus, we refer to the method as “a quasi-degenerate theory” rather than a perturbation theory. The accuracy of this method depends on the completeness of the basis eigenfunctions used for expansion in eq. (2).

2 DATA SOURCES

Four different data compilations or models are used here for computation and inversion, as examples of the effect of anisotropy on body wave and surface wave speeds and polarizations.

Data Source 1. We use the database of elastic tensor measurements of crustal rocks presented by Brownlee et al. (2017). The full elastic tensor is presented in the database for 93 samples along with the vertical transversely isotropic (VTI) or effective transversely isotropic component (Browaeys & Chevrot 2004). The VTI component of the elastic tensor for sample #20 is shown in Table 1. We use the database primarily to present examples of body wave calculations.

Table 1.

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Transversely isotropic component of the elastic tensor from sample #20, Data Source 1.

A	C	N	L	F	\|$\eta$\|	\|$\eta _K$\|	\|$\eta _X$\|	\|$\rho$\|
159.6 GPa	143.7 GPa	47.5 GPa	43.2 GPa	62.0 GPa	0.85	0.97	0.97	3 \|$\times 10^3$\| kg m⁻³

A	C	N	L	F	\|$\eta$\|	\|$\eta _K$\|	\|$\eta _X$\|	\|$\rho$\|
159.6 GPa	143.7 GPa	47.5 GPa	43.2 GPa	62.0 GPa	0.85	0.97	0.97	3 \|$\times 10^3$\| kg m⁻³

Table 1.

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Transversely isotropic component of the elastic tensor from sample #20, Data Source 1.

A	C	N	L	F	\|$\eta$\|	\|$\eta _K$\|	\|$\eta _X$\|	\|$\rho$\|
159.6 GPa	143.7 GPa	47.5 GPa	43.2 GPa	62.0 GPa	0.85	0.97	0.97	3 \|$\times 10^3$\| kg m⁻³

A	C	N	L	F	\|$\eta$\|	\|$\eta _K$\|	\|$\eta _X$\|	\|$\rho$\|
159.6 GPa	143.7 GPa	47.5 GPa	43.2 GPa	62.0 GPa	0.85	0.97	0.97	3 \|$\times 10^3$\| kg m⁻³

Data Source 2. We also use the model of the depth-dependent TTI elastic tensor in the crust and uppermost mantle at a location in western Alaska (64|$^\circ$|N, 159|$^\circ$|W), taken from Liu & Ritzwoller (2024), which is based on fitting only the isotropic Love and Rayleigh wave phase speed curves and 2|$\psi$| Rayleigh wave anisotropy. This model is used to present preliminary comparisons between surface wave observations and theoretical predictions.

Data Source 3. We use another model of the depth-dependent elastic tensor in the crust and uppermost mantle at a location in the central Pacific at the NoMelt ocean-bottom seismic array, taken from Russell et al. (2019). We revise this model and use it to compute the strength of Rayleigh–Love coupling in an oceanic setting.

Data Source 4. Finally, we use a new preliminary database of Rayleigh wave and Love wave 2|$\psi$| and 4|$\psi$| azimuthal phase speed variations measured across Alaska (Liu et al. “Observations of Rayleigh and Love wave anisotropy across Alaska”, manuscript in preparation, 2024). We apply the data primarily at the same point in western Alaska (64|$^\circ$|N, 159|$^\circ$|W) as in Data Source 2 to perform a number of inversions with different data subsets and theories, but also produce a new model in eastern Alaska for comparison (64|$^\circ$|N, 147|$^\circ$|W). We make use of the resulting models to compute the strength of Rayleigh–Love coupling in a continental setting.

3 QUASI-DEGENERATE THEORY FOR BODY AND SURFACE WAVES

3.1 Polarization and displacement basis vectors

In Cartesian coordinates |$(x_1,x_2,x_3) = (x,y,z)$|⁠, the plane wave displacement for horizontally propagating body waves at depth z can be written

$$\begin{eqnarray} \vec{\bf u}_{\mathit{ BW}}(\vec{\bf r}, t) = A \hat{\bf a} e^{i ( \vec{\bf k} \cdot \vec{\bf r} - \omega t)} , \end{eqnarray}$$

(3)

where |$\hat{\bf a}$| is the direction of particle motion or the polarization vector, the components of the position vector |$\vec{\bf r}$| are |$x_i\, \left((x_1,x_2,x_3)^T = (x,y,z)^T \right)$| and of the horizontal wavenumber vector |$\vec{\bf k}$| are |$\omega n_i/V$|⁠, where |$n_i$| is the unit vector in the direction of propagation (perpendicular to the wave front) and V is the phase speed of the wave. Surface wave displacement can be written similarly as

$$\begin{eqnarray} \vec{\bf u}_{\mathit{ SW}}(\vec{\bf r}, z, t) = A \hat{\bf s}(z) e^{i ( \vec{\bf k} \cdot \vec{\bf r} - \omega t)} , \end{eqnarray}$$

(4)

where |$z=0$| is the free surface, surface location |$\vec{\bf r} = (x,y,0)^T$|⁠, and |$\hat{\bf s}(z)$| is the vector displacement eigenfunction.

We set the basis vectors for body waves propagating horizontally at azimuth |$\psi$| relative to the x-axis to be in the the direction of motion for P, vertical for |$SV$|⁠, and perpendicular to both P and |$SV$| for |$SH$|⁠, as depicted in Fig. 3. Therefore the polarization basis vectors are

$$\begin{eqnarray} \hat{\bf a}_{P}(\vec{\bf r}, t) = \hat{\bf a}^{(1)} = (\cos \psi , \sin \psi , 0)^T , \end{eqnarray}$$

(5)

$$\begin{eqnarray} \hat{\bf a}_{SH}(\vec{\bf r}, t) = \hat{\bf a}^{(2)} = (-\sin \psi , \cos \psi , 0)^T , \end{eqnarray}$$

(6)

$$\begin{eqnarray} \hat{\bf a}_{SV}(\vec{\bf r}, t) = \hat{\bf a}^{(3)} = (0,0,1)^T \end{eqnarray}$$

(7)

which we denote with the overscript |$^{\wedge}$| and T means transpose. The displacement vectors in the reference medium are

$$\begin{eqnarray} \hat{\bf u}_{P}(\vec{\bf r}, t) = \hat{\bf a}^{(1)} f (\vec{\bf r}, t) , \end{eqnarray}$$

(8)

$$\begin{eqnarray} \hat{\bf u}_{SH}(\vec{\bf r}, t) = \hat{\bf a}^{(2)} f (\vec{\bf r}, t) , \end{eqnarray}$$

(9)

$$\begin{eqnarray} \hat{\bf u}_{SV}(\vec{\bf r}, t) = \hat{\bf a}^{(3)} f (\vec{\bf r}, t) \end{eqnarray}$$

(10)

which we also denote with an overscript |$^{\wedge}$|⁠. The propagation term for horizontal propagation is

$$\begin{eqnarray} f ( \vec{\bf r},t) = \exp \left[ i ( \vec{\bf k} \cdot \vec{\bf r} - \omega t) \right] = \exp \left[ i(k( x \cos \psi + y \sin \psi ) - \omega t) \right] , \end{eqnarray}$$

(11)

where phase speed |$V = \omega /k$|⁠. The S-wave basis vectors could be in any pair of orthogonal directions in the vertical plane perpendicular to the direction of travel of the wave, but we choose the horizontal (transverse) and vertical directions for simplicity.

$Geometry of horizontal body wave propagation in the direction defined by the azimuthal angle $\psi$ relative to the $x_1$-axis, showing the waves in the reference isotropic medium, ${P}, {SH}$ and ${SV}$, as well as the quasi-S waves ($_qS_1$, $_qS_2$) in the perturbed anisotropic medium. SV–SH coupling rotates the polarization of the quasi-shear waves through angle $\Phi$ in the plane perpendicular to the direction of propagation. We define $\Omega$ as the negative of the complement of $\Phi$ and $x_2$ is the ‘strike axis’.$

Figure 3.

Geometry of horizontal body wave propagation in the direction defined by the azimuthal angle |$\psi$| relative to the |$x_1$|-axis, showing the waves in the reference isotropic medium, |${P}, {SH}$| and |${SV}$|⁠, as well as the quasi-S waves (⁠|$_qS_1$|⁠, |$_qS_2$|⁠) in the perturbed anisotropic medium. SV–SH coupling rotates the polarization of the quasi-shear waves through angle |$\Phi$| in the plane perpendicular to the direction of propagation. We define |$\Omega$| as the negative of the complement of |$\Phi$| and |$x_2$| is the ‘strike axis’.

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Similarly, the basis vectors for surface wave displacement in the reference medium are Rayleigh and Love waves in a laterally homogeneous medium for a wave propagating at azimuth |$\psi$|⁠. The polarization vectors are

$$\begin{eqnarray} \hat{\bf a}_R(\vec{\bf r}, z, t) = \left[ (\cos \psi , \sin \psi , 0)^T V(z) + (0,0,i)^T U(z) \right] , \end{eqnarray}$$

(12)

$$\begin{eqnarray} \hat{\bf a}_L (\vec{\bf r}, z, t) = (-\sin \psi , \cos \psi , 0)^T W(z) \end{eqnarray}$$

(13)

with displacement vectors

$$\begin{eqnarray} \hat{\bf u}_R(\vec{\bf r}, z, t) = \hat{\bf a}_R(\vec{\bf r}, z, t) f( \vec{\bf r},t) , \end{eqnarray}$$

(14)

$$\begin{eqnarray} \hat{\bf u}_L (\vec{\bf r}, z, t) =\hat{\bf a}_L (\vec{\bf r}, z, t) f( \vec{\bf r},t) \end{eqnarray}$$

(15)

|$U(z)$| and |$V(z)$| are the vertical and horizontal (radial) displacement eigenfunctions for Rayleigh waves and |$W(z)$| is the Love wave horizontal (transverse) eigenfunction, which are normalized as follows:

$$\begin{eqnarray} 1 = \int _0^\infty \rho (z) W^2(z) \mathrm{ d}z , \end{eqnarray}$$

(16)

$$\begin{eqnarray} 1 = \int _0^\infty \rho (z) \left(U^2(z) + V^2(z) \right) \mathrm{ d}z . \end{eqnarray}$$

(17)

Example eigenfunctions are plotted later, in Fig. 7(a).

3.2 Coupling caused by anisotropy

In anisotropic media, the displacement of the resulting waves will be a mixture of the displacements of the basis vectors. P, SV, and SH waves will couple to produce a quasi-P wave (⁠|${_q}P$|⁠) and two quasi-S waves |$({_q}S_1, {_q}S_2)$| and Rayleigh and Love waves will couple to produce quasi-Love and quasi-Rayleigh waves |$({_q}L, {_q}R)$|⁠.

For body waves with general coupling between P, SH, and SV, the polarization vectors in the anisotropic medium will be

$$\begin{eqnarray} qP: \tilde{{\bf a}}^{(1)} = a_{11} \hat{\bf a}^{(1)} + a_{12} \hat{\bf a}^{(2)} + a_{13} \hat{\bf a}^{(3)} , \end{eqnarray}$$

(18)

$$\begin{eqnarray} qS_1: \tilde{{\bf a}}^{(2)} = a_{21} \hat{\bf a}^{(1)} + a_{22} \hat{\bf a}^{(2)} + a_{23} \hat{\bf a}^{(3)} , \end{eqnarray}$$

(19)

$$\begin{eqnarray} S_2: \tilde{{\bf a}}^{(3)} = a_{31} \hat{\bf a}^{(1)} + a_{32} \hat{\bf a}^{(2)} + a_{33} \hat{\bf a}^{(3)} . \end{eqnarray}$$

(20)

We denote quantities in the anisotropic medium with an overscript |$~$|⁠. Because the basis vectors for body waves are real and depth-independent, the expansion coefficients |$a_{ij}$| are also real; that is, |$a_{ij} \in \mathbb {R}$|⁠.

In real Earth media, the quasi-P wave phase speed is much more different from the two quasi-S wave speeds than they are from one another. Thus, we consider only coupling between the SH and SV waves and will ignore the weaker coupling between P and SV and SH. Thus, we set |$a_{11}=1$| and |$a_{12} = a_{21} = a_{13} = a_{31} = 0$|⁠. Therefore, approximately

$$\begin{eqnarray} qP: \tilde{{\bf a}}^{(1)} &\approx & \hat{\bf a}^{(1)} , \end{eqnarray}$$

(21)

$$\begin{eqnarray} qS_1: \tilde{{\bf a}}^{(2)} &\approx & a_{SH} \hat{\bf a}^{(2)} + a_{SV} \hat{\bf a}^{(3)}\\ & =& \cos \Phi \hat{\bf a}^{(2)} + \sin \Phi \hat{\bf a}^{(3)} , \end{eqnarray}$$

(22)

$$\begin{eqnarray} qS_2: \tilde{{\bf a}}^{(3)} &\approx & -a_{23} \hat{\bf a}^{(2)} + a_{33} \hat{\bf a}^{(3)} = -a_{SV} \hat{\bf a}^{(2)} + a_{SH} \hat{\bf a}^{(3)}\\ & =& -\sin \Phi \hat{\bf a}^{(2)} + \cos \Phi \hat{\bf a}^{(3)} , \end{eqnarray}$$

(23)

where we have introduced notation for the expansion coefficients |$a_{SH}$| and |$a_{SV}$|⁠, such that |$a_{SH}^2 + a_{SV}^2 = 1$|⁠. The second equalities in the latter two equations follow from the fact that the relationship between the polarizations of the quasi-S waves and the S waves in the reference medium is a rotation through polarization angle |$\Phi$|⁠, as Fig. 3 illustrates. Thus, |$a_{22} = \cos \Phi , a_{23} = \sin \Phi , a_{32} = -\sin \Phi$|⁠, and |$a_{33} = \cos \Phi$|⁠, where |$\Phi$| is the angle between the reference SH polarization vector and the polarization vector for quasi-S|$_1$|⁠. It is also the angle from the reference SV polarization vector and the polarization vector for quasi-S|$_2$|⁠. To find the polarizations of the quasi-S waves we need only find |$\Phi$|⁠.

Body wave displacement associated with the perturbed polarizations in eqs (21)–(23) is

$$\begin{eqnarray} \tilde{ {\bf u} }^{(m)} = \tilde{{\bf a} }^{(m)} f \end{eqnarray}$$

(24)

By solving the Christoffel equation (Section S.1, Supporting Information) numerically, we can compute the effect of coupling the quasi-S waves to the quasi-P wave exactly, as illustrated in Fig. 4. This shows that for the rock samples in the elastic tensor database of Brownlee et al. (2017), the average maximum tilt out of the vertical plane of the eigenvector for the quasi-S|$_2$| wave is about 3|$^\circ$|⁠. The eigenvector of the quasi-S|$_1$| wave is unaffected by coupling to the quasi-P wave.

$Numerical (non-approximate) computation of the effect of coupling with the quasi P-wave on the polarizations of the two quasi-S waves. (a) Deflection of the quasi-S$_1$ and quasi-S$_2$ eigenvectors out of the vertical plane due to coupling to the quasi-P wave, presented as a function of azimuth of propagation. Result is for the transversely isotropic component of sample #20 from the elastic tensor database in Data Source 1, tilted through a dip angle $\theta = 45^\circ$, which produces the strongest coupling to the P-wave. In this sample, the maximum effect is about 2$^\circ$ for quasi-S$_2$, with no effect on quasi-S$_1$. (b) Histogram of maximum out of vertical plane tilt angles for the quasi-S$_2$ polarizations for all 93 samples in Data Source 1 tilted by a dip angle $\theta = 45^\circ$. The mean maximum deflection is about 3$^\circ$.$

Figure 4.

Numerical (non-approximate) computation of the effect of coupling with the quasi P-wave on the polarizations of the two quasi-S waves. (a) Deflection of the quasi-S|$_1$| and quasi-S|$_2$| eigenvectors out of the vertical plane due to coupling to the quasi-P wave, presented as a function of azimuth of propagation. Result is for the transversely isotropic component of sample #20 from the elastic tensor database in Data Source 1, tilted through a dip angle |$\theta = 45^\circ$|⁠, which produces the strongest coupling to the P-wave. In this sample, the maximum effect is about 2|$^\circ$| for quasi-S|$_2$|⁠, with no effect on quasi-S|$_1$|⁠. (b) Histogram of maximum out of vertical plane tilt angles for the quasi-S|$_2$| polarizations for all 93 samples in Data Source 1 tilted by a dip angle |$\theta = 45^\circ$|⁠. The mean maximum deflection is about 3|$^\circ$|⁠.

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For surface waves, we assume the displacement for the fundamental mode in an anisotropic medium is a superposition of all modes in the reference medium. The theory we present can be applied based on any reference medium, but for simplicity we choose a effective transversely isotropic medium as the reference (Appendix B), including Rayleigh and Love waves, fundamental and overtone modes. Here, we reduce the superposition to only two modes, a Rayleigh mode and a Love mode. We consider Rayleigh–Love coupling in this subspace which is simple and valid under the assumption of weak anisotropy. For strong anisotropy, coupling between all modes needs to be considered with similar methods. We focus on fundamental modes but any pair of Rayleigh and Love modes could be used in the theory here. In this case, displacement in an anisotropic medium is the following superposition

$$\begin{eqnarray} \tilde{\bf u} = a_R \hat{\bf u}_R + a_L \hat{\bf u}_L . \end{eqnarray}$$

(25)

The expansion coefficients |$a_R$| and |$a_L$| define the Rayleigh–Love coupling and are complex mainly because the basis vectors are complex: |$a_R, a_L \in \mathbb {C}$|⁠, such that |$a_L a^{*}_L + a_R a^{*}_R =1$|⁠. Tanimoto (2004) set |$a_R$| and |$a_L$| to be real, which, as we discuss below, typically results in very weak Rayleigh–Love coupling.

Therefore, the fundamental mode displacement in an anisotropic medium for a wave propagating at azimuth |$\psi$| is:

$$\begin{eqnarray} \tilde{\bf u} (\vec{\bf r}, z, t) &=& ( a_R V(z)\cos \psi - a_L W(z) \sin \psi , a_R V(z) \sin \psi\\ && +\, a_L W(z) \cos \psi , ia_R U(z))^T f( \vec{\bf r},t) . \end{eqnarray}$$

(26)

3.3 Quasi-degeneracy

Under the quasi-degeneracy condition, waves and modes are coupled that have the same wavenumber k in the reference medium, but the resulting waves and modes will have slightly different frequencies |$\omega$| and phase speeds V than their values in the reference medium. This coupling can have a large impact on waveform and phase velocity anisotropy. Usually for the coupling Rayleigh and Love modes in the reference medium, the frequencies will be similar but not identical, which is why this is referred to as a quasi-degeneracy approximation, or in the context of perturbation theory as ‘quasi-degenerate perturbation theory’. If their frequencies or phase velocities are the same, this reduces the degenerate theory. If their frequency or phase velocity differences are much larger than their coupling, this is usually referred to as ‘non-degenerate’ and non-degenerate perturbation theory will work very well in this case. The quasi-degeneracy condition is illustrated in Fig. 5 for surface waves, presenting dashed lines with common wavenumbers (k) linking potentially coupling Rayleigh and Love modes. In particular, the figure illustrates which quasi-degenerate Rayleigh and Love modes will couple under this assumption for the Love wave at periods of 20 and 40 s.

$Phase speed curves for Rayleigh and Love wave fundamental modes and first two overtone modes for a continental and an oceanic effective transversely isotropic model, illustrating the quasi-degeneracy condition. (a) Produced using the effective transversely isotropic component of the 1-D model from Data Source 2, at (64$^\circ$N, 159$^\circ$W) in western Alaska. (b) Produced using the effective transversely isotropic component of the 1-D model from Data Source 3, southeast of Hawaii in the central Pacific. The dashed lines are lines of constant wavenumber passing through the fundamental Love wave phase speed curve at periods of 20 and 40 s. Under the quasi-degeneracy condition, modes couple along these lines.$

Figure 5.

Phase speed curves for Rayleigh and Love wave fundamental modes and first two overtone modes for a continental and an oceanic effective transversely isotropic model, illustrating the quasi-degeneracy condition. (a) Produced using the effective transversely isotropic component of the 1-D model from Data Source 2, at (64|$^\circ$|N, 159|$^\circ$|W) in western Alaska. (b) Produced using the effective transversely isotropic component of the 1-D model from Data Source 3, southeast of Hawaii in the central Pacific. The dashed lines are lines of constant wavenumber passing through the fundamental Love wave phase speed curve at periods of 20 and 40 s. Under the quasi-degeneracy condition, modes couple along these lines.

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3.4 The Lagrangian and Hamilton’s principle

For a linear elastic body, the Lagrangian density is the difference between the kinetic energy and elastic strain energy, which for body and surface waves, respectively, are given by

$$\begin{eqnarray} L _{BW} (\dot{u}_i, u_{i,j}) = T_{BW} - V_{BW} = \frac{1}{2} \omega ^2 \rho u_i u^{*}_i - \frac{1}{2} c_{ijkl} \epsilon _{ij} \epsilon ^{*} _{kl} , \end{eqnarray}$$

(27)

$$\begin{eqnarray} L _{SW} (\dot{u}_i, u_{i,j}) &=& T_{SW} - V_{SW} = \frac{1}{2} \omega ^2 \int _{0}^{\infty } \rho u_i u^{*}_i \mathrm{ d}z \\ &&-\, \frac{1}{2} \int _{0}^{\infty } c_{ijkl} \epsilon _{ij} \epsilon ^{*} _{kl} \mathrm{ d}z , \end{eqnarray}$$

(28)

where |$c_{ijk\ell }$| is the elastic tensor, |$\epsilon _{ij} = (u_{i,j} + u_{j,i})/2$|⁠, the subscript ‘|$,j$|’ represents a spatial derivative in the |$x_j$| direction, and |$*$| denotes complex conjugation. Displacement appears in eqs (27) and (28) as a product with its complex conjugate, therefore because |$ff^{*} = 1$| the propagation term f and all time-dependent terms disappear from further equations. For the anisotropic medium, |$u_i$| is replaced by |$\tilde{u}_i$|⁠.

Expressions for T and V are derived in Section S.1 (Supporting Information) for body waves and Section S.7 (Supporting Information) for surface waves.

In Section S.6 (Supporting Information), we show that Hamilton’s Principle implies that |$\partial L/\partial a_{SH} = \partial L/\partial a_{SV} = 0$| for body waves and that |$\partial L/\partial a_{L} = \partial L/\partial a_{R} = 0$| for surface waves. The latter for surface waves was first applied by Tanimoto (2004). Applying these derivatives results in an eigenvalue-eigenvector equation for the frequencies or phase speeds of the three quasi-body waves and two quasi-surface waves as well as their polarizations, which is the subject of Sections 4 and 5.

4 THE EFFECT OF SV–SH COUPLING

For purposes of comparison with Rayleigh–Love coupling, SV–SH coupling for horizontally propagating body waves is discussed in detail for a general anisotropic medium in Sections S.1 and S.2 (Supporting Information). For a TTI medium, SV–SH coupling is presented in Sections S.3 and S.4 (Supporting Information), which we summarize here.

The eigenvalues and eigenvectors for the quasi-S waves in a general anisotropic medium simplify substantially when they are considered for a TTI medium. We define tilt through dip angle |$\theta$| around the y-axis, which we refer to as the ‘strike axis’.

For the horizontally propagating quasi-S|$_1$| and quasi-S|$_2$| waves

$$\begin{eqnarray} \rho V^2_{qS_1} = C_0 + C_2 \cos 2 \psi , \end{eqnarray}$$

(29)

$$\begin{eqnarray} \rho V^2_{qS_2} &= & B_0 + B_2 \cos 2 \psi + B_4 \cos 4 \psi , \end{eqnarray}$$

(30)

where

$$\begin{eqnarray} C_0 = \frac{1}{2} \left(L(1 - \cos ^2 \theta ) + N(1 + \cos ^2 \theta ) \right) , \end{eqnarray}$$

(31)

$$\begin{eqnarray} C_2 = \frac{1}{2} (L-N) \sin ^2 \theta , \end{eqnarray}$$

(32)

$$\begin{eqnarray} B_0 &=& L +E \left(\frac{1}{2} \sin ^2 \theta \cos ^2 \theta + \frac{1}{8} \sin ^4 \theta \right)\\ & \approx& B_0^{HTI} \approx \frac{1}{8}(A + C - 2 F)(1 + \eta _X) , \end{eqnarray}$$

(33)

$$\begin{eqnarray} B_2 = \frac{1}{2} E \sin ^2 \theta \cos ^2 \theta \approx \frac{1}{2} (A+C-2F)(1-\eta _X) \sin ^2 \theta \cos ^2 \theta \\ , \end{eqnarray}$$

(34)

$$\begin{eqnarray} B_4 = - \frac{1}{8} E \sin ^4 \theta \approx -\frac{1}{8}(A + C - 2 F)(1 - \eta _X)\sin ^4 \theta \end{eqnarray}$$

(35)

and |$E \equiv A + C - 2F -4L$|⁠, as defined in the Section S.3 (Supporting Information).

The signs of |$B_2$| and |$B_4$| for quasi-S|$_2$| and their relationship to the sign of |$C_2$| for quasi-S|$_1$|⁠, will be determined in part by the sign of E. This will specify the relative phase of the azimuthal variations of quasi-S|$_1$| and quasi-S|$_2$|⁠. The sign of E will depend on the relative size of |$4L$| and |$A+C-2F$|⁠. If |$E=0$|⁠, |$4L = A+C-2F$|⁠, then quasi-S|$_2$| will show no azimuthal variation, its phase front will be spherical, and the quasi-P (⁠|$B_4 = 0$|⁠, |$E_c =0$|⁠) and quasi-S|$_1$| will both have elliptical phase fronts. This is so-called elliptical anisotropy.

As discussed further in Section S.5 (Supporting Information), this motivates the definition of a new ellipticity parameter

$$\begin{eqnarray} \eta _X \equiv \frac{4L}{A + C - 2F} \end{eqnarray}$$

(36)

which for weak anisotropy is approximately equal to the parameter |$\eta _K$| introduced by Kawakatsu (2016), as illustrated by Fig. S3 (Supporting Information). |$\eta _X = 1$| for elliptical anisotropy but is typically less than 1 for real Earth materials (Brownlee et al. 2017) as Fig. S3 (Supporting Information) shows, at least for crustal rocks.

As shown in Section S.5 (Supporting Information), the coefficients |$B_0$|⁠, |$B_2$| and |$B_4$| for quasi-S|$_2$| can be expressed approximately in terms of |$\eta _X$| according to the final expressions in eqs (33)–(35). |$A+C-2F$| is normally positive in Earth materials. The relative peak-to-peak amplitude of |$2\psi$| and |$4\psi$| anisotropy of quasi-S|$_2$| can therefore be expressed as:

$$\begin{eqnarray} \frac{|B_2|}{B_0} &\approx & 2 | 1 - \eta _X | \sin ^2 \theta \cos ^2 \theta , \end{eqnarray}$$

(37)

$$\begin{eqnarray} \frac{|B_4|}{B_0} &\approx & \frac{1}{2} |1 - \eta _X|\sin ^4 \theta . \end{eqnarray}$$

(38)

The polarization angle |$\Phi$| for the coupled quasi-S waves is derived in Section S.4 (Supporting Information) as

$$\begin{eqnarray} \tan \Phi = \tan \theta \sin \psi , \end{eqnarray}$$

(39)

where |$-\theta \le \Phi \le \theta$|⁠. |$| \Phi |$| will be no larger than the dip angle |$\theta$|⁠, and will average about |$\theta /2$|⁠.

$Azimuthal anisotropy with a dip angle $\theta = 45^\circ$ for quasi-S$_1$ and quasi-S$_2$: quasi-S$_1$ 2$\psi$ (green solid line for slow axis and orange solid line for fast axis), quasi-S$_2$ 4$\psi$ (red solid line for $\eta _X < 1$ and blue solid line for $\eta _X > 1$), and quasi-S$_2$ 2$\psi$ (purple dashed line for $\eta _X < 1$ and black dashed line for $\eta _X > 1$). The results are normalized by isotropic phase speed and $|B_2| = 4|B_4|$ by eq. (S104).$

Figure 6.

Azimuthal anisotropy with a dip angle |$\theta = 45^\circ$| for quasi-S|$_1$| and quasi-S|$_2$|⁠: quasi-S|$_1$| 2|$\psi$| (green solid line for slow axis and orange solid line for fast axis), quasi-S|$_2$| 4|$\psi$| (red solid line for |$\eta _X < 1$| and blue solid line for |$\eta _X > 1$|⁠), and quasi-S|$_2$| 2|$\psi$| (purple dashed line for |$\eta _X < 1$| and black dashed line for |$\eta _X > 1$|⁠). The results are normalized by isotropic phase speed and |$|B_2| = 4|B_4|$| by eq. (S104).

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4.1 Implications for surface waves

There are three principal implications from SV–SH coupling in a TTI medium for surface waves, which are:

2. A central argument of this paper is that Love wave 2|$\psi$| arises from Rayleigh–Love coupling. The simple results for body waves can provide a better understanding because they have similar eigenvalue problems (as we will see later in surface wave section). If we compare the equation of total 2|$\psi$| amplitude, which is |$G_c$| (eq. S55), with the 2|$\psi$| amplitude of qS|$_1$| (eq. 32) and qS|$_2$| (eq. 34), we find SV–SH coupling just splits the total 2|$\psi$| amplitude into two parts, with one going to qS|$_1$| and the other going to qS|$_2$|⁠. And their fast axes can be either parallel or perpendicular, unlike the always perpendicular case in the non-degenerate perturbation theory for surface waves. The body wave theory can be considered to be an exceptional case for surface waves (nearly degenerate and similar depth distribution of the eigenfunctions), so it provides guidance and explains a lot of observations either at global scale (e.g. Montagner & Tanimoto 1990) or regional scale like in Alaska (Liu et al. “Observations of Rayleigh and Love wave anisotropy across Alaska”, manuscript in preparation, 2024).

(3) For surface waves (derived later), the modes in an anisotropic medium are neither Rayleigh nor Love waves as they can have similar velocities and polarizations. This situation is similar to body waves as we assign qS|$_1$| and q|$S_2$| to the two quasi-shear waves, instead of qSH or qSV. However, in this paper we do not discuss these extreme cases for surface waves, so we still assign quasi-Rayleigh and quasi-Love wave to the modes in an anisotropic medium.

5 THE EFFECT OF RAYLEIGH–LOVE COUPLING

5.1 Theory

Most of the foundational equations are presented in Section 3. In Cartesian coordinates |$(x_1,x_2,x_3) = (x,y,z)$|⁠, for a laterally homogeneous isotropic or transversely isotropic reference medium, the displacements for Rayleigh and Love waves propagating at azimuth |$\psi$| are given by eqs (14) and (15) where f is given by eq. (11). Displacement |$\vec{\bf u}$| in an anisotropic medium is given by eq. (25). The displacement field in an anisotropic medium for a coupled Rayleigh and Love wave propagating at azimuth |$\psi$| is given by eq. (26). For a linear elastic body, the Lagrangian density is given by eq. (28).

Example phase speed curves for Rayleigh and Love modes are presented in Fig. 5(a). Example eigenfunctions are shown in Fig. 7(a).

$(a) Eigenfunctions for Rayleigh and Love wave fundamental modes at 20 s period computed using the effective transversely isotropic component of the 1-D model in western Alaska at (64$^\circ$N, 159$^\circ$W) (Data source 2). (b) Sensitivity kernels composing the integrals in eqs (42)–(45): $A1 = W^2$, $A2 = W^{\prime 2}/k^2$$B1 = V^2$, $B2 = (U - V^\prime /k)^2,$$B3 = V U^\prime /k$, $B4 = U^{\prime 2}/k^2$, $E1 = WV$, $E2 = (U-V^\prime /k) W^\prime /k$, $E3 = W U^\prime /k$, $X1 = V W^\prime /k$, $X2 = W (U - V^\prime /k$), $X3 = U^\prime W^\prime /k^2.$.$

Figure 7.

(a) Eigenfunctions for Rayleigh and Love wave fundamental modes at 20 s period computed using the effective transversely isotropic component of the 1-D model in western Alaska at (64|$^\circ$|N, 159|$^\circ$|W) (Data source 2). (b) Sensitivity kernels composing the integrals in eqs (42)–(45): |$A1 = W^2$|⁠, |$A2 = W^{\prime 2}/k^2$||$B1 = V^2$|⁠, |$B2 = (U - V^\prime /k)^2,$||$B3 = V U^\prime /k$|⁠, |$B4 = U^{\prime 2}/k^2$|⁠, |$E1 = WV$|⁠, |$E2 = (U-V^\prime /k) W^\prime /k$|⁠, |$E3 = W U^\prime /k$|⁠, |$X1 = V W^\prime /k$|⁠, |$X2 = W (U - V^\prime /k$|⁠), |$X3 = U^\prime W^\prime /k^2.$|⁠.

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Expressions for T and V are derived in Section S.7 (Supporting Information), and are

$$\begin{eqnarray} T = \frac{1}{2} \omega ^2 \left(a_L a_L^{*} + a_R a_R^{*} \right) , \end{eqnarray}$$

(40)

$$\begin{eqnarray} V &=& \frac{1}{2} [ a_L a_L^{*}A + a_R a_R^{*}B + (a_L a_R^{*} + a_L^{*} a_R)E \\ &&+\, i(a_L a_R^{*} - a_L^{*} a_R)X ] . \end{eqnarray}$$

(41)

In the expression for the potential energy, if |$a_L$| and |$a_R$| were real, the term in parenthesis before X would be 0. X only contributes to Rayleigh–Love coupling if |$a_R, a_L \in \mathbb {C}$|⁠. |$A, B, E$|⁠, and X are

$$\begin{eqnarray} A &=& k^2 \int _{0}^{\infty }\mathrm{ d}z [({\cal N} - E_c \cos 4 \psi + E_s \sin 4 \psi ) W^2 \\ &&+\, ({\cal L} - G_c \cos 2 \psi + G_s \sin 2 \psi )W^{\prime 2}/k^2] , \end{eqnarray}$$

(42)

$$\begin{eqnarray} B &= & k^2 \int _{0}^{\infty }\mathrm{ d}z [({\cal A} + B_c \cos 2 \psi - B_s \sin 2 \psi \\ &&+ E_c \cos 4 \psi - E_s \sin 4 \psi ) V^2 \\ &&+ ({\cal L} + G_c \cos 2 \psi - G_s \sin 2 \psi ) (U-\frac{V^\prime }{k})^2 \\ &&+ 2({\cal F} + H_c \cos 2 \psi - H_s \sin 2 \psi ) VU^\prime /k + {\cal C} U^{\prime 2}/k^2 ] , \end{eqnarray}$$

(43)

$$\begin{eqnarray} E &= & k^2 \int _{0}^{\infty }\mathrm{ d}z \bigg[\bigg(- \frac{1}{2} B_c \sin 2 \psi - \frac{1}{2} B_s \cos 2 \psi\\ && -\, E_c \sin 4 \psi - E_s \cos 4 \psi \bigg) WV \\ && + (G_c \sin 2 \psi + G_s \cos 2 \psi )\bigg(U-\frac{V^\prime }{k}\bigg)W^\prime /k \\ &&+ (- H_c \sin 2 \psi - H_s \cos 2 \psi \bigg) W U^\prime /k\bigg] , \end{eqnarray}$$

(44)

$$\begin{eqnarray} X &=& k^2 \int _{0}^{\infty }\mathrm{ d}z \lbrace [2(J_c - M_c)\sin \psi - 2(J_s + M_s)\cos \psi\\ && + D_c \sin 3 \psi - D_s \cos 3 \psi ] VW^\prime /k \\ && + (M_c \sin \psi + M_s \cos \psi + D_c \sin 3 \psi - D_s \cos 3 \psi )\\ &&\times \, W\bigg(U-\frac{V^\prime }{k}\bigg) \\ && + 2[(J_c - K_c)\sin \psi - (J_s - K_s)\cos \psi ]W^\prime U^\prime /k^2 \rbrace . \end{eqnarray}$$

(45)

Hamilton’s Principle implies that |$\partial L/\partial a_R = \partial L/\partial a_L =0$| (Section S.6.2, Supporting Information), which is used in Section S.7 (Supporting Information) to derive the following eigenvalue problem that governs Rayleigh–Love coupling:

$$\begin{eqnarray} {\begin{pmatrix}A & E + iX\\ E - iX & B \end{pmatrix}} {\begin{pmatrix}a_L ^{*}\\ a_R ^{*} \end{pmatrix}} = \omega ^2 {\begin{pmatrix}a_L ^{*}\\ a_R ^{*} \end{pmatrix}} . \end{eqnarray}$$

(46)

The solvability condition yields the coupled quasi-Love (⁠|$m=1$|⁠) and quasi-Rayleigh wave (⁠|$m=2$|⁠) eigenfrequencies given by

$$\begin{eqnarray} {\omega ^2}= \frac{A + B \pm \sqrt{(A - B)^2 + 4(E^2 + X^2)}}{2} \equiv \frac{1}{2} \left[ A + B \pm D \right]\\ \end{eqnarray}$$

(47)

or phase speed given by

$$\begin{eqnarray} {V^2}_{qL} = \frac{1}{2k^2} \left[ A + B + D \right] , \end{eqnarray}$$

(48)

$$\begin{eqnarray} {V^2}_{qR} = \frac{1}{2k^2} \left[ A + B - D \right] , \end{eqnarray}$$

(49)

where

$$\begin{eqnarray} D \equiv ((A-B)^2 + 4 (E^2 + X^2))^{1/2}. \end{eqnarray}$$

(50)

Because Love waves are consistently faster than Rayleigh waves, we assign the higher frequency or higher phase speed to the quasi-Love wave and the slower one to the quasi-Rayleigh wave.

E is typically quite small for fundamental mode Rayleigh–Love coupling, as Tanimoto (2004) discusses. When the medium is VTI or HTI (either the symmetry axis is vertical or horizontal), X is zero, which yields only weak coupling, as studied by Tanimoto (2004). The |$(E^2 + X^2)$| term satisfies reciprocity and mostly contributes to the |$2\psi$| and |$4\psi$| variations in |$V^2$|⁠. A small additional contribution to a |$6\psi$| variation is ignorable.

For clarity, we now review the assumptions we have in surface wave theory. In a general anisotropic medium, some elastic parameters can couple the eigenfunctions of Rayleigh wave and Love wave (e.g. Tromp & Dahlen 1993, eqs A.4–A.6; Tromp 1994, eqs 48–50). We assume the surface wave modes in the anisotropic medium can be expressed as a superposition of Rayleigh and Love waves in the reference medium (eq. 2) (which in our calculation is an effective transversely isotropic medium). We choose a reference medium that decouples the Rayleigh and Love waves (e.g. Tromp 1994, eqs 64–66, ignoring earth’s rotation) and is convenient to calculate eigenfunctions. The basis eigenfunctions (vectors) we use are not complete and other modes should also be included in some cases. Ignoring other modes is similar to ignoring coupling to P waves when we study S waves Fig. 4. Although this assumption causes some error when there is non-negligible coupling to other modes, later we benchmark our theory with numerical results to show that in general this assumption is valid.

5.2 Phase speeds and fast orientations

Fig. 8 presents examples of phase speeds as a function of azimuth for the 45 s Rayleigh and and 40 s Love waves computed using models at two points in Alaska with different relationships between the fast orientations for Rayleigh and Love waves. The dashed lines are Rayleigh and Love wave curves (Fig. 8a, b, d and e) computed using the non-degenerate perturbation theory (NDPT) of Smith & Dahlen (1973). Based on NDPT, the Love wave is dominated by 4|$\psi$| azimuthal variations and the Rayleigh wave variations are dominantly 2|$\psi$|⁠. The solid lines are quasi-Rayleigh and quasi-Love wave curves computed using our quasi-degenerate theory (QDT). The quasi-Rayleigh and quasi-Love wave azimuthal variations contain prominent contributions from both 2|$\psi$| and 4|$\psi$|⁠. In western Alaska, the fast axis directions of quasi-Rayleigh and quasi-Love are out of phase by 180|$^\circ$| and in eastern Alaska they are in phase.

$(Top Two Rows) Phase speed presented as a function of azimuth $\psi$ for (red lines) the 45 s Rayleigh wave and (blue lines) the 40 s Love wave using two different theories: (solid lines) the QDT presented here and (dashed lines) the NDPT of Smith & Dahlen (1973). The model of anisotropy is Model 3 (discussed in Section 5.1) using data from (left column) a point in western Alaska (64$^\circ$N, 159$^\circ$W) and (right column) a point in eastern Alaska (64$^\circ$N, 147$^\circ$W). The quasi-Love wave 2$\psi$ fast axis orientations are shown with vertical dashed grey lines. (Bottom Row) The amplitude of the 2$\psi$ component of anisotropy plotted as a function of period for (red lines) the 45 s Rayleigh wave and (blue lines) the 40 s Love wave. Solid lines are for QDT and dashed lines are for NDPT.$

Figure 8.

(Top Two Rows) Phase speed presented as a function of azimuth |$\psi$| for (red lines) the 45 s Rayleigh wave and (blue lines) the 40 s Love wave using two different theories: (solid lines) the QDT presented here and (dashed lines) the NDPT of Smith & Dahlen (1973). The model of anisotropy is Model 3 (discussed in Section 5.1) using data from (left column) a point in western Alaska (64|$^\circ$|N, 159|$^\circ$|W) and (right column) a point in eastern Alaska (64|$^\circ$|N, 147|$^\circ$|W). The quasi-Love wave 2|$\psi$| fast axis orientations are shown with vertical dashed grey lines. (Bottom Row) The amplitude of the 2|$\psi$| component of anisotropy plotted as a function of period for (red lines) the 45 s Rayleigh wave and (blue lines) the 40 s Love wave. Solid lines are for QDT and dashed lines are for NDPT.

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The phasing between the fast directions of quasi-Rayleigh and quasi-Love waves reflects the relationship between the observed quasi-Rayleigh wave fast orientations and the strike of anisotropy, which at short periods is often observed to be aligned with faults (e.g. Xie et al. 2017; Liu & Ritzwoller 2024). The fast orientation of the 2|$\psi$| component of the Love wave azimuthal variation is usually aligned with the direction of the strike of anisotropy (see Fig. 3 for definition). In western Alaska, the fast axis direction of the quasi-Rayleigh wave is perpendicular to the fast axis direction of the quasi-Love wave and therefore the strike of anisotropy, whereas in eastern Alaska it will be aligned with the strike direction. The sign of the |$G_c$| parameter (namely the relative size of |$C_{55}$| and |$C_{44}$|⁠) determines the relationship between Rayleigh wave 2|$\psi$| and Love wave 2|$\psi$| fast axes. The above and later discussion of the strike angle assume Rayleigh–Love coupling does not change the sign of the 2|$\psi$| component of the Rayleigh wave, which is usually true for fundamental mode surface waves in Alaska (Fig. 8).

5.3 Amplitudes

Figs 8(c) and (f) illustrates how the phasing between the fast axis orientations of quasi-Love and quasi-Rayleigh waves affects the amplitude of their azimuthal variations. The right column of Fig. 8 for a point in eastern Alaska is an example when the quasi-Rayleigh wave fast orientation aligns with the Love wave fast orientation. In this case, the Rayleigh–Love coupling transfers amplitude from the Rayleigh wave to the Love wave. By this we mean the amplitude of the quasi-Rayleigh wave under QDT is reduced relative to the Rayleigh wave under NDPT, whereas the quasi-Love wave amplitude is increased relative to NDPT. In contrast, when the quasi-Rayleigh and quasi-Love 2|$\psi$| fast orientations are out of phase by 180|$^\circ$|⁠, as they are in western Alaska, the amplitudes of both the quasi-Rayleigh and quasi-Love under QDT increase relative to NDPT. This transfer of 2|$\psi$| amplitude can be complicated for surface waves due to the lack of a similarly compact solution as for body waves, but the body waves provide guidance, as discussed in Section 4.

These observations provide information about the impact of applying NDPT to data that should be modelled with QDT. For example, in western Alaska (Fig. 8c), it would be very hard to fit the amplitude of azimuthal variations at long periods. The tendency would be to overestimate the amplitude of anisotropy in the mantle.

5.4 Coupling strength

The strength of coupling depends on the relative size of |$4(E^2+X^2)$| and |$(A-B)^2$| in D in eq. (50). We define the coupling strength as follows

$$\begin{eqnarray} S \equiv \frac{4(E^2+X^2)}{(A-B)^2} \end{eqnarray}$$

(51)

If |$S << 1$|⁠, Rayleigh–Love coupling will be weak. Fig. 9(a) presents an example of the relative size of the components of D at 40 s period. There is a broad range of azimuths where |$X^2 >> E^2$| and where |$4X^2$| is on the order of |$(A-B)^2$|⁠. Rayleigh–Love coupling will be strong at those azimuths, which centre on the Love wave 2|$\psi$| fast directions. The assumption here is that the Love wave is the faster surface wave, which is also assumed in the expression for the polarization of quasi-Love waves. If the Love wave were the slower one, strong Rayleigh–Love coupling would centre on the Love wave 2|$\psi$| slow axis. As discussed further in Section 6, for our seismic model in Alaska at shorter periods |$X^2$| typically is smaller than at longer periods compared to |$(A-B)^2$|⁠, so coupling weakens at the shorter periods.

$Effects of Rayleigh–Love coupling for a 45 s Rayleigh wave and a 40 s Love wave, computed with Model 3 (discussed in Section 6.1) in western Alaska (64$^\circ$N, 159$^\circ$W). (a) Comparison of $(A-B)^2$ with $4E^2$ and $4X^2$, plotted as a function of azimuth. (b) X changes sign with azimuth. (c) Tilt angle $\Phi$ of the particle motion of the quasi-Love wave out of the horizontal plane. (d) Phase angle $\phi$ between the vertical and horizontal components of the quasi-Love (and quasi-Rayleigh) wave. Vertical dashed lines are the Love wave 2$\psi$ fast axis directions, which illustrate that coupling effects maximize in these directions.$

Figure 9.

Effects of Rayleigh–Love coupling for a 45 s Rayleigh wave and a 40 s Love wave, computed with Model 3 (discussed in Section 6.1) in western Alaska (64|$^\circ$|N, 159|$^\circ$|W). (a) Comparison of |$(A-B)^2$| with |$4E^2$| and |$4X^2$|⁠, plotted as a function of azimuth. (b) X changes sign with azimuth. (c) Tilt angle |$\Phi$| of the particle motion of the quasi-Love wave out of the horizontal plane. (d) Phase angle |$\phi$| between the vertical and horizontal components of the quasi-Love (and quasi-Rayleigh) wave. Vertical dashed lines are the Love wave 2|$\psi$| fast axis directions, which illustrate that coupling effects maximize in these directions.

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5.5 Polarization and phase lag

In Section S.7 (Supporting Information), we show that for the quasi-Love and quasi-Rayleigh waves, the non-normalized eigenvectors are

$$\begin{eqnarray} (a_L, a_R)_{qL} = (1, \Gamma e^{i \phi })^T , \end{eqnarray}$$

(52)

$$\begin{eqnarray} (a_L, a_R)_{qR} =(- \Gamma e^ {-i \phi }, 1 )^T , \end{eqnarray}$$

(53)

where |$\Gamma \equiv (B-A+D)/2(E^2 + X^2)^{1/2}$|⁠. The vector eigenfunctions are therefore

$$\begin{eqnarray} {\bf {\hat{s}}}_{qL}(z) &=& (- \beta W(z) + \alpha \Gamma e^{i \phi } V(z) , \alpha W(z) + \beta \Gamma e^{i \phi } V(z),\\ && \Gamma e^{i (\phi + \pi /2)} U(z))^T \end{eqnarray}$$

(54)

$$\begin{eqnarray} {\bf {\hat{s}}}_{qR}(z) = (\alpha V(z) + \Gamma e^ {-i \phi } \beta W(z), \beta V(z) - \alpha \Gamma e^ {-i \phi } W(z), i U(z))^T \\ . \end{eqnarray}$$

(55)

The polarization vector at the surface (⁠|$z=0$|⁠) for the quasi-Love wave is rotated out of the horizontal plane by angle |$\Phi$|⁠, where

$$\begin{eqnarray} \tan \Phi = \Gamma \frac{U(0)}{W(0)} , \end{eqnarray}$$

(56)

$$\begin{eqnarray} \tan 2 \Phi = \frac{2(E^2+X^2)^{1/2}}{A-B} \frac{W(0)}{U(0)} . \end{eqnarray}$$

(57)

The quasi-Rayleigh wave is rotated from the vertical by nearly the same angle. Fig. 9(c) presents an example of |$\Phi$| at 40 s period, which maximizes near the Love wave 2|$\psi$| fast direction where coupling is strongest. In this example, the quasi-Love wave polarization will be tipped by a maximum angle |$\Phi _{\mathrm{ max}} \sim 16^\circ$| relative to the horizontal. At much shorter periods, the polarization angle away from horizontal will be smaller and would be difficult to observe. For Alaska, this example is typical.

The phase lag angle |$\phi$| between the vertical and horizontal components is plotted for the same example in Fig. 9(d). At most azimuths, the lag is about |$\pm 90^\circ$|⁠. The lag angle changes sign from 90|$^\circ$| to |$-90^\circ$| when X becomes negative, as shown in Fig. 9(b). The polarization anomalies of wave propagating in opposite directions will be opposite, therefore by observing the polarization we will be able to constrain the absolute dip direction of a medium and not just the relative dip angle. This will be revealed in numerical calculations later in the paper. For |$\phi = -90^\circ$|⁠, the vector eigenfunction for the quasi-Love wave is

$$\begin{eqnarray} {\bf {\hat{s}}}_{qL}(z) &\approx & (- \beta W(z) - i \alpha \Gamma V(z) , \alpha W(z) - i \beta \Gamma V(z) , \Gamma U(z))^T . \end{eqnarray}$$

(58)

Signs will be reversed if |$\phi = 90^\circ$|⁠.

To consider the quasi-Love particle motion it is useful to think of propagation in the |$x_1$| direction (⁠|$\alpha = 1, \beta = 0)$| such that |$(x_1,x_2,x_3)^T$| are the radial, transverse and vertical directions. In this case, the components of the vector eigenfunction become |$(-i \Gamma V, W, \Gamma U)^T$|⁠. In this case, the transverse and vertical components of the vector eigenfunction are both real and in phase. Therefore, the particle motion for the vertical and transverse components will be linear and tilted by the angle |$\Phi$|⁠, which depends on |$\Gamma$|⁠. However, the transverse and radial components will be out of phase by 90|$^\circ$|⁠, so the particle motion projected onto the horizontal plane will be an ellipse. Fig. 10 presents a visualization of this. The nearly linear particle motion in the transverse direction in the vertical plane can distinguish the quasi-Love wave from a diffracted Rayleigh wave, which will have an elliptical particle motion. Such a polarization anomaly has been observed previously (Pettersen & Maupin 2002).

$Visualization of quasi-Love particle motion when the phase angle between the vertical and horizontal components of the wave is $\phi \sim 90^\circ$, where the radial, transverse and vertical directions are denoted $r, t$ and v and wave propagation is in the r direction. (a) Horizontal slice showing that the particle motion in the radial and transverse plane is elliptical.The radial component is typically much smaller than the transverse component because $\Gamma < 1$. (b) Vertical slice showing that the particle motion in the vertical and transverse plane is approximately linear. (c) Attempt at a 3-D view, in which the plane of elliptical particle motion for the quasi-Love wave is tilted at an angle $\Phi$ relative to the transverse direction.$

Figure 10.

Visualization of quasi-Love particle motion when the phase angle between the vertical and horizontal components of the wave is |$\phi \sim 90^\circ$|⁠, where the radial, transverse and vertical directions are denoted |$r, t$| and v and wave propagation is in the r direction. (a) Horizontal slice showing that the particle motion in the radial and transverse plane is elliptical.The radial component is typically much smaller than the transverse component because |$\Gamma < 1$|⁠. (b) Vertical slice showing that the particle motion in the vertical and transverse plane is approximately linear. (c) Attempt at a 3-D view, in which the plane of elliptical particle motion for the quasi-Love wave is tilted at an angle |$\Phi$| relative to the transverse direction.

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5.6 Numerical results

Here, we benchmark our quasi-degenerate theory against numerical results using SPECFEM3D (Komatitsch & Tromp 1999). The approach we take is similar to that presented by Chen & Tromp (2007), which tested non-degenerate perturbation theory for surface waves and degenerate perturbation theory for body waves.

To simplify interpretation, we define a simple 60 km thick three-layer anisotropy model with an imposed 4 per cent anisotropy: |$M_s = -0.04(\lambda + 2 \mu )$|⁠. All other anisotropy parameters in the elastic tensor are zero. The thickness, density, |$V_p$|⁠, and |$V_s$| for the three-layer isotropic reference model are |$h_1 = 15\, \mathrm{ km}$|⁠, |$\rho _1 = 2600\, \mathrm{ kg}\,\mathrm{ m}^{-3}$|⁠, |$V_{p1} = 6.3\, \mathrm{ km}\,\mathrm{ s}^{-1}$|⁠, |$V_{s1} = 3.2\, \mathrm{ km}\,\mathrm{ s}^{-1}$|⁠; |$h_2 = 15\, \mathrm{ km}$|⁠, |$\rho _2 = 2900\, \mathrm{ kg}\,\mathrm{ m}^{-3}$|⁠, |$V_{p2} = 7.0\, \mathrm{ km}\,\mathrm{ s}^{-1}$|⁠, |$V_{s2} = 3.7\, \mathrm{ km}\,\mathrm{ s}^{-1}$|⁠; |$h_3 = 30\, \mathrm{ km}$|⁠, |$\rho _3 = 3200\, \mathrm{ kg}\,\mathrm{ m}^{-3}$|⁠, |$V_{p3} = 7.5\, \mathrm{ km}\,\mathrm{ s}^{-1}$|⁠, |$V_{s3} = 4.3\, \mathrm{ km}\,\mathrm{ s}^{-1}$|⁠. We impose a free surface boundary condition and absorbing boundary conditions on the four sides and bottom of the model. The total model size is 4000 km |$\times$| 4000 km |$\times$| 60 km and we do not consider attenuation (⁠|$Q_{\mu } = \infty$|⁠). For the numerical benchmark, we set the reference medium to be isotropic. Therefore, for the isotropic ‘reference medium’ all anisotropy parameters are zero, in particular |$M_s = 0$|⁠. The source is located at the surface and horizontal centre of the model, with 36 observing points or stations situated in a circle at a constant distance of 1800 km from the source (Fig. 11a). The source is a pure compression with the centroid moment tensor (CMT): |$M_{xx} = M_{yy} = M_{zz} = 2.5\times 10^{29}$| Nm, |$M_{xy} = M_{xz} = M_{yz} =0$|⁠. The half duration of this source is 10 s and we filter the seismograms (Fig. 11c) to measure the velocity and polarization centred at 12 s period.

$Numerical results from SPECFEM3D. (a) Vertical-component wavefield snapshot for wave propagation through the anisotropic medium 280 s after the source, where for simplicity we truncate the colour bar to positive values. The red triangles are receivers and the yellow star is the source, which is purely compressional. (b) The seismograms for all three components at the 36 receivers for the isotropic reference medium, rotated to vertical, transverse and radial components where the radial components are shifted by $\pi /2$ for visualization. (c) Similar to (b), but for seismograms in the anisotropic medium. The first arrival of surface waves is the quasi-Love wave and the following arrival is the quasi-Rayleigh wave.$

Figure 11.

Numerical results from SPECFEM3D. (a) Vertical-component wavefield snapshot for wave propagation through the anisotropic medium 280 s after the source, where for simplicity we truncate the colour bar to positive values. The red triangles are receivers and the yellow star is the source, which is purely compressional. (b) The seismograms for all three components at the 36 receivers for the isotropic reference medium, rotated to vertical, transverse and radial components where the radial components are shifted by |$\pi /2$| for visualization. (c) Similar to (b), but for seismograms in the anisotropic medium. The first arrival of surface waves is the quasi-Love wave and the following arrival is the quasi-Rayleigh wave.

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The non-degenerate perturbation theory derived by Smith & Dahlen (1973) and Tanimoto (2004) predicts that for this anisotropy model there will be no Love wave or quasi-Love wave and no azimuthal anisotropy for either surface wave, which is the same as for the isotropic reference medium (Fig. 11b). In contrast, a snapshot of the vertical-component of the numerical wavefield through the anisotropic medium is shown in Fig. 11(a). In certain directions, such as azimuth = 0|$^\circ$| or 180|$^\circ$| (measured anticlockwise from right), the coupling is the strongest (eq. 45) so there is a quasi-Love wave on the vertical component ahead of quasi-Rayleigh wave. In other directions, such as azimuth = 90|$^\circ$| or 270|$^\circ$|⁠, there is no Rayleigh–Love coupling (eq. 45), so there is no quasi-Love wave. This is also obvious in the seismograms for the 36 stations (Fig. 11c). The particle motion for both the quasi-Love and quasi-Rayleigh wave are 3-D and linear or out of phase by 180|$^\circ$| from the opposite direction, as described in Fig. 10. This raises several important points that we will discuss in detail later.

We compute the phase speed and polarization from the seismograms, with results shown in Fig. 12, comparing with our quasi-degenerate theory. The numerical results are consistent with our quasi-degenerate theory in phase speed and to first-order in polarization. The small misfit in polarization in the strong coupling directions probably results from weak coupling to other modes in the numerical results that is neglected in our theory (eq. 2).

Figure 12.

Measurements based on the waveforms in Fig. 11(c): (a) Phase velocity of quasi-Love wave. (b) Polarization anisotropy of the quasi-Love wave based on the ratio between vertical and transverse components (eqs 52 and 56). (c) Phase velocity of quasi-Rayleigh wave. (d) Polarization anisotropy of the quasi-Rayleigh wave based on the ratio between transverse and vertical components (eqs 53 and 56). The solid lines are predictions based on our quasi-degenerate theory and the red dots are numerical results from SPECFEM3D.

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6 DISCUSSION OF RAYLEIGH–LOVE COUPLING

6.1 Estimating anisotropy in the presence of Rayleigh–Love coupling

For two principal reasons, most previous inversions of observations of surface wave azimuthal anisotropy have been based exclusively on the 2|$\psi$| component of the azimuthal variation of Rayleigh waves. First, early theoretical papers on Rayleigh and Love wave azimuthal anisotropy were based on non-degenerate perturbation theory (Smith & Dahlen 1973; Montagner & Nataf 1986), which predicted only 2|$\psi$| anisotropy for Rayleigh waves and 4|$\psi$| anisotropy for Love waves. Second, for practical reasons, Love wave anisotropy and the 4|$\psi$| anisotropy for Rayleigh waves have been more difficult to observe reliably. These two factors have combined to focus efforts on inferring anisotropy from isotropic phase speeds along with the 2|$\psi$| component of azimuthal variations in Rayleigh wave anisotropy (e.g. Liu et al. 2022).

As we show in Section 5 theoretically, and has been increasingly observed in recent years (e.g. Russell et al. 2019; Liu et al. “Observations of Rayleigh and Love wave anisotropy across Alaska”, manuscript in preparation, 2024), the 2|$\psi$| component of Love wave anisotropy may be quite large and the 4|$\psi$| component of Rayleigh wave anisotropy, although smaller, may also be large enough to be observed. Fig. 1 presents an example of observations for a point in western Alaska. These signals derive from Rayleigh–Love coupling which is modelled here through a quasi-degenerate theory. 4|$\psi$| Love wave anisotropy is also expected and observable (e.g. Fig. 1), although it is uncommonly observed in practice.

Using observations at a location in western Alaska (64|$^\circ$|N, 159|$^\circ$|W), Data Source 4 in Section 2, we present three inversion results to demonstrate the effect of using new (‘unexpected’) signals (Love 2|$\psi$|⁠, Rayleigh 4|$\psi$|⁠) interpreted with and without Rayleigh–Love coupling. The three estimated models are summarized in Table 2, where the theories used are the NDPT of (Smith & Dahlen 1973) and (Montagner & Nataf 1986) in which Rayleigh–Love coupling is absent and the QDT presented here, which models Rayleigh–Love coupling. Each inversion uses a different subset of the data but is performed with the same Bayesian Monte Carlo method, which is similar to that described by Xie et al. (2015, 2017) and Liu & Ritzwoller (2024). In this method, a posterior distribution of model variables is estimated, which we summarize with the mean and standard deviation of each model variable at each depth. The crust and mantle are both modelled as depth-dependent TTI media, where the dip angle |$\theta$| can vary discontinuously with depth.

Table 2.

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Models constructed using different observations and theoretical assumptions at point (64|$^\circ$|N, 159|$^\circ$|W) in western Alaska.

Model number	Data used	Theory used
Model 1	Rayleigh 2\|$\psi$\|	NDPT
Model 2	Rayleigh 2\|$\psi$\|⁠; Love 4\|$\psi$\|	NDPT
Model 3	Rayleigh 2\|$\psi$\|⁠, 4\|$\psi$\|⁠; Love 2\|$\psi$\|⁠, 4\|$\psi$\|	QDT

Model number	Data used	Theory used
Model 1	Rayleigh 2\|$\psi$\|	NDPT
Model 2	Rayleigh 2\|$\psi$\|⁠; Love 4\|$\psi$\|	NDPT
Model 3	Rayleigh 2\|$\psi$\|⁠, 4\|$\psi$\|⁠; Love 2\|$\psi$\|⁠, 4\|$\psi$\|	QDT

Table 2.

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Models constructed using different observations and theoretical assumptions at point (64|$^\circ$|N, 159|$^\circ$|W) in western Alaska.

Model number	Data used	Theory used
Model 1	Rayleigh 2\|$\psi$\|	NDPT
Model 2	Rayleigh 2\|$\psi$\|⁠; Love 4\|$\psi$\|	NDPT
Model 3	Rayleigh 2\|$\psi$\|⁠, 4\|$\psi$\|⁠; Love 2\|$\psi$\|⁠, 4\|$\psi$\|	QDT

Model number	Data used	Theory used
Model 1	Rayleigh 2\|$\psi$\|	NDPT
Model 2	Rayleigh 2\|$\psi$\|⁠; Love 4\|$\psi$\|	NDPT
Model 3	Rayleigh 2\|$\psi$\|⁠, 4\|$\psi$\|⁠; Love 2\|$\psi$\|⁠, 4\|$\psi$\|	QDT

The estimated seismic models are shown in Fig. 13. The set of observations at this location are presented in Fig. 14 and also Fig. 2, except for the Rayleigh and Love wave isotropic phase speed curves which we do not show. Model 1 is constructed using only the 2|$\psi$| component of Rayleigh wave azimuthal anisotropy using NDPT. This is similar to the data and theory used in current observational studies to infer the TTI elastic tensor as a function of depth (e.g. Xie et al. 2015, 2017; Liu & Ritzwoller 2024). Model 2 is constructed by augmenting the observations used in Model 1 with the 4|$\psi$| component of Love wave anisotropy, where the theory is still NDPT. Model 3 further augments these observations with Love wave 2|$\psi$| anisotropy and Rayleigh wave 4|$\psi$| anisotropy, and the theory used in the inversion is the QDT presented here. The isotropic Rayleigh and Love wave phase speed curves are also used in the construction of all three models. The crust and mantle are both modelled as depth-dependent TTI media, where the dip angle |$\theta$| of the upper crust, lower crust and mantle are allowed to differ from one another. Using the same data types and the quasi-degenerate theory, we also estimate a model in eastern Alaska with observations at (64|$^\circ$|⁠, 147|$^\circ$|W), which we also refer to as Model 3 but with the identifier ‘eastern Alaska’. Examples of the azimuthal variation of phase speed for Model 3 in western and eastern Alaska are presented in Fig. 8 using both non-degenerate perturbation theory and quasi-degenerate theory.

$Four model variables presented for Models 1–3 at the location (64$^\circ$N, 159$^\circ$W) in western Alaska. $V_{SV} = \sqrt{L/\rho }$, dip angle $\theta$ for the TTI medium, S-wave anisotropy ($\gamma = (N-L)/2L$), and the ellipticity parameter $\eta _X$. Data and theory used in each inversion are listed in Table 2. The mean of the posterior distribution for Models 1 and Model 2 are shown with the blue and green dashed lines, respectively. The mean of the posterior distribution for Model 3 is shown with a solid red line, and the grey shading indicates the $\pm 1 \sigma$ corridor of the posterior distribution for Model 3.$

Figure 13.

Four model variables presented for Models 1–3 at the location (64|$^\circ$|N, 159|$^\circ$|W) in western Alaska. |$V_{SV} = \sqrt{L/\rho }$|⁠, dip angle |$\theta$| for the TTI medium, S-wave anisotropy (⁠|$\gamma = (N-L)/2L$|⁠), and the ellipticity parameter |$\eta _X$|⁠. Data and theory used in each inversion are listed in Table 2. The mean of the posterior distribution for Models 1 and Model 2 are shown with the blue and green dashed lines, respectively. The mean of the posterior distribution for Model 3 is shown with a solid red line, and the grey shading indicates the |$\pm 1 \sigma$| corridor of the posterior distribution for Model 3.

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$Comparison of observations (Data Source 4) of the amplitude of 2$\psi$ and 4$\psi$ components of Rayleigh and Love wave anisotropy (black 1$\sigma$ error bars) from 8 to 50 s period at location (64$^\circ$N, 159$^\circ$W) in western Alaska with predictions using the elastic tensor models Model 1–Model 3 constructed here (Table 2). The blue dashed line is computed using Model 1 (based on Rayleigh wave 2$\psi$ observations) and non-degenerate perturbation theory (Smith & Dahlen 1973; Montagner & Nataf 1986). The green dashed line is computed using Model 2 (based on Rayleigh wave 2$\psi$ and Love wave 4$\psi$ observations) and non-degenerate perturbation theory. The red line is computed using Model 3 (based on all observations) and using the quasi-degenerate theory we present here that includes Rayleigh-Love coupling. Using all data and the quasi-degenerate theory allows all data to be fit acceptably.$

Figure 14.

Comparison of observations (Data Source 4) of the amplitude of 2|$\psi$| and 4|$\psi$| components of Rayleigh and Love wave anisotropy (black 1|$\sigma$| error bars) from 8 to 50 s period at location (64|$^\circ$|N, 159|$^\circ$|W) in western Alaska with predictions using the elastic tensor models Model 1–Model 3 constructed here (Table 2). The blue dashed line is computed using Model 1 (based on Rayleigh wave 2|$\psi$| observations) and non-degenerate perturbation theory (Smith & Dahlen 1973; Montagner & Nataf 1986). The green dashed line is computed using Model 2 (based on Rayleigh wave 2|$\psi$| and Love wave 4|$\psi$| observations) and non-degenerate perturbation theory. The red line is computed using Model 3 (based on all observations) and using the quasi-degenerate theory we present here that includes Rayleigh-Love coupling. Using all data and the quasi-degenerate theory allows all data to be fit acceptably.

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Fig. 13 presents results from the inversions, showing four variables from the three models. These are the Love modulus L as |$V_{SV} = \sqrt{L/\rho }$|⁠, the dip angle |$\theta$| of the transversely isotropic elastic tensor, S-wave anisotropy |$(N-L)/2L$|⁠, and the ellipticity parameter |$\eta _X$| (eq. 36) which is approximately equal to the ‘new’ ellipticity parameter |$\eta _K$| of Kawakatsu (2016). All three models are represented as a posterior distribution with depth, but only the mean of the posterior distribution is shown for Models 1 and 2 whereas |$\pm 1 \sigma$| of the posterior distribution is shown for Model 3.

The introduction of observations of the 4|$\psi$| variation of Love wave phase speeds in Model 2 decreases the dip angle in the upper crust and, more significantly, reduces the ellipticity parameter in both the crust and mantle, compared to Model 1. This is illuminated by the body wave theory for a TTI medium, presented in Section 4. For example, eq. (35) shows that a large 4|$\psi$| component for quasi-S|$_2$| will only occur if the ellipticity coefficient differs strongly from 1. Thus, to fit the Love wave 4|$\psi$| observations requires |$\eta _X$| to deviate from 1, which it does not in Model 1. Thus, the use of observations of the 4|$\psi$| component of Love wave anisotropy is particularly important to estimate the ellipticity of anisotropy accurately.

Fig. 14 shows that all three models fit the Rayleigh 2|$\psi$| signal. In particular, the Rayleigh 2|$\psi$| signal can be fit with NDPT. Model 2 does fit the Love wave 4|$\psi$| signal, which shows that this signal can also be fit with NDPT. However, it typically will not be fit unless it is used in the inversion. Neither Model 1 nor Model 2 fits the Love wave 2|$\psi$| signal because quasi-degenerate theory is needed to produce large 2|$\psi$| amplitudes. Thus, applying all of the data and using the quasi-degenerate theory, which includes Rayleigh–Love coupling, allows all the data to be fit. Moreover, models produced with NDPT, such as the one presented by Liu & Ritzwoller (2024), will typically not produce strong enough Rayleigh–Love coupling to produce substantial 2|$\psi$| anisotropy for Love waves. Therefore, it is important to use quasi-degenerate theory in fitting anisotropy data to produce Rayleigh–Love coupling strong enough to produce the observed Love 2|$\psi$| signal.

Model 3 differs from Model 2 principally in the strength of anisotropy (⁠|$\gamma$|⁠), especially in the mantle. This results from the large amplitude of the Love wave 2|$\psi$| azimuthal variation. Since there is also a small observable Rayleigh wave 4|$\psi$| signal, these two models also differ somewhat in |$\eta _X$|⁠. Although olivine samples in the laboratory may produce S-wave anisotropy larger than 10 per cent (e.g. Ismaıl & Mainprice 1998), anisotropy greater than 10 per cent at the scale of seismic waves is probably not physically plausible due to spatial averaging. This calls into question the use of a TTI model to represent the elastic tensor in the mantle and highlights the need to revise the model to include a tilted orthorhombic elastic tensor in the mantle. Preliminary tests of inversions with a tilted orthorhombic elastic tensor in the mantle show that the strength of anisotropy reduces to between 4 and 10 per cent, which is physically more plausible. When inverting Rayleigh and Love wave azimuthal anisotropy simultaneously in the presence of Rayleigh–Love coupling, it is important to model the mantle as a tilted orthorhombic medium although the crust can remain as a TTI medium.

6.2 Coupling between fundamental modes and overtones

Following the publication of Tanimoto (2004), Maupin (2004) commented that in oceanic settings the coupling of the Love wave fundamental mode to the Rayleigh wave 1st-overtone may be stronger than its coupling to the fundamental Rayleigh mode. We reconsider this comment for both continental and oceanic settings in light of the quasi-degenerate theory presented here, which produces much stronger Rayleigh–Love coupling than the formalism of Tanimoto (2004).

In the foregoing, we have restricted ourselves to coupling between fundamental mode Love with fundamental mode Rayleigh waves. The quasi-degenerate theory we present can also be applied to any pair of Rayleigh and Love modes, for example coupling between the fundamental mode Love wave and the 1st-overtone Rayleigh wave, coupling between the 1st-overtone Love wave and 1st-overtone Rayleigh wave, and so on. We define coupling strength as S (eq. 51), which is plotted in Fig. 15(a) for a continental location for coupling between the fundamental Love and fundamental Rayleigh modes (red line) and the fundamental Love and 1st-overtone Rayleigh modes (blue line). The fundamental mode coupling is much stronger than the overtone coupling in this continental location as it will be for most continental locations. This is because the Love wave and overtone phase speed curves are well separated as Fig. 5(a) shows. The peak at short period (⁠|$\sim$|5 s) is caused by the near degeneracy of the fundamental mode Rayleigh and Love curves at shorter periods. The coupling between the overtone Love and overtone Rayleigh modes is much stronger than the coupling between the fundamental Love and Rayleigh modes (not shown in Fig. 15), because their phase speeds are almost degenerate. Analysis of overtones in continental areas should account for such strong coupling.

$Coupling strength S eq. (51) plotted versus period for coupling between the fundamental mode Love wave and the fundamental (red lines) and overtone (1st overtone blue lines, 2nd overtone green line) Rayleigh wave. (a) Computed in a continental setting (western Alaska, 64$^\circ$N, 159$^\circ$W) using anisotropy Model 3, aspects of which are shown in Fig. 14. (b) Computed in an oceanic setting southeast of Hawaii, using a revision of the anisotropy model from Data Source 3, aspects of which are shown in Fig. 16.$

Figure 15.

Coupling strength S eq. (51) plotted versus period for coupling between the fundamental mode Love wave and the fundamental (red lines) and overtone (1st overtone blue lines, 2nd overtone green line) Rayleigh wave. (a) Computed in a continental setting (western Alaska, 64|$^\circ$|N, 159|$^\circ$|W) using anisotropy Model 3, aspects of which are shown in Fig. 14. (b) Computed in an oceanic setting southeast of Hawaii, using a revision of the anisotropy model from Data Source 3, aspects of which are shown in Fig. 16.

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The relationship between the Rayleigh and Love phase speed curves in oceans is quite different, as Fig. 5(b) shows. To assess the effect on coupling strength we use the model of the elastic tensor in the crust and upper mantle southeast of Hawaii from Russell et al. (2019), although we revise it to increase the strength of anisotropy. We revise it by taking its effective transversely isotropic part, which is a VTI model and is included in their Supporting Information, and increase N and A, by making (N − L)/2L = (A − C)/2C = 7 per cent across all depths. We then tilt the elastic tensor by 45|$^\circ$|⁠, which produces maximal coupling. We show aspects of Russell’s model and our revisions in Fig. 16. The increase in the strength of anisotropy moves |$\eta _X$| farther from 1, making the anisotropy less elliptical. The coupling strength S between fundamental Love and Rayleigh modes is weaker than in continental areas, but the coupling between the fundamental Love and 1st-overtone Rayleigh modes is much stronger from 10–40 s period (Fig. 15b). Coupling strength between the fundamental Love and 2nd-overtone Rayleigh modes is also shown in Fig. 15(b), but strong coupling is confined to a narrower band between about 5 and 15 s period.

$Aspects of the effective transversely isotropy (VTI) component of the oceanic anisotropy model from Data Source 3 is shown with the green dashed line. The red line is our revision of this model in which the moduli A and N are increased so that $(A-C)/2C = (N-L)/2L = 7$ per cent and we tilt the elastic tensor through dip angle $\theta = 45^\circ$. (a) $V_{SV} = L/\rho ^2$. (b) Dip angle $\theta$. (c) S-wave anisotropy, $\gamma = (N-L)/2L$. (d) Ellipticity parameter $\eta _X \equiv 4L/(A+C-2F)$.$

Figure 16.

Aspects of the effective transversely isotropy (VTI) component of the oceanic anisotropy model from Data Source 3 is shown with the green dashed line. The red line is our revision of this model in which the moduli A and N are increased so that |$(A-C)/2C = (N-L)/2L = 7$| per cent and we tilt the elastic tensor through dip angle |$\theta = 45^\circ$|⁠. (a) |$V_{SV} = L/\rho ^2$|⁠. (b) Dip angle |$\theta$|⁠. (c) S-wave anisotropy, |$\gamma = (N-L)/2L$|⁠. (d) Ellipticity parameter |$\eta _X \equiv 4L/(A+C-2F)$|⁠.

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In conclusion, at most continental locations, fundamental Loves waves will be coupled principally to fundamental mode Rayleigh waves, and Love wave–overtone coupling can be safely ignored. At oceanic locations, however, fundamental mode Loves waves will be coupled principally to overtone Rayleigh waves, at least below 40 s period, and coupling to the fundamental mode Rayleigh wave will be weaker but still substantial.

6.3 Polarization

Tanimoto (2004) stressed the potential importance of measuring the polarization angle |$\Phi$|⁠, the tilt angle out of the horizontal plane of the particle motion for quasi-Love waves, as a new constraint on anisotropy. The polarization angle will vary with azimuth and maximize in the fast direction of the 2|$\psi$| quasi-Love wave (if the Love wave is faster than the Rayleigh wave). The maximum polarization angle is expected to coincide with the maximum coupling between the Rayleigh and Love waves as shown in Fig. 9 one can also find similar results in the numerical section 4.6. Its measurement, at the very least, would be a valuable consistency check on anisotropy constrained by phase speeds, with its maximum aligning with the quasi-Love 2|$\psi$| fast direction. Polarization measurements, however, could be used directly in inversions for the depth-dependent elastic. As mentioned in section 4.5, a unique constraint from polarization anisotropy is to infer the absolute tilt direction of a medium.

Fig. 17 presents the maximum polarization angle plotted as a function of period for Model 3 in western Alaska for the quasi-Love wave coupled to the fundamental mode Rayleigh wave and the 1st overtone Rayleigh wave, respectively. Not surprisingly, these curves look similar to the coupling strength plotted in Fig. 15(a). A polarization anomaly of 15|$^\circ$| is expected at this location at periods longer than about 30 s. The polarization anomaly for coupling the Love wave to the first-overtone Rayleigh wave is much smaller and we believe it can be safely ignored in most cases. We believe this is a typical result for Alaska and probably for other continental locations as well.

$Maximum polarization angle $\Phi$ plotted versus period for coupling between the fundamental mode Love wave and the fundamental mode Rayleigh wave (red line) and 1st-overtone Rayleigh wave (blue line). Computed in a continental setting (western Alaska, 64$^\circ$N, 159$^\circ$W) using anisotropy Model 3, aspects of which are shown in Fig. 14.$

Figure 17.

Maximum polarization angle |$\Phi$| plotted versus period for coupling between the fundamental mode Love wave and the fundamental mode Rayleigh wave (red line) and 1st-overtone Rayleigh wave (blue line). Computed in a continental setting (western Alaska, 64|$^\circ$|N, 159|$^\circ$|W) using anisotropy Model 3, aspects of which are shown in Fig. 14.

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7 CONCLUSIONS

We present a quasi-degenerate theory of Rayleigh–Love coupling based on the application of Hamilton’s Principle to Rayleigh and Love waves. This theory explains the observation of 2|$\psi$| phase velocity anisotropy for Love waves and 4|$\psi$| anisotropy for Rayleigh waves. Previous theories based on non-degenerate perturbation theory (Smith & Dahlen 1973; Montagner & Nataf 1986) do not explain these observations, and for this reason we refer to 2|$\psi$| anisotropy for Love waves and 4|$\psi$| anisotropy for Rayleigh waves as ‘unexpected’. The reason for this is that these theories do not model the coupling of Rayleigh and Love waves by anisotropy. The quasi-degenerate theory we present here does model Rayleigh–Love coupling and succeeds to explain observations of 2|$\psi$| anisotropy for Love waves. In addition, it allows for these observations to be included in inversions simultaneously with ‘expected’ observations, such as the 2|$\psi$| anisotropy for Rayleigh waves and the 4|$\psi$| anisotropy for Love waves. We also benchmark our theory against numerical results from SPECFEM3D (Komatitsch & Tromp 1999) and the discrepancy is small.

For comparison, we also present a theory of SV–SH coupling for horizontally propagating body waves to help illuminate Rayleigh–Love coupling. We apply Hamilton’s Principle to develop this theory, too, which generates the same results as the degenerate perturbation theory of Jech & Pšenčík (1989). However, we specialize the results by applying them to a TTI medium, which is commonly assumed in inversions for anisotropy (e.g. Montagner & Nataf 1988; Xie et al. 2015; Liang et al. 2024), and present simple expressions for the anisotropy of the quasi-S waves based on the dip angle |$\theta$| of anisotropy and the ellipticity parameter |$\eta _X$|⁠, which we introduce here. Through these body wave results, we motivate how observations of Love wave 4|$\psi$| azimuthal anisotropy can be used to infer |$\eta _X$| and |$\theta$| and how coupling splits the total 2|$\psi$| amplitude.

We present examples that illustrate that when the unexpected 2|$\psi$| anisotropy for Love waves is included in inversions for a depth-dependent TTI medium along with observations of expected anisotropy, better constraints are placed on the ellipticity parameter |$\eta _X$|⁠, but the amplitude of anisotropy in the mantle may become so large as to be physically unrealistic. We find that using an orthorhombic tensor in the mantle reduces the amplitude of anisotropy, and advise that future inversions should use a tilted orthorhombic tensor in the mantle.

Tanimoto (2004) suggested that polarization measurements for coupled quasi-Love and quasi-Rayleigh waves should be considered as new information to constrain anisotropy within the Earth. We would like to second this suggestion, particularly because the quasi-degenerate theory we present predicts stronger Rayleigh–Love coupling and therefore stronger polarization anomalies than the theory presented by Tanimoto (2004). We present evidence that polarization anomalies, or tilts of the quasi-Love wave’s particle motion out of the horizontal plane, of 15|$^\circ$| should be common in a continental setting, in particular at periods sensitive to the mantle.

Maupin (2004) raised the important point that the coupling between the fundamental mode Love wave and the first and higher overtone Rayleigh waves may also be important, particularly in oceanic settings. We provide evidence that coupling between the fundamental Love wave and Rayleigh overtones can probably be ignored in continental settings. However, coupling between the fundamental Love wave and both fundamental and overtone Rayleigh waves are likely to be strong in oceanic settings and can be modelled with the theory we present although only for coupling between two modes at a time.

Our results indicate that greater efforts are needed in both continental and oceanic settings to observe unexpected anisotropy such as Love wave 2|$\psi$| anisotropy. Such observations would be important to improve models of anisotropy that are deriving from the inversion of isotropic Rayleigh and Love wave phase speeds along with the 2|$\psi$| component of Rayleigh wave anisotropy (e.g. Xie et al. 2015, 2017; Liu & Ritzwoller 2024).

The theory presented in this paper is derived in Cartesian coordinates and ignores rotation, self-gravitation and finite frequency effects, for example arising from SV–SH coupling (e.g. Coates & Chapman 1990) and Rayleigh–Love coupling away from the receiver (e.g. Maupin 2001; Sieminski et al. 2007, 2009). Non-degenerate perturbation theory has been derived in spherical coordinates (e.g. Larson et al. 1998) based on the study of Tromp (1994), which also includes the effects of rotation, self-gravitation and some other effects based upon the JWKB approximation. The typical method to deal with finite-frequency effects is the first Born approximation (e.g. Snieder 1986; Snieder et al. 1987). However, due to the strong mode coupling between Rayleigh and Love waves discussed in this paper, this standard Born approximation needs to be revised to account accurately for strong interactions caused by quasi-degeneracy. Some shortcomings of the first-order Born approximation have been studied before (e.g. Romanowicz et al. 2008) and this problem is solved in normal modes by considering coupling between multiplets and higher order Born series (e.g. Park 1990; Tromp & Dahlen 1990; Su et al. 1993). Future efforts on this topic should consider extension to spherical coordinates, the inclusion of finite frequency effects, and coupling between multiple modes (⁠|$> 2$|⁠) because surface waves can strongly couple to fundamental modes and overtone surface waves at the same time.

ACKNOWLEDGMENTS

We thank Prof. Valérie Maupin, Prof. Jeroen Tromp and Editor Huajian Yao for helpful comments to improve the manuscript. We are grateful to Sarah Brownlee for valuable conversations and for providing the database of elastic tensor measurements (Brownlee et al. 2017). We also thank Chuanming Liu for many helpful conversations and for providing his model of the depth-dependent elastic tensor beneath Alaska (Liu & Ritzwoller 2024). XL also thanks Chuanming Liu for guidance in data processing. We greatly appreciate help from IRIS Data Services, which are funded through the Seismological Facilities for the Advancement of Geoscience and EarthScope (SAGE) Proposal of the National Science Foundation under Cooperative Agreement EAR-1851048. Aspects of this research were supported by EAR-1537868, EAR-1928395 and EAR-1952209 at the University of Colorado Boulder.

DATA AVAILABILITY

Original seismic waveform data were obtained from the Data Management Center of IRIS (www.iris.edu). The model based on Rayleigh wave azimuthal anisotropy alone is available at the EarthScope Earth Model Collaboration repository (https://ds.iris.edu/ds/products/emc-earthmodels/). ObsPy (Beyreuther et al. 2010) is used in data processing.

REFERENCES

Aki

Richards

P.G.

2002

Quantitative Seismology

, 2nd edn,

University Science Books

Beyreuther

Barsch

Krischer

Megies

Behr

Wassermann

2010

Obspy: A python toolbox for seismology

Seismol. Res. Lett.

(

530

–

533

10.1785/gssrl.81.3.530

Browaeys

J.T.

Chevrot

2004

Decomposition of the elastic tensor and geophysical applications

J. Geophys. Int.

159

(

667

–

678

10.1111/j.1365-246X.2004.02415.x

Brownlee

S.J.

Schulte-Pelkum

Raju

Mahan

Condit

Orlandini

O.F.

2017

Characteristics of deep crustal seismic anisotropy from a compilation of rock elasticity tensors and their expression in receiver functions

Tectonics

(

1835

–

1857

Červenỳ

Pšenčík

2020

Seismic ray theory

Encyclopedia of Solid Earth Geophysics

, pp.

–

Cambridge University Press

Google Scholar

Google Preview

OpenURL Placeholder Text

WorldCat

Chapman

2004

Fundamentals of Seismic Wave Propagation

Cambridge University Press

10.1017/CBO9780511616877

Chen

Tromp

2007

Theoretical and numerical investigations of global and regional seismic wave propagation in weakly anisotropic earth models

J. Geophys. Int.

168

(

1130

–

1152

10.1111/j.1365-246X.2006.03218.x

Coates

Chapman

1990

Quasi-shear wave coupling in weakly anisotropic 3-D media

J. Geophys. Int.

103

(

301

–

320

10.1111/j.1365-246X.1990.tb01773.x

Dahlen

Tromp

2020

Theoretical global seismology

, in

Theoretical Global Seismology

Princeton University Press

Google Scholar

Google Preview

OpenURL Placeholder Text

WorldCat

Ekström

2011

A global model of love and rayleigh surface wave dispersion and anisotropy, 25–250 s

J. Geophys. Int.

187

(

1668

–

1686

10.1111/j.1365-246X.2011.05225.x

Forsyth

D.W.

1975

The early structural evolution and anisotropy of the oceanic upper mantle

J. Geophys. Int.

(

103

–

162

10.1111/j.1365-246X.1975.tb00630.x

Ismaıl

W.B.

Mainprice

1998

An olivine fabric database: an overview of upper mantle fabrics and seismic anisotropy

Tectonophysics

296

(

1–2

145

–

157

Google Scholar

OpenURL Placeholder Text

WorldCat

Jech

Pšenčík

1989

First-order perturbation method for anisotropic media

J. Geophys. Int.

(

369

–

376

10.1111/j.1365-246X.1989.tb01694.x

Kawakatsu

2016

A new fifth parameter for transverse isotropy

J. Geophys. Int.

204

(

682

–

685

Komatitsch

Tromp

1999

Introduction to the spectral element method for three-dimensional seismic wave propagation

J. Geophys. Int.

139

(

806

–

822

10.1046/j.1365-246x.1999.00967.x

Larson

E.W.

Tromp

Ekström

1998

Effects of slight anisotropy on surface waves

J. Geophys. Int.

132

(

654

–

666

10.1046/j.1365-246X.1998.00452.x

Lévěque

Debayle

Maupin

1998

Anisotropy in the Indian ocean upper mantle from Rayleigh-and Love-waveform inversion

J. Geophys. Int.

133

(

529

–

540

10.1046/j.1365-246X.1998.00504.x

Liang

Zhao

Hua

Y.-G.

2024

Big mantle wedge and intraplate volcanism in Alaska: insight from anisotropic tomography

J. Geophys. Res. Solid Earth

129

(

e2023JB027617

Lin

F.-C.

Ritzwoller

M.H.

Snieder

2009

Eikonal tomography: surface wave tomography by phase front tracking across a regional broad-band seismic array

J. Geophys. Int.

177

(

1091

–

1110

10.1111/j.1365-246X.2009.04105.x

Lin

F.-C.

Ritzwoller

M.H.

Yang

Moschetti

M.P.

Fouch

M.J.

2011

Complex and variable crustal and uppermost mantle seismic anisotropy in the western United States

Nat. Geosci.

(

–

Liu

Ritzwoller

M.H.

2024

Seismic anisotropy and deep crustal deformation across Alaska

J. Geophys. Res. Solid Earth

129

(

e2023JB028525

Liu

Zhang

Sheehan

A.F.

Ritzwoller

M.H.

2022

Surface wave isotropic and azimuthally anisotropic dispersion across Alaska and the Alaska-Aleutian subduction zone

J. Geophys. Res. Solid Earth

127

(

e2022JB024885

Maupin

Park

2015

1.09—theory and observations—seismic anisotropy

T. Geophys.

277

–

305

10.1016/B978-0-444-53802-4.00007-5

Maupin

1989

Surface waves in weakly anisotropic structures: on the use of ordinary or quasi-degenerate perturbation methods

J. Geophys. Int.

(

553

–

563

10.1111/j.1365-246X.1989.tb02289.x

Maupin

2001

A multiple-scattering scheme for modelling surface wave propagation in isotropic and anisotropic three-dimensional structures

J. Geophys. Int.

146

(

332

–

348

10.1046/j.1365-246x.2001.01460.x

Maupin

2004

Comment on ‘The azimuthal dependence of surface wave polarization in a slightly anisotropic medium’ by T. Tanimoto

J. Geophys. Int.

159

(

365

–

368

10.1111/j.1365-246X.2004.02390.x

Montagner

J.-P.

Jobert

1988

Vectorial tomography—II. Application to the Indian Ocean

J. Geophys. Int.

(

309

–

344

10.1111/j.1365-246X.1988.tb05904.x

Montagner

J.-P.

Nataf

H.-C.

1986

A simple method for inverting the azimuthal anisotropy of surface waves

J. Geophys. Res. Solid Earth

(

511

–

520

10.1029/JB091iB01p00511

Montagner

J.-P.

Nataf

H.-C.

1988

Vectorial tomography—I. theory

J. Geophys. Int.

(

295

–

307

10.1111/j.1365-246X.1988.tb05903.x

Montagner

J.-P.

Tanimoto

1990

Global anisotropy in the upper mantle inferred from the regionalization of phase velocities

J. Geophys. Res. Solid Earth

(

4797

–

4819

10.1029/JB095iB04p04797

Nishimura

C.E.

Forsyth

D.W.

1988

Rayleigh wave phase velocities in the Pacific with implications for azimuthal anisotropy and lateral heterogeneities

J. Geophys. Int.

(

479

–

501

10.1111/j.1365-246X.1988.tb02270.x

Park

1993

Seismic determination of elastic anisotropy and mantle flow

Science

261

(

5125

1159

–

1162

10.1126/science.261.5125.1159

Park

1990

The subspace projection method for constructing coupled-mode synthetic seismograms

J. Geophys. Int.

101

(

111

–

123

10.1111/j.1365-246X.1990.tb00761.x

Pettersen

Ø.

Maupin

2002

Lithospheric anisotropy on the Kerguelen hotspot track inferred from Rayleigh wave polarisation anomalies

J. Geophys. Int.

149

(

225

–

246

10.1046/j.1365-246X.2002.01646.x

Romanowicz

Panning

Gung

Capdeville

2008

On the computation of long period seismograms in a 3-D earth using normal mode based approximations

J. Geophys. Int.

175

520

–

536

Google Scholar

OpenURL Placeholder Text

WorldCat

Russell

J.B.

Gaherty

J.B.

Lin

P.-Y.P.

Lizarralde

Collins

J.A.

Hirth

Evans

R.L.

2019

High-resolution constraints on Pacific upper mantle petrofabric inferred from surface-wave anisotropy

J. Geophys. Res. Solid Earth

124

(

631

–

657

Sieminski

Liu

Trampert

Tromp

2007

Finite-frequency sensitivity of surface waves to anisotropy based upon adjoint methods

J. Geophys. Int.

168

(

1153

–

1174

10.1111/j.1365-246X.2006.03261.x

Sieminski

Trampert

Tromp

2009

Principal component analysis of anisotropic finite-frequency sensitivity kernels

J. Geophys. Int.

179

(

1186

–

1198

10.1111/j.1365-246X.2009.04341.x

Smith

M.L.

Dahlen

1973

The azimuthal dependence of Love and Rayleigh wave propagation in a slightly anisotropic medium

J. geophys. Res.

(

3321

–

3333

10.1029/JB078i017p03321

Snieder

Nolet

1987

Linearized scattering of surface waves on a spherical Earth

J. Geophys.

(

–

Google Scholar

OpenURL Placeholder Text

WorldCat

Snieder

1986

3-D linearized scattering of surface waves and a formalism for surface wave holography

J. Geophys. Int.

(

581

–

605

10.1111/j.1365-246X.1986.tb04372.x

Park

1993

Born seismograms using coupled free oscillations: the effects of strong coupling and anisotropy

J. Geophys. Int.

115

(

849

–

862

10.1111/j.1365-246X.1993.tb01497.x

Tanimoto

Anderson

D.L.

1985

Lateral heterogeneity and azimuthal anisotropy of the upper mantle: Love and Rayleigh waves 100–250 s

J. Geophys. Res. Solid Earth

(

1842

–

1858

10.1029/JB090iB02p01842

Tanimoto

2004

The azimuthal dependence of surface wave polarization in a slightly anisotropic medium

J. Geophys. Int.

156

(

–

10.1111/j.1365-246X.2004.02130.x

Trampert

Woodhouse

J.H.

2003

Global anisotropic phase velocity maps for fundamental mode surface waves between 40 and 150 s

J. Geophys. Int.

154

(

154

–

165

10.1046/j.1365-246X.2003.01952.x

Tromp

Dahlen

1990

Summation of the Born series for the normal modes of the Earth

J. Geophys. Int.

100

(

527

–

533

10.1111/j.1365-246X.1990.tb00704.x

Tromp

Dahlen

1993

Surface wave propagation in a slowly varying anisotropic waveguide

J. Geophys. Int.

113

(

239

–

249

10.1111/j.1365-246X.1993.tb02542.x

Tromp

1994

Surface-wave propagation on a rotating, anisotropic earth

J. Geophys. Int.

117

(

141

–

152

10.1111/j.1365-246X.1994.tb03308.x

Xie

Ritzwoller

M.H.

Brownlee

Hacker

2015

Inferring the oriented elastic tensor from surface wave observations: preliminary application across the western United States

J. Geophys. Int.

201

(

996

–

1021

Xie

Ritzwoller

M.H.

Shen

Wang

2017

Crustal anisotropy across eastern Tibet and surroundings modeled as a depth-dependent tilted hexagonally symmetric medium

J. Geophys. Int.

209

(

466

–

491

Google Scholar

OpenURL Placeholder Text

WorldCat

Yao

van Der Hilst

R.D.

Montagner

J.-P.

2010

Heterogeneity and anisotropy of the lithosphere of SE Tibet from surface wave array tomography

J. Geophys. Res. Solid Earth

115

(

B12

10.1029/2009JB007142

Google Scholar

OpenURL Placeholder Text

WorldCat

Crossref

Yuan

Romanowicz

2010

Lithospheric layering in the North American craton

Nature

466

1063

–

1068

7310

APPENDIX A: ELASTIC TENSOR IN VARIOUS MEDIA

The elastic tensor |$c_{ijk\ell }$| can be written in abbreviated or Voigt notation as a symmetric |$6\times 6$| matrix |$C_{mn}$| such that each pair of indices |$(ij)$| is replaced with a single index m according to the following rule: if |$i=j$| then |$m=i$| and if |$i\ne j$| then |$m=9-(i+j)$|⁠. A general elastic tensor can then be visualized as follows:

$$\begin{eqnarray} C_{mn} = {\begin{bmatrix}C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\ C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\ C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \\ \end{bmatrix}} . \end{eqnarray}$$

(A1)

For an isotropic elastic tensor

$$\begin{eqnarray} c_{ijk\ell }^{\mathrm{ isotropic}} = \lambda \delta _{ij} \delta _{k \ell } + \mu ( \delta _{ik} \delta _{j\ell } + \delta _{i\ell }\delta _{jk} ), \end{eqnarray}$$

(A2)

the elastic tensor can be visualized as follows:

$$\begin{eqnarray} C^{\mathrm{ isotropic}}_{mn} = {\begin{bmatrix}\lambda + 2 \mu & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda + 2 \mu & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & \lambda + 2 \mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \\ \end{bmatrix}} . \end{eqnarray}$$

(A3)

Similarly, the elastic tensor for a transversely isotropic medium with a vertical symmetry axis, or a VTI medium, can be written as

$$\begin{eqnarray} C^{\mathrm{ VTI}}_{mn} = {\begin{bmatrix}A & A - 2N & F & 0 & 0 & 0\\ A - 2N & A & F & 0 & 0 & 0\\ F & F & C & 0 & 0 & 0 \\ 0 & 0 & 0 & L & 0 & 0 \\ 0 & 0 & 0 & 0 & L & 0 \\ 0 & 0 & 0 & 0 & & N \end{bmatrix}} , \end{eqnarray}$$

(A4)

To produce a tilted transversely isotropic medium, the symmetry axis of the VTI medium is rotated through a dip angle |$\theta$| around the y-axis as follows

$$\begin{eqnarray} {\bf C}^{TTI} = {\bf B C}^{VTI} {\bf B}^T \end{eqnarray}$$

(A5)

where B is the Bond matrix and |${\bf B}^T$| is its transpose. Sometimes we refer to the y-axis as the ‘strike axis’. The components of the elastic tensor for the TTI medium are

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{11} = A \cos ^4 \theta + C \sin ^4 \theta + (2F + 4L)\sin ^2 \theta \cos ^2 \theta , \end{eqnarray}$$

(A6)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{22} = A , \end{eqnarray}$$

(A7)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{33} = A \sin ^4 \theta + C \cos ^4 \theta + (2F + 4L)\sin ^2 \theta \cos ^2 \theta , \end{eqnarray}$$

(A8)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{44} = L \cos ^2 \theta + N \sin ^2 \theta , \end{eqnarray}$$

(A9)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{55} = (A + C - 2F)\sin ^2 \theta \cos ^2 \theta + L (\cos ^2 \theta - \sin ^2 \theta )^2 , \end{eqnarray}$$

(A10)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{66} = L \sin ^2 \theta + N \cos ^2 \theta , \end{eqnarray}$$

(A11)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{12} = C^{\mathrm{ TTI}}_{21} = (A - 2N)\cos ^2 \theta + F \sin ^2 \theta , \end{eqnarray}$$

(A12)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{13} &=& C^{\mathrm{ TTI}}_{31} = (A + C - 4L)\sin ^2 \theta \cos ^2 \theta \\ &&+ F(\sin ^4\theta + \cos ^4\theta ) , \end{eqnarray}$$

(A13)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{15} &=& C^{\mathrm{ TTI}}_{51} = (F + 2L - A)\sin \theta \cos ^3 \theta \\ &&- (F + 2L - C)\sin ^3 \theta \cos \theta , \end{eqnarray}$$

(A14)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{23} = C^{\mathrm{ TTI}}_{32} = (A - 2N)\sin ^2 \theta + F \cos ^2 \theta , \end{eqnarray}$$

(A15)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{25} = C^{\mathrm{ TTI}}_{52} = (F + 2N - A)\sin \theta \cos \theta , \end{eqnarray}$$

(A16)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{35} &=& C^{\mathrm{ TTI}}_{53} = (F + 2L - A)\sin ^3 \theta \cos \theta \\ &&- (F + 2L - C)\sin \theta \cos ^3 \theta , \end{eqnarray}$$

(A17)

$$\begin{eqnarray} C_{46}^{\mathrm{ TTI}} = C^{\mathrm{ TTI}}_{64} = (L - N)\sin \theta \cos \theta , \end{eqnarray}$$

(A18)

$$\begin{eqnarray} C^{\mathrm{ TTI}}_{14} &=& C^{\mathrm{ TTI}}_{16} = C^{\mathrm{ TTI}}_{24} = C^{\mathrm{ TTI}}_{26} = C^{\mathrm{ TTI}}_{34} = C^{\mathrm{ TTI}}_{36} \\ &=& C^{\mathrm{ TTI}}_{45} = C^{\mathrm{ TTI}}_{56} = 0 . \end{eqnarray}$$

(A19)

Only 13 of the 21 components of the elastic tensor for a TTI medium are independent. These 13 components form a monoclinic elastic solid.

For a transversely isotropic medium with a horizontal symmetry axis, |$\theta = 90^\circ$|⁠, so

$$\begin{eqnarray} C^{\mathrm{ HTI}}_{mn}= {\begin{bmatrix}C& F & F & 0 & 0 & 0 \\ F & A & A-2N & 0& 0 & 0 \\ F & A-2N & A & 0 & 0 & 0 \\ 0 & 0 & 0 & N &0 & 0 \\ 0 & 0 & 0 & 0 & L&0 \\ 0 & 0 & 0 & 0 & 0 & L \end{bmatrix}} \end{eqnarray}$$

(A20)

APPENDIX B: THE 21 ANISOTROPIC PARAMETERS

Montagner & Nataf (1986) introduced linear recombinations of the elastic tensor components for surface waves. Chen & Tromp (2007) introduced others that also are needed for body waves. We follow Chen & Tromp (2007) by including the negative sign in the definition of |$G_s, B_s, H_s$| and |$E_s$|⁠:

$$\begin{eqnarray} {\cal A} = \frac{1}{8} (3C_{11} + 3C_{22} + 2C_{12} + 4C_{66}) , \end{eqnarray}$$

(B1)

$$\begin{eqnarray} {\cal C} = C_{33} , \end{eqnarray}$$

(B2)

$$\begin{eqnarray} {\cal N} = \frac{1}{8} (C_{11} + C_{22} - 2C_{12} + 4C_{66}) , \end{eqnarray}$$

(B3)

$$\begin{eqnarray} {\cal L} = \frac{1}{2} (C_{44} + C_{55}) , \end{eqnarray}$$

(B4)

$$\begin{eqnarray} {\cal F} = \frac{1}{2} (C_{13} + C_{23}) , \end{eqnarray}$$

(B5)

$$\begin{eqnarray} J_c = \frac{1}{8} (3C_{15} + C_{25} + 2C_{46}) , \end{eqnarray}$$

(B6)

$$\begin{eqnarray} J_s = \frac{1}{8} (C_{14} + 3C_{24} + 2C_{56}) , \end{eqnarray}$$

(B7)

$$\begin{eqnarray} K_c = \frac{1}{8} (3C_{15} + C_{25} + 2C_{46} - 4C_{35}) , \end{eqnarray}$$

(B8)

$$\begin{eqnarray} K_s = \frac{1}{8} (C_{14} + 3C_{24} + 2C_{56} - 4C_{34}) , \end{eqnarray}$$

(B9)

$$\begin{eqnarray} M_c = \frac{1}{4} (C_{15} - C_{25} + 2C_{46}) , \end{eqnarray}$$

(B10)

$$\begin{eqnarray} M_s = \frac{1}{4} (C_{14} - C_{24} - 2C_{56}) , \end{eqnarray}$$

(B11)

$$\begin{eqnarray} G_c = \frac{1}{2} (C_{55} - C_{44}) , \end{eqnarray}$$

(B12)

$$\begin{eqnarray} G_s = - C_{45} , \end{eqnarray}$$

(B13)

$$\begin{eqnarray} B_c = \frac{1}{2} (C_{11} - C_{22}) , \end{eqnarray}$$

(B14)

$$\begin{eqnarray} B_s = - (C_{16} + C_{26}) , \end{eqnarray}$$

(B15)

$$\begin{eqnarray} H_c = \frac{1}{2} (C_{13} - C_{23}) , \end{eqnarray}$$

(B16)

$$\begin{eqnarray} H_s = - C_{36} , \end{eqnarray}$$

(B17)

$$\begin{eqnarray} D_c = \frac{1}{4} (C_{15} - C_{25} - 2C_{46}) , \end{eqnarray}$$

(B18)

$$\begin{eqnarray} D_s = \frac{1}{4} (C_{14} - C_{24} + 2C_{56}) , \end{eqnarray}$$

(B19)

$$\begin{eqnarray} E_c = \frac{1}{8} (C_{11} + C_{22} -2C_{12} - 4C_{66}) , \end{eqnarray}$$

(B20)

$$\begin{eqnarray} E_s = - \frac{1}{2} (C_{16} - C_{26}) . \end{eqnarray}$$

(B21)

We use the script notation for |$\cal A, C, N, L$| and |$\cal F$| [using these five parameters to construct a VTI medium (eq. A4) is so called an effective transversely isotropic medium] to distinguish them from the Love moduli |$A, C, N, L$| and F that define a VTI medium, which is the basis for producing the elastic tensor for a TTI medium in Appendix A.

|$J_c$| (⁠|$J_s$|⁠), |$K_c$| (⁠|$K_s$|⁠) and |$M_c$| (⁠|$M_s$|⁠) are body wave 1|$\psi$| azimuthal anisotropy parameters and |$D_c$| (⁠|$D_s$|⁠) is the body wave 3|$\psi$| azimuthal anisotropy parameter, which were not included by Montagner & Nataf (1986). |$G_c$| (⁠|$G_s$|⁠), |$B_c$| (⁠|$B_s$|⁠) and |$H_c$| (⁠|$H_s$|⁠) are 2|$\psi$| azimuthal anisotropic parameters for both body waves and surface waves. |$E_c$| (⁠|$E_s$|⁠) is the 4|$\psi$| azimuthal anisotropic parameter for both body waves and surface waves.

For a TTI medium, all parameters with the ‘s’ subscript are zero, so 13 of the anisotropic parameters are non-zero, forming a medium with monoclinic symmetry.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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Article Contents

The effect of Rayleigh–Love coupling in an anisotropic medium

SUMMARY

1 INTRODUCTION

2 DATA SOURCES

3 QUASI-DEGENERATE THEORY FOR BODY AND SURFACE WAVES

3.1 Polarization and displacement basis vectors

3.2 Coupling caused by anisotropy

3.3 Quasi-degeneracy

3.4 The Lagrangian and Hamilton’s principle

4 THE EFFECT OF SV–SH COUPLING

4.1 Implications for surface waves

5 THE EFFECT OF RAYLEIGH–LOVE COUPLING

5.1 Theory

5.2 Phase speeds and fast orientations

5.3 Amplitudes

5.4 Coupling strength

5.5 Polarization and phase lag

5.6 Numerical results

6 DISCUSSION OF RAYLEIGH–LOVE COUPLING

6.1 Estimating anisotropy in the presence of Rayleigh–Love coupling

6.2 Coupling between fundamental modes and overtones

6.3 Polarization

7 CONCLUSIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

APPENDIX A: ELASTIC TENSOR IN VARIOUS MEDIA

APPENDIX B: THE 21 ANISOTROPIC PARAMETERS

Supplementary data

Citations

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Article Contents

The effect of Rayleigh–Love coupling in an anisotropic medium

SUMMARY

1 INTRODUCTION

2 DATA SOURCES

3 QUASI-DEGENERATE THEORY FOR BODY AND SURFACE WAVES

3.1 Polarization and displacement basis vectors

3.2 Coupling caused by anisotropy

3.3 Quasi-degeneracy

3.4 The Lagrangian and Hamilton’s principle

4 THE EFFECT OF SV–SH COUPLING

4.1 Implications for surface waves

5 THE EFFECT OF RAYLEIGH–LOVE COUPLING

5.1 Theory

5.2 Phase speeds and fast orientations

5.3 Amplitudes

5.4 Coupling strength

5.5 Polarization and phase lag

5.6 Numerical results

6 DISCUSSION OF RAYLEIGH–LOVE COUPLING

6.1 Estimating anisotropy in the presence of Rayleigh–Love coupling

6.2 Coupling between fundamental modes and overtones

6.3 Polarization

7 CONCLUSIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

APPENDIX A: ELASTIC TENSOR IN VARIOUS MEDIA

APPENDIX B: THE 21 ANISOTROPIC PARAMETERS

Supplementary data

Citations

Views

Altmetric

Email alerts

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Citing articles via

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