Point-source radiation in attenuative anisotropic media

Transcription

1 GEOPHYSICS, VOL. 79, NO. 5 (SEPTEMBER-OCTOBER 4); P. WB5 WB34, 4 FIGS., TABLE..9/GEO3-47. Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at Point-source radiation in attenuative anisotropic media Bharath Shear Ilya Tsvanin ABSTRACT Important insights into point-source radiation in attenuative anisotropic media can be gained by applying asymptotic methods. Analytic description of point-source radiation can help in the interpretation of the amplitude-variation-withoffset response physical-modeling data. We derive the asymptotic Green s function in homogeneous, attenuative, arbitrarily anisotropic media using the steepest-descent method. The saddle-point condition helps describe the behavior of the slowness group-velocity vectors of the far-field P-wave. Numerical results from the asymptotic analysis compare well with those obtained by the ray-perturbation method for P-waves in transversely isotropic media. INTRODUCTION Velocity attenuation anisotropy significantly influence the radiation pattern of seismic waves excited by a point source. A proper correction for the source directivity can help improve the robustness of amplitude-variation-with-offset (AVO) attenuation analysis. Point-source radiation in homogeneous anisotropic media has been mostly studied for nonattenuative materials using asymptotic numerical methods (e.g., Tsvanin Chesnoov, 99; Gajewsi 993; Wang Achenbach, 994; Červený, ). Zhu (6) presents an analytic numerical study of point-source radiation in D homogeneous, attenuative, transversely isotropic (TI) media. Vavryču (7) derives the asymptotic Green s function for homogeneous, attenuative media with arbitrary anisotropic symmetry by formally extending the results of Wang Achenbach (994) obtain for elastic media. In attenuative media, the Christoffel matrix becomes complexvalued because the stiffnesses acquire an imaginary part (Carcione, 7; Borcherdt, 9). Although many results derived for elastic models can be generalized for attenuative media, there are several important differences. In particular, the saddle-point condition involves complex-valued slowness group-velocity vectors, whose real imaginary parts can have different directions; hence, the properties of these vectors have to be clearly defined. Here, we present a rigorous derivation of the saddle-point condition the Green s function in attenuative anisotropic media. We start by reviewing the definitions of the attenuation coefficient, group velocity, other ey signatures in attenuative media. Then the integral expression for the Green s function in homogeneous attenuative anisotropic media is evaluated by the steepestdescent method. The saddle-point condition is used to study the influence of attenuation on the properties of the far-field P-wave. Finally, we compare the P-wave group-velocity, polarization, slowness vectors obtained from our asymptotic analysis for TI media with those found from the ray-perturbation theory (Červený Pšenčí, 9). BASIC DEFINITIONS In attenuative media, the density-normalized stiffness tensor ~a ijl is complex-valued (complex quantities are denoted here by the tilde sign on top): ~a ijl a R ijl iai ijl. () The components of the frequency-domain displacement vector ~u for plane waves propagating in attenuative media can be written as ~u i ~g i Ue iωðt ~p xþ ; () where ~g represents the polarization vector, U is the scalar amplitude, the slowness vector ~p p R þ i p I consists of the real-valued propagation (p R ) attenuation (p I ) vectors. The wave vector ~ is related to the slowness vector by ~ ω ~p. The orientations of p R p I can be different, the angle between p R p I is called the inhomogeneity (or attenuation) angle ξ (Červený Pšenčí, Manuscript received by the Editor November 3; revised manuscript received 3 March 4; published online 6 September 4. Formerly Colorado School of Mines, Center for Wave Phenomena, Golden, Colorado, USA; presently Indian Institute of Technology Bombay, Department of Earth Sciences, Mumbai, India. bshear@iitb.ac.in. Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, Golden, Colorado, USA. ilya@mines.edu. 4 Society of Exploration Geophysicists. All rights reserved. WB5

2 WB6 Shear Tsvanin Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at 5; Behura Tsvanin, 9; Tsvanin Grecha, ). Plane waves can satisfy the wave equation for a range of values of ξ, except for those corresponding to certain forbidden directions (Krebes Le, 994; Červený Pšenčí, 5; Carcione, 7). For waves excited by point sources, however, the inhomogeneity angle is determined by medium properties boundary conditions (Zhu, 6; Vavryču, 7). In reflection/transmission problems for plane waves, the inhomogeneity angle for reflected transmitted modes is constrained by Snell s law can reach values of up to 9 (Hearn Krebes, 99; Červený, 7; Behura Tsvanin, 9). Zhu Tsvanin (6) define the phase attenuation coefficient A ph as A ph ji j j R j : (3) The angle-dependent phase quality factor Q ph is related to A ph by Q ph : (4) Aph Following Carcione (7), Červený Pšenčí (8) derive the following expression for the group quality factor: Q gr pr F R p I F R ; (5) where F R denotes the real part of the Poynting vector. The real part of the complex-valued Poynting vector is parallel to the groupvelocity vector. The time-averaged energy flux is given by F R i κ Re½ ~a ijl ~p l ~g j ~g Š; (6) where the asteris denotes the complex conjugate, κ is a constant, the components of the complex-valued polarization vector ~g are normalized using the condition ~g ~g. Equation 5 involves a projection of the attenuation propagation vectors onto the Poynting vector, so Q gr quantifies the energy decay along the raypath. Definitions similar to that in equation 5 have been used to introduce the group attenuation coefficient (e.g., Behura Tsvanin, 9). For wealy attenuative media, the quality factor Q gr is independent of the inhomogeneity angle coincides with the quality factor corresponding to the phase attenuation coefficient for ξ (Q ph, see equation 4). This result was derived independently by Behura Tsvanin (9) Červený Pšenčí (8, 9). Behura Tsvanin (9) also show that the group attenuation coefficient remains practically independent of the inhomogeneity angle even for strong attenuation. ASYMPTOTIC GREEN S FUNCTION IN HOMOGENEOUS ATTENUATIVE ANISOTROPIC MEDIA Here, we derive the asymptotic Green s function in the frequency domain for a homogeneous, attenuative, arbitrarily anisotropic medium. The analysis is valid for all three wave modes (P, S, S ), but it breas down in the vicinity of S-wave singularities where the Christoffel equation has two equal or close eigenvalues. According to the causality principle, the stiffnesses in attenuative media should be, in general, frequency-dependent (Ai Richards, 98). Although we do not explicitly account for velocity dispersion, the following analysis is valid for a frequency-dependent stiffness tensor. The exact Green s function for each mode can be found as the solution of the viscoelastic wave equation (Appendix A,equationA-9): G n ðx;x ;ωþ iω Z ðπþ Z ~S n ½detð ~Γ IÞŠ p 3 p 3 ~p r 3 e ω R ~ϕ dp dp ; (7) where the x 3 -axis points in the source-receiver direction ~ϕ i ~p r 3 : (8) Here, R is the source-receiver distance, p j are the slowness components, ~Γ i ~a ijl p j p l is the Christoffel matrix, I is the identity matrix, ~S n are the cofactors of the matrix ~Γ I, ~p r 3 ~p 3ðp ;p Þ is the solution of the complex-valued Christoffel equation det½ ~a ijl p j p l δ i Š corresponding to the mode of interest. If we assume that ωr v (v is the average of the group velocity over all angles), G n (equation 7) can be evaluated by iterative application of the steepest-descent method (Bleistein, ). The saddle-point condition satisfied at ½ ~p s ; ~ps Š is The iterative steepest-descent method yields: ~ϕ ~ϕ : (9) p p G n ðx; x ; ωþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πr j det ~Φ j ½detð ~Γ IÞŠ p 3 exp iωr ~p r 3 i arg½det ~Φ Š ; () where ~Φ is the Hessian matrix of the second-order partial derivatives of ~ϕ with respect to p p ; all quantities are obtained at the saddle point. We now discuss the identification of the saddle point ½ ~p s ; ~ps Š. Equation 9 implies that ~p3 ðp ;p Þ () p ~p s ; ~ps ~p3 ðp ;p Þ p ~p s ; ~ps ~S n : () Each eigenvalue ~λ ðmþ of the Christoffel matrix ~Γ i ~a ijl p j p l is a function of the slowness vector (with ~p 3 ~p r 3 ) (see Appendix B, equation B-9): ~λ ðmþ λð ~a ijl ; ~p j Þ ~a ijl ~p j ~p l ~g ðmþ i ~g ðmþ ; (3)

3 Point-source radiation WB7 Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at where m taes values from to 3 ~g ðmþ denotes the corresponding unit polarization vector introduced in Appendix B. The largest eigenvalue ~λ ðþ is obtained for the slowness vector of the fastest mode (P-wave). The partial derivatives in equations can be calculated from the function λð ~a ijl ; ~p j Þ (equation 3) using the analytic implicit function theorem (Krantz Pars, ): ~p3 ðp ;p Þ λ p (4) p ~p s ; λ p ~ps 3 ~p s ; ~ps ~p3 ðp ;p Þ p ~p s ; ~ps λ p λ p 3 ~p s ; ~ps ; (5) where λ p j is given by equation B-; λ p 3 because the x 3 -axis points in the source-receiver direction. Next, we introduce the energy-velocity vector ~U (Vavryču, 7) as ~U j λ : (6) p j Equations 4 5 imply that at the saddle point, ~U ~U. (7) Hence, the real imaginary parts of ~U are parallel to the vector connecting the source receiver (i.e., in our case to the x 3 -axis). Note that Červený et al. (8) arrived at the same conclusion for wealy dissipative media using perturbation analysis. Equations 6 7 can be used to constrain the slowness vector ~p that corresponds to the plane wave that maes the most significant contribution to the wavefield at the receiver location. It is convenient to parametrize ~p in the following way (Červený Pšenčí, 5): ~p ~σ n þ idm; (8) where n is a unit vector in the phase direction, ~σ is a complex-valued quantity that corresponds to the projection of the slowness vector onto the vector n, m is chosen to be perpendicular to n (i.e., n m ), D is called the inhomogeneity parameter (Červený Pšenčí, 5). For plane waves, ~σ is found as one of the solutions of the Christoffel equation corresponding to a particular mode, whereas D can vary between zero infinity. The parameters ~σ D are determined by the saddle-point condition for point-source radiation. Therefore, we use the condition in equation 7 to set up the following constrained optimization problem for ~σ D: Minimize ~ U þ ~ U ; subject to ~ λ ðþ : (9) This problem is nonlinear, the values of ~σ D at the global minimum yield the slowness vector at the saddle point. Equation can be simplified once the saddle point has been found. From equations 6, 7, B-5, we have ~p 3 ~ U3 ~ λ ðþ : () The phase function at the saddle point (equation 8) can, therefore, be written as ~ϕ i ~p r 3 i ~ U3 : () In elastic media, equation 6 defines the components of the groupvelocity vector. In attenuative media, the real imaginary parts of ~U determine the traveltime energy dissipation, respectively. Using equations B-6 6, we have ~S n ½detð ~Γ IÞŠ p 3 ~S n Equation can then be rewritten as ~gðþ n n ~S ij ð ~Γ ij p 3 Þ ~gðþ. () ~U 3 exp i ωr i ~U 3 arg½det ~Φ Š ; G n ðx;x ;ωþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4πR ~U 3 jdet ~Φ j (3) where the components of the matrix ~Φ are computed at the saddle point using the implicit function theorem: ~Φ MN λ ð p M p N Þ ; (4) λ p 3 ~p s ; ~ps the indices M N tae values from to. The second-order partial derivatives of λð ~a ijl ; ~p j Þ can be evaluated using equation B-. Equation 3 was derived for a rotated coordinate frame with the x 3 -axis pointing in the source-receiver direction. The Green s function in a general (global) Cartesian coordinate frame taes the following form: ~gðþ n G n ðx;x ;ωþ q 4πRj ~Uj exp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jdet ~Φ j i ωr j ~Uj i arg½det ~Φ Š ; (5) where ~U is the energy-velocity vector in the source-receiver direction j ~Uj denotes its complex-valued magnitude (j ~Uj ~U 3 from equation 6. For TI media, equation 5 reduces to the expression for the Green s function derived by Zhu (6). Although the asymptotic analysis carried out above is similar to that presented by Vavryču (7),we proved (rather than assumed) that at the saddle point, the real imaginary parts of the energy-velocity vector are parallel to each other. RAY-PERTURBATION ANALYSIS FOR ANISOTROPIC ATTENUATIVE MEDIA Here, we briefly review the ray-tracing methodology of Červený Pšenčí (9), which is applicable in wealy attenuative, anisotropic, heterogeneous media with smooth spatial variations of the stiffness tensor. We provide expressions for the ray-theoretical Green s function analyze the orientation of the slowness vector in homogeneous attenuative models. These results will be compared to those obtained in the previous section. Following Klimeš (), Červený Pšenčí (9) treat the traveltime as a complex-valued quantity, with the real part

4 Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at WB8 contributing to the phase (arrival time) the imaginary part to the dissipation along the ray. The traveltime its spatial gradients are computed as perturbations of the corresponding real-valued quantities obtained along the ray traced in the reference elastic medium. In the density-normalized stiffness tensor given by equation, the real part a R ijl is treated here as corresponding to the reference elastic medium, the imaginary part a I ijl is the attenuation-related perturbation. We consider the linear perturbation Hamiltonian HðαÞ defined in Červený Pšenčí (9): with HðαÞ H þ α H; (6) H ~H H ; (7) where H ~H are the Hamiltonians corresponding to the (elastic) reference (viscoelastic) perturbed medium, respectively. The perturbation parameter is denoted by α; α corresponds to the reference elastic medium α to the perturbed attenuative medium. The reference Hamiltonian H can be expressed through the real-valued slowness (p ) polarization (g ) vectors computed for the reference model: H N ½aR ijl p j p l g i g ŠN ; (8) where N (integer) is the degree of the Hamiltonian. The perturbed Hamiltonian ~H is given by ~H N ½ ~a ijlp j p l ~g i ~g Š N ; (9) where the complex polarization vector ~g is computed from the complex Christoffel matrix ~Γ i ~a ijl p j p l using equation B-8. The traveltime its spatial gradients can be exped into a perturbation series in terms of the parameter α: τðαþ τ þ α τ þ (3) α τ τ τ þ α þ ; (3) x i x i x i α α where all terms correspond to the reference medium. The secondorder partial derivatives in equation 3 can be computed using dynamic ray tracing (Červený Pšenčí, 9). The first-order approximation for the traveltime the traveltime gradients in attenuative media can be obtained by substituting α retaining the first two terms of the expansion in equations 3 3. The real part of the traveltime contributes to the phase (i.e., the arrival time of the wave) the imaginary part is responsible for the amplitude decay along the ray. The real part of the traveltime gradient corresponds to p R the imaginary part to p I : Shear Tsvanin p R i τ Re x i p i þ Re τ x i α α (3) τ p I i Im τ Im : (33) x i x i α α The inhomogeneity angle can be computed from cos ξ pi p R jp I jjp R j : (34) An approximate ray-theoretical Green s function in homogeneous, wealy attenuative media can be found by substituting the complex-valued traveltime into the expression for the reference elastic model (Gajewsi Pšenčí, 99): G n ðx; x g ; ωþ g p n ffiffiffiffiffiffi exp iω Re½~τŠþi π 4π R juj jkj s expð ω Im½~τŠÞ; (35) where g is the polarization vector, R is the source-receiver distance, juj is the magnitude of group velocity, K is the Gaussian curvature of the slowness surface, s quantifies the phase shift due to K. Except for the complex traveltime ~τ, all quantities are computed for the reference elastic medium. The group quality factor responsible for energy decay along the ray corresponding to the linear perturbation Hamiltonian (equation 6) can be found from ðq gr Þ Im ~ H; (36) where ~H is defined in equation 9. Červený Pšenčí (9) also provide an approximation for the group-velocity components ~U i in attenuative media: ~U i i Q gr U i ; (37) where U i is computed in the reference elastic medium. Hence, in homogeneous, wealy attenuative media, ~τ R j ~Uj, equation 35 can be rewritten as G n ðx;x g ;ωþ g n pffiffiffiffiffiffiexp iω R 4π R juj jkj j ~Uj þ iπ s : (38) Note that the complex-valued group velocity obtained from the perturbation analysis appears only in the exponential function, whereas the polarization vector the Gaussian curvature K of the slowness surface, which control the magnitude of the Green s function, are computed for the reference elastic medium. NUMERICAL EXAMPLES In this section, the analytic results presented above are used to study the Green s function the behavior of the slowness vector

5 Point-source radiation WB9 Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at in homogeneous, attenuative VTI media. Table shows the parameters of the velocity attenuation functions for four VTI models similar to those in Zhu (6). First, we compare the P-wave group attenuation coefficients obtained from the asymptotic analysis discussed above perturbation theory. The asymptotic coefficient is computed from equation 5 with the slowness ~p the corresponding Poynting vector F obtained from the asymptotic analysis. Equation 36 is used to find the attenuation coefficient from perturbation analysis with the Hamiltonians of degrees N. The example in Figure shows that the group quality factor Q gr (equation 5) obtained from the perturbation theory is close to the asymptotic value; also, the quality factors for the two choices of N coincide. Although the attenuation-anisotropy coefficients ϵ Q δ Q for models 3 4 are equal to zero, there is a slight angular variation in Q gr for model 3; this is due to the combined effect of velocity anisotropy the difference between the values of Q P Q S (Zhu Tsvanin, 6). Substituting the slowness vector computed from the asymptotic analysis into the Christoffel matrix using equation B-8 yields the plane-wave polarization vector that corresponds to the saddlepoint condition. Similarly, substituting the slowness vector obtained by ray tracing in the reference medium into the Christoffel matrix yields the polarization vector in the perturbation analysis. We found that the asymptotic perturbation approaches produce the polarization vectors with close magnitudes. Figure displays the phase of the vertical component of the polarization vector ~g 3 (arg ~g 3 ) computed from the asymptotic perturbation methods for models 4. The function arg ~g 3 monotonically increases with the group angle ϕ for models 3, whereas it is negligible for model 4 because the attenuation function is isotropic. Note that the magnitude of ~g 3 for angles approaching 9 (near the isotropy plane) is small, which distorts the phase of ~g 3. For all models, the asymptotic perturbation methods yield similar values of arg ~g 3, with some deviations only for.5 large angles ϕ. Next, we analyze the component G 33 of the Green s function obtained from asymptotic.5 analysis (equation 3): 3 ~gðþ 3 G 33 ðx;x ;ωþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4πRj ~Uj jdet ~Φ j exp i ωr j ~Uj i arg½det ~Φ Š : (39) The same component produced by the perturbation approach has the form (equation 38): g ðþ 3 gðþ 3 G 33 ðx;x ;ωþ pffiffiffiffiffiffi 4π RjUj jkj exp iω R j ~Uj þ i π s :(4) The polarization vector, group velocity, Hessian of the slowness surface in equation 39 are complex-valued. The only complex-valued.5 quantity in equation 4 is the perturbed group-velocity vector ~U, which determines the attenuation along the ray. Our tests show that the magnitude argument of the energy-velocity vector ~U computed from the asymptotic analysis (equation 6) are close to those computed from perturbation analysis (equation 37). Also, the difference between the magnitudes of the Hessian of the slowness surface (det ~Φ ) in equation 39 of the Gaussian curvature K in equation 4 is small. Although the magnitudes of the complex-valued quantities (the polarization vector, group velocity, the Hessian of the slowness Table. Homogeneous TI models with anisotropic velocity attenuation functions. The parameters V P V S are the P- S-wave symmetry-direction velocities, Q P Q S are the P-wave S-wave symmetry-direction quality factors, ϵ Q δ Q are the attenuation-anisotropy parameters defined in Zhu Tsvanin (6) Tsvanin Grecha (). Model 3 4 V P (m s) V S (m s) ϵ δ Q P Q S 6 6 ϵ Q..45 δ Q..5 a) b) c) d) Figure. Comparison of the P-wave group quality factors (equation 5) as a function of the group angle ϕ with the vertical obtained from the asymptotic (blue circles) perturbation (red stars) analysis for (a) model, (b) model, (c) model 3, (d) model 4. The models are defined in Table.

6 WB3 Shear Tsvanin Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at surface) in equation 39 are close to their real-valued counterparts in equation 4, the phase of these quantities influence the phase of the Green s function. The attenuation-induced phase distortion ϕ d of the component G 33 in equation 39 can be expressed as ϕ d arg½ 3 Š arg½det ~Φ Š arg½j ~UjŠ: (4) The values of ϕ d for the models in Table range between 5 5, hence do not significantly distort the phase of the Green s Figure. Phase of the vertical component ~g 3 of the polarization vector computed from the asymptotic (blue circles) perturbation (red stars) analysis for (a) model, (b) model, (c) model 3, (d) model 4. Figure 3. Asymptotic (blue) perturbation (red) component G 33 of the Green s function convolved with a Ricer wavelet of pea frequency Hz for (a) model, (b) model, (c) model 3, (d) model 4. The source-receiver line maes an angle of 45 with the symmetry axis, the propagation time is s function (Figure 3). The other components of the Green s function exhibit properties similar to those of G 33. Figure 4 compares the inhomogeneity angle ξ computed from the perturbation approach (equation 34) the asymptotic analysis. The values of ξ obtained by the two methods are close to one another for models 3. There is a discrepancy for model, which can be expected because P-wave attenuation for that model is strongly anisotropic. Although both methods used here are approximate, the asymptotic analysis is expected to be more accurate for models with strong attenuation pronounced attenuation anisotropy. Although model 4 has substantial attenuation an anisotropic velocity function, the inhomogeneity angle for that a) b) c) d) a) b) c) d)

7 Point-source radiation WB3 Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at a) b) (degree) c) model vanishes because the components Q ij of the phase quality factor (defined in Zhu Tsvanin, 6) are identical (see Appendix C). CONCLUSIONS We presented a rigorous derivation of the asymptotic Green s function in homogeneous, attenuative, arbitrarily anisotropic media using the steepest-descent method. Application of the saddle-point condition helps identify the plane wave that maes the most significant contribution to the displacement field of each mode. Our results mae it possible to evaluate the inhomogeneity angle describe the complex-valued group-velocity vector in the highfrequency (far-field) approximation. P-wave signatures obtained from our asymptotic analysis for TI media were compared with the same quantities computed by rayperturbation theory. The asymptotic energy-velocity vector that describes the traveltime attenuation along the ray is close to the perturbed group-velocity vector. The inhomogeneity angles computed from the saddle-point condition perturbation theory differ only for strongly attenuative models. Although complex-valued quantities change the phase of the Green s function in attenuative media, the magnitude of the phase distortion is small even for models with strong attenuation anisotropy. ACKNOWLEDGMENTS We are grateful to the members of the A(nisotropy)-Team of the Center for Wave Phenomena (CWP), Colorado School of Mines, for fruitful discussions. The authors would lie to than I. Pšenčí for stimulating discussions his rigorous critique. C. Thomson (Schlumberger), P. Martin (CSM), M. Reynolds (CU Boulder) provided valuable help with asymptotic analysis linear algebra. We are grateful to the three anonymous reviewers for their constructive suggestions. This wor was supported by the Consortium 4 3 (degree) (degree) Figure 4. P-wave inhomogeneity angle computed from the asymptotic (blue circles) perturbation (red stars) analysis for (a) model, (b) model, (c) model 3. Project on Seismic Inverse Methods for Complex Structures at CWP. APPENDIX A EXACT GREEN S FUNCTION FOR ATTENUATIVE ANISO- TROPIC MEDIA The viscoelastic wave equation in the frequencywavenumber domain for a homogeneous, attenuative, anisotropic medium can be written as (Zhu, 6; Carcione, 7) ð ~a ijl j l ω δ i Þ ~U ð; ωþ ~f i ð; ωþ; (A-) where ω is the frequency, ~a ijl are the components of the density-normalized stiffness tensor, j are the wavenumbers, ~U is the displacement vector, ~fð; ωþ is the body force per unit volume (source term). Summation over repeated indices (changing from to 3) is implied. The frequency-domain displacement can be found as the triple Fourier integral: Z ~u ðx; ωþ ðπþ 3 where d d d d 3 ~U ð; ωþe i j x j d; (A-) B ~U ð; ωþ ~ i ~f i ð j ; ωþ : (A-3) det ~D The matrix ~D ( ~D i ~a ijl j l ω δ i ) with cofactors ~B i is closely related to the Christoffel matrix. The source in equation A- can be defined as a point impulsive force applied at location x parallel to the x n -axis: ~f i ð; ωþ δ in e i j x j : (A-4) The particle displacement from this source is the Green s function G n : G n ðx;x ;ωþ Z ~B i δ in ðπþ 3 det ~D ei jðx j x j Þ d: (A-5) Following Červený (), we rotate the coordinate frame to align the x 3 -axis with the source-receiver direction. Equation A-5 in the rotated coordinate frame now reads G n ðx; x ; ωþ Z ~B i δ in ðπþ 3 det ~D ei3r d; (A-6) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where R ðx x Þ þðx x Þ þðx 3 x 3 Þ is the sourcereceiver distance. The Bond transformation has to be applied to the stiffness tensor to account for the coordinate rotation. Note that

8 WB3 Shear Tsvanin Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at the components of the wave vector j also correspond to the rotated coordinate frame. For convenience, here we retain the symbols introduced in the previous equations, which were defined in the unrotated coordinates. The integral over 3 in equation A-6 can be extended into the complex plane by representing the vertical wavenumber as ~ 3 Re 3 þ i Im 3. The closed contour includes the real axis a semicircle with an infinitely large radius in the upper half-plane (because R>). The integral can then be evaluated by the residue theorem (i.e., by computing the residues at the poles), as described by Ai Richards (98), Tsvanin Chesnoov (99), Tsvanin (995). The poles correspond to the roots of the Christoffel equation for 3 : det ~D det½ ~a ijl j l ω δ i Š. (A-7) Equation A-7 is a sixth-order polynomial in 3 with complex coefficients that can have at most six distinct roots corresponding to the up- downgoing P-, S -, S -waves. For homogeneous (nondecaying) waves in unbounded nonattenuative media, the roots of 3 lie on the real 3 -axis. Then, the integral i nequationa-6 can be evaluated by introducing small attenuation, moving the roots to the complex plane, applying the residue theorem (Tsvanin, 995). Alternatively, the integral over 3 can be evaluated using Cauchy s principal value (Bleistein, 984). In the presence of attenuation, the roots of equation A-7 lie away from the real axis. The pole ~ r 3 ~ 3 ð ; Þ, which corresponds to a certain mode (e.g., P-waves) is located inside the integration contour, yields the residue for that mode. Hence, the integral over 3 in equation A-6 can be evaluated using the residue at the pole, the Green s function is expressed as the following double integral: G n ðx;x ;ωþ i Z ðπþ Z ~B n ðdet ~DÞ 3 3 ~ r 3 e i ~ r 3 R ^x 3 d d. (A-8) Substituting j ωp j, where p j denotes the components of the slowness vector, yields det ~D ω 6 detð ~Γ i δ i Þ, where ~Γ i ~a ijl p j p l. The cofactors of ~Γ I (I is the identity matrix) are denoted by ~S n ~B n ω 4 ~S n. Equation A-8 can then be written as G n ðx; x ; ωþ i ω Z ðπþ Z ~S n ½detð ~Γ IÞŠ p 3 p 3 ~p r 3 e i ω R ~pr 3 dp dp : (A-9) APPENDIX B PROPERTIES OF THE CHRISTOFFEL MATRIX IN ATTENUATIVE ANISOTROPIC MEDIA Here, we summarize the properties of the eigenvalues eigenvectors of the Christoffel matrix in attenuative anisotropic media. These properties are used in the asymptotic perturbation analyses presented in the main text. The results in this appendix are based on section 4.4 of Horn Johnson (99), which discusses complex symmetric matrices. The components of the Christoffel matrix ~Γ in attenuative media are given by (see the main text) ~Γ i ~a ijl ~p j ~p l ; (B-) where ~a ijl is the density-normalized complex stiffness tensor ~p is the complex-valued slowness vector. The eigenvector-eigenvalue problem for matrix ~Γ can be written as ~Γ ~V ~V ~ Λ; (B-) where ~ Λ is the diagonal matrix of the eigenvalues, the columns of ~V contain the corresponding eigenvectors. The matrix ~V is nonsingular (Horn Johnson, 99), so ~Γ ~V ~ Λ ~V (B-3) ~Λ ~V ~Γ ~V : (B-4) If the eigenvalues of the Christoffel matrix are distinct, the matrix ~V satisfies ~V T ~V ~D; (B-5) where ~D is a diagonal matrix. The Christoffel matrix can be diagonalized in the following way: with ~Γ ~ G ~ Λ ~ G T ; (B-6) ~Λ ~ G T ~Γ ~ G ; (B-7) ~G ~V ~D : (B-8) The matrix G ~ is complex orthonormal; i.e., G ~ T G ~ G ~ G ~ T I. In purely elastic media ~D I, consequently ~V T ~V ~V G. ~ The eigenvector matrix ~V then includes the polarization vectors of the three wave modes. Choosing the columns of G ~ as the polarization vectors helps extend expressions derived for elastic media to attenuative models. Next, we provide expressions for quantities related to the complex Christoffel matrix used throughout this paper. The results below are based on sections of Červený (), with the real-valued stiffness coefficients slowness vector replaced by the corresponding complex-valued quantities. We denote the eigenvalues (elements of Λ)by~λ ~ ðþ, ~λ ðþ, ~λ ð3þ, the corresponding polarization vectors (columns of G)by~g ~ ðþ, ~g ðþ, ~g ð3þ. Using equation B-7, the eigenvalue ~λ ðþ can be found as

9 Point-source radiation WB33 Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at Then, the derivatives ~λ ðþ ~p n form: λ ðþ i ~a ijl ~p j ~p l : (B-9) ~λ ðþ ~p n ~Γ i ~p n λ ~ ðþ ~p n ~p q tae the i λ ~ ðþ ~Γ i i þ ~Γ i ~p n ~p q ~p n ~p q ~p n ~p q ; (B-) i ~Γ i ~p n ð~a inq þ ~a iqn Þ ~p q ; ~Γ i ~p n ~p q ~a inq þ ~a iqn ; Γ in ~λ ðþ ~λ ðþ ~p q ~Γ þ in ~λ ðþ ~λ ð3þ ~p q ~g ðþ i n ~g ðþ From equations B-9 B-, it follows that ~p n λ ðþ ~p n λ ðþ : Finally, the product j can be expressed as j S ~ j Tr½ SŠ ~ ; ~p q ; (B-) (B-) (B-3) ~g ð3þ i n ~g ð3þ : (B-4) (B-5) (B-6) where ~S j represent the components of the cofactor matrix of ½ ~Γ ~λ ðþ IŠ, Tr½ SŠ ~ is the trace of the cofactor matrix. APPENDIX C PERTURBATION ANALYSIS OF THE INHOMOGENEITY ANGLE In this section, we present expressions for the propagation attenuation vectors (, hence, the inhomogeneity angle) in homogeneous, attenuative, anisotropic media using the method of perturbation Hamiltonians introduced by Červený Pšenčí (9). We also derive the conditions under which the inhomogeneity angle vanishes. The second-order partial derivative in equation 3 can be evaluated using quadratures along the reference rays, as shown by Klimeš () Červený Pšenčí (9): with the vector TðαÞ given by τ x i α ~T ðαþ½q ray i Š ; (C-) ~T K ðαþ ~T KðαÞþτ W ~ i P ray ik ; K ; ; (C-) ~W i ~H H ; (C-3) p i p i ~T 3 ðαþ H ~H: (C-4) The index K changes from to, the lowercase index from to 3. The real-valued matrices Q ray i (not to be confused with the quality-factor matrix) P ray i are computed using dynamic ray tracing in the reference elastic medium: Q ray i x i γ ; P ray i p i γ ; (C-5) γ denotes a certain ray parameter (e.g., the initial phase angle or the traveltime along the ray). In equation C-, the initial conditions ~T KðαÞ are set to zero for a point source; for plane-wave propagation, ~T KðαÞ may be chosen arbitrarily (Klimeš, ). We now derive the conditions under which the inhomogeneity angle vanishes in homogeneous media. Substituting equations 8 9 into equation C-3 yields ~W i ~a ijl p ~g j ~g l a R ijl p g j g l : (C-6) For wealy dissipative media, we can use the approximation ~g g reduce equation C-6 to ~W i ia I ijl p g j g l : (C-7) For the special case of identical components of the phase quality factor (i.e., a I ijl a R ijl Q), we have ~W i i ar ijl Q p g j g l i U i Q ; (C-8) where U i are the components of the group-velocity vector in the reference elastic medium. Substituting equation C-8 into equation C-, we obtain ~T Kα ðγ 3 Þ iðγ 3 γ 3 Þ U i Q Pray ik ; K ; (C-9) because the group-velocity vector is orthogonal to the first two columns of the matrix P ray U i P ik (Červený, ). Equations 3 33 for p R p I then tae the form: p R i p i þ Re½ ~T 3α Š p i (C-)

10 WB34 Shear Tsvanin Downloaded /7/4 to Redistribution subject to SEG license or copyright; see Terms of Use at p I i Im½ ~T 3α Š p i ; (C-) where p i ½Q ray 3i Š (Červený, ). From equations C- C-, it follows that p R p I are parallel to p. Hence, the inhomogeneity angle vanishes in the case of identical Q ij components, i.e., when the attenuation coefficients of all three modes are equal isotropic. Note that the velocity function may still be angledependent (anisotropic). Furthermore, the inhomogeneity angle also vanishes for isotropic velocity attenuation functions with different values of the quality factor for P- S-waves. This can be proved by considering the expression for the Hamiltonian in isotropic media. REFERENCES Ai, K., P. G. Richards, 98, Quantitative seismology: Theory methods, vol. : W.H. Freeman Company. Behura, J., I. Tsvanin, 9, Role of the inhomogeneity angle in anisotropic attenuation analysis: Geophysics, 74, no. 5, WB77 WB9, doi:.9/ Bleistein, N., 984, Mathematical methods for wave phenomena: Academic Press. Bleistein, N.,, Saddle point contribution for an n-fold complex-valued integral: Center for Wave Phenomena Research Report 74. Borcherdt, R. D., 9, Viscoelastic waves in layered media: Cambridge University Press. Carcione, J., 7, Wave fields in real media: Theory numerical simulation of wave propagation in anisotropic, anelastic, porous electromagnetic media: Elsevier. Červený, V.,, Seismic ray theory: Cambridge University Press. Červený, V., 7, Reflection/transmission laws for slowness vectors in viscoelastic anisotropic media: Studia Geophysica et Geodaetica, 5, 39 4, doi:.7/s Červený, V., L. Klimeš, I. Pšenčí, 8, Attenuation vector in heterogeneous, wealy dissipative, anisotropic media: Geophysical Journal International, 75, , doi:./j x x. Červený, V., I. Pšenčí, 5, Plane waves in viscoelastic anisotropic media I: Theory: Geophysical Journal International, 6, 97, doi:./j x x. Červený, V., I. Pšenčí, 8, Quality factor Q in dissipative anisotropic media: Geophysics, 73, no. 4, T63 T75, doi:.9/ Červený, V., I. Pšenčí, 9, Perturbation Hamiltonians in heterogeneous anisotropic wealy dissipative media: Geophysical Journal International, 78, , doi:./j x.9.48.x. Gajewsi, D., 993, Radiation from point sources in general anisotropic media: Geophysical Journal International, 3, 99 37, doi:./j x.993.tb889.x. Gajewsi, D., I. Pšenčí, 99, Vector wavefield for wealy attenuating anisotropic media by the ray method: Geophysics, 57, 7 38, doi:.9/ Hearn, D. J., E. S. Krebes, 99, On computing ray-synthetic seismograms for anelastic media using complex rays: Geophysics, 55, 4 43, doi:.9/ Horn, R. A., C. R. Johnson, 99, Matrix analysis: Cambridge University Press. Klimeš, L.,, Second-order higher-order perturbations of travel time in isotropic anisotropic media: Studia Geophysica et Geodaetica, 46, 3 48, doi:.3/a: Krantz, S. G., H. R. Pars,, The implicit function theorem: Springer. Krebes, E. S., L. H. T. Le, 994, Inhomogeneous plane waves cylindrical waves in anisotropic anelastic media: Journal of Geophysical Research, 99, , doi:.9/94jb6. Tsvanin, I., 995, Seismic wavefields in layered isotropic media (course notes): Samizdat Press. Tsvanin, I., E. M. Chesnoov, 99, Synthesis of body wave seismograms from point sources in anisotropic media: Journal of Geophysical Research, 95, 37 33, doi:.9/jb95ib7p37. Tsvanin, I., V. Grecha,, Seismology of azimuthally anisotropic media seismic fracture characterization: SEG. Vavryču, V., 7, Asymptotic Green s function in homogeneous anisotropic viscoelastic media: Proceedings of the Royal Astronomical Society, 463, , doi:.98/rspa Wang, C. Y., J. D. Achenbach, 994, Elastodynamic fundamental solutions for anisotropic solids: Geophysical Journal International, 8, , doi:./j x.994.tb397.x. Zhu, Y., 6, Seismic wave propagation in attenuative anisotropic media: Ph.D. thesis, Colorado School of Mines. Zhu, Y., I. Tsvanin, 6, Plane-wave propagation in attenuative transversely isotropic media: Geophysics, 7, no., T7 T3, doi:.9/.8779.