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GEOPHYSICS, VOL. 73, NO. 4 �JULY-AUGUST 2008�; P. T63–T75, 6 FIGS.10.1190/1.2937173

uality factor Q in dissipative anisotropic media

lastislav Červený1 and Ivan Pšenčík2

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ABSTRACT

In an isotropic dissipative medium, the attenuation propertiesof rocks are usually specified by quality factor Q, a positive, di-mensionless, real-valued, scalar quantity, independent of the di-rection of wave propagation. We propose a similar, scalar, but di-rection-dependent quality Q-factor �also called Q� for time-har-monic, homogeneous or inhomogeneous plane waves propagat-ing in unbounded homogeneous dissipative anisotropic media.We define the Q-factor, as in isotropic viscoelastic media, as theratio of the time-averaged complete stored energy and the dissi-pated energy, per unit volume. A solution of an algebraic equa-tion of the sixth degree with complex-valued coefficients is nec-essary for the exact determination of Q. For weakly inhomoge-neous plane waves propagating in arbitrarily anisotropic, weakly
dissipative media, we simplify the exact expression for Q con-
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iderably using the perturbation method. The solution of thequation of the sixth degree is no longer required. We obtain aimple, explicit perturbation expression for the quality factor, de-oted as Q. We prove that the direction-dependent Q is related tohe attenuation coefficient � measured along a profile in the di-ection of the energy-velocity vector �ray direction�. The qualityactor Q does not depend on the inhomogeneity of the plane wavender consideration and thus is a convenient measure of the in-rinsic dissipative properties of rocks in the ray direction. In allther directions, the quality factor is influenced by the inhomoge-eity of the wave under consideration. We illustrate the peculiari-ies in the behavior of Q and its accuracy on a model of anisotrop-c, weakly dissipative sedimentary rock. Examples show inter-sting inner loops in polar diagrams of Q in regions of S-waveriplications.

INTRODUCTION

The dissipative properties of rocks are commonly specified by theuality factor Q �also called Q�. In isotropic dissipative media, Q is aositive, dimensionless, scalar quantity, independent of the direc-ion of wave propagation. A review of definitions, terminology andbservations in seismology and seismic exploration devoted to iso-ropic dissipative media can be found in Aki and Richards �1980�,ohnston and Toksöz �1981�, Cormier �1989�, and Carcione �2001,006, 2007�. The most common definition of the Q-factor of time-armonic plane P- and S-waves propagating in homogeneous, weak-y dissipative �low-loss� isotropic media is Q � �Re�M�/Im�M�,here M is the complex-valued P- or S-wave viscoelastic modulus.lternatively, the definition may take the form Q � �Re�vc

2�/m�vc

2�, where vc is the complex-valued P- or S-wave phase velocitye.g., Johnston and Toksöz, 1981; Moczo et al., 1987�. The signs inhe expressions for Q depend on the sign convention used for time-armonic waves. With the sign convention used in this paper �the ex-

Manuscript received by the Editor 21August 2007; revised manuscript rec1Charles University, Faculty of Mathematics and Physics, Department of G2Academy of Sciences of the Czech Republic, Institute of Geophysics, Pra2008 Society of Exploration Geophysicists.All rights reserved.

onential time factor exp��i�t��, Im�M� and Im�vc2� are negative;

hus, expressions for Q contain negative signs.We are interested in the direction-dependent Q-factor for aniso-

ropic dissipative media. We consider only those dissipation mecha-isms that can be described within the framework of linear vis-oelasticity. Classically, the Q-factor of time-harmonic plane wavesas been defined by the relation Q�1 � Wd/E �Buchen, 1971�,here Wd is the density of the time-averaged dissipated energy and E

s the density of the time-averaged complete stored energy, E � KU. Quantities K and U are densities of the time-averaged kinetic

nd strain energies, respectively. For perfectly elastic media, Uquals K; but for viscoelastic media, U generally differs from K.odified versions of the definition Q�1 � Wd/E sometimes have

een used in seismic literature. For example, Carcione �2001, p. 58–9� uses the definition Q�1 � Wd/2U. This definition is adopted byervený and Pšenčík �2006� and by Zhu and Tsvankin �2006, 2007�.ere, we consider systematically the classical definition Q�1

Wd/E of Buchen �1971�. The definition is applicable to homoge-

February 2008; published online 8 July 2008.ics, Prague, Czech Republic. E-mail: [email protected].

ech Republic. E-mail: [email protected].

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T64 Červený and Pšenčík

eous and inhomogeneous plane waves propagating in isotropic asell as anisotropic viscoelastic media.For time-harmonic plane waves propagating in homogeneous, an-

sotropic, viscoelastic media, the Q-factor has often been introducedn the same way as for isotropic media, using the definition Q �

Re�vc2�/Im�vc

2�, where vc is the complex-valued phase velocity ofhe plane wave under consideration. Individual approaches to deter-

ining the Q-factor, based on this definition, differ in the way ofomputing complex-valued phase velocities, in the specification ofhe complex-valued slowness vector, and in the treatment of inho-

ogeneous waves �e.g., Carcione, 1994, 2001, 2007; Krebes and Le,994; Chichinina et al., 2006�. However, the applicability of thisefinition of Q to anisotropic media is problematic, particularly fornhomogeneous plane waves. For example, Krebes and Le �1994�how that this definition of Q “gives a generally unreliable measuref anelasticity” for inhomogeneous plane waves propagating in an-sotropic viscoelastic media. This might, however, be the conse-uence of inappropriate parameterization of the slowness vectors ofnhomogeneous plane waves using the attenuation angle.

Inhomogeneous waves play an important role in the propagationf waves in viscoelastic isotropic or anisotropic media. The slow-ess vector p of waves propagating in viscoelastic media is alwaysomplex valued: p � P � iA, where P is the propagation vector, Ahe attenuation vector, and i the imaginary unit. For A parallel to P oror A � 0, the wave is homogeneous; otherwise, it is inhomoge-eous. The advantage of plane waves propagating in homogeneousiscoelastic media is that their inhomogeneity may be chosen arbi-rarily. This is not the case for waves generated by a point source in aiscoelastic medium. The inhomogeneity of these waves resultsrom the solution of the problem under consideration. In most cases,ncluding homogeneous anisotropic viscoelastic media, wavesropagating from a point source to a receiver are inhomogeneousČervený et al., 2008�. Waves generated by a point source are homo-eneous only in unbounded homogeneous isotropic viscoelastic me-ia. In general, the waves propagating in viscoelastic anisotropicedia are inhomogeneous, although their inhomogeneity may beeak. We show that the attenuation of the wave does not depend on

he strength of its inhomogeneity only for a certain privileged direc-ion, the direction of energy flux.

Inhomogeneous waves play an important role in many branchesf physics. Declercq et al. �2005� give an excellent review of ultra-onic inhomogeneous waves, their generation, and applications inondestructive testing. Applications of inhomogeneous waves ineismology and acoustics are studied by Borcherdt and Wennerberg1985�, Borcherdt et al. �1986�, and Hosten et al. �1987�.

In viscoelastic media, the stiffnesses cijkl, relating the stress andtrain tensors, are not real valued as in perfectly elastic media but areomplex valued. We call them viscoelastic moduli. The imaginaryarts of viscoelastic moduli are responsible for the dissipative prop-rties of waves, propagating in viscoelastic media. The sign conven-ion for the imaginary parts of the viscoelastic moduli must corre-pond to the sign convention used to describe the time-harmonicave under consideration — particularly to the sign in the exponen-

ial time factor exp��i�t�. We use the exponential time factorxp��i�t�, for which cijkl � cijkl

R � icijklI . To simplify certain rela-

ions, we use the density-normalized viscoelastic moduli aijkl

c /�:
ijkl
aijkl � aijklR � iaijkl

I . �1�

he density-normalized viscoelastic moduli aijkl are frequency de-endent in viscoelastic media, but we do not write aijkl�� becausee consider an arbitrary but fixed � �0. We assume that aijkl satisfy

he symmetry relations

aijkl � ajikl � aijlk � aklij . �2�

Because of the symmetry relations 2, the number of density-nor-alized viscoelastic moduli reduces from 81 to 21. They can be de-

cribed by the 6�6 symmetric matrix in Voigt notation, A � AR

iAI, with elements A��,� � 1,2, . . . ,6�. The elements can bexpressed as A�� A��

R � iA��I . We use bars in A, AR, and AI to dis-

inguish the matrix A from attenuation vector A. Hereinafter, we call

ijklI and A��

I the dissipation moduli, and we call the 6�6 matrix AI

he dissipation matrix. We do not use the so-called Q matrix with ele-ents Q�� A��

R /A��I , which is sometimes seen in the literature.

he direct application of dissipation matrix AI simplifies all expres-ions.

We assume that matrix AR is positive definite �Fedorov, 1968;uld, 1973; Helbig, 1994; Carcione, 2001�. Similarly, we assume AI

s positive definite or zero �Červený and Pšenčík, 2006, p. 1302�.hese assumptions are valid for both sign options of the exponential

ime factor. In the literature, both signs are used. Consequently, oneust be careful when comparing the signs in equations given by dif-

erent authors.There are few studies of the Q-factor of plane waves propagating

n homogeneous, viscoelastic anisotropic media. Among them arehe important contributions by Carcione �1992, 1994, 2000, 2001,007� and Carcione and Cavallini �1993, 1995a�. For plane SH-aves propagating in the plane of symmetry of an anisotropic medi-m, we refer to Krebes and Le �1994�. For plane P-waves propagat-ng in transversely isotropic �TI� and orthorhombic media, we refero Zhu and Tsvankin �2006, 2007�, who base their concept of attenu-tion exclusively on homogeneous plane waves. Their concept is notsed in this paper. The expression for the Q-factor for homogeneouslane SH-waves propagating in homogeneous media of unrestrictediscoelasticity and anisotropy is found in Carcione �2001, 2007�.he expression for Q, valid for homogeneous and inhomogeneouslane waves propagating in media of unrestricted anisotropy andiscoelasticity, is found in Červený and Pšenčík �2006�. Using Car-ione’s �2001� definition of Q �Q�1 � Wd/2U� and the so-calledixed specification of the slowness vector, Červený and Pšenčík

2006� derive an exact formula in which Q is expressed in terms ofhe viscoelastic moduli and complex-valued slowness and polariza-ion vectors of homogeneous or inhomogeneous waves. To obtainxact Q, one tone must solve an algebraic equation of the sixth de-ree with complex-valued coefficients. Červený and Pšenčík �2008�how that solving the algebraic equation of the sixth degree can bevoided if weakly inhomogeneous plane waves propagating ineakly dissipative anisotropic media are considered and the pertur-ation approach is applied. We denote the corresponding quality fac-or as Q and derive and discuss the relevant perturbation formula fort.

The first expression for the direction-dependent quality factor forenerally anisotropic, weakly dissipative media, mathematicallydentical with Q, is given by Gajewski and Pšenčík �1992�. To deriveˆ , Gajewski and Pšenčík �1992� use a quite different approach,ased on modifying the asymptotic ray-series method. The inhomo-

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Q-factor in dissipative anisotropic media T65

eneous waves are not explicitly considered in the derivation. Theame expression for Q is also derived by Červený �2001, equation.5.21�, using the first-order perturbation equation for traveltime.either of these publications provides answers to two importantuestions: �1� How is the expression derived for Q related to the def-nition Q�1 � Wd/E? �2� Is the result also valid for inhomogeneousaves when inhomogeneous waves were not considered in the deri-ation?

We answer these two questions in this paper. We show that the ex-ression for the quality factor, given by Gajewski and Pšenčík1992� and Červený �2001�, is consistent with the definition Q�1

Wd/E. In addition, it is valid not only for homogeneous but alsoor weakly inhomogeneous waves. From formulas given by Gajew-ki and Pšenčík �1992�, Q for anisotropic weakly dissipative mediaan also be expressed in terms of a complex-valued energy velocityray velocity� vr: Q�1 � � 2 Im�vr�/Re�vr�.

We investigate only the Q-factor of plane waves propagating innbounded homogeneous, dissipative anisotropic media. Vavryčuk2007a, 2007b� investigates the attenuation of waves generated by aoint source in a similar medium. He defines the so-called ray-quali-y factor by the relation Qray � �Re�vr

2�/Im�vr2�, where vr is the

omplex-valued energy velocity �Vavryčuk, 2007b, his equation4�. For weakly dissipative anisotropic media, this relation immedi-tely yields the expression Q�1 � �2 Im�vr�/Re�vr�, given by Ga-ewski and Pšenčík �1992�, valid locally even for heterogeneous me-ia.Auniform treatment of the Q-factor in weakly dissipative aniso-ropic media, valid for both plane waves and waves generated byoint sources, can be found in Červený et al. �2007�.

In this paper, we speak of velocity isotropy or anisotropy when theeferenced perfectly elastic medium, specified by the moduli aijkl

R , issotropic or anisotropic. We speak of Q-factor isotropy or anisotropyhen the Q-factor is or is not directionally dependent. For nonvan-

shing dissipation moduli aijklI , the velocity anisotropy generally im-

lies Q-factor anisotropy. Only for a special choice of dissipationatrix AI related to AR by the relation AI � AR/q, where q is a real-

alued positive constant, do we obtain strict Q-factor isotropy.Our paper begins with a brief review of the theory of time-har-onic, homogeneous and inhomogeneous plane waves propagating

n general viscoelastic anisotropic media. We use the mixed specifi-ation of slowness vector p �Červený and Pšenčík, 2005a�, which re-uces the problem of determining p to the solution of an algebraicquation of the sixth degree with complex-valued coefficients. Oncehe equation is solved, the exact expression for the Q-factor can bevaluated for all P, S1, and S2 homogeneous and inhomogeneouslane waves. For weakly inhomogeneous plane waves propagatingn weakly dissipative anisotropic media, we substitute the numericalolution of the algebraic equation by an approximate but simplerst-order perturbation formula. The expression for Q-factor thenimplifies considerably. We show that the perturbation formula forˆ , which follows from Buchen’s �1971� definition Q�1 � Wd/E, iselated explicitly to the direction of the energy flux and is indepen-ent of the inhomogeneity of the considered plane wave. It is in-ersely proportional to the intrinsic attenuation factor Ain; Q�1

Ain, where Ain is a scalar quantity, characterizing intrinsic dissi-ative properties of the medium.

Finally, we study the relation of Q to attenuation coefficient �; wehow that both are independent of the inhomogeneity of the planeave under consideration only if they are considered along the rayirection. We illustrate the behavior of Q�1 and Q�1 on numerical

xamples for a model of a viscoelastic anisotropic medium. Appen-ix A contains explicit formulas for Q for weakly dissipative mediaf higher-symmetry anisotropy, both for arbitrarily strong anisotro-y as well as for the weak-anisotropy approximation.

Throughout our paper, we use Cartesian coordinates xi and time t.he lower- case italic indices i, j, . . . take the values 1,2,3; the Greek

ndices �,� , . . . take the values 1,2,…, 6. The Einstein summationonvention over repeated indices is used. For time-harmonic waves,e consider the exponential time factor exp��i�t�, where � is axed, real-valued, positive, circular frequency. The dots above let-

ers denote partial derivatives with respect to time �ui � � 2ui/� t2�,nd the index following the comma in the subscript indicates the par-ial derivative with respect to the relevant Cartesian coordinateui,j � �ui/�xj�.

TIME-HARMONIC PLANE WAVES INVISCOELASTIC ANISOTROPIC MEDIA

Time-harmonic plane waves propagating in an unbounded homo-eneous viscoelastic anisotropic medium can be described by thexpression

ui�xj,t� � aUi exp��i��t � pnxn�� , �3�

here � is a circular frequency; a is the scalar amplitude; and ui, Ui,nd pi are Cartesian components of the complex-valued displace-ent vector u, the normalized polarization vector U, �U ·U � 1�,

nd the slowness vector p, respectively. The values U, p, and a arendependent of x and t. The vector u satisfies the equation of motion

ij,j � �ui, where � is density and � ij is the stress tensor, related tohe infinitesimal strain tensor eij � �ui,j � uj,i�/2 by the generalizedooke’s law � ij � cijklekl. Quantities cijkl are complex-valued vis-

oelastic moduli.It follows from the equation of motion that equation 3 represents a

lane wave if and only if U1, U2, and U3 satisfy the system of threeinear equations:

��ik � ik�Uk � 0, i � 1,2,3, �4�

here �ik is the generalized complex-valued Christoffel matrix, giv-n by the relation

�ik � aijklpjpl, �5�

nd where ik is the Kronecker symbol. System 4 can be solved fork only if

det��ik � ik� � 0. �6�

quation 6 represents a constraint relation for complex-valued p.We express p using the so-called mixed specification, proposed

y Červený �2004� �see also Červený and Pšenčík, 2005a�:

p � n � iDm, n · m � 0. �7�

he two real-valued, mutually orthogonal unit vectors n and m andhe real-valued scalar quantity D uniquely specify the plane wavender consideration and can be chosen arbitrarily. Vector n specifieshe direction of vector Re p, and m is perpendicular to n. Vectors nnd m specify the propagation-attenuation plane. The quantity D isalled the inhomogeneity parameter because it controls the inhomo-eneity of the plane wave. For D � 0, the plane wave is homoge-eous, for D�0 it is inhomogeneous, and for �D� small, it is weaklynhomogeneous.

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The complex-valued quantity in equation 7 can be determinedrom the constraint relation 6, which, after inserting equation 7,akes the form �Červený and Pšenčík, 2005a�

det�aijkl�nj � iDmj��nl � iDml� � ik� � 0. �8�

quation 8 is an algebraic equation of the sixth degree for withomplex-valued coefficients. It has six complex-valued roots corre-ponding to P, S1, and S2 plane waves, propagating in the directionsf n and �n. Quantity is a function of aijkl, n, m, and D.For certain simple cases, the algebraic equation of the sixth degree

an be factorized to two equations: one of the fourth degree and thether of the second degree. Quantity of a plane wave propagatingn an isotropic viscoelastic medium or of an SH-wave propagating inplane of symmetry of a monoclinic �orthorhombic, hexagonal� me-ium is the solution of the quadratic equation resulting from factor-zing equation 8. If we know , formula 7 for the slowness vectoran be expressed in a closed analytical form.

It is often useful to express p as

p � P � iA , �9�

here P and A are real-valued vectors, called the propagation and at-enuation vectors, respectively. Vector P is perpendicular to theavefront defined by equation Pixi � const. Both P and A are situat-

d in the propagation-attenuation plane formed by n and m. It fol-ows from equations 7 and 9 that P and A are given by the relations

P � n Re , A � n Im � Dm . �10�

he real-valued phase-velocity C in the direction of n and attenua-ion angle � �the angle between P and A� are then expressed as fol-ows:

C �1

�P��

1

�Re �, cos � �

� Im

��Im �2 � D2�1/2 ,

�11�

here � � Re /�Re � � �1. Once is found, we can determinefrom equation 4 and from the normalization condition U ·U � 1.We use D to specify the inhomogeneity of the wave and equation 7

o describe p. We call the inhomogeneous plane waves with smallD� �not with small � � weakly inhomogeneous plane waves. Čer-ený and Pšenčík �2005a, 2005b� show that attenuation angle � isot a suitable parameter for characterizing the inhomogeneity oflane waves, particularly in weakly dissipative media. One reason ishat in weakly dissipative media, with �aijkl

I � small, � is very unstable.t can take any value between 0° and � max, where � max depends onelocity anisotropy. It is usually close to 90° and may exceed it. Forxample, in a perfectly elastic isotropic medium, � is always 90°.owever, weakly inhomogeneous waves, including homogeneousaves �� 0o�, can propagate in a weakly dissipative isotropic me-ium �arbitrarily close to a perfectly elastic isotropic medium�.hese observations follow from equation 11, considering Im mall �weak dissipation�.

Another reason that � is not a suitable parameter for characteriz-ng the inhomogeneity of plane waves is that so-called forbidden di-ections can be generated. In these directions, the square of the phaseelocity C2 is negative �Krebes and Le, 1994; Carcione and Caval-ini, 1995b; Carcione, 2007�, which is, of course, physically unac-eptable. Červený and Pšenčík �2005a, 2005b� explain this phenom-non in detail.

EXACT EXPRESSIONS FOR Q-FACTOR

We use the classical definition of Q-factor, given by Buchen1971�:

Q�1 �Wd

E. �12�

f and U are known, we can compute the densities of the time-aver-ged energy-related quantities, e.g., the kinetic K, strain U, complete� K � U and dissipated Wd energies, the complex-valued Poyn-

ing vector F �also called the complex-valued energy flux�, and theeal-valued Poynting vector S � Re F �also called the energy flux�.

The most important expression is for F. Its Cartesian componentsi for a plane wave as defined in equation 3 are given by the relation

Červený and Pšenčík, 2006, their equation 24�

Fi � caijklplUj*Uk, �13�

here

c �1

2��2�a�exp�� 2�Anxn� . �14�

n equation 14, � is the density and a is the scalar amplitude. Conse-uently, c is a positive real-valued constant, and the time-averagedector S has the form

Si � c Re�aijklplUj*Uk� . �15�

In equation 12, we can use the relations

E � P · S, Wd � 2A · S , �16�

erived by Carcione and Cavallini �1993�; see also Červený andšenčík �2006�. In equations 16, P and A are given by equation 10nd S by equation 15. Inserting equations 16 into equation 12, we ob-ain the expression for Q:

Q�1 �2A · S

P · S. �17�

quation 17 for Q, with equation 10 for P and A and equation 15 for, is exact. It is valid for homogeneous and inhomogeneous planeaves propagating in isotropic or anisotropic media, either perfectly

lastic or viscoelastic. No approximation is used in its derivation,nd no restrictions are imposed on the anisotropy or viscoelasticityf the medium and on the inhomogeneity of the plane wave. It is ob-ious from equation 17 that Q does not depend on c, specified inquation 14.

Figure 1 illustrates the most important quantities related to propa-ating an inhomogeneous plane wave in a homogeneous viscoelas-ic anisotropic medium.

THE Q-FACTOR FOR WEAKLYINHOMOGENEOUS PLANE WAVES IN

ANISOTROPIC, WEAKLY DISSIPATIVE MEDIA

In this section, we use the first-order perturbation method to de-ive an approximate expression for the quality factor of Q of a weak-y inhomogeneous plane wave propagating in an anisotropic, weaklyissipative medium ��aijkl

I � small�. In practical problems of seismolo-y and seismic exploration, the dissipation is usually small and the

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nhomogeneity of the considered waves is also weak. In such a case,he numerical solution of the algebraic equation 8 of the sixth degreean be substituted by its approximate explicit solution, and the quali-y factor can be expressed in a simple analytical form.

As usual in perturbation methods, we use the word “weakly” onlys a qualitative description in the terms weakly inhomogeneouslane waves and weakly dissipative media. The accuracy of pertur-ation equations increases with decreasing inhomogeneity of thetudied plane waves and with decreasing dissipation of the medium.uantitative estimation of the error of the perturbation equationsould not be simple. The simplest estimation is to compare the re-

ults of exact and perturbation equations for several test situations,s we do in the section on numerical examples. This is one of the rea-ons for presenting exact equations.

The problem of weakly inhomogeneous plane waves propagatingn weakly dissipative media is treated by Červený and Pšenčík2008�, who use the first-order perturbation method to solve it. As aeference �unperturbed� case, a homogeneous plane wave with a re-l-valued slowness vector �D � 0� propagating in a perfectly elasticedium �aijkl

I � 0� is considered. The quantities corresponding tohis reference case are denoted below by upper indices 0. Therefore,

ijkl0 � aijkl

R .The solutions for the reference case are well known. For 0, P0,

nd A0, we have

0 �1

C0 , P0 �n

C0 , A0 � 0 . �18�

ymbol C0 denotes the phase velocity in the direction of n in the ref-rence medium. The polarization vector U0 of the reference homo-eneous wave can be determined from equation 4, specified for theerfectly elastic reference medium:

�� ik0 � ik�Uk

0 � 0, i � 1,2,3. �19�

s in equation 4, we consider U0 to be normalized, Ui0Ui

0 � 1. Theeal-valued energy-velocity vector U 0 of a homogeneous planeave propagating in the perfectly elastic reference medium is giveny the relation

Ui0 � aijkl

0 pl0Uj

0Uk0. �20�

ultiplying equation 19 by Ui0, we obtain the relation

ijkl0 pj

0pl0Ui

0Uk0 � 1 and the relations

Ui0pi

0 � 1, Ui0ni � C0. �21�

The inhomogeneity of the studied plane wave �specified by D�nd the imaginary parts of the viscoelastic moduli �aijkl

I � are smallerturbations. Unit vectors n and m remain fixed and are not per-urbed. Under these assumptions, Červený and Pšenčík �2008� de-ive the following approximate expression for quantity :

� �C0��1�1 � iAin

2� iD�U 0 · m�� . �22�

e call quantity Ain the intrinsic attenuation factor because it de-ends only on the intrinsic dissipation of the medium, not on D. It isiven by the relation

Ain � aI p0p0U 0U 0. �23�
ijkl j l i k
or the perturbed propagation vector P and attenuation vector A, weet from equation 10

P �n

C0 , A �nAin

2C0 � D�m � n�U 0 · m�

C0 � . �24�

hus, the perturbed P is the same as the unperturbed, i.e., P � P0.he perturbed A, however, differs from the unperturbed significant-

y. The vector A0 is zero �equation 18�, but A depends on dissipationoduli aijkl

I as well as on inhomogeneity parameter D.Now we take the exact relation 17 for Q�1 and use it to derive the

rst-order perturbation relation for Q�1. Because A0 � 0, we canubstitute S and P in equation 17 by their values S0 and P0 in the ref-rence medium. Equation 15 yields

Si0 � c Re�aijkl

0 pl0Uj

0Uk0� � cUi

0, �25�

here c is a real-valued positive constant given in equation 14 and0 is the energy-velocity vector in the reference medium �equation

0�. Multiplying equation 25 successively by A and P � P0, we ob-ain AjSj

0 � cAjU j0 and Pj

0Sj0 � cpj

0U j0 � c. We insert this into rela-

ion 17 and get the simple first-order perturbation formula:

Q�1 � 2A · U 0. �26�

he caret over Q emphasizes that Q is derived by the first-order per-urbation method and is valid for weakly inhomogeneous planeaves propagating in weakly dissipative media only.Inserting A from equation 24 into equation 26, we get the final ex-

ression for Q�1:

Q�1 � Ain � aijklI pj

0pl0Ui

0Uk0. �27�

he term depending on D in the expression for A in equation 24 van-shes when multiplied by U 0. Thus, the Q-factor in equation 27 does

igure 1. Diagram showing the most important quantities related tonhomogeneous plane-wave propagation in anisotropic dissipative

edia. Two real-valued, mutually orthogonal unit vectors n and mre used to define complex-valued slowness vector p, given by equa-ion 7. Vector n is perpendicular to the wavefront and specifies theirection of propagation-vector P. In turn, P and the attenuation vec-or A define propagation attenuation plane. Vectors P and A make anttenuation angle � . Projections of A into n and m are Im and thenhomogeneity parameter D, respectively.As indicated, the real-val-ed time-averaged Poynting vector S, given by equation 15, mayoint out of the propagation-attenuation plane.

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ot depend on D, regardless of whether inhomogeneous plane waveswith D�0� are considered.

We have shown thus far that for weakly inhomogeneous planeaves propagating in weakly dissipative anisotropic media, the ex-

ct Q-factor Q � E/Wd reduces to the approximate form Q given byquation 27. It does not depend on D of the plane wave under consid-ration; it depends only on the intrinsic dissipation parameters of theedium. In the next section, we show that we should associate Qith the direction of unit vector t � U 0/�U 0�. Then Q is inverselyroportional to attenuation coefficient ��t�, which does not dependn D. The plane wave under consideration can be inhomogeneous,ith attenuation vector A depending on D �see equation 24�.The equation for quality factor, mathematically identical to equa-

ion 27, is derived by Gajewski and Pšenčík �1992, their equation 7�.hey use modification of the asymptotic ray-series method, in which

hey assume that dissipation moduli aijklI are small quantities of the

rder of ��1 for �→ . The method is valid locally for high-fre-uency elementary waves propagating in smoothly heterogeneous,eakly dissipative, generally anisotropic media. The resulting di-

ection-dependent quality factor corresponds to the attenuationlong the reference ray direction. Consequently, it corresponds to thettenuation along energy-velocity vector U 0 �not along propagationector P0�. Although Gajewski and Pšenčík �1992� consider neitheruchen’s �1971� definition of Q nor inhomogeneity of waves, theirquation 7 is fully consistent with our equation 27. Thus, it is alsoonsistent with definition 12 of Buchen �1971�, and it is valid evenor weakly inhomogeneous waves. Also identical to equation 27 iservený’s �2001� equation 5.5.28, obtained using the first-order per-

urbation equation for traveltime.The intrinsic attenuation factor Ain is equivalent to Q�1. Conse-

uently, using classical terminology, it could also be called the lossactor. However, we prefer the term intrinsic attenuation factor tomphasize the independence of Ain on D and its strict definition byquation 23.

For some special cases, equation 27 for Q simplifies. For example,e can consider the 6�6 matrix A of viscoelastic moduli, given by

he relation

A � AR � iAI � AR�1 �i

q , �28�

here q is an arbitrary real-valued positive constant. This means wean express any aijkl

I as aijklI � aijkl

R /q. Equation 27 then yields

Q�1 �aijkl

R pj0pl

0Ui0Uk

0

q�

1

q. �29�

quation 29 is the consequence of relations 20 and 21. Gajewski andšenčík �1992� obtain a similar result; we too can conclude that un-er condition 28, Q is direction independent. We can thus speak of-factor isotropy.In anisotropic dissipative media, Q is generally direction depen-

ent if condition 28 is not satisfied; Q is then direction dependentven if dissipation moduli aijkl

I have an isotropic structure:

aijklI � ��I

� ij kl � ��I

�� ik jl � il jk� , �30�

here � �I and � �I are the imaginary parts of Lamé’s viscoelasticoduli. Inserting equation 30 into equation 27 yields

Q�1 ��I � �I

��pi

0Ui0��pj

0Uj0� �

�I

��pi

0pi0� . �31�

f the perfectly elastic reference medium is anisotropic, i.e., if weeal with velocity anisotropy, then Q is direction dependent �even ifhe matrix of dissipation moduli has an isotropic structure 30�. In thisase, we can speak of Q-factor anisotropy. For the dissipation modu-i given in equation 30, Q becomes approximately direction indepen-ent in weakly anisotropic media with weak anisotropy parametersf the same order as the weak dissipation parameters �see equations-7–A-9�.If the reference medium is isotropic, equation 31 yields the well-

nown results for isotropic viscoelastic media:

Q�1 ��I � 2�I��R � 2�R�

� �Im�vP

2 �Re�vP

2 �for P-waves

nd

Q�1 ��I

�R � �Im�vS

2�Re�vS

2�for S waves,

here vP and vS are the complex-valued P- and S-wave phase veloci-ies and where �R, �R and � �I, � �I are real and imaginary parts ofomplex-valued Lamé viscoelastic moduli. Moreover, n and t coin-ide in the reference isotropic medium. Thus, equation 27 is fullyonsistent with the most commonly used definition of the Q-factor insotropic dissipative media and represents its natural generalizationor anisotropic dissipative media.

The specifications of formula 27 for anisotropic media of higherymmetry and for weakly anisotropic media are given in Appen-ix A.

RELATION OF Q TO ATTENUATIONCOEFFICIENT �

Attenuation coefficient � is another important parameter charac-erizing dissipation. In this section, we study its relation to Q. Weiscuss � of an arbitrary time-harmonic homogeneous or inhomoge-eous plane wave defined in equation 3, specified by n, m, and Dlong an arbitrarily oriented straight-line profile in a homogeneous,iscoelastic, anisotropic medium. Using equation 9 in formula 3 forhe plane wave, we can alter the formula to the following form:

ui�xj,t� � aUi exp�� i��t � Pnxn��exp��Anxn� .

�32�

e now introduce the plane-wave dissipation filter D�� by the rela-ion

D�� exp��Anxn� . �33�

quation 32 can then be expressed as

ui�xj,t� � aUiD��exp��i��t � Pnxn�� . �34�

The physical meaning of the plane-wave dissipation filter is obvi-us. It describes the variation of amplitudes along an arbitrarily cho-en straight profile. We specify the profile under consideration by thenit real-valued vector l. We further introduce distance s � s0, mea-ured along profile l with an arbitrarily selected initial point s on the
0

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rofile so that s � s0 is positive in the direction of l. The plane-waveissipation filter 33 along profile l is then given by

D�� exp�� s � s0�A · l� � exp�� l��s � s0�� ,

�35�

here we introduce the quantity ��l�:

��l� � �A · l . �36�

e call ��l� the attenuation coefficient �or absorption coefficient� ofhe studied homogeneous or inhomogeneous plane wave along pro-le l.Attenuation coefficient ��l� is real valued, and its dimension islength��1. For certain directions of l, the attenuation coefficientay be negative and thus may represent the exponential growth of

mplitude �for example, if profile l is oriented against the directionf propagation�.

Inserting A from equation 10 into equation 36, we obtain a for-ally simple expression for ��l� along profile l:

��l� � ��n · l�Im � �m · l�D� . �37�

his expression for ��l� is exact and may be used for any homoge-eous or inhomogeneous, plane wave �P, S1, S2� propagating in anrbitrary, perfectly elastic or viscoelastic isotropic or anisotropicedium along any profile l. The plane wave is specified by n, m, and. To determine ��l� exactly using equation 37, we must solve alge-raic equation 8 of the sixth degree in . For viscoelastic isotropicedia, the algebraic equation of the sixth degree factorizes into qua-

ratic algebraic equations for plane P- and S-waves.For weakly inhomogeneous plane waves propagating in weakly

issipative media, we can simplify equation 37 by inserting the first-rder perturbation formula 24 for attenuation vector A:

��l� � � �

2C0�Ain�n · l� � 2D�C0�m · l�

� �n · l��U 0 · m�� . �38�

n equation 38, we use Ain given by equation 23. Expression 38 is ap-roximate, but otherwise it is quite general. We assume, of course,hat dissipation moduli aijkl

I and inhomogeneity parameter D aremall. The effects of aijkl

I and D are coupled in equation 38 even foreakly inhomogeneous waves propagating in weakly dissipativeedia.In seismological structural studies, in seismic exploration, or in

ondestructive testing of materials by ultrasonic techniques, re-earchers primarily are interested in intrinsic dissipation, caused ex-lusively by aijkl

I . The attenuation coefficients, related fully or partlyo D, are usually of no interest because they are not related to theroperties of the tested rocks only but also to the properties of theave used for testing. Therefore, a very important question arises:oes a profile l exist, along which attenuation coefficient ��l� doesot depend on D?

An exact answer requires studying equation 37 numerically be-ause Im in equation 37 also depends on D. However, for weaklynhomogeneous plane waves propagating in weakly dissipative an-sotropic media, we can use equation 38 and obtain the answer im-

ediately: attenuation coefficient ��l�, given by equation 38, doesot depend on D if

C0�m · l� � �n · l��U 0 · m� � 0. �39�

his equation is satisfied only if l is taken parallel to U 0:

l U 0. �40�

hus, � does not depend on D if l is parallel to the unperturbed ener-y-velocity vector U 0 �the ray direction in the reference medium�.ereinafter, we consider ��t� along profile t, where t is the real-val-ed unit vector along U 0, t � U 0/�U 0�. Consequently, for profile laken along t, attenuation coefficient ��t� is given by the simple rela-ion

��t� ��

�2QU 0�, �41�

here Q is the direction-dependent quality factor given by equation7.

Equation 41 shows that the attenuation coefficient measuredlong the direction of U 0 depends only on intrinsic attenuation pa-ameters aijkl

I and does not depend on D of the studied plane wave.ttenuation coefficient ��t� is inversely proportional to Q. We ob-

ain this result using the first-order perturbation method. Conse-uently, the result is approximate and valid only for weakly inhomo-eneous plane waves propagating in weakly dissipative media. Aimilar conclusion is drawn by Deschamps and Assouline �2000� inheir numerical study of plane waves in orthorhombic anisotropic

edia.Using equation 41, we can define Q in terms of ��t� along profile

� t:

Q�1 ��t�U 0

� f, �42�

here f is the frequency �in hertz�. This relation for Q is formally theame as the relation known for isotropic viscoelastic media, namely,

�1 � �V/� f , where V is the velocity of the relevant plane P- or-wave �Johnston and Toksöz, 1981, their equation 36�. In homoge-eous, isotropic viscoelastic media, �, V, and Q are direction inde-endent. In homogeneous, anisotropic viscoelastic media, ��t�, U 0,nd Q depend on the direction of the real-valued vector t. In this way,quation 42 represents a generalization of the relation Q�1 � �V/f for anisotropic viscoelastic media.To calculate Q corresponding to the direction of t using equation

7, one must know U 0 and the corresponding slowness vector p0,oth in the reference medium. The real-valued vectors p0 and U 0 areutually related �see equation 20�. We can proceed in two ways.The first possibility is to specify p0 and to calculate the corre-

ponding U 0 from equation 20. This straightforward procedure isxtremely simple. The S-wave singularities and the multivaluednessf S-wave surfaces cause no problems. It is typical for ray tracing ineterogeneous, anisotropic, weakly dissipative media �Gajewskind Pšenčík, 1992�. A certain disadvantage is that U 0 and thus t areot known in advance but must be computed. Consequently, we can-ot choose the profile of measurements in advance. The situationimplifies considerably for computations in any plane of symmetry.

The second possibility consists of directly computing Q for a giv-n t. In this case, p0 appearing in equation 27 is not known andhould be calculated from a given U 0. This procedure is consider-bly more complicated than the previous one. It is very sensitive to-wave singular regions and to the multivaluedness of the wave sur-aces when several values of p0 can be obtained for one given U 0.

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he procedure resembles the two-point ray-tracing procedure. Toolve the problem, p0 is sought iteratively so that it yields the given

0. Alternatively, the method based on the numerical solution of aystem of polynomial equations, proposed by Vavryčuk �2006�, cane used.

If we use profile l along energy-velocity vector U 0 in equations 35nd 41, the final expression for the dissipation filter in weakly dissi-ative anisotropic media is as follows:

D�� exp��s � s0�

2QU 0 � , �43�

here Q is given by equation 27 and U 0 � �U 0� is given by equation0. For weakly inhomogeneous plane waves, ��t�, Q, and dissipa-ion filter D�� do not depend on D of the wave. Equations 41 and 43lso remain valid for isotropic media, where U 0 and Q are directionndependent.

Equation 38 also can be used to investigate attenuation coeffi-ients ��l� in directions l other than the direction of energy-velocityector U 0. In all these directions, however, ��l� depends on the in-omogeneity of the considered plane wave. For example, considerhe case of the profile in the direction of vector n, perpendicular tohe wavefront in the reference medium and oriented in the directionf wave propagation. Equation 38 then yields

��n� � � �

2C0�Ain � 2D�U 0 · m�� . �44�

n anisotropic media, U 0 ·m is generally different from zero so thatttenuation coefficient ��n� is practically always influenced by D of

a) b)

c) d)

igure 2. Polar diagrams of �a, c� phase velocity C and �b, d� energyelocity U in a plane of symmetry of the medium defined in equa-ions 46 and 47 for D � 0 and 0.15 s/km: red is the fastest wave �P-ave�, black is the S-wave with intermediate phase velocity, andlue is the slower S-wave. The velocity diagrams of the SH-wave aregg shaped. The SH-wave is faster than the SV-wave in the direc-ions around the horizontal and, for D � 0.15 s/km, very close tohe vertical.

he considered wave. For certain n, attenuation coefficient ��n� mayven be negative �Červený and Pšenčík, 2005b�. Consequently, pro-le n �perpendicular to the wavefront in the reference medium� isnsuitable for measuring attenuation coefficient and Q-factor in in-omogeneous anisotropic media.

NUMERICAL EXAMPLES

In this section, we illustrate the behavior of Q in a model of sedi-entary rock and estimate the accuracy of the perturbation formula

or Q�1 � Ain by comparing its results with the exact results for�1. The model is a modification of the one used by Zhu and Ts-ankin �2006� and Vavryčuk �2007b�. The 6�6 matrix A of com-lex-valued, density-normalized viscoelastic moduli A�� in Voigtotation �measured in �km/s�2� is as follows:

A � AR � iAI, �45�

here

AR ��14.40 4.40 4.50 0 0 0

14.40 4.50 0 0 0

9.00 0 0 0

2.25 0 0

2.25 0

5.00

��46�

nd

AI ��0.48 0.15 0.30 0 0 0

0.48 0.30 0 0 0

0.45 0 0 0

0.15 0 0

0.15 0

0.66

� . �47�

oth matrices AR and AI are positive definite. Matrix AR corre-ponds to an anisotropic medium of hexagonal symmetry with a ver-ical axis of symmetry.

Although the formulas derived in previous sections can be useduite generally, for simplicity we concentrate on a symmetry planef the above medium. We can choose it so that it is situated in thex1,x3�-plane. In this case, the components of n �unit vector parallelo the propagation vector� and t �unit vector parallel to the energy-elocity vector� can be expressed as

n1 � sin iph, n2 � 0, n3 � cos iph,

t1 � sin ien, t2 � 0, t3 � cos ien. �48�

e call angle iph the phase angle and ien the ray angle.The following figures show polar diagrams parameterized by

hase and ray angles, with i � 0� upward and i � 90� to the right.he polar diagrams in Figures 2–5 are plotted in the form used byarcione �1992�.Figure 2 shows the polar diagrams of the exact phase velocity C

left column� and of the exact energy velocity U �right column� in thelane of symmetry of the medium defined in equations 46 and 47.

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he phase-velocity diagrams are parameterized by the phase angle;he energy-velocity diagrams are parameterized by the ray angle.ed denotes the fastest wave �P-wave�. Black corresponds to the-wave with higher phase velocity, and blue corresponds to thelower S-wave. We see that different S-waves �SH or SV� are fastern different directions. From the polarization diagrams, we can iden-ify the egg-shaped curves corresponding to the SH-waves.

The velocity diagrams are shown for two values of D: for D0 s/km �homogeneous wave� and for D � 0.15 s/km. We later

rove that D � 0.15 s/km corresponds to rather strongly inhomoge-eous waves. Because of strong S-wave velocity anisotropy, loopsre formed on the SV-wave energy-velocity diagrams, even for D

0. The size of the loops increases with increasing inhomogeneityf the wave. Figure 2 shows that the SV- and SH-wave phase-veloci-y curves intersect each other close to the vertical axis and close tohase angles iph � � /4, 3� /4, . . . . They also show the mutual at-raction of P- and SV-wave phase-velocity curves for increasing D.hese interesting phenomena are described by Červený and Pšenčík

D D

D D

D D

igure 5. Comparison of polar diagrams of Q�1�Q�1 � Ain�, calcu-ated from equation 27 �red� with exact polar diagrams of Q�1 calcu-ated from equation 17 �black� as a function of the ray angle. The val-es Q and Q are dimensionless. The P-wave �left� and S-wave dia-rams �right� are in a plane of symmetry of the medium defined inquations 46 and 47 for inhomogeneity parameters D � 0, 0.1, and.15 s/km. Neither Q�1 �red� nor � depends on inhomogeneity pa-ameter D; see equation 41.

) b)

igure 3. Polar diagrams of the inverse value of Q �dimensionless;ˆ �1 � Ain� calculated from equation 27 as a function of the ray an-le. The �a� P-wave and �b� S-wave diagrams are in a plane of sym-etry of the medium defined in equations 46 and 47. Note the inner

mooth loops on the curve corresponding to the SV-wave. The loopsre situated at the same ray angles as the triplications on the SV-wavenergy-velocity curves; see Figure 2. The loops correspond to theowest attenuation.

D D

D D

D D

igure 4. Comparison of polar diagrams of Q�1�Q�1 � Ain�, calcu-ated from equation 27 �red� with polar diagrams of exact Q�1 calcu-ated from equation 17 �black� as a function of the phase angle. Thealues Q and Q are dimensionless. The P-wave �left� and S-wave di-grams �right� are in a plane of symmetry of the medium defined inquations 46 and 47 for inhomogeneity parameters D � 0, 0.1 and.15 s/km. The value of Q�1 �red� does not depend on inhomogene-ty parameter D, but attenuation coefficient � does; see equation 44.

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2005b�. The analogous plots of the phase and energy velocities for aerfectly elastic reference medium AR from equation 46 are effec-ively identical with the plots in Figure 2.

In Figure 3, we show the polar diagrams of Q�1 calculated fromormula 27 as a function of the ray angle �specified by the energy-ve-ocity vector in the reference medium�. The simpler plot �a� corre-ponds to the P-wave; the more complicated plots �b� correspond tooth S-waves. The egg-shaped curve corresponds to the SH-wave,nd the curve with internal loops corresponds to the SV-wave. Be-ause of low P-wave attenuation anisotropy, the behavior of Q�1 forhe P-wave is, in comparison with the SV-wave, rather simple. Thetrongest attenuation of P-wave can be observed in directions closeo diagonals; in the SH-wave, in directions close to the horizontal; inhe SV-wave, in directions close to both the vertical and the horizon-al. The latter wave displays the strongest directivity of all threeaves.The smooth inner loops on the curve in the S-wave diagram, cor-

esponding to the SV-wave, are the most remarkable phenomenon inigure 3. The loops are situated at exactly the same ray angles athich we observe the outer loops on the SV-wave energy-velocity

urves in Figure 2.At these loops, the values of Q�1 attain minimumalues. Consequently, the attenuation of SV-waves is minimal in theegions of the loops. This important phenomenon, which we observen other models, certainly deserves more investigation.

The profile of measurement corresponds to the direction of ener-y-velocity vector in the reference medium l � t, so both Q and �re independent of D of the wave under consideration and are relatedy equation 41. Consequently, they both are suitable for measuringntrinsic dissipative properties of anisotropic media.

In Figure 4, we compare the polar diagrams of Q�1 � Ain �in red�ith the polar diagrams of exact Q�1 �in black�. The diagrams are

gain calculated in the plane of symmetry of the above medium. Theolar diagrams are displayed as functions of phase angle iph, i.e., asunctions of n parallel to the propagation vector. The left columnhows the plots corresponding to the P-wave; the right column, to the-waves. In both columns, plots are shown for inhomogeneity pa-ameters D � 0, 0.1, and 0.15 s/km. The factor Q �calculated usingquation 27� is independent of D, but the exact values of Q �calculat-d using equation 17� depend on D. Only for D � 0 s/km are Q and

igure 6. Particle motion polar diagrams in the plane of symmetry ofhe medium defined in equations 46 and 47 for D � 0 and.15 s/km. The colors are the same as in Figure 2. The SH-wave isolarized perpendicular to the plane of the plots. Note the effectiveinearity and strong ellipticity of the particle motion of the P-wavend SV-waves for D � 0 and 0.15 s/km, respectively. The elliptici-y is caused primarily by the inhomogeneity of the consideredaves. The value of D � 0.15 s/km thus corresponds to rather

the same. The difference between them increases with increasing. Because D � 0.15 s/km corresponds to a rather strong inhomo-eneity of the wave under consideration �see Figure 6�, we can con-lude that Q�1 is a good approximation of the exact Q�1 for weaklynhomogeneous waves. The red curves, corresponding to Q�1, arelotted over the black curves, so that the red dominates with an exactt. Consequently, it would be possible to use Q as a good approxima-

ion of Q in the direction of n, Q�n�. The problem, however, is that�n� along the profile in the direction of n is not intrinsic and de-ends on D �see equation 44�. Consequently, profile l � n is unsuit-ble for measuring the intrinsic dissipative properties of rocks in an-sotropic media.

In Figure 5, we show the same comparison as Figure 4, the differ-nce being the use of the ray angle ien in Figure 5 instead of the phasengle iph in Figure 4. Consequently, both Q and � are independent of

and are suitable for measuring the intrinsic dissipative propertiesf anisotropic media. The system of plots and the use of colors is theame as in Figure 4. There is a remarkable difference of behavior of-waves between Figures 4 and 5. The most remarkable phenome-on is the inner smooth loops on the SV-wave curves. Their positions the same for the exact and perturbation results. The width of thexactly calculated loops increases slightly with increasing inhomo-eneity of the wave, whereas the perturbation result does not dependn the inhomogeneity of the wave.

To estimate the strength of the inhomogeneity of the studiedaves, we present the particle motion diagrams for D � 0 �homoge-eous wave� and for D � 0.15 s/km in the plane of symmetry of thebove medium in Figure 6. The colors are the same as in Figure 2. Inigure 6, for D � 0 s/km, the polarization of all waves is effectively

inear. The SH-wave is polarized linearly in both plots, perpendicu-ar to the planes of the plots. In contrast to the homogeneous waves,he polarization of the inhomogeneous P- and SV-waves for D �.15 s/km is strongly elliptical, caused primarily by the inhomoge-eity of the waves. This indicates that D � 0.15 s/km correspondso the rather strong inhomogeneity of the wave. Good fit of Q and Qn Figure 5 thus implies that the applicability of perturbation formula7 is rather broad. It may yield sufficiently accurate results even foraves of relatively strong inhomogeneity.The above tests seem to indicate that equation 27 can be applied to

nhomogeneous waves of considerable strength, propagating in an-sotropic media of arbitrary symmetry and strength and of rathertrong Q-factor anisotropy.

CONCLUSIONS

We have derived the simple perturbation formula 27 for Q. It haseveral interesting and important properties.

First, expression 27 for Q is valid for homogeneous and weaklynhomogeneous time-harmonic P, S1, and S2 plane waves propagat-ng in homogeneous, isotropic, or generally anisotropic weakly dis-ipative media. The factor Q is a positive scalar quantity, both in iso-ropic and anisotropic dissipative media. It does not depend on inho-

ogeneity parameter D of the considered plane wave, although itsttenuation vector A might depend on D. Consequently, Q describeshe intrinsic dissipative properties of the medium.

In anisotropic media, expression 27 for Q is fully consistent withhe most common definitions of Q used in isotropic dissipative me-ia and represents their natural generalization. It is generally direc-ion dependent, the appropriate direction being specified by the di-
trongly inhomogeneous waves.

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ection of the energy-velocity vector �ray direction� and not by theirection of the propagation vector. This is because Q is inverselyroportional to attenuation coefficient ��t�, which does not dependn the inhomogeneity of the wave and corresponds to the profile par-llel to the energy-velocity vector. The factor Q is direction depen-ent even if dissipation matrix AI in equation 45 has a structure thatorresponds to an isotropic medium, i.e., Q-factor anisotropy is al-ays related to velocity anisotropy. The Q-factor isotropy can exist

trictly only in anisotropic media with property 28 or approximatelyn weakly anisotropic, weakly dissipative media.

Expression 27 for Q can be specified easily for an anisotropy ofigher symmetry �orthorhombic, hexagonal� as well as for weak an-sotropy. The computation of Q in the vicinity of S-wave singulari-ies or in the case of the multivaluedness of wave surfaces does notause any difficulties, as shown by the examples presented. The ex-mples display certain interesting phenomena, especially in the re-ions of triplication of the S-wave wavefront, not known from previ-us studies.

Expression 27 for Q remains locally valid even for high-frequen-y elementary waves generated by point sources and propagating ineterogeneous, isotropic, or anisotropic weakly dissipative media.

ACKNOWLEDGMENTS

The authors are grateful to José Carcione, Associate Editorengliang Gao, Jyoti Behura, and an anonymous referee for valu-

ble comments and recommendations. The research was supportedy the Seismic Waves in Complex 3-D Structures Consortiumroject; by research projects 205/05/2182, 205/07/0032, and 205/8/0332 of the GrantAgency of the Czech Republic; and by researchroject MSM 0021620860 of the Ministry of Education of the Czechepublic.

APPENDIX A

Q FOR HIGHER-ANISOTROPY SYMMETRIESAND WEAK ANISOTROPY

Equation 27 for Q�1 holds for all three waves — P, S1, and S2 —ropagating in weakly dissipative anisotropic media of arbitraryymmetry and strength. Equations for the individual waves can bebtained from equation 27 by specifying the appropriate polariza-ion vector U0 in it. Although it is expressed in terms of the unit vec-or n, perpendicular to the wavefront in the perfectly elastic refer-nce medium, Q in equation 27 corresponds to the direction of refer-nce ray t, t � U 0/�U 0�. Consequently, equation 27 must always beupplemented by equation 20, which provides the relation for deter-ining t from n. Equation 27 can be used without problems even in

he case of wavefront triplication.For models with higher material symmetry, it is more suitable to

xpress dissipation moduli aijklI in Voigt notation A��

I . Here, A��I are

lements of the 6�6 real-valued, symmetric, positive definite dissi-ation matrix AI. The material symmetry of AI may, in general, differrom the material symmetry of AR. For simplicity, we consider theaterial symmetry of AI to be equal to or higher than the symmetry

f AR.

Let us consider orthorhombic symmetry with its axes of symme-ry coinciding with the axes of the Cartesian coordinate system, inhich the x1- and x2-axes are horizontal and the x3-axis is vertical,ositive downward. The coordinate system is right-handed. In this

ase, dissipation matrix AI �similarly to matrix AR� is specified by

ine nonzero dissipation moduli: A11I , A22

I , A33I , A44

I , A55I , A66

I , A12I ,A13

I ,

nd A23I . We can then express equation 27 for Q�1 as follows:

Q�1 � Ain

� A11I �p1

0U10�2 � A22

I �p20U2

0�2 � A33I �p3

0U30�2

� A44I �p2

0U30 � p3

0U20�2 � A55

I �p10U3

0 � p30U1

0�2

� A66I �p1

0U20 � p2

0U10�2 � 2A12

I p10p2

0U10U2

0

� 2A13I p1

0p30U1

0U30 � 2A23

I p20p3

0U20U3

0. �A-1�

For transverse isotropy with a vertical axis of symmetry �VTI�,

e have A11I � A22

I , A44I � A55

I , A13I � A23

I , and A12I � A11

I � 2A66I .

or Q�1, equation A-1 then yields

Q�1 � Ain

� A11I �p1

0U10 � p2

0U20�2 � A33

I �p30U3

0�2 � A44I ��p2

0U30

� p30U2

0�2 � �p10U3

0 � p30U1

0�2� � A66I �p1

0U20

� p20U1

0�2 � 2A13I p3

0U30�p1

0U10 � p2

0U20� . �A-2�

quations A-1 and A-2 hold for velocity anisotropy of unlimitedtrength and for all three waves: P, S1, and S2. By using U0 of one of

he waves in equation A-1 or A-2, Q can be specified for that wave.Equation 27 simplifies considerably if weak velocity anisotropy

s considered. In this case, we can substitute the polarization vectorsy determinable unit vectors �Farra and Pšenčík, 2003�. For exam-le, let us consider a P-wave with Ui

0 ni, where ni is the unit vectorerpendicular to the wavefront in the reference medium. Equation7 is then given by the relation

QP�1 � AP

in � �CP0 ��2aijkl

I njnlnink. �A-3�

n equation A-3, CP0 is the first-order P-wave phase velocity in the di-

ection of vector n. For P-waves in a weakly orthorhombic medium,quation A-1 yields

QP�1 � AP

in

� �CP0 ��2�A11

I n14 � A22

I n24 � A33

I n34

� 2�A12I � 2A66

I �n12n2

2 � 2�A13I � 2A55

I �n12n3

2

� 2�A23I � 2A44

I �n22n3

2� . �A-4�

S

Wpr

a

wpwdocsF

dtt

E

It

i

daen

A

A

B

B

B

C

—

—

—

—

—

C

—

—

Č—

Č

Č

—

—

—

C

C

D

D

F

FG


imilarly, for P-waves in a weakly VTI medium, equation A-2 yields

QP�1 � AP

in

� �CP0 ��2�A11

I �1 � n32�2 � A33

I n34

� 2�A13I � 2A55

I �n32�1 � n3

2�2� . �A-5�

e can see that the velocity anisotropy, expressed by direction-de-endent CP

0 , always affects the Q-factor anisotropy expressed by di-ection-dependent QP.

We can express the square of first-order P-wave phase velocity CP0

s

�CP0 �2 � �2�1 � 2Y�ni�� , �A-6�

here Y�ni� stands for the linear combination of weak-anisotropyarameters, with coefficients containing powers of ni. By symbol �,e denote a formal parameter with the dimension of velocity, used toefine dimensionless weak-anisotropy parameters. We use a barver � to distinguish it from attenuation coefficient �. We canhoose the nonzero parameter � arbitrarily. For the explicit expres-ion for Y�ni� for P-waves and for more details, refer to Pšenčík andarra �2005, their equation 12�.

If the weak-anisotropy parameters are of the same order as theissipation parameters, we can simplify equations A-4 and A-5 fur-her. We substitute the phase velocity CP

0 by � and then express equa-ion A-4 as

QP�1 � AP

in

� �QPiso��1�1 � 2��x

In14 � �y

I n24 � �z

In34 � x

In12n3

2

� yI n2

2n32 � z

In12n2

2�� . �A-7�

quation A-5 becomes

QP�1 � �QP

iso��1�1 � 2��xI�1 � n3

2�2 � �zIn3

4

� xI�1 � n3

2�n32�� . �A-8�

n this case, the velocity anisotropy does not affect Q-factor aniso-ropy.

We define the coefficients in equations A-7 andA-8 in a way sim-lar to weak-anisotropy parameters �Pšenčík and Farra, 2005�:

� xI �

�A11I � �2�QP

iso��1�2�2 ,

� yI �

�A22I � �2�QP

iso��1�2�2 ,

� zI �

�A33I � �2�QP

iso��1�2�2 ,

xI �

�A13I � 2A55

I � �2�QPiso��1�

�2 ,

yI �

�A23I � 2A44

I � �2�QPiso��1�

�2 ,

zI �

�A12I � 2A66

I � �2�QPiso��1�

�2 . �A-9�

The value QPiso is a quality factor corresponding to an isotropic

issipative medium. The value QP reduces to QPiso if parameters A-9

re zero. By inserting equation A-9 into equation A-7 or A-8, we canasily prove that QP does not depend on QP

iso if parameters A-9 areonzero.

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