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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
8/11/2019 Weak Eastic Anisotropy
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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
=
8/11/2019 Weak Eastic Anisotropy
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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.
8/11/2019 Weak Eastic Anisotropy
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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
8/11/2019 Weak Eastic Anisotropy
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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)
8/11/2019 Weak Eastic Anisotropy
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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)
8/11/2019 Weak Eastic Anisotropy
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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
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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.
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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.
8/11/2019 Weak Eastic Anisotropy
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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