Weak Eastic Anisotropy

8/11/2019 Weak Eastic Anisotropy

1/13

GEOPHYSICS, VOL. 51, NO. 10 (OCTOBER 1986); P. 1954-1966, 5 FIGS., 1 TABLE.

e k el stic nisotropy

eon Thomsen

ABSTRACT

Most bulk elastic media are weakly anisotropic. The

equations governing weak anisotropy are much simpler

than those governing strong anisotropy, and they are

much easier to grasp intuitively. These equations indi

cate that a certain anisotropic parameter (denoted 8)

controls most anisotropic phenomena of importance in

exploration geophysics, some of which are nonnegligible

even when the anisotropy is weak. The critical parame

ter 8 is an awkward combination of elastic parameters,

a combination which is totally independent of horizon

tal velocity and which may be either positive or nega

tive in natural contexts.

INTRODUCTION

In most applications of elasticity theory to problems in pe

troleum geophysics, the elastic medium is assumed to be iso

tropic. On the other hand, most crustal rocks are found exper

imentally to be anisotropic. Further, it is known that if a

layered sequence ofdifferent media (isotropic or not) is probed

with an elastic wave of wavelength much longer than the typi

cal layer thickness i.e., the normal seismic exploration con

text), the wave propagates as though it were in a homoge

neous, but anisotropic, medium (Backus, 1962 . Hence, there is

a fundamental inconsistency between practice on the one hand

and reality on the other.

Two major reasons for the continued existence of this in

consistency come readily to mind:

(1) The most commonly occurring type of anisotropy

(transverse isotropy) masquerades as isotropy in near

vertical reflection profiling, with the angular dependence

disguised in the uncertainty of the depth to each reflec

tor

(cf.,

Krey and Helbig, 1956 .

(2) The mathematical equations for anisotropic wave

propagation are algebraically daunting, even for this

simple case.

The purpose of this paper is to point out that in most cases of

interest to geophysicists the anisotropy is weak (10-20 per

cent), allowing the equations to simplify considerably. In fact,

the equations become so simple that certain basic conclusions

are immediately obvious:

(1) The most common measure of anisotropy (con

trasting vertical and horizontal velocities) is not very

relevant to problems of near-vertical P-wave propaga

tion.

(2) The most critical measure of anisotropy (denoted

8) does not involve the horizontal velocity at all in its

definition and is often undetermined by experimental

programs intended to measure anisotropy of rock sam

ples.

(3) A common approximation used to simplify the

anisotropic wave-velocity equations (elliptical ani

sotropy) is usually inappropriate and misleading for

P-

and SV-waves.

(4)Use of Poisson s ratio, as determined from vertical

and S velocities, to estimate horizontal stress usually

leads to significant error.

These conclusions apply irrespective of the physical cause of

the anisotropy. Specifically, anisotropy in sedimentary rock

sequences may be caused by preferred orientation of aniso

tropic mineral grains (such as in a massive shale formation),

preferred orientation of the shapes of isotropic minerals (such

as flat-lying platelets), preferred orientation of cracks (such as

parallel cracks, or vertical cracks with no preferred azimuth),

or thin bedding of isotropic or anisotropic layers. The con

clusions stated here may be applied to rocks with any or all of

these physical attributes, with the sole restriction that the re

sulting anisotropy is weak (this condition is given precise

meaning below).

To establish these conclusions, some elementary facts about

anisotropy are reviewed in the next section. This is followed

by a presentation of the simplified angular dependence of

wave velocities appropriate for weak anisotropy. In the fol

lowing section, the anisotropic parameters thus identified are

used to analyze several common problems in petroleum geo

physics. Finally, further discussion and conclusions are pre

sented.

REVIEW OF ELASTIC ANISOTROPY

A linearly elastic material is defined as one in which each

component of stress

j

is linearly dependent upon every com

ponent of strain (Nye, 1957 . Since each directional index

may assume values of 1, 2, 3 (representing directions x,

y

z

Manuscript received by the Editor September 9, 1985; revised manuscript received February 24, 1986.

*Amoco Production Company, P.O. Box 3385,Tulsa, OK 74102.

() 1986 Society of Exploration Geophysicists. All rights reserved.

95


2/13

Weak lastic nisotropy

9

so that the 3 x 3 x 3 x 3 tensor Cijkt may be represented by

the 6 x 6 matrix

C.

p

.

Each symmetry class has its own pat

tern of nonzero, independent components C.

p

. For

example,

for isotropic media the matrix assumes the simple form

6a

6b

6c

C

1

1--- C

3 3

}

C

6 6

---+ C

4 4

isotropy.

C

l

C

3 3

- 2C

4 4

The elastic modulus matrix in equation 5 may be used to

reconstruct the tensor

Cijkt

using equat ion 2 , so

that

the

constitutive relation in equation 1 is known for the aniso

tropic medium.

The relation may be used in the equat ion of motion e .g.,

Daley and Hron, 1977; Keith and Crampin, 1977a, b, c , yield

ing a wave equation. There are three independent

solutions-

where the three-direction z is taken as the unique axis. is

significant

that

the generalization from isotropy to anisotropy

introduces three new elast ic moduli, rather than jus t one or

two. If the physical cause of the ani sotropy is known, e.g.,

thin layering of certain isotropic media, these five moduli may

not be independent after all. However, since the physical cause

is rarely determined, the general treatment is followed here. A

comparison of the isotropic matrix, equation 3 ,

with the an

isotropic matrix, equation 5 , shows how the former is a de

generate special case of the latter, with

2

1

12 = 21

t

6

31 = 13

t

5

32 = 23

t

4

22 33

l t

2 3

where the 3 x 3 x 3 x 3 elastic modulus tensor

C

i jk i

com

pletely characterizes the elasticity of the medium. Because of

the symmetry of stress

crij

=


3/13

9 6

homsen

The ut il ity of the factors of two in def in itions

SaH8d)

will be

evident short ly. The definit ion of

equation

(Sc) is not unique,

and it may be justified only as in the case considered next

where it leads eventually to simplification. The vertical

sound

speeds for P- and S-waves are, respectively,

(7d)

Before considering the case of weak anisotropy, it is impor

tant to clarify the dis tinc tion between the phase angle 8 and

the ray angle

I

(along which energy propagates). Referring to

Figure 1, the wavefront is local ly perpendicular to the propa-

gation vector k, since k poin ts in the di recti on of maximum

rate of increase in phase. The phase velocity v 8 is also called

the wavefront velocity, since it measures the velocity of ad

vance of the wavefront along

k(8).

Since the wavefront is non

spherical , it is c lear that 0 (also called the wavefront-normal

angle) is different from < > the ray angle from the source point

to the wavefront . Following Berryman (1979), these relat ion

ships may be stated (for each wave type) in terms of the wave

vector

Phase

Wavefront

Angle e

and

Group Ray Angle

->

w ve vector

k

FIG.

1.

This figure graphical ly indica tes the def in it ions of

phase (wavefront) angle and group (ray) angle.

D 8 is

compact notation

for the

quadratic combination:

D 8 == {C

33

- C

4 4

2

+

2[2 C 13

+

C

4 4

2 C

33

- C

44 C 11

+

C

33

2C

44

]

sin?

8

[ C

11

+C

3 3

- 2C

4 4

2

-4 C

I 3

+C

44

Z] sin 8} I/Z

(note mispr in t in the corresponding expression in Daley and

Hron, 1977).

is the a lgebra ic complexity of D which is a

primary

obs ta cl e to use of an isotrop ic models in analyzing

seismic

exploration

data.

.

is u s ~ f u t.o recast

equations

7aH7d) (involving five elas

tic

moduli)

using

notation

involving only two elast ic moduli

(or equivalently, vertical P

and

S-wave velocities) plus three

measures of anisotropy. These three anisotropies

should

be

appropriate

combinations

of elastic moduli which (1) simplify

equations (7); (2) are nondimensional, so

that

one may speak

of X percent P anisotropy, etc.; and 3 r educe to zero in the

degenerate case of i so tropy, as indicated by relat ions

(6),

so

that materia ls with small values (

< i

1)

of

anisotropy may be

denoted weakly

anisotropic.

Some

suitable

combinations are

suggested by the form of

equations (7):

and

= JC

44/P.

Then, equations (7) become (exactly)

v ~ 8 =

u6

[

+ E sin' 8 +

D* 8 }

v ~ v 8

=

[1 +

i E

sin? 8 -

i D* 8 }

v ~ H 8 = [1

+

2y

sin?

8

1

with

* 1

~ 6 { [

40*

D (8) == 2: 1 - U6 1

+

(1 _ ~ 6 u 6 2 sirr' 8 cos

z

8

+

4 1 -

~ U u +

E)E . 4

]1/2

}

_ ~ 6 U 6 Z sm e

1

where the

components

are clearly

k

x

= k 8 sin 8;

k; =

k 8

cos

8;

(9a)

(9b)

(lOa)*

(lOb)*

(lOc)*

(10d)*

)*

(11a)*

b)*

(Sa)

(8b)

and

and the scalar length is

and

k 8

=

k +

k; = O/v 8 ,

llc)*

(8c)

where

is

angular

frequency. The ray velocity

V

is then given

lThis,and other expressions belowwhichare marked with an asterisk

are valid for arbitrary (notjust weak)anisotropy.


4/13

Weak lastic nisotropy 9 7

by

15

8*

D

. 2 8 2 8 . 4 8

1 _

~ ~ a ~ sm cos +

sm .

obtained from four measurements (at least one measurement

at an oblique angle, preferably 45 degrees, is required). As is

shown below, this omitted datum is the most

important

one

for most applications in petroleum geophysics. Hence, these

partial studies are omitted from Table

1.

In addit ion to intrinsic anisotropy, one must consider ex

trinsic anisotropy, for example, due to fine layering of iso

tropic beds.

Many

examples could be listed, but it is not clear

how to pick representative examples. This table has been lim

ited to the particular examples defined by Levin (1979) (these

choices are discussed further below).

With Table 1 as justification of the approximation of weak

anisotropy, it now makes sense to expand equations

10

in a

Taylor series in the small parameters 10 0*, and y at fixed 8.

Retaining only terms linear in these small parameters, the

quadratic

D

is approximately

12 *

(13b)*

(13a)*

=

v

sin

8

+ cos

8) v

cos

8 -

sin 8)

(

1

dV)

tan 8 dV

= tan 8 +; d8 1 - d8

tan


5/13

9 8

homs n

Table

1.

Measured anisotropy in sedimentary rocks. This table compiles and condenses virtually all published data on

anisotropy of sedimentary rocks, plus some related materials.

Sample

Conditions

V f s)

V

f s)

6*

6

p g/cm

3

p m/s)

s m/ s

Tay1or

1

P

=

0

Pa

11 050

6000

0.110 -00127

-0.035

0.255 2.500

sandstone

urated 3 368

1 829

Mesaverde 4903)2 P

=

27.58 Pa

14860 8869 0.034 0.250 0.211 0.046 2.520

mudsha1e s ~ undrnd 4 529

2 703

Mesaverde 4912 2 P

27.58

Pa

14 684 9232 0.097 0.051 0.091 0.051 2.500

immature

sandstone s ~ H

undrnd

4476

2814

Mesaverde 4946 2

P

27.58

Pa

13449 7696 0.077

-0.039 0.010

0.066 2.450

immature

sandstone

s ~ undrnd

4099

2 346

Mesaverde 5469.5)2

P

=

27.58

Pa 16 312 9 512 0.056 -0.041 -0.003

0.067 2.630

s ty limestone

s ~ f a

undrnd

4972

2899

Mesaverde

5481.3)2

P

27.58

Pa

14 270

8434

0.091 0.134 0.148

0.105

2.460

immature sandstone

s ~

undrnd 4 349 2 571

Mesaverde

5501 2

P

27.58

Pa

12 887 6 742 0.334

0.818 0.730

0.575

2.590

c1aysha1e s ~ undrnd

3 928 2 055

Mesaverde 5555.5)2

P e ~ =

27.58

Pa

14 891

8877

0.060 0.147

0.143

0.045 2.480

immature

sandstone

sa

,undrnd

4 539 2 706

Mesaverde

5566.3)2

P

=

27.58

Pa

14596

8482

0.091

0.688 0.565 0.046 2.570

laminated

si l tstone s ~

undrnd

4449

2 585

Mesaverde

5837.5)2

P

27.58

Pa

15 327

9 294 0.023

-0.013

0.002 0.013 2.470

immature

sandstone

s ~ f t

undrnd

4672

2 833

Mesaverde

5858.6)2

P

27.58

Pa

12448

6 804

0.189 0.154 0.204 0.175

2.560

clayshale s ~ undrnd

3 794

2 074

Mesaverde

6423.6)2

P

27.58

Pa

17 914

10 560

0.000

-0.345

-0.264

-0.007

2.690

calcareous sandstone

s ~ undr nd

5460

3 219

Mesaverde

6455.1)2

P

=

27.58 Pa 14496

8487

0.053 0.173

0.158 0.133 2.450

immature sandstone

s ~ undrnd

4418

2 587

Mesaverde

6542.6)2

P

=

27.58 Pa 14451

8339

0.080

-0.057

-0.003

0.093

2.510

immature

sandstone

s ~ undrnd

4405

2 542

Mesaverde

6563.7)2

P

=

27.58

Pa 16644 9837

0.010 0.009

0.012

-0.005

2.680

mudshale

sHa,

undrnd

5073

2998

Mesaverde

7888.4)2

P 27.58

Pa 15 973

9549

0.033 0.030

0.040

-0.019

2.500

sandstone

s ~

undrnd

4869

2911

Mesaverde

7939.5)2

P

ef a

=

27.58

Pa

14096

8106

0.081

0.118

0.129

0.048

2.660

mudshale sa

,undrnd 4296

2471

1Rai and

Frisi l lo, 1982

2Kelley,

1983 number in

parentheses is

depth label)


6/13

Weak lastic nisotropy

9 9

Table I. Continued

Sample

Conditions

V f l s V fls

6*

y

p g cm

p m/s

s m/s

Mesaverde P

20.00 MPa

11 100

8000 0.065

-0.003

0.059 0.071 2.35

shale 350 3

d

e

ff

3383 2438y

Mesaverde P

=

20.00

MPa

12 100

9100

0.081

0.010 0.057 0.000 2.73

sandstone

l582 3

deff

3688

2 774

y

Mesaverde

P 20.00 MPa 12800

8800

0.137

-0.078 -0.012

0.026 2.64

shale l599 3

d

e

ff

3901 2682y

Mesaverde P 50.00 MPa 13 900 9900 0.036

-0.037 -0.039

0.030 2.69

sandstone

l958 3

d

e

ff

4237

3018

y

Mesaverde P 50.00 MPa

15 900 10400

0.063

-0.031 0.008

0.028

2.69

shale l968 3

d

e

ff

4846 3 170

y

Mesaverde

P 50.00

MPa

15200

10600

-0.026 -0.004 -0.033

0.035 2.71

sandstone

3512 )3

d

e

ff

4633

3 231

y

Mesaverde P

50.00 MPa 14300

10000

0.172

-0.088

0.000 0.157 2.81

shale 3511 3

d

e

ff

4359 3048

y

Mesaverde P

=

20.00 MPa

13000 9600

0.055

-0.066 -0.089

0.041

2.87

sandstone

3805

3

d

e ff

3962

2926

y

Mesaverde

P

eff =

50.00

MPa

12 300

8600

0.128

-0.025

0.078

0.100 2.92

shale

3883 3

dry

3 749

2 621

Dog

Creek

4

in s i tu ,

6 150

2 710

0.225 -0.020

0.100

0.345

2.000

shale

z

=

143.3 m

430 f t

1 875 826

Wills Point

4

in

situ,

3470

1 270 0.215

0.359 0.315 0.280 1.800

shale

z

=

58.3 m 175 t

1 058 387

P

=

0

13 550

7810

0.085 0.104

0.120 0.185

2.640

s ~ f t u ~ r

4 130 2 380

Cotton

Va

ey

5

P = 111.70

MPa

15 490

9480

0.135 0.172 0.205

0.180

2.640

shale

s ~ undrnd

4 721 2 890

Pierre

6

1n situ,

6 804 2850 0.110 0.058

0.090 0.165 2.25?

shale

z = 450 m

2 074 869

in s i tu ,

6 910

2910

0.195 0.128 0.175

0.300 2.25?

z = 650 m

2 106 887

in situ,

7 224 3 180

0.015

0.085 0.060

0.030 2.25?

z = 950 m

2 202 969

shale

5000)7

P = 0

10 000

4890

0.255 -0.270

-0.050

0.480

2.420

c

satd,

undrnd

3 048 1490

3Lin, 1985 number in parentheses is

depth

label

4Robertson and

Corrigan,

1983

5 To saya, 1982

6White,

et

a l . , 1982

7Jones

and Wang 1981

depth

of

core

shown)


7/13

96

homs n

Table 1. ontinued

Sample

Conditions V (f ls) V ( f l s ) t

8* 8

I

p g/cm

3

}

p m/s} S m/s}

P

= 101. 36 Pa

11080

4890

0.200

-0.282 -0.075 0.510

2.420

s ~ t

undrnd

3377 1490

Oil

Sha1e

8

unknown

13880

8330

0.200 0.000 0.100 0.145

2.370

4231 2539

Green

River

9

P

= 0

13670

7980 0.040

-0.013

0.010 0.030

2.310

c

shale

satd, undrnd

4 167

2432

P = 68.95 MPa, 14450 8470 0.025

0.056 0.055 0.020

2.310

c

4404 2 582atd , undrnd

Berea

l o

P

f

= 68.95

Mpa,

13800 8740 0.002 0.023

0.020 0.005

2.140

sandstone

s ~ t u r t e

4206

2664

Bandera

l o

Peff =

68.95

Mpa,

12 500

7 770

0.030

0.037 0.045 0.030

2.160

sandstone

sa urated

3810

2 368

Green

Ri ve r l

P

= 202.71

MPa,

10800

5800

0.195 -0.45 -0.220 0.180

2.075

shale

a r r

dr y

3292

1 768

Lance II

P

= 202.71 Mpa,

16 500 9800 -0.005 -0.032

-0.015 0.005 2.430

c

sandstone a ir

dr y

5 029 2 987

F t. Un

i

on

P

= 202.71 Mpa,

16 000

9650

0.045 -0.071 -0.045

0.040

2.600

c

s i l t s t o n e

a I r dr y

4877

2 941

Timber

Mtn 11

P

=

202.71

Mpa, 15 900

6090 0.020

-0.003

-0.030

0.105

2.330

t u f f

c

4846

1856

Ir dr y

Muscovite

l 2

P

=

0

14500

6860 1.12

-1.23

-0.235

2.28

2.79

crystal

c

4420

2091

Quartz

crysta1

12

P

e f f

0 20000

14700

-0.096

0.169 0.273

-0.159

2.65

hexag.

approx.)

6 096

4481

Calcite

crysta1

12

P

e ff

a

17 500

000

0.369 0.127

0.579

0.169

2.71

hexag.

approx.)

5 334

3353

B i o t i t e

crystal

l 2

P

e ff

a

13 300 4400

1.222

-1.437 -0.388

6.12

3.05

4054

1 341

Apatite

c r y s t a l

12

Pe ff

0

20800 14400

0.097

0.257

0.586

0.079

3.218

6 340

4389

Ic e

I

crystal

l 2

P

=

a

11 900

5

500

-0.038 -0

.1 0

-0

.164

0.031 1.064

e f f

40F

3 627 1 676

A1uminum-1ucite

l 3

clamped; oi 1

9410

4430

0.97

-0.89 -0.09

1. 30

1.86

composite between layers 2 868 1 350

8Kaarsberg,

1968

9podio et

a l . , 1968

I OKi ng , 1964

11S

chock

e t a l . , 1974

12Simmons and Wang, 1971


8/13

Weak Elastic Anisotropy 9

able ontinued

Sample

Conditions

V

(f ls)

V

(f ls)

8*

y

p(g/cm:J)

P(m/s)

s

m l

s )

Sandstone-

hypothetical

9 871

5 426

0.013

-0.010 -0.001

0.035

2 31

sha1e 14

50-50

3 009 1 654

SS-anisotropic

hypothetical

9 871 5 426

0.059 -0.042

-0.001

0.163

2

shale 14

50-50

3 009

1 654

Limestone-

hypothetical 10845

5 968

0.134

-0.094 0.000 0.156

2.44

shale 14

50-50

3 306 1 819

LS-anisotropic

hypothetical 10845 5968 0.169 -0.123

0.000 0.271

2.44

sha1e 14

50-50

3 306 1 819

Anisotropic hypothetical 9 005

4 949 0.103

-0.073

-0.001 0.345

2.34

shale 14

50-50

2 745 1 508

Gas sand- hypothetical

4624

2560

0.022

-0.002

0.018 0.004

2.03

water

sand 14 50-50 1 409 780

Gypsum-weathered

hypothetical

6 270

2 609 1.161

-1.075 -0.140 2.781

2.35

material } 1

50-50 1 911

795

13Dalke, 1983

14Levin, 1979

that

(1Sb)

is, in fact, the fractional difference between vertical and hori

zontal P velocities, i.e., it is the parameter usually referred to

as

the

anisotropy of a rock. [Without the factor of 2 in

equation (Sa), s would not correspond to this common usage

of the

term

anisotropy. J

However, the parameter 8 which controls the near-vertical

anisotropy is a different combination of elastic moduli, which

does

not

include (i.e., the horizontal velocity) at all. Since

the

E

term is negligible for near-vertical propagation, most of

one s intuitive understanding of E is irrelevant to such prob

lems. For example, it is normally true that horizontal

P

veloci

ty is greater than vertical

P

velocity, i.e., E > 0 (Table 1).How

ever, this is of little use in understanding anisotropy in near

vertical reflection problems, because E is multiplied by sin 6

in equation (16a). The near-vert ical anisotropic response is

dominated by the 8 term,

and

few can claim intuitive

familiarity with this combination of parameters [equation

(17)]. In fact, Table 1 shows a substantial fraction of cases

with negative 8.

Figures 2

and

3 show

P

wavefronts radiating from a point

source into two uniform half-spaces, each with positive

E

but

different values of 8, one positive and one negative. These are

jus t plots of

V

p

in polar coordinates. is clear from the

figures that quite complicated wavefronts may occur. Similar

complications arise with

V

wavefronts, although no actual

cusps or triplications are present in the limit of weak ani

sotropy, (The term V

N O

in these figures ;s discussed in the

next section.)

At this point, where the linearization procedure has identi

fied 8 as the crucial anisotropic parameter for near-vert ical

P-wave propagation, it is appropriate to discuss a special case

of transverse isotropy which has received much attention: el

liptical anisotropy. An elliptically anisotropic medium is

characterized by elliptical

P

wavefronts emanating from a

point source. is defined cf. Daley and Hron, 1979) by the

condition

8

E

elliptical anisotropy.

Notable for its algebraic simplicity, this special case, is, of

course, defined by a mathematical restriction of the parame

ters which has no physical justification. Accordingly, one may

expect the occurrence of such a case in nature to be van

ishingly rare. In fact, Table 1 shows that 8

and

E are not even

well-correlated (being frequently of opposite sign), so that the

assumption of their equali ty may lead to serious error. This

point is reinforced by Figure 4, which plots 8 versus E for the

rocks of Table I and shows for comparison the elliptic con

dition defined above. The inadequacy of the elliptic assump

tion is immediately obvious.

These results are for intrinsic anisotropy. Berryman (1979)

and Helbig (1979) show that if anisot ropy is caused by fine

layering of isotropic materials, then strictly

8 E isotropic layers.


9/13

1962

Thomsen

W V FRONTS

For completeness, note from equation (l6c) that

SOME APPLICATIONS OF WEAK ANISOTROPY

so that

y

corresponds to the conventional meaning of ..SH

anisotropy. Also note that in the elliptical case

0 = E,

the

functional form of equation (16a) becomes the same as that of

equation (l6c). This demonstrates that SH wavefronts are el

liptical in the general case; this is true even for strong ani

sotropy.

Returning to the central point of 0 as the crucial parameter

in near-vertical anisotropic P-wave propagation, some dis

cussion is necessary regarding the reliability of its measure

ment.

is clear from equations (16a) and (18b)that

0

may be

found directly (in the case of weak anisotropy) from a single

set of measurements at e

=

0,45 and 90degrees:

1)

=

p 1 t /4 l /Vp 0

-

-

[V

p 1 t /2 l /Vp 0

-1 ]

Because of the factor of 4, errors in V

p 1t /4 /V

p 0

propagate

into 0 with considerable magnification. In fact, if the relative

standard error in each velocity is 2 percent, then the (indepen

dently) propagated absolute standard error in 1) is of order .12,

which is of the same order as 0 itself (Table

1 .

The propaga

tion of this error through equation (17) implies that the rela

tive error in C 13 is even larger. To reduce these errors to

within acceptable limits requires many redundant experiments,

of both V

p

e and V

sv

e). Since the measurement at 45 degrees

may involve cutting a separate core, questions of sample het

erogeneity (as distinct from anisotropy) naturally arise. The

data ofTable 1 should be viewed with appropriate caution.

ISOTROPIC

.

.

ANISOTROPIC:

e

= 0.20

6

= 0.20

ISOTROPIC:

y

............

e

= 0.20

---- ---

6 = 0.20

W V FRONTS

FIG.

3. This figure indicates a plausible anisotropic wavefront

0

=

E .

The curve marked

V

N O

is a segment of the wave

front that would be inferred from isotropic moveout analysis

of reflected energy. V

N O

< v ert since

0

< o.

FIG. 2. This figure indicates an elliptical wavefront 0

=

E . The

curve marked V

N O

is a segment of the wavefront that would

be inferred from isotropic moveout analysis of reflected

energy. V

N O

> v ert.

Group

velocity

i.e., it is linear in anisotropy. Therefore, the group velocity

[equation (l4)J expanded in such terms,

(19)

(20c)

(20a)

(20b)

For

the quasi-P-wave, the derivative in equation (14) IS

given for the case of weak anisotropy [equation (16a)] by

1

av [

J

e

=

sin e cos e 0 +

2 E

- 0 sin? e ,

v

p

v

p

V

p

= vp e .

Similarly for the other wave types,

and

Note that equations (20)do not say that group velocity equals

phase velocity (or equivalently, that ray velocity equals wave

front velocity). These equations do say tha t at a given ray

is quadratic in anisotropy. Therefore, this term is neglected in

the linear approximation

As a final remark, note that, for small

e

the last term in

equation (l6a), E sin

e,

might be comparable to a neglected

quadratic term in

0

2

sin e, or OE sin? e. However, all neglect

ed terms quadratic in anisotropy are multiplied by trigono

metric terms of order sin e cos eor smaller, and hence are in

fact negligible, even for small

Berryman (1979) writes a perturbation approximation to

equation (10) in which the small parameter is a combination

of anisotropic parameters and trigonometric functions. His

derivation, which is also valid for strong anisotropy at small

angles, reduces to the present equations (l6a) and (16b) for

weak anisotropy (at any angle). His approximation is less re

strictive than the present one, but it yields formulas which are

less simple (and which do not readily disclose the crucial role

of the parameter 0, or contrast it with

E .

is therefore an

approximation intermediate between the exact expressions

(10)and the intuitively accessible approximation 16 .

Backus (1965) treats the case of weak anisotropy of arbi

trary symmetry, defining anisotropy differently than is done

here, without implementing criteria (1) and (2) which follows

equation (7d).

Consideration of the linearized

SV

result equation (16b)

immediately confirms the well-known special result that ellip

tical P wavefronts (0 = E) imply spherical wavefronts (no e

dependence). However, equation (16b) shows that the more

general case of weakly anisotropic but nonelliptical media is

still algebraically tractable.


10/13

Weak Elastic Anisotropy

1963

group angle

4>

if the corresponding wavefront normal

phase angle e is calculated using equations 22 below, then

equat ions 16 and 20 may be used to find the ray group

velocity.

r up angle

velocity formulas [equation 16 ], does not constitute a viola

tion of the linearization process, even though products of

small quantities implicitly appear. In linearizing equations 10

in terms of anisotropy, the angle S was held constant, i.e.,was

not part of the linearization process. The linear dependence of

S on anisotropy, at fixed 4> is then given by equations

22 .

The relationship

13

between group angle

4>

and phase

angle

e

is, in the linear approximation,

Polarization angle

*

The particle motion of a quasi-P-wave is polarized in the

direction of the eigenvector gp Daley and Hron, 1977 ,where

Since this is not parallel to the propagation vector k

p

[equa

tion

11

], the wave is said to be quasi-longitudinal, rather

than strictly longitudinal; similar remarks apply to the quasi

SV-wave. The angle l,p between k, and gpisgiven by

I

2 2

cos

l,p

=

k k

p

gp

=

p

sin

Sp+ m

p

cos

p

). *

pY

p

kpg

p

Expressions for the scalars

t

p and

m

p

are given by Daley and

Hron 1977 ;in the case of weak anisotropy, these expressions

reduce to

21

22c

22b

tan SH = tan 8

sH 1

+ 2y .

tan

4>sv

= tan

sv

[1 + 2 ; s -

0 1

- 2 sin? Ssy

J

and for SH-waves,

Similarly, for SV-waves,

tan

= tan

8[

+ sin 8

1

COS

8

1

For P-waves, use of equation 19 fully linearized in equation

21 leads to

tan >p = tan

[

+

20

+

4 - 0 sin?

e

p

} 22a

These expressions, along with equations 16 and 20 , define

the group velocity, at any angle, for each wave type.

Note tha t inclusion of the anisotropy terms in the angles

[equations 22 ], when used in conjunction with the phase-

and

Comparison of

P nisotropies

0.8

Data on

Crustal

Rocke

Lab Field

~ ; ~ o ~

v ot

0

.

, .

~ -

-

,

-

,

.

.

0.4

0 2

0.8

O O r : : _ =

0 2

CD

G

E

t

Q

o

o

GO

C

Weak Eastic Anisotropy

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