I N T RO D U C T I O N

Based on non-degenerate per turbation theor y, Smith & Dahlen ( 1973 ) showed that the azimuthal variation of Rayleigh and Love wave phase and group speeds at angular frequency ω in a slightly anisotropic medium is of the well-known form

where ψ 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 Ra yleigh wa ve speeds will be dominated by the 2 ψ terms in eq. ( 1), whereas the Love wave phase speeds will be dominated by the 4 ψ 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 inte gral e xpressions for each of the coefficients in eq. ( 1) to be used to inver t obser vational 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 ψ component of Ra yleigh wa ve anisotropy and to a lesser extent the 4 ψ component of Lov e wav e anisotropy. Many studies have presented and interpreted the 2 ψ 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 ;L év ěque 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 ψ component of Love wave anisotropy are much more rare (e.g. Montagner & Tanimoto 1990 ;Trampert & Woodhouse 2003 ;Ekstr öm 2011 ;Russell et al. 2019 ). Much less effort has been devoted to observing the 2 ψ component of Love wave anisotropy or the 4 ψ component of Rayleigh wave anisotropy. We refer to the 2 ψ component for Rayleigh waves and the 4 ψ component for Love waves as 'expected' anisotropy, according to non-degenerate perturbation theory . Similarly , the 4 ψ component for Rayleigh waves and the 2 ψ 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 ψ component of Love wave anisotropy is observed and its amplitude is commensurate with the 4 ψ component of Lov e wav e anisotropy and the 2 ψ 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 Lov e wav e anisotropy across Alaska", manuscript in preparation, 2024). Strong 2 ψ Love wave anisotropy is observed at 20 s period as well as the weaker 4 ψ component of Rayleigh wave anisotropy. As expected, the 2 ψ component of the Rayleigh wave and the 4 ψ component of the Lov e wav e anisotropy are also observed at this point.

Such strong Lov e wav e 2 ψ and Rayleigh wave 4 ψ anisotropy cannot be explained by the non-degenerate per turbation theor y 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 Ra yleigh wa ve 2 ψ 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 ψ component of Ra yleigh wa ve anisotropy w ell but strongly underpredict the observed amplitude of the 2 ψ component of Love wave anisotropy.

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-de generac y 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-de generac y condition that coupling Love and Ra yleigh wa ves ha ve the same wavenumber but slightl y dif ferent 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 ( ˆ a R , ˆ a L ):

where a L and a R are coupling coefficients following the notation of T animoto ( 2004 ). T animoto ( 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 quasidegenerate theory also is unable to explain observations of strong 2 ψ Lov e wav e or 4 ψ Ra yleigh wa ve anisotropy and types of anisotropy remained unexpected.

In this paper, we present a revised quasi-degenerate theory that does explain observations of strong 2 ψ Love wave and 4 ψ Rayleigh wave anisotropy. When Rayleigh-Love coupling is strong enough, significant 2 ψ Lov e wav e anisotropy is expected although the 4 ψ Ra yleigh wa ve 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 deri v ati ves 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-de generac y 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 & P šen č ík ( 1989 ) (also Chapman 2004 ;Chen & Tromp 2007 ; Červen ỳ & P šen č ík 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, K omatitsch & T romp 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 ψ and 4 ψ and Ra yleigh wa v e 4 ψ observations in the inv ersion 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-Lov e wav es to constrain anisotropy, which was a point emphasized by Park & Yu ( 1993 ), Tanimoto ( 2004 ) and Maupin & Park ( 2015 ). Principal deri v ations 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 ef fecti v e transv ersely 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).

D ATA S O U RC E S

Four different data compilations or models are used here for computation and inv ersion, as e xamples 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 v ertical transv ersely isotropic (VTI) or ef fecti v e transv ersely 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.

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 • N, 159 • W), taken from Liu & Ritzwoller ( 2024 ), which is based on fitting only the isotropic Love and Rayleigh wave phase speed curves and 2 ψ 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 Ra yleigh wa v e and Lov e wav e 2 ψ and 4 ψ 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 • N, 159 • 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 • N, 147 • W). We make use of the resulting models to compute the strength of Rayleigh-Love coupling in a continental setting.

Q U A S I -D E G E N E R A T E T H E O RY F O R B O DY A N D S U R FA C E WAV E S

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

where ˆ a is the direction of particle motion or the polarization vector, the components of the position vector r are x i ( x 1 , x 2 , x 3 ) T = ( x , y , z) T and of the horizontal wavenumber vector k are ωn 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

where z = 0 is the free surf ace, surf ace location r = ( x , y , 0) T , and ˆ s ( z) is the vector displacement eigenfunction. We set the basis vectors for body waves propagating horizontally at azimuth ψ 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 S H , as depicted in Fig. 3 . Therefore the polarization basis vectors are Figure 3. Geometry of horizontal body wave propagation in the direction defined by the azimuthal angle ψ relative to the x 1 -axis, showing the waves in the reference isotropic medium, P, S H and SV , as well as the quasi-S waves ( q S 1 , q S 2 ) in the perturbed anisotropic medium. SV-SH coupling rotates the polarization of the quasi-shear waves through angle in the plane perpendicular to the direction of propagation. We define as the ne gativ e of the complement of and x 2 is the 'strike axis'.

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X. Liu and M. H. Ritzwoller

which we denote with the overscript ∧ and T means transpose. The displacement vectors in the reference medium are

which we also denote with an overscript ∧ . The propagation term for horizontal propagation is

where phase speed V = ω/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.

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 ψ. The polarization vectors are

U ( z ) and V ( z ) are the vertical and horizontal (radial) displacement eigenfunctions for Rayleigh waves and W ( z) is the Lov e wav e horizontal (transverse) eigenfunction, which are normalized as follows:

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

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

We denote quantities in the anisotropic medium with an overscript . Because the basis vectors for body waves are real and depthindependent, the expansion coefficients a i j are also real; that is, a i j ∈ 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

where w e ha ve introduced notation for the expansion coefficients a S H and a SV , such that a 2 S H + a 2 SV = 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 , as Fig. 3 illustrates. Thus, a 22 = cos , a 23 = sin , a 32 = -sin , and a 33 = cos , where 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 .

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

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 • . The eigenvector of the quasi-S 1 wave is unaffected by coupling to the quasi-P wave.

For surface wa ves, w e 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 ef fecti ve transversely isotropic medium as the reference (Appendix B), including Rayleigh and Lov e wav es, fundamental and ov ertone 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

The expansion coefficients a R and a L define the Rayleigh-Love coupling and are complex mainly because the basis vectors are 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 ψ is:

(26)

Quasi-degeneracy

Under the quasi-de generac y condition, wav es 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 ω 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 quaside generac y 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-de generac y 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.

T he Lag rangian 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

where c i jkℓ is the elastic tensor, i j = ( u i, j + u j,i ) / 2 , the subscript ' , j' represents a spatial deri v ati ve 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 f f * = 1 the propagation term f and all time-dependent terms disappear from further equations. For the anisotropic medium, u i is replaced by ˜ 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 ∂ L/∂ a S H = ∂ L/∂ a SV = 0 for body waves and that ∂ L/∂ a L = ∂ L/∂ a R = 0 for surface waves. The latter for surface waves was first applied by Tanimoto ( 2004 ). Applying these deri v ati ves results in an eigenv alue-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 . 42)-( 45

T H E E F F E C T O F S V -S H C O U P L I N G

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 (Suppor ting Infor mation). 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 θ around the y-axis, which we refer to as the 'strike axis'.

For the horizontally propagating quasi-S 1 and quasi-S 2 waves

where

and E ≡ A + C -2 F -4 L , as defined in the Section S.3 (Suppor ting Infor mation).

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 4 L and A + C -2 F . If E = 0 , 4 L = A + C -2 F , 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 moti v ates the definition of a ne w ellipticity parameter

which for weak anisotropy is approximately equal to the parameter η K introduced by Kawakatsu ( 2016 ), as illustrated by Fig. S3 (Suppor ting Infor mation). η 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 η X according to the final expressions in eqs ( 33 )-( 35 ). A + C -2 F is normall y positi ve in Earth materials. The relative peak-to-peak amplitude of 2 ψ and 4 ψ anisotropy of quasi-S 2 can therefore be expressed as:

The polarization angle for the coupled quasi-S waves is derived in Section S.4 (Supporting Information) as tan = tan θ sin ψ,

where -θ ≤ ≤ θ . | | will be no larger than the dip angle θ , and will average about θ/ 2 . Figs 6 (a) and (b) illustrate how changing the value of the ellipticity parameter η X changes the azimuth of the fast directions. For the 2 ψ component of the quasi-S 2 wave, the orientation of the fast directions rotates 90 • when 1η X changes sign. For the 4 ψ component, the rotation is 45 • . Fig. 6 (c) includes how the variation of quasi-S 1 and quasi-S 2 wave with azimuth depends on the relationship with L -N and η X .

Implications for surface w av es

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

(1) Lov e wav e 4 ψ phase speed would be gi ven b y a depth integral of eq. ( 35) with associated eigenfunctions. This implies that observations of Love wave 4 ψ in a TTI medium would imply that anisotropy is non-elliptical ( η X = 1 ). If η X < 1 , then the Love wave 4 ψ fast axis will have a 45 • difference relative to the Rayleigh wave 2 ψ fast axis, as was observed in the central Pacific (Russell et al. 2019 ) and in Alaska (Liu et al. "Observations of Rayleigh and Love wave anisotropy across Alaska", manuscript in preparation, 2024). If η X > 1 , the two fast axes will be parallel. The depth-averaged amplitude of Love wave 4 ψ is reflected in eq. ( 38). Interpretting the fast axis and amplitude of the Love wave 4 ψ together constrains the ellipticity parameter η X in the TTI inversion. Thus, observations of the Love wave 4 ψ component is extremely useful.

2. A central argument of this paper is that Love wave 2 ψ 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 ψ amplitude, which is G c (eq. S55), with the 2 ψ amplitude of qS 1 (eq. 32 ) and qS 2 (eq. 34 ), we find SV-SH coupling just splits the total 2 ψ 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 al wa ys 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 w aves (deri ved 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 wa ves as w e assign qS 1 and q S 2 to the two quasi-shear waves, instead of qSH or qSV. Ho wever , 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.

T H E E F F E C T O F R AY L E I G H -L OV E C O U P L I N G

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 ψ are given by eqs ( 14) and ( 15) where f is given by eq. ( 11). Displacement u in an anisotropic medium is given by eq. ( 25). The displacement field in an anisotropic medium for a coupled Rayleigh and Lov e wav e propagating at azimuth ψ is gi ven b y 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).

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

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 ∈ C. A, B, E, and X are

We refer to the products of eigenfunctions in A, B, E and X as 'sensitivity kernels '. Fig. 7 (b) shows examples of the 12 sensitivity kernels at 20 s period. The kernels

Hamilton's Principle implies that ∂ L/∂ a R = ∂ L/∂ a L = 0 (Section S.6.2, Supporting Information), which is used in Section S.7 (Suppor ting Infor mation) to derive the following eigenvalue problem that governs Rayleigh-Love coupling:

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

or phase speed given by

where

Because Lov e wav es 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 ψ and 4 ψ variations in V 2 . A small additional contribution to a 6 ψ variation is ignorable.

For clarity, we now re vie w 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 ef fecti ve transversely isotropic medium). We choose a reference medium that decouples the Rayleigh and Lov e wav es (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.

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 wav e curv es (Fig . 8 a, b, d and e) computed using the non-degenerate per turbation theor y (NDPT) of Smith & Dahlen ( 1973 ). Based on NDPT, the Love wave is dominated by 4 ψ azimuthal variations and the Rayleigh wave variations are dominantly 2 ψ. 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 ψ and 4 ψ. In western Alaska, the fast axis directions of quasi-Rayleigh and quasi-Love are out of phase by 180 • and in eastern Alaska they are in phase.

The phasing between the fast directions of quasi-Rayleigh and quasi-Lov e wav es reflects the relationship between the observed quasi-Ra yleigh wa ve 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 ψ 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-Ra yleigh wa ve 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 ψ and Love wave 2 ψ fast axes. The above and later discussion of the strike angle assume Rayleigh-Love coupling does not change the sign of the 2 ψ component of the Rayleigh wave, which is usually true for fundamental mode surface waves in Alaska (Fig. 8 ).

Amplitudes

Figs 8 (c) and (f) illustrates how the phasing between the fast axis orientations of quasi-Love and quasi-Rayleigh w aves af fects 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-Ra yleigh wa ve under QDT is reduced relative to the Ra yleigh wa ve under NDPT, whereas the quasi-Love wave amplitude is increased relative to NDPT. In contrast, when the quasi-Rayleigh and quasi-Love 2 ψ fast orientations are out of phase by 180 • , 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 ψ 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. 8 c), 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.

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

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 4 X 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 ψ fast directions. The assumption here is that the Lov e wav e is the f aster surf ace 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 Lov e wav e 2 ψ 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.

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

where , ≡ ( B -A + D) / 2( E 2 + X 2 ) 1 / 2 . The vector eigenfunctions are therefore

,e i( φ+ π/ 2) U ( z)) T (54)

ˆ s

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

The quasi-Rayleigh wave is rotated from the vertical by nearly the same angle. Fig. 9 (c) presents an example of at 40 s period, which maximizes near the Lov e wav e 2 ψ fast direction where coupling is strongest. In this example, the quasi-Love wave polarization will be tipped by a maximum angle max ∼ 16 • 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 φ between the vertical and horizontal components is plotted for the same example in Fig. 9 (d). At most azimuths, the lag is about ±90 • . The lag angle changes sign from 90 • to -90 • when X becomes ne gativ e, as shown in Fig. 9 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 φ = -90 • , the vector eigenfunction for the quasi-Lov e wav e is

(b). The polarization

Signs will be reversed if φ = 90 • .

To consider the quasi-Love particle motion it is useful to think of propagation in the x 1 direction ( / = 1 , . = 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,V , W, ,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 , which depends on ,. Ho wever , the transverse and radial components will be out of phase by 90 • , 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 Ra yleigh wa ve, which will ha ve an elliptical particle motion. Such a polarization anomaly has been observed pre viousl y (Pettersen & Maupin 2002 ).

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 per turbation theor y for surface waves and degenerate perturbation theory for body waves.

To simplify interpretation, we define a simple 60 km thick threelayer anisotropy model with an imposed 4 per cent anisotropy: M s = -0 . 04( λ + 2 µ) . 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 km , ρ 1 = 2600 kg m -3 , V p1 = 6 . 3 km s -1 , V s1 = 3 . 2 km s -1 ; h 2 = 15 km , ρ 2 = 2900 kg m -3 , V p2 = 7 . 0 km s -1 , V s2 = 3 . 7 km s -1 ; h 3 = 30 km , ρ 3 = 3200 kg m -3 , V p3 = 7 . 5 km s -1 , V s3 = 4 . 3 km 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 × 4000 km × 60 km and we do not consider attenuation ( Q µ = ∞ ). For the numerical benchmark, we set the reference medium to be isotropic. Therefore, for the 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. 11 b). 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 • or 180 • (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 • or 270 • , 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. 11 c). The particle motion for both the quasi-Love and quasi-Rayleigh wave are 3-D and linear or out of phase by 180 • 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 quasidegenerate 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 ).

D I S C U S S I O N O F R AY L E I G H -L OV E C O U P L I N G

Estimating anisotropy in the presence of Ra yleigh-Lo ve coupling

For two principal reasons, most previous inversions of observations of surface wave azimuthal anisotropy have been based exclusi vel y on the 2 ψ component of the azimuthal variation of Rayleigh w aves. First, earl y 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 ψ anisotropy for Rayleigh waves and 4 ψ anisotropy for Love waves. Second, for practical reasons, Love wave anisotropy and the 4 ψ 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 ψ component of azimuthal variations in Ra yleigh wa ve 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 Lov e wav e anisotropy across Alaska", manuscript in preparation, 2024), the 2 ψ component of Lov e wav e anisotropy may be quite large and the 4 ψ 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 ψ Lov e wav e 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 • N, 159 • W), Data Source 4 in Section 2 , we present three inversion results to demonstrate the effect of using new ('unexpected') signals (Love 2 ψ, Rayleigh 4 ψ) interpreted with and without Rayleigh-Love coupling. The three estimated models are summarized in Tab le 2 , w here 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. ( 2015Xie et al. ( , 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 θ can vary discontinuously with depth.

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 Lov e wav e isotropic phase speed curves which we do not show. Model 1 is constructed using only the 2 ψ 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. 2015Xie et al. , 2017 ; ;Liu & Ritzwoller 2024 ). Model 2 is constructed by augmenting the observations used in Model 1 with the 4 ψ component of Love wave anisotropy, where the theory is still NDPT. Model 3 further augments these observations with Lov e wav e 2 ψ anisotropy and Ra yleigh wa ve 4 ψ anisotropy, and the theory used in the inversion is the QDT presented here. The isotropic Rayleigh and Lov e wav e phase speed curv es are also used in the construction of all three models. The crust and mantle are both modelled as depthdependent TTI media, where the dip angle θ 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 • , 147 • 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.

Fig. 13 presents results from the in versions, sho wing four variables from the three models. These are the Love modulus L as V SV = √ L/ρ, the dip angle θ of the transversely isotropic elastic tensor, S -wave anisotropy ( N -L ) / 2 L , and the ellipticity parameter η X (eq. 36 ) which is approximately equal to the 'new' ellipticity parameter η 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 ±1 σ of the posterior distribution is shown for Model 3. The introduction of observations of the 4 ψ 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 ψ component for quasi-S 2 will only occur if the ellipticity coefficient differs strongly from 1. Thus, to fit the Love wave 4 ψ observations requires η X to deviate from 1, which it does not in Model 1. Thus, the use of observations of the 4 ψ component of Lov e wav e anisotropy is par ticularly impor tant to estimate the ellipticity of anisotropy accurately.

Fig. 14 shows that all three models fit the Rayleigh 2 ψ signal. In particular, the Rayleigh 2 ψ signal can be fit with NDPT. Model 2 does fit the Love wave 4 ψ signal, which shows that this signal can also be fit with NDPT. Ho wever , it typically will not be fit unless it is used in the inversion. Neither Model 1 nor Model 2 fits the Lov e wav e 2 ψ signal because quasi-degenerate theory is needed to produce large 2 ψ 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 ψ 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 ψ signal.

Model 3 differs from Model 2 principally in the strength of anisotropy ( γ ), especially in the mantle. This results from the large amplitude of the Lov e wav e 2 ψ azimuthal variation. Since there is also a small observable Rayleigh wave 4 ψ signal, these two models also differ somewhat in η 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.

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 Ra yleigh wa v e 1st-ov ertone 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 wav es. The quasi-de generate theory we present can also be applied to any pair of Rayleigh and Lov e modes, for e xample coupling between the fundamental mode Love wave and the 1st-overtone Ra yleigh wa ve, coupling betw een the 1st-o vertone Lo ve wave and 1st-overtone Ra yleigh wa ve, and so on. We define coupling strength as S (eq. 51 ), which is plotted in Fig. 15 (a) for a continental lo- 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 )/2 L = ( A -C )/2 C = 7 per cent across all depths. We then tilt the elastic tensor by 45 • , 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 η 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 1stovertone Rayleigh modes is much stronger from 10-40 s period (Fig. 15 b). 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.

In conclusion, at most continental locations, fundamental Lov es wav es will be coupled principally to fundamental mode Rayleigh waves, and Love wave-overtone coupling can be safely ignored. At oceanic locations, ho wever , fundamental mode Lov es wav es will be coupled principally to overtone Ra yleigh wa ves, at least below 40 s period, and coupling to the fundamental mode Ra yleigh wa ve will be weaker but still substantial.

Polarization

Tanimoto ( 2004 ) stressed the potential importance of measuring the polarization angle , 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 ψ quasi-Love wave (if the Lov e wav e is faster than the Ra yleigh wa ve). 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 ψ fast direction. Polarization measurements, ho wever , could be used directly in inversions for the depthdependent 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-Lov e wav e coupled to the fundamental mode Ra yleigh wa ve and 2 ). The blue dashed line is computed using Model 1 (based on Rayleigh wave 2 ψ 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 ψ and Love wave 4 ψ 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. the 1st overtone Ra yleigh wa v e, respectiv ely . Not surprisingly , these curves look similar to the coupling strength plotted in Fig. 15 (a). A polarization anomaly of 15 • is expected at this location at periods longer than about 30 s. The polarization anomaly for coupling the Lov e wav e to the first-ov ertone Ra yleigh wa ve 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.

C O N C L U S I O N S

We present a quasi-degenerate theory of Rayleigh-Love coupling based on the application of Hamilton's Principle to Rayleigh and Lov e wav es. This theory e xplains the observation of 2 ψ phase velocity anisotropy for Love waves and 4 ψ anisotropy for Rayleigh w aves. Pre vious 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 ψ anisotropy for Lov e wav es and 4 ψ 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 ψ anisotropy for Love waves. In addition, it allows for these observations to be included in inversions simultaneously with 'expected' observations, such as the 2 ψ anisotropy for Ra yleigh wa ves and the 4 ψ 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 & P šen č ík ( 1989 ). Ho wever , 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 θ of anisotropy and the ellipticity parameter η X , which we introduce here. Through these body wave results, we moti v ate how observ ations of Love wave 4 ψ azimuthal anisotropy can be used to infer η X and θ and how coupling splits the total 2 ψ amplitude.

We present examples that illustrate that when the unexpected 2 ψ anisotropy for Lov e wav es is included in inversions for a depth-dependent TTI medium along with observations of expected anisotropy, better constraints are placed on the ellipticity parameter η 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 • 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 impor tant, par ticularly in oceanic settings. We provide evidence that coupling between the fundamental Love wave and Rayleigh overtones can probably be ignored in continental settings. Ho wever , coupling between the fundamental Lov e wav e 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 Lov e wav e 2 ψ 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 ψ component of Rayleigh wave anisotropy (e.g. Xie et al. 2015Xie et al. , 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. 2007Sieminski et al. , 2009 ) ). Nondegenerate 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 ). Ho wever , 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-de generac y. Some shor tcomings of the first-order Bor n 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.

A P P E N D I X A : E L A S T I C T E N S O R I N VA R I O U S M E D I A

The elastic tensor c i jkℓ can be written in abbreviated or Voigt notation as a symmetric 6 × 6 matrix C mn such that each pair of indices ( i j) is replaced with a single index m according to the following rule: if i = j then m = i and if i = j then m = 9 -( i + j) . A general elastic tensor can then be visualized as follows:

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         . (A1) For an isotropic elastic tensor c isotropic i jkℓ = λδ i j δ kℓ + µ( δ ik δ jℓ + δ iℓ δ jk ) , (A2)