Anisotropic Parameters and P-wave Velocity for Orthorhombic Media

8/10/2019 Anisotropic Parameters and P-wave Velocity for Orthorhombic Media

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G EOP HYSI CS, VOL . 62, NO. 4 (JULY-AU G UST 1997); P. 12921309, 6 FI G S.

Anisotropic parameters and P-wavevelocity for orthorhombic media

Ilya Tsvankin

ABSTRACT

Although orthorhombic (or orthotropic) symmetry is

believed to be common for fractured reservoirs, the dif-

ficulties in dealing with nine independent elastic con-

stants have precluded this model from being used in seis-

mology. A notation introduced in this work is designed

to help make seismic inversion and processing for or-

thorhombic media mo re practical by simplifying the de-

scription o f a wide range of seismic signatures. Taking

advantage of the fact that the Christoffel equation has

the same form in the symmetry planes of orthorhombic

and transversely isotropic (TI) media, we can replace

the stiffness coeffi cients by tw o vertical (Pa nd S) veloc-

ities and seven d imensionless parameters tha t represent

an extension of Thomsens anisotropy coeffi cients to or-

thorhombic mod els. B y design, this notat ion provides a

uniform description of anisotropic media with both or-

thorhombic a nd TI symmetry.

The dimensionless anisotropic parameters introduced

here preserve all attractive features of Thomsen nota -

tion in treating wave propagation and performing 2-D

processing in the symmetry planes of ort horhombic me-

dia. The new notation has proved useful in describing

seismic signaturesoutsidethe symmetry planes as well,

especially for P-waves. Linearization of P-wave phasevelocity in the anisotropic coeffi cients leads to a concise

weak-anisotropy approximation that provides good ac-

curacy even for mod els with pronounced polar and a z-

imuthal velocity variations. This approximation can be

used efficiently to build analytic solutions for various

seismic signature s.

One of the most important advantages of the new no-

tation is the reduction in the number of parameters re-

sponsible for P-wave velocitiesa nd traveltimes. All kine-

matic signatures of P-wa ves in orthorhomb ic media de-

pend on just the vertical velocity VP0a nd fiveanisotropic

parameters, with VP0 serving as a scaling coefficient in

homogeneous media. This conclusion, which holds even

for orthorhombic modelswith strong velocitya nisotropy,provides an analytic basisfor application of P -wa ve trav-

eltime inversion and data processing algorithms in or-

thorhombic media.

INTRODUCTION

The vast majority of existing studies of seismic anisotropy

are limited t o tra nsversely isotropic (TI) mo dels with diff erent

orientation o f the symmetry axis. It has been shown in the liter-

ature (e.g., Sayers, 1994a) that TI (or hexagonal) symmetry ad-

equa tely describest he elastic properties of shales, which repre-

sent the major sourceo f anisotropy in sedimentary basins. Most

shale formations are horizonta lly layered, yielding an effective

transversely isotropic medium with a vertical symmetry axis

(the so-called VTI medium). In some cases, shale layers may

be dipping (e.g., near flanks of salt domes), which leads to an

azimuthally anisotropic medium with a tilted TI symmetry a xis

[for moveout analysis in such a model, see Tsvankin (1997a)].

However, the transversely isotropic model becomes much

more restrictive when applied to the d escription of cracked

Manuscript received by the Editor June 21, 1996; revised manuscript received January 22, 1997.Ce nter fo r Wave P henom ena, D ept. of G eophysics, Co lora do Scho ol of M ines, G olden, Co lora do 80401-1887. E-mail: [email protected] 1997 Society of Exploration G eophysicists. All rights reserved.

media that contain small (compared to the predominant

wavelength) fracturesor microcracks. Indeed, only the simplest

fractured model with a single system of parallel vertical circular

( penny-shaped ) cracks embedded in an isotropic matrix ex-

hibits transverse isotropy with a horizontal symmetry axis(H TI

media).D eviationsfrom the circular crackshape, misalignment

of the crackplanes, the addition of a second cracksystem, or the

presence of a nisotropy or thin layering in the matrix lower the

symmetry of the effective medium to orthorhombico r less. One

of the most common reasons for orthorhombic anisotropy in

sedimentary ba sins is a combination o f para llel vertical cracks

and vertical transverse isotropy in the background medium

(e.g., Wild and Crampin, 1991; Schoenberg and Helbig, 1997)

as illustrated by Figure 1. O rthorhombic symmetry can also

be caused by two or three mutually orthogonal crack systems

or tw o identical systems of cracks making a n a rbitrary angle

1292


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Parameters for Orthorhombic Media 1293

with each other. H ence, orthorhombic anisotropy may be the

simplest realistic symmetry for many geophysical problems.

Wave propagation in orthorhombic media has been stud-

ied in a number of publications including Musgrave (1970),

Tsvankin and Chesno kov (1990a, b), Wild and Cra mpin (1991),

B rown et al. (1991), Sayers (1994b) and Schoenberga nd Helbig

(1997). Such numerical tools a s fi nite-difference o r refl ectivity

methods generalized for anisotropic media make the forward

modeling of seismic wavefields in orthorhombic models a rel-

atively straightforward (albeit a costly) procedure. H owever,

extending inversion and processing techniques to ort horhom-

bic media is a much more difficult endeavor.

Seismic methods of fracture detection, extensively devel-

oped since the early 1980s, are based on the analysis of the

traveltimes and reflection coefficients of split shear wa ves at

near-vertical incidence (Crampin, 1985; Thomsen, 1988). If

azimuthal anisotropy is caused by a single system o f penny-

shaped vertical cracks, the relative time dela y or t he difference

inthe normal-incidence reflection amplitude between the shear

modes provides a direct estimate of the crack density, while the

polarization of the fast S-wave1 determines the crack orienta-

tion. These well-known shear-wa ve splitting method s remain

valid in some o rthorhombic mo dels such as tho se in Figure 1.

H owever, vertically traveling shear waves can yield only a sin-

gle anisotropic parameter (the splitting coefficient), which is

not sufficient, for instance, to separate empty and fluid-filled

cracks. In more complicated orthorhombic models contain-

ing two crack systems or noncircular cracks, interpretation of

the shear-wa ve splitting para meter becomes ambiguous, unless

some additional data a re available.

Therefore, seismic characterization of orthorhomb ic media

should include, if possible, analysis of azimuthally dependent

P-wave signatures and shear da ta a t oblique incidence angles.

Recently, Lynn et al. (1996a, b) presented field-data exam-

plesshow ing the feasibility of detecting azimuthal anoma lieso f

P-wave velocities and amplitudes over typical fra ctured reser-

voirs. Interpretation and inversion of such a nomalies is im-

possible without a good understanding of the dependence of

FIG. 1. An orthorhombic model caused by parallel verticalcracks embedded in a medium composed of thin horizontallayers. O rthorhombic media have three mutually orthogonalplanes of mirror symmetry.

1For brevity, the qualifi ers in q uasi-P-wave and quasi-S-wave willbe omitted.

seismic signature s on the anisotro py param eters. The complex-

ity of this problem is explained by the fact that general or-

thorhomb ic media are described by nine independent stiffness

components.

This work is aimed at making seismic treatment of or-

thorhombic media more practical by identifying the combina-

tions of the stiffness coefficients responsible for a wide range

of seismic signatures. The notation that enables us to reach

this goal is based on the same principle as that of Thomsen

(1986), devised for VTI media. A unified description of ver-

tical transverse isotropy and orthorhombic media provides a

convenient bridge between the two models and makes it pos-

sible to take full advanta ge of recent developments in the ana-

lytic description of seismic signatures in VTI media (Tsvankin,

1996). This approach has a lready been succesfully a pplied by

R uger (1997) and Tsvankin (1997b) to TI models with a hori-

zontal symmetry axis. For instance, Tsvankin (1997b) showed

that the anisotropicparameter that governsazimuthally depen-

dent P-wave normal-moveout (NMO) velocity in HTI media

represents the samecombination of the stiffness coeffi cients

as do es the Thomsens VTI paramet er . A lso, Thomsen-style

parameters have been used in physical-modeling studies by

B rown et a l. (1991) and C headle et a l. (1991).

Introduction of the new notation is followed by a n analytic

description of seismic signatures in the symmetry planes of

orthorhombic media ba sed on a limited analogy w ith vertical

transverse isotropy. Linearization of the exact phase-velocity

equa tion in the dimensionless anisotropic parameters leads to

a simple weak-anisotropy approximation fo r P -wave phase ve-

locityoutsidethe symmetry planes. Further analysis shows that

P-wa ve kinematic signatures in orthorhomb ic media are deter-

mined by the vertical velocity and only five anisotropic coeffi-

cients.

ORTHORHOMBIC STIFFNESSTENSOR

The stiff ness tensor ci jkfor orthorhomb ic media can be rep-resented in a two-index notat ion (e.g., Musgrave, 1970) often

called the Voigt recipe as

corthor=

c11 c12 c13 0 0 0

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

0 0 0 0 0 c66

. (1)

This tensor ha s the same null components as that for tra ns-

versely isotropic (TI) media with the symmetry axis aligned

with one of the coordinate directions, but the TI tensor hasonly fi ve independent coefficients. For certain subsets of or-

thorhombic models (such as o rthorhombic media caused by

vertical cracks in a VTI background), not all nine stiffnesses in

the tensor (1) are independent (Schoenberg and H elbig, 1997),

but here we d o not restrict ourselves to a ny specific type o f or-

thorhomb ic anisotropy.

The pha se velocity Va nd the displacement vector U of planewaves in arbitrary homogeneous anisotropic media satisfy the

Christoffel equa tion (Musgrave, 1970):G ik V2ik

Uk= 0, (2)


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1294 Tsvankin

where is the density,ikis the K ronekers symbolic , andG

is the symmetric Christoffel matrix,

Gi k= ci jknjn. (3)Here, n is the unit vector in the slowness direction; summa-tion over repeated indices is implied. The Christoffel equa-

tion (2) describesa standa rd eigenvalue (V2)-eigenvector (U)

problem for the matrix G, w ith the eigenvalues determinedby

de tG ik V2i k

= 0. (4)

Solving the cubic equa tion (4)a t any specific slownessdirection

n (for d etails, see Appendix A) yields three positive values of

c11n

21 + c55n23 V2 0 (c13 + c55)n1n3

0 c66n21 + c44n23 V2 0

(c13 + c55)n1n3 0 c55n21 + c33n23 V2

U1

U2

U3

= 0. (11)

the squared phase velocity V2, which correspond to theP -waveand two S-wa ves (strictly, the qua si-P-wave and quasi-S-

wa ves). For certain orientations of the vector n, the velocitiesof the split S-waves coincide with each other, which leads to

the shear-wa ve singularities. Af ter the eigenvalues have been

determined, the associated displacement (polarization) vectors

U for each mode can be found from equation (2). Since theChristoffel matrix

Gis symmetric, the polarization vectors of

the three modes are alway s orthogonal to each other, but none

of t hem is necessarily para llel or perpendicular to n.U sing equa tions (1) and (3) and the Voigt recipe for the two -

index notation yields the Christoffel matrix for orthorhombic

media:

G11 = c11n21 + c66n22 + c55n23, (5)

G22 = c66n21 + c22n22 + c44n23, (6)

G33 = c55n21 + c44n22 + c33n23, (7)

G12 = (c12 + c66)n1n2, (8)

G13 = (c13 + c55)n1n3, (9)

c11 sin2+ c55 co s2 V2 (c13 + c55) sinco s

(c13 + c55) sinco s c55 sin2 + c33 cos2 V2U1

U3 = 0. (12)

a nd

G23 = (c23 + c44)n2n3. (10)WAVE PROPAGATION IN THE SYMMETRY PLANES

Orthorhombic models have three mutually orthogonal

planes of mirror symmetry that coincide with the coordinate

planes [x1,x2], [x1,x3], and [x2,x3] (Figure 1). Because of the

relative simplicity of the basic equa tions, symmetry-plane anal-

ysis provides important insights into wave propagation in or-

thorhombic media. Anticipating future applications in reflec-

tion seismology, we assume that the symmetry plane [x1,x2] is

horizontal. Then information o btained from refl ection seismic

experimentswill be mostly related to phase-and group-velocity

variations near the (vertical) x 3-axis. If none of the symmetry

planes is horizontal, the treatment of reflected waves becomes

more complicated; however, the eq uations derived below for

a homogeneousorthorhombic medium remain entirely valid.

For a w ave propagating in the [x1,x3] plane, the projection

of the slowness vector onto the x 2-axis vanishes (n2= 0), theChristoffel matrix

G [equations (5)(10)] simplifies, and equa -

tion (2) takes the following form:

Eq uation (11) reduces to the corresponding C hristoffel

equation for the transversely isotropic model with the sym-

metry axis in the x3-direction (V TI medium), if c44= c55. (Inthe orthorhombic model, however, the stiffness components

c44 a nd c55 a re no tequal to each other, which leads to shear-

wa ve splitting at vertical incidence.) This ana logy with VTI me-

dia has already been mentioned in the literature (Musgrave,

1970; C headle et al., 1991; Schoenberg and H elbig, 1997);

here, how ever, it is used in a systematic fashion to describe

wave propagation in the symmetry planes and to introduce

new dimensionless anisotropy parameters for orthorhombic

media.

The Christoffel eq uation (11) also reduces to that o f an H TI

medium w ith the symmetry a xis pointing in the x1-direction.

The properties of the HTI model and a limited equivalence

betw een vertical and horizontal tra nsverse isotropy w ere stud-

ied in detail by Tsvankin (1997b) and R uger (1997).

Exactly as in VTI media, equation (11) splits into two inde-

pendent equations for the in-plane (P-SV ) motion (U2= 0)and the pure transverse ( SH) motion (U1= U3= 0). Ex-pressing the slowness components in equation (11) in terms

of the phase angle with the vertical (x3) a xis (n1 = sin;n3= cos ) yields for the in-plane pola rized wa ves

Equation (12) is identical to the well-known Christoffel

equation for the P-SVw aves in VTI media that ha s been thor-

oughly d iscussed in the literature (e.g., Pa yton, 1983). H ence,

the phase velocities of the P-wa ve and the in-plane polarized

shear wave (SV-wave) in the [x1,x3] plane of orthorhombic

media represent the same f unctions of the stiffness coeffi cients

ci j and the phase a ngle w ith vertical as do the P-SV phase

velocities in VTI media. As in VTI media, the P-wa ve phase


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(and group) velocity in the vertical (x3) a nd horizontal (x1)

directions is determined as

c33/ a nd

c11/, respectively,

while the vertical and horizontal S V-wave velocity is equal toc55/.

The phase-velocity function of each wave in the [x1,x3]sym-

metry plane is sufficient to obtain the corresponding group

(ray) velocity and group angle and, consequently, all other

kinematic signatures, such as norma l-moveout (NMO ) veloc-

ity. This means that the kinematics of wave propagation in

the [x1,x3] plane of orthorhombic media is described fully by

known VTI equa tions. The only exception to this ana logy iscus-

poidalS-wa ve group-velocity surfaces (wavefront s) formed in

the symmetry planes of orthorhombic media near shear-wave

point singularities. Some in-plane branches of the cusps may

be caused by slowness vectors that lie outsideof the symme-

try planes (G rechka a nd O bolentseva, 1993); clearly, cuspoidal

features of this nature cannot exist in VTI media.

Since the displacement (polarization) vectors U of the in-plane polarized plane waves are determined from the same

equation (12), they are also given by the corresponding expres-

sions for VTI media. Furthermore, analysis of the bo undary

conditions shows that the eq uivalence with vertical transverse

isotropy holds for plane-wave reflection coefficients as well

(provided the symmetry planes have the same orientation

above and below the boundary).

Ho wever, body-wave amplitudes in general d o not comply

with this equivalence principlebecausethey depend on the 3-

D shape of the slowness surface, not just in the symmetry plane

itself, but also in its vicinity. Out-of-plane velocity variations

lead to focusing and defocusing of energy and may have a sig-

nificant infl uence on the distribution of energy along the wa ve-

front within the symmetry planes (Tsvankin and Chesnokov,

1990a).

Eq uation (11) also has a solution corresponding to the

shea r w a ve pola rized orthogona lly t o t he [x1,x3] pla ne

(U1=U3

=0,U2

=0):

c66 sin2+ c44 co s2 V2 = 0. (13)

E qua tion (13) describes a pure shear mod e with an ellip-

tical slowness curve and an elliptical wavefront in the [x1,x3]

planea nd isidentical to the phase-velocity equation for theSH-

wa ve in VTI media . Ho wever, since in orthorhombic media c44is not equal to c55, the vertical velocities of the shear waves

are different. In fact, the two shear waves in the [x1,x3] plane

of an orthorhombic medium are fully decoupled because they

depend on different sets of elastic constants [compare equa-

tions ( 12) a nd (13)].

Similarconclusionscanbedrawnforwavepropagationinthe

[x2,x3]symmetry plane.Substituting n1

=0into the Christoffel

matrix [equations (5)(10)]a nd introducing the in-plane phaseangle with vertical (n2= sin; n3= cos ) yields for P-SV-waves:

c22 sin

2+ c44 co s2 V2 (c23 + c44) sinco s

(c23 + c44) sinco s c44 sin2 + c33 cos2 V2

U2

U3

= 0. (14)

Eq uation (14) becomes identical to the corresponding

Christoffel eq uation (12) fo r the [x1,x3] plane o f o rthorhom-

bic media if we replace U2withU1and interchange t he indices

1 and 2 in the appropriate components of the stiffness tensor

ci jk, i.e.,

c22 c11; c44 c55; c23 c13. (15)Therefore, to ob tain the phase velocity of b oth in-plane po-

larized modes in the [x2,x3] plane a s a function of the phase

angle with vertical, it is sufficient to make the above substitu-

tions of the elastic constants in the known phase-velocity equa-

tions for VTI media (or for the [x1,x3] plane of orthorhombic

media).

It can be demonstrated exactly in the same fashion that the

Christoffel equation in the third ([x1,x2]) symmetry plane be-

comesidentical to the VTI equations with the appropriate sub-

stitution of elastic constants, but the point has already been

made.

ANISOTROPIC PARA METERSFOR

ORTHORHOMBIC MEDIA

To take full advantage of the analogy w ith VTI media,

the stiffness coefficients ci j ca n be repla ced w it h a s et of

anisotropic parameters (combinations of stiffnesses) that con-

cisely characterize a wide range of seismic signatures for or-

thorhombic anisotropy. Indeed, since seismic signatures for

transversely isotropic media are especially convenient to de-

scribe in terms of dimensionless Thomsen (1986) parameters

(Tsvankin, 1996), it is natural to extend that notation to the

symmetry planes of orthorhombic media. As w e w ill see be-

low, these parameters turn out to be helpful in studying wave

propagation outside the symmetry planes as well.

Another, similar version of Thomsen-style notation de-

signed for arbitrary anisotropic media was proposed simul-

taneously (during the 7th Workshop on Seismic Anisotropy

in Miami, 1996) by G ajewski and P senck (1996) and Mensch

and Rasolofosaon (1997). Their approach, however, is based

on the weak-anisotropy approximation for phase velocity and

leads to a different (linearized in ci j ) definition of the coeffi-

cients. One of the main points substantiated below is that the

notation introduced here is advantageous for media with any

strength of the anisotropy.

Definitionsof theparameters

First, follow ing Thomsens recipe, we defi ne tw o vertical ve-

locities (for P- a n d S-wa ves) of the reference isotropic model.

For orthorhombic media, we can choose either of the two ve-

locities of split shear waves at vertical incidence. Here, the

preference is given to the S-wave po larized in the x1-direction

to make the new nota tion for the P -SV -wa ves in the [x1,x3]plane identical to Thomsens notation in the VTI case. Thus,

the two isotropic velocities are defined b y


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1296 Tsvankin

VP0 c33

, (16)

VS0 c55

. (17)

Since the Christoffel equation (12) for the waves polarized

in the [x1,x3] plane is identical to the corresponding equationfor vertical transverse isotropy,w e introduce the dimensionless

coefficients (2) a nd (2) [the superscript (2) refers to the x2-axis

direction, which definesthe orientation of the[x1,x3]symmetry

plane] through the same equations as those used in Thomsen

(1986) for VTI media:

(2) c11 c332c33

, (18)

(2) (c13 + c55)2 (c33 c55)2

2c33(c33 c55) . (19)

Note that in the definition of for VTI media, the coeffi cient

c44rather than its equal (c55) has often been used. Since these

two parameters differ for orthorhombic media, we should al-wa ys use c55in eq uation (19).

As in VTI media, the coefficient (2) from equation (19)

provides the exact second derivative of P-wave phase veloc-

ity at vertical incidence in the [x1,x3] plane: (d2V/d2)|=0=

2VP0(2) . Since this derivative is needed to ob tain the NMO ve-

locity (Tsvankin, 1995) and small-angle reflection coeffi cient,

our definition of (2) is particularly suitable for describing re-

flection seismic signatures.

The analogy with vertical transverse isotropy implies that

we can obtain kinematic signatures and polarizations of the

P-SV-waves in the [x1,x3] plane of orthorhombic media just

by substitutingVP0,VS0,(2) , and (2) into the known equations

for VTI media expressed in terms of Thomsen parameters VP0,

VS0, , and . For instance, using the exactphase-velocity equa -tion for VTI media expressed through Thomsen parameters

(Tsvankin, 1996) and replacing a nd with (2) a nd (2), re-

spectively, we find the phase velocities of the P-SV-wa ves in

the [x1,x3] plane a s

V2()

V2P0= 1 + (2) sin2 f

2

f2

1 + 2

(2) sin2

f

2 2

(2) (2)

sin2 2

f,

(20)

f 1 V2S0V

2P0. (21)

The plus sign in front of the ra dical corresponds to the P -wave,

while the minus sign corresponds to the SV-wave. In the w eak-

anisotropy approximation, the P-wave phase velocity in the

[x1,x3] plane can be linearized in the small para meters(2) a nd

(2) to obtain the well-known expression (Thomsen, 1986)

VP()= VP0

1 + (2) sin2 co s2 + (2) sin4 . (22)

Pha se-velocity equat ions (20) and (22) demonstrat e how we

can use the new a nisotropic parameters to ta ke advantage of

the analogy between the symmetry planes of orthorhombic

media and vertical transverse isotropy (for a more detailed

discussion, see the next section).

Next, using the equivalence between equation (13) and the

corresponding expression for the SH-wave in VTI media, we

introduce the parameter

(2) c66 c442c44

. (23)

Here, (2) isidenticalto Thomsenscoefficientfor VTI media:

it isresponsible for the velocity variationsof theSH-waveinthe

[x1,x3] plane. The coeffi cients(2) , (2) , and (2) coincide with

the parameters (V) , (V), and (V) (respectively) introduced in

Tsvankin (1997b) and R uger (1997) for transversely isotropic

media with the symmetry axis along the x1-direction.

In principle, it would be convenient to specify the parame-

ters, , and in the same fashion for the other tw o symmetry

planes as well. H owever, if we did so, some of the anisotropy

coefficients would not be independent, and the new notation

would suffer from redundancy. In defining the anisotropic pa-

rameters, we put emphasis on simplifying seismic signatures in

the t wo vertical planes of symmetry. To describe P-SV-waves

in the [x2,x3] symmetry plane (the plane normal to thex1-axis),

we introduce the para meters (1) a nd (1) defined analogously

to (2) a nd (2). U sing the recipe in equa tion (15) yields

(1) c22 c332c33

, (24)

(1) (c23 + c44)2 (c33 c44)2

2c33(c33 c44) , (25)

(1) c66 c552c55

. (26)

Now, for instance, we can obtain the phase velocities of P-

SV-waves in the [x2,x3] plane from equa tion (20) by substitut-

ing (1) fo r (2) a nd (1) fo r (2) ; a lso, f should be replaced byf1 1 V2S1/V2P0, where

VS1 =c44/ (27)

is the vertical velocity of the S-wave polarized in the x2-

direction.

The two vertical velocities and six anisotropy parameters

introduced above can be used instead of eight original stiff-

ness coeffi cients:c11,c22,c33, c44, c55, c66, c23, and c13. The o nly

remaining stiffness c12 can b e replaced with a dimensionless

anisotropic parameter analogous to the coefficients in the

vertical planes of symmetry

(3)

(c12

+c66)

2

(c11

c66)

2

2c11(c11 c66) . (28)The coeffi cient (3) plays the role of Thomsens in TI

equations written for the [x1,x2] symmetry plane, with the

x1-direction substituted for the symmetry axis. Note that the

quantities (3) a nd (3) would be redundant.

Ha ving introduced the new parameters,it is convenient to

list all of them, along w ith their brief descriptions:

VP0the vertical velocity of the P-wave;

VS0 the vertical velocity of the S-wa ve polarized in the x1-

direction;


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(2) the VTI parameter in the symmetry plane [x1,x3]

normal to the x2-axis (close to the fractional differ-

ence between the P-wave velocities in the x1- and x3-

directions);

(2) the VTI parameter in the [x1,x3] plane (responsi-

ble for near-vertical P-wave velocity variations, also

influencesSV-wave velocity anisotropy);

(2)

the VTI parameterin the [x1,x3] plane (close to thefractional difference between the SH-wa ve velocities

in the x1- a n d x3-directions);

(1) the VTI parameter in the [x2,x3] plane;

(1) the VTI parameter in the [x2,x3] plane;

(1)the VTI parameter in the [x2,x3] plane;

(3) the VTI parameter in the [x1,x2] plane (x1is used as

the symmetry a xis).

While the new parameters are determined uniquely by the

nine independent stiffness coeffi cients of orthorhombic me-

dia, the inverse transition is unique only for the coefficients

associated w ith the velocities along the coordinate axes (c11,

c22,c33,c44,c55, and c66). To ob tain the o ther three coeffi cients

(c12, c13, c23) from the corresponding values, it is necessaryto specify the sign of the sums (c13+ c55) [equation (19)],(c23+c44) [equation (25)], and (c12+c66) [equation (28)]. As dis-cussed in Tsvankin(1996),exactly the same problem arisesw ith

Thomsen parameters in transversely isotropic media. Ho w-

ever, since the stability condition (Musgrave, 1970) requires

the coefficientsc55,c44, and c66to b e alw ays positive, the sums

under considerat ion can be negative only fo r uncommon large

negative values of c13, c23, or c12. Therefore, fo r pra ctical pur-

poses of seismic processing and interpretation, w e can assume

that (c13+ c55), (c23+ c44), and (c12+ c66) are positive. Thatwould correspond to one of the conditions of so-called mild

anisotropy as specified in Schoenberg and H elbig (1997) and

ensures the absence of anomalous body-wave polarizations in

the symmetry planes (Helb ig and Schoenb erg, 1987). Note tha tthe term mild anisotropy doesnotmean that the magnitude

of velocity variations is small.

Although the nine parameters introduced above are suffi-

cient to characterize general orthorhombic media, one may

need to use different combina tions of these coeffi cients in spe-

cific a pplications. For instance, shear-wa ve splitting at vertical

incidence is described conventionally by the fractional differ-

ence between the parameters c44a nd c55a s

(S) c44 c552c55

= (1) (2)

1 + 2(2) VS1 VS0

VS0. (29)

The pa rameter (S) represents a direct measure of the time

delay betw een two split shear waves at vertical incidence and is

identical to the generic Thomsenscoefficientfor transversely

isotropic media with a horizontalsymmetry axis that coincides

with the x1-direction.

Special cases:VTI andHTI media

B oth vertical and horizontal tra nsverse isotropy can be con-

sidered as degenerate special casesof orthorhombic media. An

orthorhomb ic medium reduces to the VTI model if the proper-

ties in all vertical planes are identical, and the velocity of ea ch

mode in the [x1,x2] plane (the so-called isotropy plane ) is

constant (although the velocities of the two S-wa ves generally

differ from one another). H ence, the VTI constraints, w hich

reduce the number of independent parameters from nine to

five, can be rewritten in terms of the new parameters as

(1) = (2) = ,(1)

=(2)

=,

(1) = (2) = ,a nd

(3) = 0,where , , and are Thomsens VTI anisotropy coefficients.

Anot her special case is transverse isotropy w ith a horizonta l

axis of symmetry. If the symmetry axis is oriented along the

x1-direction, then the axes x2 a nd x3 form the isotropy plane,

a nd

(1) = 0,

(1)

=0,

(1) = 0.The parameters (2) , (2) , and (2) in this case coincide with

the coefficients (V) , (V), and (V) (respectively) introduced

in Tsvankin (1997b) and R uger (1997). Wave propagation in

HTI media is described fully b y these three dimensionless

anisotropic coefficients and tw o vertical velocitiesVP0a nd VS0.

(The la st anisotropic para meter, (3) , in this model is not inde-

pendent.)

APPLICATION OF THE NEWNOTATIONI N THE

SYMMETRY PLANES

By design, the new parameters provide a simple way of

describing seismic signatures in the symmetry planes of or-

thorhombic media using the known equations for VTI me-

dia expressed through Thomsen parameters (Thomsen, 1986;

Tsvankin, 1996). The coefficients (i),(i) , and(i) conveniently

quantify the magnitude of velocity anisotropy in orthorhom-

bic media, both within and outside the symmetry planes. The

parameters (2) a nd (1) are close to the fractional difference

between vertical and horizontalP -wa ve velocities in the planes

[x1,x3] a nd [x2,x3] (respectively) and, therefore, yield an over-

all measure of the P -wave anisotropy in these planes. Sim-

ilarly, the coefficients (2) a nd (1) govern the magnitude of

the velocity variation of the elliptical SH-wave in the vertical

symmetry planes.

One of t he most important a dvanta ges of Thomsen notation

is the reduction in the number of parameters responsible for

P-wa ve kinematic signatures. The exact P -wave phase-velocity

equa tion for VTI media [analogous to equa tion (20)] contains

four independent parameters (VP0, VS0, , and ), but the in-

fluence of the shear-wave vertical velocity VS0 is practically

negligible, even for strong anisotropy (Tsvankin and Thomsen,

1994). As discussed in Tsvankin (1996), the stiffn ess coeffi cient

that determines the shear-wave vertical velocity (c44= c55inVTI media) does make a contribution to the P-wa ve velocity

equa tions, butonlythrough the parameter . The ana logy with

VTI media implies that P-wave kinematic signatures in each


7/18

1298 Tsvankin

symmetry plane are de termined by justthr eeindependent coef-

fi cients. In the vertical symmetry planes the needed parameters

are the vertical P-wa ve velocity VP0 (the scaling factor) and a

pair of the a nisotropic coefficients introduced above: (2) a nd

(2) ([x1,x3] plane) or (1) a nd (1) ([x2,x3] plane). As in VTI

media, the coefficients (i) a nd (i) are responsiblefor phaseand

group velocity in different ra nges of phase a ngles, which is ex-

tremely convenient for purposes of seismic processing and in-

version. Specifica lly, the coef fi cients (2) (plane [x1,x3]) and(1)

(plane [x2,x3]) determine near-vertical P-wave velocity vari-

ations [see equation (22)], as well as the anisotropic term in

the expression for normal-moveout velocity from horizontal

reflectors (discussed in more detail below). Also, the new no-

tation simplifies the elliptical condition in both vertical symme-

try planes: for example, the P-wave anisotropy in the [x1,x3]

plane is elliptical if (2) = (2) (theSV-wa ve velocity in this caseis constant).

To obtain any kinematic signature (e.g., phase and group

velocity, moveout from horizontal reflectors, etc.), polariza-

tion vector, and reflection coefficients of the P - a n d S V-waves

in the [x1,x3] plane of orthorhombic media, it is sufficient to

substitute VP0, VS0, (2) , and (2) into VTI eq uations expressed

through VP0,VS0, , and , respectively. The same analogy with

vertical transverse isotropy holdsfor P -SV-wavesin the[x2,x3]

plane, if we use VTI eq uations with the coeffi cients VP0, VS1,

(1) , and (1) . Adaptation of VTI signatures to the symmetry

planes of orthorhombic media using the new a nisotropic pa-

rameters is discussed b riefly b elow. The refl ection coeffi cients

in the symmetry planes of orthorhombic media are described

in R uger (1996).

Phaseandgroupvelocity

The exact phase velocity of the P-SV-waves in the [x1,x3]

plane and the weak-anisotropy approximation for the P-wave

phase velocity were expressed through the new parameters

in equations (20) and (22), respectively (analogous expres-

sions hold in the [x2,x3] plane). In a similar way, we can

adapt Thomsens (1986) weak-anisotropy approximation for

the phase velocity of the SV-wa ve in VTI media . The SV-wave

phase velocity in the [x1,x3] plane, linearized in the anisotropic

parameters, takes the form

VSV(){[x1,x3]plane} = VS0

1+(2) sin2 co s2 , (30)

where(2) wa s introd uced in Tsvankin and Thomsen (1994) as

(2) VP0

VS0

2 (2) (2)

. (31)

The pa rameter (2) largely determines the kinematic signa-

tures of the SV-wave in the [x1,x3] symmetry plane, although

the influence of the individual values of the VP0/VS0 ratio,

(2) , and (2) on SV propagation becomes more pronounced

with increasing valueso f anisotropic coeffi cients. Although the

weak-anisotropy approximations for phase velocity become

less accurate with increasing velocity anisotropy, they provide

a valuable analytic insight into a wide range of seismic signa-

tures (Tsvankin, 1996). It is also possible to improve the accu-

racy of equations (22) and (30) by adding terms quadra tic in

the anisotropic parameters. For instance, Tsvankin (1996) pre-

sented the q uadratic weak-anisotropy approximation for the

P-wave velocity in VTI media that can be directly applied in

the symmetry planes of orthorhombic media.

U s in g t he a n al og y w it h t he SH-w a ve in V TI media

(Thomsen, 1986), we can obtain the exact SH-wa ve phase ve-

locity in the vertical symmetry planes in the following form:

VSH()

{[x1,x3]plane

} =VS11 +

2(2) sin2

= VS0

1 + 2(1)1 + 2(2)

1 + 2(2) sin2 ;

(32)

VSH(){[x2,x3]plane} = VS0

1 + 2(1) sin2 . (33)

The gro up velocity vgrin the symmetry planes of orthorhom-

bic media represents the same function of phase velocity as in

VTI media (e.g., B erryma n, 1979):

vgr

=V1 +

1

V

dV

d

2

, (34)

where is the phase angle with one of the coordinate axes.

[Eq uation (34), however, will not describe cuspoidal b ranches

of S-wa ve group-velocity surfaces caused by out-of-plane slow -

ness vectors.] Hence, to compute the group velocity in any of

the symmetry planes, we just have to substitute the appropri-

ate phase-velocity function into eq uation (34) (e.g., from eq ua-

tion (20) for the P-SV-waves in the [x1,x3] plane). A s is the

case for TI media, in the linearized weak-anisotropy approxi-

mation the group velocity expressed through the phase angle

coincides with the phase velocity. H owever, the gro up velocity

corresponds to the energy propagating a t the group angle

given in VTI med ia b y (B erryman, 1979)

tan=ta n+ 1

V

dV

d

1 ta nV

dV

d

. (35)

Again, equation (35) can beused to calculate the groupa ngle

w ith vertical in the symmetry planes of orthorhombic me-

dia by substituting the corresponding phase-velocity functions.

[The phase angle in equation (35) should be computed with

respect to the vertical (x3) a xis]. Also, the w eak-anisotropy ap-

proximations for the group angles can be adapted easily from

the corresponding VTI expressions (Thomsen, 1986). For in-

stance, the P-waves group angle w ith vertical in the [x1,x3]

plane can be represented as

ta n= ta n

1 + 2(2) + 4(2) (2)

sin2

. (36)

Note that the difference between the group and phase angles

does contain termslinear in the anisotropiccoeffi cients. An ob -

vious modificat ion of eq uation (36) provides a similar expres-

sion in the [x2,x3] symmetry plane. For wa ve propagat ion along

the coordinate a xes of orthorhombic media the d erivative of

phase velocity vanishes, and the group-and phase-velocity vec-

tors for any wave type coincide with each other.


8/18


Polarization vector

For a given slowness direction, the polarization vectors of

the three plane waves (the eigenvectors of the Christoffel

equation) are always mutually orthogonal. This orthogonal-

ity does not hold, however, for nonplanar wa vefronts excited

by point sources because the three body waves recorded at

any fixed point in space correspond to different slowness di-

rections. Also, the polarization vector of the plane P-wa ve isnot necessarily aligned with either phase (slowness) or group

(ray) vector. Likewise, in general the split shear waves are no t

polarized orthogonally to the phase- or group-velocity vector

(this explains the qualifi er qua si usually add ed to the names

of body waves in anisotropic media). In some directions, how-

ever, the polarization vector of the P-wave is parallel to the

slowness vector; Helbig (1993) described such directions as

longitudinal. In the symmetry planes of orthorhombic me-

dia the longitudina l directions include (but are not limited to)

the coordina te axes (Schoenberg a nd H elbig, 1997).

U sing the known expression for the polarization vector in

transversely isotropic media (e.g., H elbig a nd Schoenberg,

1987; Tsvankin, 1996), w e ca n w rite the polarization angle of

the plane P-wave in the [x1,x3] plane of orthorhombic mediaa s

ta n U1U3

= sinco s (c13 + c55)V2 c11 sin2 c55 co s2

, (37)

where Vis the phase velocity of the P-wave. E qua tion (37) is

valid for any strength of the anisotropy and determines the SV-

wa ve polarization angle as well (the polarization vectors of the

P- and SV-waves are orthogonal). The second shear wave in

the [x1,x3] plane represents a pure shear (SH) mode polarized

in the x2-direction.

Ad apt ing the results of R omme l (1994) and Tsvankin (1996)

obtained for TI media, we find the following weak-anisotropyapproximation for equation (37)

ta n= tan

1 + B

2(2) + 4(2) (2)

sin2

,

(38)

B 12f

= 12

1 V2S0V2P0

.With the appropriate substitutions of the elastic constants

[see rela tions (15)] or anisotropic coeffi cients, eq uations (37)

and (38) can be used to obt ain the polarization angles for P -SV

waves in the [x2,x3] plane.

As d iscussed in Tsvankin (1996), comparison of the weak-

anisotropy approximations for the group angle (36) and the

polarization angle (38) shows that for weak and moderate

anisotropy, the P-wave polarization vector lies closer to the

corresponding ray (group) vector than to the phase vector.

To obtain body-wave polarizations in the fa r-field of a point

source, it is necessary to find the phase angle corresponding

to a given group (source-receiver) direction and substitute it

into eq uations (37) or (38). Nea r-field polarizations, however,

depend on the relative amplitudes of several terms of the ray

seriesexpansion and, therefore, can be influenced by azimuthal

velocity variations in orthorhombic media.

Moveoutfromhorizontal reflectors

Suppose a common-midpoint (CMP) line is parallel to the

x1-axis of a horizontal orthorhombic layer. Then the phase-

and group-velocity vectors of the waves reflected from the

bottom of the layer stay in the [x1,x3] symmetry plane, and

the short-spread normal-moveout (NMO) velocity can be ob-

tained from the known VTI equations (e.g., Thomsen, 1986;

Tsvankin, 1996). The substitutions described above yield

V(2)nmo [P-wave] = VP0

1 + 2(2), (39)

V(2)nmo [SV-wave] = VS0

1 + 2(2), (40)

V(2)nmo [SH-wave] = VS0

1 + 2(2), (41)

where(2) was defined in equation (31).

Similarly, the P -wave NM O velocity on a line parallel to the

x2-axis is given by

V(1)

nmo[P-wave]

=VP01 + 2(1). (42)

Moveout velocity is one of the most important parameters

in reflection data processing, a nd the simplicity of the a bove

expressions is a good illustration of the advantages of the no-

tation introduced here. Note that equations (39)(42) remain

valid for any strength of velocity anisotropy. Evidently, if the

symmetry-plane directions and the vertical velocity is known,

short-spread P-wa ve moveout in the vertical symmetry planes

makes it possible to obtain both coeffi cients.

To fi nd norma l-moveout velocity in symmetry planes of mul-

tilayered latera lly homogeneous orthorhomb ic models, it is suf-

fi cient just to apply the conventional D ix equa tion. This conclu-

sion follows from the general NMO formula for any symmetry

plane in anisotropic media given in Alkhalifah and Tsvankin

(1995). Of course, for a stack of orthorhombic layers to havea thro ughgoing vertical symmetry plane, the horizonal coord i-

nate axes in all layers have to be aligned.

For relatively large spreadlengths (exceeding the reflector

depth) in anisotropic media, reflection moveout becomes non-

hyperbolic and can be described by the following expression

developed by Tsvankin and Thomsen (1994) for vertically in-

homogeneous VTI media :

t2(x)= t20+ A2x2 +A4x

4

1 + Ax 2 , (43)

where t0is the zero-offset reflection traveltime, A2= 1/V2nmo ,A4is the quartic moveout coefficient, and

A = A41

V2P90 A2

,

whereVP90is the horizonta l velocity. The norma lization of the

quartic moveout term makes equation (43) numerically accu-

rate for P -wa ve reflection moveout on long spreads (23t imes,

and more, the reflector depth), even in VTI models with pro-

nounced velocity a nisotropy (Tsvankin a nd Thomsen, 1994).

The analogy with vertical transverse isotropy implies that the

nonhyperbolic moveout eq uation (43) can be used without a ny


9/18

1300 Tsvankin

modifi cation on CM P lines parallel to one of the horizontal co-

ordinate axes of an orthorhombic layer (or in the symmetry

planes of a layered orthorhomb ic medium). For instance, for a

line coinciding with the x1-axis we have to make the following

substitutions in equa tion (43) (for the P-wave)

A2 =1

V2P01 +2(2)

,

VP90 = VP0

1 + 2(2),a nd

A4 = 2(2) (2)

t20V

4P0

1 + 2(2)f

1 + 2(2)4,

with f given by eq uation (21). While equation (43) as a whole

is an approximation for long-spread moveout, the a bove ex-

pression for the quartic coefficient A4is exact fo r any strength

of the anisotropy.

Obvious modifications of the above eq uations provide sim-

ilar expressions for t he [x2,x3] symmetry plane. The beha vior

of the exact normal-moveout velocity outside the symmetry

planes is discussed in G rechka and Tsvankin (1996).

Dippingreflectorsand2-D processing

The equivalence between orthorhombic media and vertical

transverse isotropy breaks dow n if the phase-or group-velocity

vectors of seismic waves deviate from the vertical symmetry

planes. Evidently, 3-D seismic processing in orthorhombic me-

dia cannot be carried out by just adapting the known VTI al-

gorithms. However, if the subsurface can be approximated by

a 2-D structure with the strike parallel to one of the horizonta l

coordinate axis of an orthorhombic model, the incident and

reflected rays on the dip line (normal to the strike) will be

confined to one of the vertical symmetry planes. (For more

complicated 3-D models it may be possible to suppress out-of-plane events.) Normal-moveout velocity of in-plane dipping

events can be described by the following equation valid for

pure modes in any symmetry plane of arbitrary anisotropic

media (Tsvankin, 1995):

Vnmo ()=V()

co s

1 + 1

V()

d2V

d2

=

1 ta nV()

dV

d

=

, (44)

where is the phase angle with vertical, and is the dip of

the reflector. Equation (44) can be considerably simplified by

using the wea k-anisotropy a pproximation. Ada pting the result

obtained by Tsvankin (1995) for vertical transverse isotropy in

the limit of small a nd , we get the following expression for

th e P -wave NMO velocity in the [x1,x3]plane of orthorhombic

media (|(2)| 1, |(2)| 1):

V(2)

nmo () cos

V(2)

nmo (0)= 1 + (2) sin2

+ 3(2) (2)

sin2 (2 sin2 ). (45)

In reflection dat a processing,N MO velocity isusually treated

as a function of the ray parameter p() that can be determined

from the slope of reflections recorded o n zero-offset (or CM P-

stacked) seismic sections. For vertical t ransverse isotropy, P-

wave NMO velocity from dipping reflectors [equation (44)]

expressed as a function of pis determined by just t woparam-

eters: the NMO velocity from a horizontal reflector [Vnmo (0)]

and a n anisotropic coefficient defi ned as ( )/(1 + 2)(Alkhalifah and Tsvankin, 1995). The parameter ca n be

considered as a measure of anellipticity of a particular TI

medium. Alkhalifah and Tsvankin (1995) have shown tha t

Vnmo (0) and control not just the d ip-dependent NM O veloc-

ity, but also all time-related P-wa ve processing steps (NMO,

D MO, prestack and poststacktime migration) in homogeneous

or vertically inhomogenous VTI media.

This conclusion remains entirely valid for P -wave refl ections

confined to one of the vertical symmetry planes of orthorhom-

bic media. For instance, the kinematics of 2-D time processing

in the [x1,x3] symmetry plane is governed by the correspond-

ing P-wave NMO velocity from a horizontal reflector [equa-

tion (39)] and a pa rameter (2) given by

(2) (2) (2)

1

+2(2)

. (46)

As an example, the nonhyperbolic moveout equat ion (43) in

the [x1,x3] plane can be rewritten as a function of just these two

effective parameters and the vertical time t0(the contribution

of VS0to the q uartic term A4can be ignored):

t2(x)= t20+x2

V(2)

nmo (0)2

2(2)x4

V(2)

nmo (0)2

t20V

(2)nmo (0)

2 + 1 + 2(2)x2 .(47)

I f (2) = 0, the P-wave anisotropy in the [x1,x3] plane is ellip-tical and the reflection moveout is purely hyperbolic.

Alkhalifah and Tsvankin (1995) show that both paramet ers

needed for P-wave time processing in VTI media can be ob-

tained from surface P-wave data using NMO velocities from

two reflectors w ith different d ips. This algorithm holds for pa -

rameter estimation using in-plane dipping events in the vertical

symmetry planes of orthorhombic media. For instance, if the

strike of the structure is parallel to the x2-axis, NMO velocities

of reflections from horizontal and dipping interfaces recorded

on the dip line (parallel to the x 1-axis) can be used to recover

the zero-dip NMO velocity and the parameter (2) from equa-

tion (44). In principle, the coefficient(2) can be estimated using

the long-spread (nonhyperbolic) P-wave moveout from hori-

zontal reflectors [see equation (47)], although with lower ac-

curacy (Tsvankin and Thomsen, 1995). Once the two effective

paramet ers have been det ermined, 2-D time processing in the

[x1,x3] symmetry plane can be carried out using the methodol-

ogy d eveloped for vert ical tra nsverse isotropy. To perfo rm 2-D

depth processing in either o f the vertical symmetry planes, it is

also necessary to know the P-wa ve vertical velocity.

In genera l, all 2-D dat a-processing steps (NMO, D MO, time

and depth migration) in the vertical symmetry planes of or-

thorhomb ic media can be performed by VTI algorithms. B ody-

wa ve amplitudes, however, will be influenced by a zimuthal ve-

locity varia tions, which ma y ca use distortions in the so-called

true-amplitude D MO and migration methods.


10/18


P-WAVE KINEMATICSOUTSIDE THE SYMMETRY PLANES

While the new notation has obvious advantages in describ-

ing symmetry-plane seismic signatures, an open question is

whether it can be useful in treating wave propagation outside

the symmetry planes. The focushere willbe on phase velocity

the most fundamental function that determines all other kine-

matic signaturesfor a particular mode.Since Thomsen notation

does not provide any tangible simplification of the exact phase-velocity eq uations for transversely isotropic media (Tsvankin,

1996), our not ation can ha rdly be expected to accomplish this

task for orthorhombic models. Indeed, the exact phase veloc-

ity in terms of the new coefficients still has to be computed

numerically (see Appendix A).

However, as demonstrated by the symmetry-plane analy-

sis in the previous section, the new notat ion may be helpful in

developing concise weak-anisotropy approximations for phase

and group velocity, reducing the number of paramet ers respon-

sible for P -wave kinematics for any strength of the a nisotropy,

and simplifying the exact a nalytic expressions for reflection

seismic signatures (such as NMO velocity). Hence, our next

step here is to transform the phase-velocity equations in the

limit of weak anisotropy.

Weak-anisotropyapproximation for phasevelocity

Since the dimensionless anisotropic parameters introduced

above should be small in weakly anisotropicmedia, the new no-

tation is particularly suitable for developing weak-anisotropy

approximations by expanding seismic signatures in powers of

(i) , (i) , and (i) (usually only the linear terms are retained).

For the planes of symmetry it is sufficient just to adapt the

known expressions for w eak tra nsverse isotropy (see the pre-

vious sections). Approximate P-wave phase velocity outside

the symmetry planes is obta ined in Appendix B b y linearizing

the exact equations (A-8) and (A-9) in the anisotropic coeffi-

cients:

V2P= V2P0

1 + 2n41(2) + 2n42(1) + 2n21n23(2)

+ 2n22n23(1) + 2n21n22

2(2) + (3). (48)

It is convenient to replace the directional cosinesof the slow-

ness (or phase-velocity) vector by the polar () and a zimuthal

() pha se angles,

n1 = sinco s, n2 = sinsin , n3 = cos .Then, taking the square-root of equation (48), we obtain the

phase velocity exactly in the same form a s in VTI media

[equation (22)], but with azimuthally dependent coefficients

a nd :

VP(, )= VP0[1+() sin2 co s2 +() sin4 ]; (49)

()= (1) sin2 + (2) cos2 ,

() = (1) sin4 + (2) cos4 +

2(2) + (3)

sin2 co s2 .

In both vertical symmetry planes, equation (49) reduces to

Thomsens (1986) weak-anisotropy approximation for TI me-

dia.

The structure of eq uation (49) is similar to the expansion of

P-wave pha se velocity in a serieso f spherical harmonics devel-

oped by Sayers (1994b). Here, however, instead of describing

the medium by perturbations of the stiffness coefficients (as

was done by Sa yers), we ha ve obtained a concise approxima-

tion in terms of the dimensionless anisotropic parameters (i)

a nd (i) . Since Sayers (also see Jech and P senck, 1989; M ensch

and Ra solofosaon, 1997) neglects terms of order (ci j/ci j )2,

whereas we neglect terms of order 2 (etc.), this linearization

is slightly different. B ecause of the structure of the C hristoffel

equation (2) for orthorhombic media, it is convenient to use

nonlinearcombinations of elastic moduli ((i) ) to characterize

the anisotropy. As mentioned above, the coeffi cients defined

here yield theexactsecond derivative of P -wave phase velocity

at vertical incidence that is needed to describe such reflection

seismic signatures as NM O velocity and small-angle refl ection

coefficient.

Parametersresponsiblefor P-wavevelocity

It should be emphasized that equation (49) does not con-

tain any o f the three parameters (VS0, (1) , (2) ) that describe

the shear-wave velocities in the directions of the coordinate

axes. Evidently, kinematic signatures of P-waves in weakly

anisotropicorthorhom bic media depend on justfiveanisotropic

coefficients((1) , (1) , (2) , (2) , a n d(3) ) and the vertical velocity.

Similar conclusions for weakly anisotropic models of various

symmetries were drawn by Chapman and Pratt (1992), Sayers

(1994b), and Mensch and Rasolofosaon (1997).

The equivalence between the symmetry planes of or-

thorhombic media and transverse isotropy, discussed in de-

tail a bove, implies that these six parameters are sufficient to

describe P-wave phase velocity in all three symmetry planes,

even forstrongvelocity anisotropy. An important question, to

be add ressed next, is whether or not P-wave phase velocity de-

pends only on the same six parameters outsidethe symmetry

planes and in the presence of significantvelocity va riations.

In the numerical examples below, we examine the infl uence

of the parameters VS0 a nd (i) on the exact phase velocity of

P-wave for orthorhombic models with moderate and strong

velocity anisotropy. It is sufficient to consider the velocity func-

tion in a single octant, for instance, the one corresponding to

0 90, 0 90. Figure 2 shows the dependence ofP-wa ve phase-velocity variations in fo ur vertical planes on t he

shear-wa ve vertical velocityVS0(or the VP0/VS0ratio), with all

other model parameters being fi xed. Clearly, the influence of

VS0within a wide range of the VP0/VS0ratios is negligible not

only in the [x1,x3] symmetry plane (= 0), as was expected(Tsvankin, 1996), but outside the symmetry planes as well.

This conclusion holds for two models with more pronounced

phase-velocity variations shown in Figure 3. Note that even

for a medium with uncommonly strong velocity anisotropy

(Figure 3b) the difference between the phase-velocity curves

corresponding to the two extreme values of the VP0/VS0 ratio

remains ba rely visible. The contribution of VS0to the P-wave

phase velocity becomes somewhat more noticeable only for

uncommon models with close values of the P -and shear-wave

velocities in one of the coordina te directions.

It should be emphasized that if the conventional notation is

used, the influence of the stiffness coefficientsc44, c55, a n dc66 on

P-wa ve velocity cannot b e ignored. For instance, c55doesmake


11/18

1302 Tsvankin

a contribution to the value of (2) [equation (19)]. As in VTI

media, our nota tion makes it possible to reduce the number

of parameters that govern P-wave velocity by combining the

stiffness coeffi cientsc55a nd c13in the single para meter (2) ; the

same holds fo r the stiffnessesc44a nd c66as well.

The P-wave phase velocity at two extreme values of the pa-

rameter (1) is shown in Figure 4. Since none of the elastic

constants governing P-wave propagation in the [x1,x3] planedepends on(1) (with VS0being fixed), the plot starts at an az-

imuth of 20. Although the influence of (1) increases slightlyin the vicinity of the [x2,x3] plane (= 90), the contributionof this parameter can b e safely ignored at a ll azimuthal angles.

Similarly,P -wa ve velocity isindependent of the coeffi cient(2) .

FIG.2. Infl uence of the vertical shear-wa ve velocity VS0on the exact P-wave phase velocity. The velocity is calculated as a functionof the phase angle with vertical at the four azimuthal phase angles shown on top of the plots. The solid curve corresponds toVP0/VS0= 1.5 (VP0= 3 km/s, VS0= 2 km/s), the d ott ed curve to VP0/VS0= 2.5 (VP0= 3 km/s, VS0= 1.2 km/s). The othe r m od elparameters are: (1) = 0.25, (2) = 0.15, (1) = 0.05, (2) = 0.1, (3) = 0.15,(1) = 0.28,(2) = 0.15.

FIG.3. P-wave phase velocity for differentVP0/VS0 ratiosin media with stronger anisotropy than inFigure2.The velocity iscalculatedas a function of the phase angle with vertical at an a zimuthal phase angle of 45. The solid curve corresponds to VP0/VS0= 1.5(VP0= 3 k m/s,VS0= 2 km/s), the d ott ed curve to VP0/VS0= 2.5 (VP0= 3 km/s,VS0= 1.2 km/s). The o ther m ode l para meters a re:(a ) (1) = 0.25, (2) = 0.45, (1) = 0.1, (2) = 0.2, (3) = 0.15,(1) = 0.28,(2) = 0.15; (b) (1) = 0.45, (2) = 0.6, (1) = 0.15,(2) = 0.1, (3) = 0.2,(1) = 0.7,(2) = 0.3.

Thus, P -wa ve phase-velocity variations in orthorhomb ic me-

dia are governed by just five anisotropic parameters ((1) , (1) ,

(2) , (2) , a nd (3) ), with the vertical P-wave phase velocity

serving as a scaling coefficient (in homogeneous media). The

3-D phase-velocity (or slowness) function fully determines the

group-velocity (ray) vector a nd, therefore, all other kinematic

signatures (e.g., reflection t raveltime). This means that the fi ve

anisotropic parameters listed ab ove and the vertical velocity

are sufficient to describe all kinematic properties of P-waves

in orthorhombic media.

Furthermore, P -waver eflection tr avel ti mesfrom mildly dip-

ping reflectors in orthorhombic models w ith a horizontal

symmetry plane should be determined largely by the four


12/18


anisotropic parameters defined in the vertical planes of sym-

metry [(1) , (1) , (2) , a n d(2) ]a nd the symmetry-plane azimuths.

Indeed, the influence of the fifth coefficient [(3) ] is mostly lim-

ited to near-horizontal velocity variations outside the vertical

symmetry planes [see equation (49)].

Accuracyof theweak-anisotropyapproximation

Now tha t we have shown that the w eak-anisotropy approx-imation (49) contains al l parameters responsible for P-wave

velocity in orthorhombic media, we can study the dependence

of the error of equation (49) on these anisotropic coefficients.

As illustrated by Figure 5, equation (49) provides a good

approximation (especially near vertical) for the exact pha se ve-

locity in media with mod erate velocity anisotropy, both within

and outside the symmetry planes. The model in Figure 5, taken

from Schoenberg and Helbig (1997), corresponds to an effec-

tive orthorhombic medium formed by parallel vertical cracks

embedded in a VTI background medium.

Even in t he model w ith uncommonly st rong velocit y

anisotropy shown in Figure 6, equation (49) deviates from the

exact solution by no more than 10%. Also, note that for this

example the weak-anisotropy approximation does not deteri-orate outside the symmetry planes; in fact, it is in the symmetry

planes where we o bserve the maximum error for some models.

Although the accuracy of the weak-anisotropy approxima-

tion becomes lower with increasing anisotropic coeffi cients,

the error remains relatively small for polar angles up to

about 70. Higher errors of equation (49) near horizontalare not surprising since our notation is designed to provide

a better approximation for near-vertical velocity variations.

To increase the a ccuracy of the wea k-anisotropy a pproxima-

tion at near-horizontal directions (which may be important in

FIG.4. Influence of the parameter(1) on the P-wave phase velocity. The solid curve corresponds to (1) = 0.1, the dotted curveto (1) = 0.65. The other model parameters are: VP0= 3 km /s, VS0= 1.5 km/s, (1) = 0.25, (2) = 0.45, (1) = 0.1, (2) = 0.2,(3) = 0.15,(2) = 0.15.

cross-borehole studies), the isotropic parameter can be cho-

sen as one of the horizonta l velocities.

E qua tion (49) can be used to build ana lytic weak-anisotropy

solutions for such signatures as group velocity, polarization

angle,point-sourcera diation pattern,etc. Ho wever,since these

solutionsreq uire multiple add itional linearizations and involve

the derivatives of phase velocity, their accuracy may be much

lower t han t hat of the phase-velocity expression itself.

DISCUSSION AND CONCLUSIONS

Analytic description of wave propagation in orthorhombic

media can be significantly simplified by replacing the stiff-

ness coefficients with tw o isotropic vertical velocities and

a set of anisotropy parameters similar to the coefficients ,

, and suggested by Thomsen (1986) for vertical transverse

isotropy. The definitions of the anisotropy parameters intro-

duced here are based on the analogous form of the Christoffel

equation in the symmetry planes of orthorhombic and trans-

versely isotropic media. This ana logy impliest hat a ll kinematic

signatures of bod y waves, plane-wave polarizations, and reflec-

tion coefficients in the symmetry planesof orthorhombicmedia

are given by the same equations (with the appropriate substi-tutions of ci j s) as for vertical transverse isotropy. (The only

exception is cuspoidal bra nches of shear-wa ve gro up-velocity

surfaces formed in symmetry planes of orthorhombic media by

out-of-plane slowness vectors.) Since Thomsens notation pro-

vides particularly concise expressions for seismic signatures in

VTI media (Tsvankin, 1996), the new paramete rs can b e con-

veniently used to obtain phase, group, and normal-moveout

velocity, nonhyperbo lic (long-spread) refl ection moveout, and

plane-wave polarization a ngle in the symmetry planes of or-

thorhombic media by adapting the known VTI results. The


13/18

1304 Tsvankin

equivalence with VTI media can even be used to get analytic

expressions for reflection moveo ut in throughgoing symmetry

planes of layered orthorhombic media.

All 2-D seismic processing steps in t he vertical symmetry

planes can be carried out using the algortihms developed for

vertical transverse isotropy, although the a mplitude treatment

in ortho rhomb ic media should be diff erent . To perfo rm 2-D P-

wa ve time processing (NMO, DM O, time migration) in either

symmetry plane, it is necessary to obtain the corresponding

zero-dip NMO velocity and the anisotropy coefficient intro-

duced in Alkhalifah and Tsvankin (1995).

It should be mentioned, however, that the kinematic equiv-

alence b etween the symmetry planes of orthorhombic media

and vertical transverse isotropy does notapply to bod y-wave

FIG. 5. Comparison between the exact P-wave phase velocity (solid curve) and the weak-anisotropy approximation from equa-tion (49) (dashed curve) for the standard orthorhomb ic model taken from S choenberg and H elbig (1997). The velocities are cal-culated a t the six azimuthal phase angles shown on top of the plots. The para meters are VP0= 2.437 k m/s,(1) = 0.329, (2) = 0.258,(1) = 0.083, (2) = 0.078, (3) = 0.106 (the other coefficients that do not influence P-wave velocity are VS0= 1.265 km /s,(1) = 0.182,(2) = 0.0455).

amplitudes in general. In particular, point-source radiation

patterns in the symmetry planes still depend on azimuthal

velocity variations and, therefore, should be studied using

analytic and numerical methods developed for azimuthally

anisot ropic media (Tsvankin and Che snokov, 1990a; G ajewski,

1993). The influence of out-of-plane velocity variations on

body-wave amplitudes also means that near-field polariza-

tions in the symmetry planes of orthorhombic media may be

different from those in VTI media. Indeed, polarization in

the nongeometrical region close to the source is generally

nonlinear a nd depends on the relative amplitudes of at least

the first two terms of the ray-series expansion (Tsvankin and

Chesnokov, 1990a). In contra st, far-fi eld polarizat ions are a d-

equately described by the geometrical-seismics (plane-wave)


14/18


approximation, which can be obta ined by analogy with vertical

transverse isotropy.

The dimensionless anisotropic coefficients conveniently

characterize the magnitude of anisotropy and represent a

natural tool for developing wea k-anisotropy approximations

outside the symmetry planes of orthorhombic media. The

linearized weak-anisotropy approximation for P-wave phase

velocity in terms of the new parameters has the same form as

for vertical transverse isotropy, but with azimuthally depen-

dent coeffi cients a nd . This expression provides suffi cient

accuracy even in models with pronounced velocity anisotropy

and can be used to develop weak-anisotropy solutions for the

group (ray) angle, polarization vector, etc.

The phase-velocity approximation also shows tha t the kine-

matics of P -wa ves in wea kly anisotropic orthorhombic models

FIG. 6. Comparison between the exact P-wave phase velocity (solid curve) and the weak-anisotropy approximation from equa-tion (49) (da shed curve). The mod el para meters are VP0 = 3 km/s, (1) = 0.2, (2) = 0.6, (1) = 0.15, (2) = 0.15, (3) = 0.2.

is independent of the three parameters responsible for shear-

wa ve velocities along the coordinat e axes (the influence ofc44,

c55, a n dc66 isabsorbed by the coefficients).Although the error

of the weak-anisotropy approximation grows with increasing

anisotropy, the exact phase velocity is still governed by only

five anisotropic parameters ((1) , (1) , (2) , (2) , and (3) ) and

the vertical P -wave velocity. This means that all kinematic sig-

natures of P-waves in orthorhombic media depend on six pa-

rameters (rather than nine in the conventional notation), one

of w hichthe vertical P -wave velocityrepresents just a scal-

ing coefficient in homogeneo us media. Furthermore, reflection

traveltimes from mildly dipping interfaces in orthorhombic

media with a horizontal symmetry plane are expected to be

only weakly dependent on the coefficient (3) . In each symme-

try plane, the kinematics of P-waves is determined by three


15/18

1306 Tsvankin

parametersa pair of the corresponding coefficients a nd

and the P-wave velocity along one of the in-plane coordinate

axes. Clea rly, the new pa rameters capture the combina tions of

the stiffness coeffi cients responsible for P -wave kinematic sig-

naturesin orthorhombicmedia. Hence,P -wave inversion algo-

rithms for orthorhombic media should ta rget these anisotropic

coeffi cients instead of t he stiffness components.

It is important to emphasize that the notation introduced

here is convenient to use in orthorhombic media with any

strength of the anisotropy. The most important advantages of

the new anisotropic coefficients, such as the reduction in the

number of parameters responsible for P-wave velocity and

the simple expressions for seismic signatures in the symme-

try planes, remain valid even in strongly anisotropic models. As

shown b y G rechka and Tsvankin (1996), theexactNMO veloc-

ity outside the symmetry planes of a horizontal orthorhombic

layer is a simple function of the two coefficientsdefi ned in the

vertical symmetry planes. Hence, concise weak-anisotropy ap-

proximationscan be regarded as just another advantage of this

notation, as is the case with Thomsen notation in VTI media

(Tsva nkin, 1996).

The parameters introduced here are defined with respect to

the coordina te system associated w ith the symmetry planes. In

seismic experiments, the orientation of the symmetry planes

may be unknown and should generally be determined from

the data. The number of unknown coefficients in this case in-

creases from nine to twelve. I n ma ny typical fractured reser-

voirs, however, one of the symmetry planes is horizontal and

the only additional (tenth) parameter is the azimuth of one of

the vertical symmetry planes.

While the new notation is clearly superior to the stiffness

coefficients in the inversion and processing of seismic data,

the generic parameter set described above may require some

modification for specific acquisition geometries. For example,

in the inversion of cross-borehole da ta it may b e convenient to

replace (1) a nd (2) by similar parameters that govern phase-

velocity variations near horizontal.

The new parameters make up a unified framework for an

analytic description of seismic signatures in orthorhombic,

VTI, and H TI mod els. Bo th vertical and horizonta l transverse

isotropy can be characterized by specific subsets of the di-

mensionless anisotropic paramet ers defined here for the more

complicated o rthorhombic model. For instance, the parame-

ters (V) , (V) , and (V) introduced for H TI med ia by Tsvankin

(1997b) and R uger (1997) are equivalent to the set of the

anisotropic coefficients in one of the vertical symmetry planes

(depending on the orienta tion of the symmetry a xis). This uni-

formity of notation simplifies comparative analysis of seismic

signatures and tra nsition between different anisotropic models

in seismic inversion. On the whole, the new parameters pro-

vide an analytic basis for extending inversion and processing

algorithms to media with orthorhombic symmetry, which is be-

lieved to b e typical fo r realistic fractured reservoirs.

ACKNOWLEDGMENTS

This work was first presented at the 7th International

Workshop on Seismic Anisotropy in Miami in February 1996;

a short version was published in the Expanded A bstracts of the

SE G Annual Interna t. Meeting in Denver, 1996. I am gra teful

to Leon Thomsen (Amoco), D irk G ajewski (University

of Hamburg), Francis Muir (Stanford U niversity), Patrick

Ra solofosaon (IFP) and members of the A (nisotropy)-Team

at the C enter for Wave Phenomena (CWP), particularly

Vladimir G rechka, for useful d iscussions. R eviews by Jack

Cohen, Vladimir G rechka, Ken Larner (Colorado School of

Mines), Sabhashis Mallick (Western) and Leon Thomsen

helped to improve the manuscript. The support for this work

was provided by the members of the C onsortium P roject

on Seismic Inverse Methods for Complex Structures at

CWP, Colorado School of Mines and by the United States

D epartment of Energy (project Velocity Analysis, Paramet er

Estimation, and Constraints on Lithology fo r Transversely

Isotropic Sediments within the framework of the Advanced

Co mputational Technology Initiat ive).

REFERENCES

Alkhalifah, T., and Tsvankin, I., 1995, Velocity ana lysis for tra nsverselyisotropic media: G eophysics, 60, 15501566.

B erryman, J. G ., 1979, L ong-wave elastic anisotropy in tra nsverselyisotropic media: G eophysics, 44, 896917.

B rown, R . J., Law ton, D . C., and Chea dle, S. P., 1991, Scaled physicalmodeling of anisotropic wave propaga tion: Multioffset profiles overan ort horhombic medium: G eoph. J. Int.,107, 693702.

Chapman, C . H ., and Pratt, R. G., 1992, Traveltime tomography inanisotropic mediaI. Theory: G eoph. J. Int., 109, 119.Cheadle, S. P., Brown, R. J., and Lawton, D. C., 1991, Orthorhombic

anisotropy: A physical modeling study: G eophysics, 56, 16031613.C rampin, S., 1985, Evidence for a ligned cracks in the earths crust: First

Break, 3, No. 3, 1215.G ajewski,D., 1993, Ra diationfrom point sourcesingeneralanisotropic

media: G eophys. J. Int., 113, 299317.G ajewski, D., a nd P senck, I., 1996, Weak elastic anisotropy, a per-

turbation approach: 7th I nternat. Workshop on Seismic Anisotropy,Paper no. 73.

G rechka, V., and Obo lentseva, I., 1993, G eometrical structure of shear-wave surfaces near singularity directions in anisotropic media: G eo -phys. J. Int., 115, 609616.

G rechka,V.,a nd Tsvankin,I., 1996, 3-D description of normal moveoutin anisotropic media: 66th Ann. Internat. Mtg., Soc. Expl. G eophys.,Expanded Abstracts, 14871490.

Helbig, K., 1993, Lo ngitudinal directions in media with arbitraryanisotropy: G eophysics,58, 680691.

Helbig, K., and Schoenberg, M., 1987, Anomalous polarization of elas-ticwa vesin transversely isotropicmedia: J.A coust. Soc.Am., 81,No.5, 12351245.

Jech, J., and Psenck, I., 1989, First order approximation method foranisotropic media: G eo phys. J. I nt.,99, 369376.

Korn, G ., and Korn, T., 1968, Ma thematical handb ook f or scientistsand engineers: McG raw-Hill Book Co.

Lynn,H .,Simon, K., and Bates, C., 1996a, Correlation betweenP -waveAVOA and S-wave tra veltime anisotropy in a naturally fractured gasreservoir: The L eading E dge,15, No. 8, 931935.

Lynn, H., Simon, K ., B ates, C ., and Van D oc, R., 1996b, Azimuthalanisotropy in P-wave 3-D (multiazimuth) data : The Lea ding E dge,15, No. 8, 923928.

Mensch, T., and Ra solofosaon, P., 1997, Elastic-wave velocitiesin anisotropic media of arbitrary symmetrygeneralization ofThompsensparameters, , a n d: G eophys. J.I nternat., 128, 4364.

Musgrave, M. J. P., 1970, C rystal a coustics: Holde n D ay.Pa yton, R . G ., 1983, E lastic wave propagat ion in transversely isotropic

media: M atinus Nijhoff P ublishers.Rommel, B. E., 1994, Approximate polarization of plane waves in

a medium having weak transverse isotropy: G eophysics, 59, 16051612.

R uger, A., 1996, Variation of P-wave reflectivity with offset and az-imuth in anisotropic media: Research R eport, Center for Wave Phe-nomena (CWP-218).

1997, P-wave reflection coefficients for transversely isotropicmodels with vertical and horizontal a xis of symmetry: G eophysics,62, 713722.

Say ers, C. M., 1994a, The elastic a nisotropy of sha les: J. G eophys. Res.,99, No . B 1, 767774.

1994b, P -wave propagation in w eakly anisotropic media: G eo-phys. J. Int., 116, 799805.

Schoenberg, M., and Helbig, K., 1997, Orthorhombic media: Model-ing elastic wave behavior in a vertically fractured ea rth: G eophysics(tentatively scheduled fo r No vemberD ecember).


16/18


Thompsen, L ., 1986, Weak elastic a nisotropy : G eophysics, 51, 19541966. 1988, R eflection seismology over azimuthally anisotropic media:

G eophysics,53, 304313.Tsvankin,I., 1995, Normal moveout from dipping reflectors in anisotropic

media: G eophysics,60, 268284. 1996, P-wave signatures and notation for transversely isotropic

media: A n o verview: G eophysics,61, 467483. 1997a, Moveout analysis in tra nsversely isotropic media with a

tilted symmetry axis: G eophysical Prospecting, 45, 479512.1997b, Reflection moveout and parameter estimation for hori-

zontal transverse isotropy: G eophysics,62, 614629.Tsvankin, I., and Chesnokov, E., 1990a, Synthesis of body-wave seis-

mograms from point sources in anisotropic media: J. G eophys. Res.,95(B 7), 1131711331.

1990b, Synthetic waveforms and polarizat ions at the freesurface of an anisotropic halfspace: G eophys. J. Int., 101, 497505.

Tsvankin, I., and Thomsen, L., 1994, Nonhyperbolic reflection moveoutin anisotropic media: G eophysics, 59, 12901304.

1995, Inversion o f refl ection traveltimes for transverse isotropy:G eophysics,60, 10951107.

Wild, P., and C ram pin, S., 1991, The range of ef fects of azimut hal isotropy

and E DA anisotropy in sedimentary basins: G eophys. J. Int., 107, 513529.

APPENDIX A

EXACT PHASE VELOCITY FOR ORTHORHOMBIC MEDIA

The goal of this Appendix is to review the known phase-

velocity equations for orthorhombic media used to derive the

weak-anisotropy approximation for P-wa ve phase velocity in

Appendix B. It is also shown how the exact phase velocities of

all three waves can be computed using the anisotropic param-

eters defined in the main text.

Equation (4) of the main text leads to the following cubic

equation for the phase velocity valid for any homogeneous

anisotropic medium (e.g., Schoenberg and H elbig, 1997):

x3 + ax 2 + bx + c = 0, (A-1)wherex= V2 a nd

a= (G11 + G22 + G33), (A-2)

b = G11G22+G11G33+G22G33G212G213G223, (A-3)

c= G 11G223 + G22G213 + G33G212G11G22G33 2G12G13G23. (A-4)

Through a change of variables (x= y a/3), we can elimi-nate the quadratic term in equation (A-1) and reduce it to the

following form (e.g., Korn and Korn, 1968):

y3 + dy + q= 0, (A-5)with the coefficients

d= a2

3+ b, (A-6)

q= 2a

3

3 ab

3+ c. (A-7)

U sing the fact that the matrixG is symmetric, it can be shown

that the coefficient d is negative (Schoenberg and Helbig,1997). For roots of equation (A-5) to be real, the following

combination o f da nd q should be nonpositive:

Q=d

3

3+q

2

2 0.

Then the three solutions of equation (A-5) can be repre-

sented a s (Korn a nd K orn, 1968)

y1,2,3 = 2

d3

cos

3+ k2

3

, (A-8)

wherek= 0,1, 2,a nd

cos = q2

(d/3)3; 0 .

The phase velocity is found f rom

V2

=y

a/3. (A-9)

The largest ro ot (k= 0) of eq uation (A -8) yields the phasevelocity of the P -wa ve, while the other two roots (k= 1,2) cor-respond to the split shear waves. For equation (A-5) to have

three d istinct ro ots,Q should be negat ive; ifQ= 0, two of theroots are identical. In the case of the Christoffel equation, the

two smaller root s may coincide with each other, which leads to

shear-wave singularities in certain slowness directions. In prin-

ciple, the two bigger roots may also become identical (then it

no longer makes sense to talk about P- a n d S-waves), but this

is an unusual o ccurrence that requires a large magnitude of

velocity a nisotropy.

Equation (A-8) is valid for an arbitrary anisotropic medium

provided the a ppropriate C hristoffel ma trix is substituted into

equations (A-2)(A-4). Here, we are interested in evaluatingthe phase velocity in orthorhombic media as a function of the

new anisotropic parameters defined in the main text. It can be

accomplished by calculating the stiffness coefficients through

the new parameters and substituting them into equations (5)

(10) for the components of the Christoffel matrix. Alterna-

tively, we can express the Christoffel matrix explicitly as a

function of the anisotropic paramet ers introduced above. Rep-

resenting the stiffness coefficients through the new paramet ers

[using eq uat ions (1619), (23), (2428)] and substituting th e re-

sults into the expressions for the components of the Christoffel

matrix [equa tions (5)(10)], w e fi nd

G11

= n2

1V

2

P0

1 + 2(2)+ n22V2S01 + 2(1)+ n23V2S0,

(A-10)

G22

= n21V2S0

1 + 2(1)

+ n22V2P0

1 + 2(1)

+ n23V2S0

1 + 2(1)1 + 2(2) , (A-11)

G33

= n21V2S0 + n22V2S0

1 + 2(1)1 + 2(2)+ n

23V

2P0, (A-12)


17/18

1308 Tsvankin

G12

= n1n2V2P0

1 + 2(2)

D

1 + 2(3)

D;

D= 1 V2S0

V2P0

1 + 2(1)1 + 2(2),

(A-13)

G13

= n1n3V2P0f1 + 2

(2)

f, (A-14)G23

= n2n3V2P0E

1 + 2(1)

E; (A-15)

E= 1 V2S0

V2P0

1 + 2(1)1 + 2(2) .

In equations (A-13)(A-15) it is assumed that c12 + c66 0,c13 + c55 0, and c23 + c44 0, which corresponds to the con-ditions of mild anisotropy (Schoenberg and Helbig, 1997)

discussed above. After computing the Christoffel matrix at a

given slowness direction, we can calculate the coefficients ofthe cubic equation (A-1) and obtain the phase velocity [equa-

tion (A-9)] using the trigonometric solution (A-8).

APPENDIX B

THE WEA K-ANISOTROPY APPROXIMATION FOR P-WAVE PHASE VELOCITY

The exact P -wave pha se velocity in orthorho mbic media can

be ob tained from equations (A-8) (with k= 0) and (A-9) ofAppendix A:

V2P= 2

d3

co s

3

a

3, (B -1)

wherecos = q

2

(d/3)3; 0 ;

da nd q are d efi ned in equa tions (A-6) and (A-7),a is given by

equa tion (A -2).

Our goal is to obtain the linearized wea k-anisotropy approx-

imation for V2 by expanding p, , and a in the anisotropic pa-

rameters (i) , (i), and (i) and dropping quadratic and higher-

order t erms. The cosine function in equa tion (B -1) can be rep-

resented as

co s

3

1

2

18+ 1

4!

4

81 .

Straightforward algebraic transformations show that in the ab-

sence of anisotropy = 0. This implies that , after beingexpanded in the anisotropic parameters, is composed of linear

and higher-order a nisotropic terms with no pure isotropic

contribution. Therefore, 2 contains only quadratic or higher-

order terms in the anisotropic coeffi cients, and in the linearized

wea k-anisotropy a pproximation cos(/3) can be replaced with

unity. Then the pha se-velocity eq uat ion (B -1) reduces to

V2p 2

d3

a3. (B -2)

The next step is to obtain the weak-anisotropy approxi-

mation for the termd/3. U sing equa tions (A -2), (A-3),

and (A-6), we can expressdin the following way through the

components of the Christoffel matrix:

d= 13

G211 + G222 + G233 + 3G212 + 3G213 + 3G223

G11G22 G11G33 G22G33. (B -3)

Substituting the expressions for Gik given in the main text

[equations (A -10)(A-15)] into eq uation (B -3) and retaining

the terms linear in the anisotropic coefficients, we find

3d= n41A1 +n42A2 +n43A3 +n21n22A4 +n21n23A5 +n22n23A6,(B -4)

where

A1= V4P0

1 + 4(2)+ V4S0

1 + 2(1)

2V2P0V2S0

1 + 2(2) + (1)

, (B -5)

A2= V4P0

1 + 4(1

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Anisotropic Parameters and P-wave Velocity for Orthorhombic Media