Frequency-domain ray series for viscoelastic waves with a non-symmetric stiffness matrix

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1 Frequency-domain ray series for viscoelastic waves with a non-symmetric stiffness matrix Ludě Klimeš Department of Geophysics, Faculty of Mathematics Physics, Charles University, Ke Karlovu 3, Praha 2, Czech Republic ( Received: September 8, 2017; Revised: November 29, 2017; Accepted: December 14, 2017 ABSTRACT In an elastic medium, it was proved that the stiffness tensor is symmetric with respect to the exchange of the first pair of indices the second pair of indices, but the proof does not apply to a viscoelastic medium. In order to indicate which phenomena could be observed in the wave field if the stiffness matrix were non symmetric, we propose the frequency domain ray series for viscoelastic waves with a non symmetric stiffness tensor in this paper. K e y wo r d s: viscoelastic media, stiffness tensor, wave propagation, ray theory, ray series, travel time, amplitude 1. INTRODUCTION The complex valued frequency domain stiffness tensor (elastic tensor, tensor of elastic moduli) c ijl = c ijl (x m, ω), projecting the strain tensor onto the stress tensor, depends on spatial coordinates x m circular frequency ω. It is symmetric with respect to the first pair of indices with respect to the second pair of indices c ijl = c jil (1) c ijl = c ijl. (2) It is thus frequently expressed in the form of the 6 6 stiffness matrix whose lines correspond to the first pair of indices columns to the second pair of indices according to the Voigt notation. In an elastic medium, it was proved that the stiffness tensor is symmetric with respect to the exchange of the first pair of indices the second pair of indices, c ijl = c lij. (3) The 6 6 stiffness matrix is thus symmetric in an elastic medium. The ray theory for a 3 D smoothly heterogeneous anisotropic elastic medium with a symmetric Stud. Geophys. Geod., 62 (2018), , DOI: /s c 2018 The Authors 261

2 L. Klimeš stiffness matrix is well developed (Babich, 1961; Červený, 1972; 2001; Chapman, 2004; Červený et al., 2007). However, the above mentioned proof does not apply to a viscoelastic medium, we do not now whether symmetry (3) holds in a viscoelastic medium. It is then reasonable to indicate which phenomena could be observed in the wave field if the stiffness matrix were non symmetric. Analogously to Thomson (1997), we thus do not assume symmetry (3) in this paper, propose the frequency domain ray series for viscoelastic waves with a non symmetric stiffness matrix, c ijl c lij. (4) The lower case Roman indices tae values 1, 2 3. The Einstein summation over repetitive lower case Roman indices is used throughout the paper. 2. STANDARD RAY SERIES In the frequency domain, the viscoelastodynamic equation for complex valued displacement u i = u i (x m, ω) reads (c ijl u l, ),j (iω) 2 u i = 0, (5) where lower case Roman subscript, following a comma denotes the partial derivative with respect to the corresponding spatial coordinate x. Here c ijl = c ijl (x m, ω) is the complex valued frequency domain stiffness tensor, = (x m ) is the density ω is the circular frequency. We express the displacement in terms of its complex valued vectorial amplitude U i = U i (x m, ω) complex valued travel time τ = τ(x m ) as u i = U i exp(iωτ). (6) We exp the frequency dependent amplitude into high frequency asymptotic series U i = (iω) n U [n] i, (7) where U [n] i n=0 = U [n] i (x m, ω) is the n th order vectorial amplitude. We insert displacement (6) into viscoelastodynamic equation (5) obtain equation (iω) 2 N i (U m, τ,u ) + iωm i (U m, τ,u ) + L i (U m ) = 0, (8) where the differential operators are defined as Here the Christoffel matrix, defined as N i (U m, τ,u ) = [Γ il (x m, τ,u )U l U i ], (9) M i (U m, τ,u ) = (c ijl τ, U l ),j + c ijl τ,j U l,, (10) L i (U m ) = (c ijl U l, ),j. (11) Γ il (x m, p u ) = a ijl (x m ) p j p, (12) 262 Stud. Geophys. Geod., 62 (2018)

3 Frequency-domain ray series for viscoelastic waves with a non-symmetric stiffness matrix is a function of six phase space coordinates x m, p u formed by three spatial coordinates x m three slowness vector components p u which represent the spatial gradient of travel time τ, p u = τ,u. In definition (12), a ijl (x m ) = c ijl (x m ) [ (x u ) ] 1 is the density reduced stiffness tensor. In order to obtain the stard ray series, we insert series (7) into viscoelastodynamic equation (8) sort the terms according to the order of iω, analogously to Babich (1961) or Červený (1972; 2001, Sec. 5.7). We then obtain the system of equations N i( U [n], τ,l ) + M i ( U [n 1], τ,l ) + L i ( U [n 2] (13) ) = 0 (14) for each order n = 0, 1, 2,... Here U [ 1] = 0 U [ 2] = 0, i.e. operator M i is missing in this equation for n = 0 operator L i is missing in this equation for n = 0, 1. Note that the system of equations (14) does not follow necessarily from viscoelastodynamic equation (8). The above sorting according to the order of iω is our assumption leading to the stard ray series. If we wish to obtain another ind of ray series, e.g., the coupling ray series, we may choose a solution of viscoelastodynamic equation (8) different from the system of equations (14), see Klimeš (2013). 3. CHRISTOFFEL EQUATION AND EIKONAL EQUATION Equation (14) for n = 0 constitutes the matrix Christoffel equation Γ il (x m, τ,u )U [0] l U [0] i = 0. (15) The three eigenvalues of Christoffel matrix (12) correspond to three waves: the P wave two S waves. Unlie as in the elastic case, the Christoffel matrix is not symmetric. We select one of three eigenvalues of Christoffel matrix (12) denote it as G = G(x m, τ,u ). We denote the corresponding right h eigenvector of the Christoffel matrix as g i = g i (x m, τ,u ), Γ il g l = Gg i, (16) the corresponding left h eigenvector of the Christoffel matrix as g i = g i (x m, τ,u ), g i Γ il = g l G. (17) The three right h eigenvectors of the Christoffel matrix the three left h eigenvectors of the Christoffel matrix are mutually biorthogonal. The lengths of the three right h eigenvectors of the Christoffel matrix are not determined, but we choose the lengths of the corresponding left h eigenvectors so that the three right h eigenvectors the three left h eigenvectors are mutually biorthonormal, g g = 1. (18) Stud. Geophys. Geod., 62 (2018) 263

4 L. Klimeš The zero order vectorial amplitude then reads U [0] i = U [0] g i, (19) where the zero order ray theory amplitude U [0] will be determined by the transport equation in Section 6. Whereas the two S waves, which propagate with different velocities, are linearly polarized in elastic media, they may be elliptically or even circularly polarized in viscoelastic media. Whereas the two elliptically polarized S waves always display equal hedness for a symmetric stiffness matrix, they display opposite hedness for a sufficiently non symmetric stiffness matrix. In order to satisfy Christoffel equation (15), selected eigenvalue G must be unit, G(x m, τ,u ) = 1. (20) Nonlinear first order partial differential equation (20) for travel time τ is called the Hamilton Jacobi equation. In wave propagation problems, it is also often referred to as the eional equation. The methods for solving the Hamilton Jacobi equation are already mostly developed (Hamilton, 1837; Červený, 1972; Klimeš, 2002; 2010; 2016). Hamilton Jacobi equation (20) generates the equations of rays (Hamilton equations, equations of geodesics) the related equations such as the Hamiltonian equations of geodesic deviation (dynamic ray tracing equations). The equations of rays may be expressed in various forms, e.g., as dx i dγ = 1 G, (21) 2 p i dp i dγ = 1 G 2 x i, (22) where parameter γ along rays coincides with the values of travel time τ. Refer to Section 4 for the case of complex valued eigenvalue G = G(x m, p u ) parameter γ. G Hereinafter, x G i p i denote the partial derivatives of function G(x m, p u ) of six phase space coordinates x m, p u. Using this notation, the partial derivatives of any function G ( x m, τ,u (x a ) ) of three spatial coordinates read Since G,j = G x j + G p τ,j. (23) G(x m, p u ) = g r (x m, p u )Γ rs (x a, p b )g s (x c, p d ), (24) the first order phase space derivatives of eigenvalue G(x m, p u ) can be expressed as G p i (x m, p u ) = g r (x m, p u ) Γrs p i (x a, p b )g s (x c, p d ) (25) G x i (xm, p u ) = g r (x m, p u ) Γrs x i (xa, p b )g s (x c, p d ). (26) 264 Stud. Geophys. Geod., 62 (2018)

5 Frequency-domain ray series for viscoelastic waves with a non-symmetric stiffness matrix Note that the phase space derivatives of the left h right h eigenvectors in the above derivatives (25) (26) of relation (24) vanish because of definitions (16) (17) normalization condition (18). We insert definition (12) into phase space derivatives (25) (26), obtain expressions Note that G p i (x m, p u ) = g r (x m, p u ) [ a ris (x a ) + a ris (x a ) ] p g s (x c, p d ) (27) G x i(xm, p u ) = g r (x m, p u ) arjs x i (x a )p j p g s (x c, p d ). (28) V i (x m, p u ) = 1 2 g r(x m, p u ) [ a ris (x a ) + a ris (x a ) ] p g s (x c, p d ) (29) is the ray velocity vector, which is sometimes also referred to as the group velocity vector, e.g., by Červený (2001). 4. COMPLEX VALUED AND REAL VALUED RAYS The above relations (20) (29) are applicable both to real valued travel time τ complex valued travel time τ. The methods for their solution are obvious if eigenvalue G = G(x m, p u ) in Hamilton Jacobi equation (20) is a real valued function of real valued slowness vector p u. However, eigenvalue G = G(x m, p u ) in Hamilton Jacobi equation (20) may be a complex valued function may generate complex valued rays. Since we usually do not now the values of the frequency domain stiffness tensor c ijl = c ijl (x m, ω) at imaginary coordinates x m, we can instead trace the reference real valued rays which are close to the complex valued rays defined by the equations (21) (22) of rays, to calculate travel time τ = τ(x m ) by means of the perturbation expansion by Klimeš Klimeš (2011) along the reference real valued rays. In this case, the slowness vector p u = τ,u in definition (29) is also approximated by the perturbation expansion. 5. PRINCIPAL AND ADDITIONAL AMPLITUDE COMPONENTS Analogously to the zero order vectorial amplitude (19), we define the amplitude components with respect to the three right h eigenvectors of Christoffel matrix (12): eigenvector g i corresponding to the selected eigenvalue G other two eigenvectors gi corresponding to other two eigenvalues G. We decompose each vectorial amplitude coefficient U [n] i in series (7) into the principal amplitude component U [n] two additional amplitude components U [n], U [n] i = U [n] g i + U [n] g i, (30) Stud. Geophys. Geod., 62 (2018) 265

6 L. Klimeš where the summation is performed over two superscripts. Considering expression (19), we assume that both U [0] = 0. Definition (9) with decomposition (30) yields N i( U m [n] ), τ,u = U [n] ( G 1 ) gi. (31) We multiply viscoelastodynamic equation (14) by left h eigenvector g i Christoffel matrix, consider relation (31), obtain expression of the U [n] = 1[ g i Mi( U [n 1] ), τ,u + g i L i( U [n 2] ) ] ( G 1 ) 1 (32) for the additional amplitude components in terms of lower order amplitudes. Expression (32) differs from the analogous expression for the symmetric stiffness matrix (Červený, 2001, Eq ) just by left h eigenvectors g i. 6. TRANSPORT EQUATION We multiply viscoelastodynamic equation (14) by left h eigenvector g i of the Christoffel matrix, consider relation (31), obtain transport equation g i M i( U [n], τ ),u + gi L i( U [n 1] ) = 0 (33) for the principal amplitude components. We separate the terms with higher order principal amplitude components from the terms containing higher order additional amplitude components lower order amplitude components, g i M i( U [n] ) g, τ,l = g i M i( U [n] g ), τ,l gi L i( U [n 1] ). (34) Relations (33) (34) differ from the analogous relations for the symmetric stiffness matrix (Červený, 2001, Eqs ) just by left h eigenvector g i. We express the left h side of transport equation (34) as g i M i (U [n] g m, τ,l ) = 2 V j U [n],j + ( V j ),j U [n] 2 S U [n], (35) where ray velocity vector V j is given by definition (29), which can also be expressed as (x a )V i (x m, p u ) = 1 2 g r(x m, p u ) [ c ris (x a ) + c ris (x a ) ] p g s (x c, p d ). (36) The first two terms on the right h side of relation (35) are well nown from the ray series with a symmetric stiffness matrix (Červený, 2001, Eq ). Quantity S in the rightmost term of transport equation (35) can be determined using definition (10) as S = 1 { 1 ( [ g i c ijl + c ijl) ] } τ, g l 2 2 g ( i c ijl ) τ, g l,j,j g ic ijl τ, g l,j, (37) see (36). Considering (35), we express transport equation (34) as V j U [n],j ),j U [n] = S U [n] + Z [n 1], (38) 266 Stud. Geophys. Geod., 62 (2018)

7 Frequency-domain ray series for viscoelastic waves with a non-symmetric stiffness matrix where Z [n 1] = 1 [ 2 g i M i( U [n] g, τ ),u + gi L i( U [n 1] ) ]. (39) The solution of transport equation (38) for n = 0 reads ( τ ) U [0] = U [0] 0 ( 0 J 0 ) 1 2 ( J) 1 2 exp dγ S τ 0, (40) where subscript 0 denotes the initial conditions. Squared geometrical spreading ( ) x i J = det γ a (41) (Babich, 1961, Eq. 3.7; Červený, 2001, Eq ) represents the Jacobian of transformation from ray coordinates γ 1, γ 2, γ 3 to spatial coordinates x i. Here γ 1 γ 2 are the ray parameters, γ 3 = γ. Factor exp ( τ τ 0 dγ S ) in (40) is present due to the sew part of the stiffness matrix, see the next section. The solution of transport equation (38) for n>0 reads (Červený, 2001, Eq ) [ U [n] = U [0] U [n] τ ] 0 Z [n 1] + dγ U [0] U [0] 0 τ 0. (42) 7. NON RECIPROCITY DUE TO A NON SYMMETRIC STIFFNESS MATRIX The only difference of expressions (40) (42) for the principal amplitudes from the analogous expressions derived for a symmetric stiffness matrix is exponential term exp ( τ τ 0 dγ S ) in expression (40). We shall now derive various expressions for quantity S = S(x m ). We express definition (37) as S = 1 { 1 ( [ g i c ijl +c ijl) ] } τ, g l 2 2 ( g i c ijl ) τ, g l,j,j + g i,jc ijl τ, g l g i c ijl τ, g l,j. (43) After summation, we obtain expression S = 1 2 ] [ g i,j a ijl τ, g l g i a ijl τ, g l,j 1 ( [ g i a ijl a ijl) ] τ, g l 4,j, (44) where a ijl is the density reduced stiffness tensor given by definition (13). The last term on the right h side differs from 1 j ( V ) 2,j = 1 ( [ g i a ijl + a ijl) ] τ, g l (45) 4,j Stud. Geophys. Geod., 62 (2018) 267

8 L. Klimeš just by the subtraction. Note also that [ ( a ijl a ijl) τ, ] τ,j = 0. (46) We differentiate the product in the rightmost term of expression (44) obtain expression S = 1 4 g i,j( a ijl +a ijl) τ, g l 1 4 g ( i a ijl +a ijl) τ, g l,j 1 [ 4 g i (a ijl a ijl)] τ,g l.,j (47) We differentiate characteristic equation (16) for the right h eigenvector with respect to spatial coordinates, consider the biorthonormality of the left h right h eigenvectors, obtain relation g i,j = g i g Γ l,jg l ( G G ) 1 + gi g g,j (48) for the spatial gradient of the right h eigenvector. The rightmost term in relation (48) accounts for the undefined changes of the length of the right h eigenvector g i. We differentiate characteristic equation (17) for the left h eigenvector with respect to spatial coordinates, obtain analogous relation g i,j = g Γ l,jg l g i ( G G ) 1 + gi g g,j (49) for the spatial gradient of the left h eigenvector. The rightmost term in relation (49) accounts for the undefined changes of the length of the left h eigenvector g i, satisfies identity g g,j = g g,j (50) obtained by differentiating normalization condition (18). We insert the gradients (48) (49) of the eigenvectors of the Christoffel matrix into expression (47), consider identity (50) arrive at expression S = 1 4 ( g Γ l,jg l g r Γ rs g s g r Γ rs g s g Γ l,jg l ) ( G G ) g ( i c ijl c ijl),j τ,g l g g,j V j, (51) where ray velocity vector V j is given by definition (29). We now express the spatial derivatives Γ l,j of the Christoffel matrix in terms of its phase space derivatives as Γ l,j = Γl x j + Γl p s τ,sj. (52) The partial derivatives of the Christoffel matrix (12) with respect to phase space coordinates x m p u read Γ l x j (xm, τ,u ) = a rsl,j τ,r τ,r (53) 268 Stud. Geophys. Geod., 62 (2018)

9 Frequency-domain ray series for viscoelastic waves with a non-symmetric stiffness matrix Γ l (x m, τ,u ) = ( a jrl + a rjl) τ,r. (54) Quantity (51) with identity (52) finally reads S = 1 Γ ( g l 4 x j g l g r Γ rs Γ l g s g gl g r 1 4 g ( i c ijl c ijl),j τ dg i,g l g i dγ The last term g i dg i dγ Γ rs ) ( x j g s G G ) 1. (55) in expression (55) represents just the correction of principal amplitude U [n] in decomposition (30) due to the undefined length of right h eigenvector g i, vanish if we put g i dg i dγ = 0 (56) along each ray. Each element of matrix (c ijl c ijl ),j τ, in the last but one term of relation (55) represents the divergence of a vector tangent to the wavefront, see identity (46). Note also that g Γ l g l τ,j = 0 (57) Vectors g Γ l g l τ,j = 0. (58) Γ l g gl (59) g Γ l g l (60) thus represent two sets of the contravariant basis vectors of the ray centred coordinate system. Expression (55) for quantity S may be singular at slowness surface singularities, but is regular at spatial caustics. Quantity S vanishes for a symmetric stiffness matrix. For a non symmetric stiffness matrix, quantity S vanishes in a homogeneous medium. Quantity S which expresses the difference between the ray series for symmetric non symmetric stiffness matrices is thus generated by a combination of a non symmetric stiffness matrix heterogeneities, is generally non zero in a heterogeneous medium with the non symmetric stiffness matrix. Expression (55) with identities (46), (57) (58) suggest that quantity S may mostly be influenced by the wavefront tangent component of the gradient of the sew part of the stiffness matrix. Stud. Geophys. Geod., 62 (2018) 269

10 L. Klimeš 8. CONCLUSIONS We have derived the anisotropic ray theory series for viscoelastic waves with a non-symmetric stiffness matrix. These ray series enable us to estimate which phenomena could be observed in the wave field if the stiffness matrix were non symmetric. Whereas the two S waves, which propagate with different velocities, are linearly polarized in elastic media, they may be elliptically or even circularly polarized in viscoelastic media. Whereas the two elliptically polarized S waves always display equal hedness for a symmetric stiffness matrix, they display opposite hedness for a sufficiently non symmetric stiffness matrix, similarly as electromagnetic waves in optically active media. The ray theory amplitudes corresponding to a non symmetric stiffness matrix are not reciprocal in the same way as the ray theory amplitudes corresponding to a symmetric stiffness matrix. This non reciprocity is expressed in terms of quantity (55) in the expression (40) for the zero order ray theory amplitude. Refer to Klimeš (2017, Eq. 18) for the sense in which the ray theory Green function corresponding to a non symmetric stiffness matrix is reciprocal. Acnowledgements: The suggestions by Einar Iversen, Edward S. Krebes two anonymous reviewers made it possible for me to improve the paper considerably. The research has been supported by the Grant Agency of the Czech Republic under contract S, by the members of the consortium Seismic Waves in Complex 3 D Structures (see ). References Babich V.M., Ray method of calculating the intensity of wavefronts in the case of a heterogeneous, anisotropic, elastic medium. In: Petrashen G.I. (Ed.), Problems of the Dynamic Theory of Propagation of Seismic Waves, Vol. 5. Leningrad Univ. Press, Leningrad, (in Russian, English translation: Geophys. J. Int., 118(1994), ). Červený V., Seismic rays ray intensities in inhomogeneous anisotropic media. Geophys. J. R. Astr. Soc., 29, Červený V., Seismic Ray Theory. Cambridge University Press, Cambridge, U.K. Červený V., Klimeš L. Pšenčí I., Seismic ray method: Recent developments. Adv. Geophys., 48, Chapman C.H., Fundamentals of Seismic Wave Propagation. Cambridge University Press, Cambridge, U.K. Hamilton W.R., Third supplement to an essay on the theory of systems of rays. Trans. Roy. Irish Acad., 17, Klimeš L., Second order higher order perturbations of travel time in isotropic anisotropic media. Stud. Geophys. Geod., 46, Stud. Geophys. Geod., 62 (2018)

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