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A SONIC METHOD FOR
PETROGRAPHIC ANALYSIS

by

Stephen Edward Tilmann

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE

Department of Geology

197%

ACKNOWLEDGMENTS

To Dr. Hugh Bennett, my thesis advisor, for his patience
and guidance; to Dr. Tom Vogel, for first arousing my
curiosity in this subject, and his time spent with me in
the field; to Dr. Bob Ehrlich, for giving me a perspec-
tive; to Adrian Bass, for moral encouragment; and to those
friends of mine in the Department, for making this exper-

ience, I gratefully extend my acknowledgments.

VOL. 73,

NO. 35 JOURNAL OF GEOPHYSICAL RESEARCH

A Sonic Method for Petrographic Analysis

STEPHEN E. TILMANN AND HUGH F. BENNETT

Department 0/ Geology, Michigan State University, East Lansing, Michigan 48824

A sonic method as a. tool for detecting and describing preferred crystallographic orienta-
tion has been proposed by Bennett (1972). The Q ellipsoid is a theoretical surface whose
magnitude for any direction is the sum of the squares of the three seismic wave phase ve-
Iocities for that direction. The orientation of the ellipsoid relative to the sample is controlled
by crystal orientation and structural effects of the sample. For completely isotropic samples
the Q surface is a sphere; for anisotropic samples the Q surface is ellipsoidal. Sample
homogeneity is testable by the closeness of ﬁt of the velocity data. to the ellipsoidal surface.
In this respect, crystal aggregates can be considered to behave as elastic long-wave equivalents
to single crystals. The baraboo quartzite, a Grenville marble, and a. plastically deformed
granite boulder are analyzed according to the Q ellipsoid technique. Optical analysis is per-
formed on the quartzite and marble. The oriented Optical indicatrixes for the individually
measured crystals are summed, an ellipsoidal surface characterizing the preferred crystal
orientation direction of the sample thus being produced. For the quartzite, which is nearly
isotropic, the optical surface and the sonic surface closely coincide. This situation is evidence
that the sonic orientation accurately reﬂects the subtle crystallographic orientation. The
marble displays a strong crystallographic orientation. as well as a pronounced micaceous
layering. The orientation of the Q ellipsoid reﬂects the net effect of this structural fabric
and the crystallographic fabric. The granitic boulder was plastically deformed into an
ellipsoidal shape. The shape axes and the Q ellipsoid axes closely coincide, the indication
being that the Q ellipsoid technique may be useful in describing regional tectonic forces.

DECEMBER 10, 1973

The concept of the Q ellipsoid as a tool for
detecting and describing preferred crystal-
lographic orientations has been developed by
Bennett [1972]. The purpose of this paper is
to test this seismic model with empirical data
gathered from several rock types

The Q ellipsoid is a theoretical surface whose
value for any particular direction is the sum
of the squares of the three seismic wave type
phase velocities in that direction. It has been
proved that the principal axes of the Q ellipsoid
always coincide with the optical indicatrix axes
for a single crystal in the cubic through
orthorhombic systems [Bennett, 1972]. For
cubic crystals the Q surface reduces to a sphere.
For uniaxial crystals the Q surface is an el-
lipsoid of revolution, and for biaxial crystals
the Q surface is a triaxial ellipsoid.

If a crystal aggregate is considered as an
elastic long-wave equivalent to a single crystal,
then the locus of values, of which each value
is the sum of the squares of the three
seismic wave velocities for any particular di-
rection, should be represented by an ellipsoidal

Copyright © 1973 by the American Geophysical Union.

surface; i.e., the material behaves as a homoge-
neous pseudosingle crystal. Further, the prin-
cipal axes of this ellipsoid would be controlled
by preferred crystallographic“ orientation and
structural effects within the rock material [Ben-
nett, 1972]. Thus, if the Q surface is ellipsoidal.
then (1) the material is homogeneous and
anisotropic, (2) the principal anisotropic direc-
tions are described by the ellipsoid principal
axes, and (3) the percent difference between
the ellipsoid principal axes is a measure of the
degree of elastic anisotropy, which is controlled
by anisotropic crystal orientation and structural
effects. ,

The concept of the Q ellipsoid need not be
restricted to rock materials of a single phase.
Indeed an advantage of the Q ellipsoid concept
is that a multiphase material can be treated
as a pseudosingle crystal in terms of elastic
behavior and crystallographic orientation.

It should be pointed out that the P wave
velocity surface (also the S, and S. surfaces)
need not conform to any simple geometric
shape. For rock materials the three velocity
surfaces could be controlled primarily by struc-
tural effects. such as microfractures, and ap-

8463

8464
pear to be unrelated to any preferred crystal
orientation within the material. This condition
may be more noticeable where orientations are
weak and contribute less to the anisotropy. Also,
since the P wave or 8 wave velocity surfaces
may be quite complex in shape, the maximum
value chosen from just a few measurements
may not be the true velocity surface maximum.

MODEL

The calculated value Q.’ of the Q ellipsoid
for the ith direction is given by

Qt = Qa’/P = (V12 ‘l" V:2 + V32).‘ (1)
where p is the material density, V1 is the P
wave velocity in the ith direction, and V,
and V, are the velocities of the two orthogonally
polarized shear waves for the ith direction [Ben-
nett, 1972]. By deﬁning a polarization plane as
the plane that contains the propagation direction
and shear wave particle motion, it can be stated
that the two polarization planes are nearly
orthogonal for any propagation direction. Thus
the values of V, and V. can usually be measured
uttambiguously for any 'particular direction
[Tilmann and Bennett, 1973]. Since the density
tetm is constant, it may be incorporated into
the Q.’ term without affecting the shape or the
orientation of the Q ellipsoid. Thus the calcu—
lated value of the Q ellipsoid in the ith direction
will be. referred to as Q,.

‘ The least squares value 01.3, of the Q
ellipsoid for the ith direction is given by

' 2 2 2
QLS.‘ = It an “l" m,- aza + 7h 033

+ 2mtnia23 + zniliasl ‘l' 2ltmia12 (2)

where (ll, mi, 12.) are the directional cosines of
the. ith direction relative to any orthogonal set
of axes 2:, y, and 2. In practice the 1:, y, and z
axes are conveniently chosen relative to the
sample being analyzed. The or terms are the
elements of a 3 X 3 symmetric ellipsoid ma-
trix.

The elements of the a matrix are determined
by the least squares method outlined by Nye
[1957]. This procedure is based on the matrix
equation relating the Q. values to the directional
cosine matrix 0 and the a matrix by

Q=9a (3)

TILMANN AND BENNETT: SONIC PETROGRAPHIC ANALYSIS

The Q matrix elements are the measured Q(
values from (1). The 0 matrix is constructed
by using the l, m, and n coefﬁcients of (2). The
a matrix is then determined by solving (3)
for a, resulting in

a = (GM-'01) (4)

This is the computational form for determin-
ing the a matrix. (See Nye [1957, pp. 164-
165] for a more complete treatment of this
procedure.)

The principal axes of the Q ellipsoid are
found by the successive approximation method
[Nye, 1957]. The procedure entails successive
relocation of a vector normal to the surface
of the ellipsoid until it corresponds to the minor
axis. By inverting the a matrix the major axis
is similarly located. The intermediate axis is
the cross product of the major and minor axes.
By using these directional cosines of the major,
minor, and then intermediate axes in (2), the
magnitude of these axes is easily determined.
(Also see Nye [1957, pp. 165-168] for a more
complete treatment of this procedure.)

Comparison of the measured ellipsoidal values
Q, and the calculated ellipsoidal values Q”,
provides a test for sample homogeneity. The
values of interest are, ﬁrst,

n 1/2
a; = [2 (Qt — Q;,,)2/n] (5)
where Q,- is the measured ellipsoidal value in the
ith direction (1), Q; is the arithmetic mean
measured value, and n is the number of propaga-
tion directions i measured; second,

0;. = [:2 (01.3.

— est/n] (6)

where Q”, is the calculated ellipsoidal value (2)
in the ith direction; and, third,

0. = [2”: (Qt _ QI.S:‘)2/n] (7)

The standard deviation 0: can be thought of
as the deviation of the measured ellipsoidal
values from the best-ﬁt sphere to the measured
values, 0;. is the deviation of the calculated
ellipsoidal values from the best-ﬁt sphere, and
a, is the deviation between the measured and the
calculated ellipsoidal values. If all the data

TILMANN AND BENNETT: SONIC PETROGRAPHIC ANALYSIS

points fall exactly on the ellipsoidal surface,
then a, = 0, and 0;, = a;

Sample homogeneity and elastic behavior as a
pseudosingle crystal are indicated by the rela-
tionship

era-20;.>a. (8)

Sample inhomogeneity is indicated by the re-
lationship

6: > a. > 6;. (9)

The inhomogeneity may be in the form of
variance of the preferred crystal orientation
within the sample or irregular compositional or
structural differences within the sample. In-
homogeneity is not consistent with the concept
of elastic behavior as a pseudosingle crystal.

TEST

The baraboo quartzite, a Grenville marble,
and a plastically deformed granitic boulder
were analyzed according to the Q ellipsoid
technique. The measuring apparatus used for
determining the elastic velocities is described by
Tilmann and Bennett [1973]. In addition,
optical petrofabric analyses were performed
on the quartzite and marble.

The quartzite is an essentially pure quartz
rock that has undergone slight metamorphism.
In hand sample and thin section, no obvious
structure was observed that would inﬂuence the
elastic anisotropy. The marble comprises cal-
cite and a well-deﬁned micaceous layering. The
granite boulder has been plastically deformed
during metamorphism. The shape of the boulder
is roughly that of a triaxial ellipsoid, with the
minor axis normal to the plane of outcrop folia-
tion.

 

 

TABLE 1. Optical Ellipsoid Axes
Magnitude Directional Cosines Symbol
quartzite
41.98 (-0.986, 0.155, 0.058) hm
41.82 (0.161, 0.983, 0.087) mo
41.94 (0.043, -0.095, 0.995) Io

humble
12.46 (0.398, -0.859, 0.322) Mb
12.10 (0.150, 0.285, 0.947) Mg
12.20 (0.965, 0.425, 0.015) I0

8465
Bamboo

Ouuune

 

a}; -1159 tonic optic
69,-1.601 M A .
0'e - .72s "I A (.3
I A o
XIVP mu

Fig. l. Equal-area projection of 200 quartz c

axes for the baraboo quartzite. The direction of
the observed maximum P wave velocity, the 0p-
tical surface, and the sonic Q ellipsoid principal
axes are also plotted, M, m, and I being the el-
lipsoidal major, minor, and intermediate princi-
pal axes, respectively. Close coincidence between
the axes of the two surfaces indicates that the
orientation of the Q ellipsoid describes the pre—
ferred crystallographic orientation.

Baraboo quartzite. The results of the optical
petrofabric analysis on the quartzite are pre-
sented in Figure 1. This diagram is an equal-
area projection of the measured quartz c axes.
This projection was not contoured in order to
emphasize the diffuse nature of the orientation.
The oriented optical indicatrixes of the indi-
vidually measured crystals were summed in
order to produce an ellipsoid analogous to an
optical indicatrix surface. This ellipsoidal sur-
face provides a convenient parameter that de-
scribes the preferred crystallographic orienta-
tion. The magnitude and directional cosines of
the ellipsoidal axes are listed in Table l and are
shown in Figure l.

The three velocity measurements V1, V3, and
V., which are referred to as a data set, were
taken over nine directions. Each data set was rep-
licated 4 times for each direction, the result

8466 TILMANN AND BENNETT: SONIC PETROGRAPHIC ANALYSIS

TABLE 2 .

Oi Values

 

Propagation Direction

 

Replicate 1 2 3 4

g

5 6 7 8 9

Quartzite (F a 50.11)

1 49.581 52.443 53.968 54.851 53.629 55.214 50.649 55.575 51.071
2 50.943 53.084 54.484 55.009 54.027 55.250 52.278 55.643 51.134
3 51.021 53.103 54.248 55.331 54.113 55.668 52.578 55.748 51.989
4 50.872 53.396 54.960 55.132 54.746 55.725 52.900 56.458 51.679
Marble (F I 64.75)
1 48.133 38.617 48.049 41.396 37.184 41.716 43.176 44.424 46.309
2 48.089 39.060 45.401 42.308 37.476 41.501 42.953 44.080 42.994
3 49.645 38.136 45.844 42.160 37.999 41.411 42.803 45.482 47.332
4 49.465 38.576 45.989 42.068 37.850 42.058 43.576 45.837 47.171
Granite (F I 73.20)
1 55.303 60.130 58.411 59.496 60.608 57.881 57.322 59.896 59.282
2 56.157 60.051 58.994 59.227 60.330 57.057 56.915 60.098 59.828
3 56.115 59.936 58.217 58.947 59.809 57.142 57.936 59.895 59.812
4 55.851 59.623 58.954 58.872 60.308 58.098 57.447 60.271 59.888

 

being 36 data sets. The order in which the data
sets were measured was randomized. The Q,-
values from these data sets were determined
according to (1) and are listed in Table 2. The
mean velocities for each direction (Table 3) were
used in computing the Q ellipsoid. The magnitude
and directional cosines of the least squares Q

TAILS 3. Directional Cosinea and Hean Velocities

 

Mean Velocity, kaisec

 

 

ellipsoid axes are listed in Table 4 in the section
on quartzite. The plot of these axes is shown in
Figure 1. The mean Q,- values with their re-
spective Q”, values are listed in Table 5. This
table also presents the 0;, 0;" and a. values. The
maximum P wave velocity observed Vp m is
shown in Figure l.

Grenville marble. The results of the optical
petrofabric analysis on the marble are shown in
Figure 2. Since the marble displayed a strong
preferred orientation, the c axes, (0001), were
contoured according to the Mellis method
[Turner and Weiss, 1963]. The observed mica-
ceous layering is in the N -S vertical plane. The
c axes were space-averaged (Table 1, section on

TABLE 4. Q Ellipsoid Axes

 

 

Prepagatlon
Direction Directional Cosine: V1 V2 V3
Guartaite
1 (l, 0, 0) 5.110 3.436 3.561
2 (0, l, 0) 5.206 3.659 3.537
3 (0, 0, 1) 5.332 3.598 3.611
4 {-0.602, 0.0, 0.800) 5.377 3.517 3.715
5 (0.574, 0.0, 0.819) 5.357 3.561 3.571
6 (0.0, -0.652, 0.758) 5.399 3.719 3.533
7 (-0.602, -0.800, 0.0) 5.261 3.548 3.440
8 (0.0, 0.663, 0.749) 5.389 3.668 3.655
9 (0.663, -0.749, 0.0) 5.325 3.553 3.239
Marble
1 (1, 0, 0) 5.575 3.059 2.897
2 (o, 1, 0) 4.711 2.335 2.392
3 (0, 0,.1) 5.387 2.852 3.026
4 (0.0, 0.707, 0.707) 5.028 2.792 2.985
5 (0.0, —0.707, 0.707) 4.634 2.799 2.885
6 (-0.707, 0.707, 0.0) 5.009 2.826 2.931
7 (0.707, 0.707, 0.0) 5.114 2.855 2.970
8 (-0.707, 0.0, 0.707) 5.273 3.029 2.823
9 (0.707, 0.0, 0.707) 5.397 3.019 2.773
Granite
1 (1, 0, 0) 5.684 3.425 3.438
2 (0, 1, 0) 5.882 3.505 3.612
3 (0, 0, 1) 5.775 3.498 3.614
4 (0.0, 0.623, 0.783) 5.819 3.544 3.555
5 (0.0, -0.643, 0.766) 5.931 3.543 3.540
6 (0.469, 0.0, 0.883) 5.728 3.571 3.461
7 (~O.530, 0.0, 0.848) 5.786 3.556 3.358
8 (0.415, 0.910, 0.0) 5.880 3.537 3.600
9 (-0.446, 0.895, 0.0) 5.860 3.507 3.614

Magnitude Directional Cosines Symbol

Quartsite

56.010 (-0.097, -0.082, 0.992) Mg

50.539 (0.975, -0.184, 0.127) m8

53.243 (0.172, 0.979, 0.097) I5

Ahrble

48.048 (0.988, 0.088, 0.128) Mg

36.679 0.062, -0.970, 0.236) ms

44.947 {-0.145, 0.225, 0.964) 13
Granite

60.644 (-0.034, -0.995, 0.090) Mg

55.792 (0.994, 0.015, 0.112) ms

58.306 (-0.113, 0.079, 0.988) 1b

 

 

 

'I‘n.uANN AND BnNNm'r: SONIC Psmooaapmc ANALvsIs

Tm 5. 0" Gui)

and Deviatiai Values

 

Quartaite' Marble?
Propagation

Granite‘

 

Direction at 059i:

Q.

t

013‘

 

47.887
37.308
44.478
43.071
38.716
41.870
43.833

”.NOM‘MNH
one
N
.

46:671

55.856
59.935
58.643
59.135
60.263
37.543
57.401

592702

55.753
60.558
38.528
38.777
59.910
57.919
37.747

59:501

 

marble), and the axes of the optical indicatrix
type surface are plotted in Figure 2.

Twelve velocity measurements in nine in-
dependent directions were taken, and the Q,-
values were calculated (Table 2). From the mean
velocities (Table 3) the Q ellipsoid was deter-
mined. The directional cosines and the magni-
tudes of the ellipsoidal axes are listed in Table 4
in the motion on marble. The plot of these axes
is shown in Figure 2. The Q. and Q“, values were
derived as 0:, 0;” and a. were and are listed in
Table 5. The observed maximum P wave velocity
is shown in Figure 2.

Granite boulder. Velocity measurements of
the granite were used to compute the Q,- values
(Table 2, section on granite). From the mean
velocities (Table 3) the Q ellipsoid was deter-
mined (Table 4, section on granite). The axes of
the Q ellipsoid are plotted in Figure 3. The Q,-
and Q”, values, along with 0:, 0;" and c,, are
listed in Table 5. The observed maximum P
wave velocity is plotted in Figure 3.

The orientation of the Q ellipsoid relative to
the boulder shape is shown in Figure 3. The
axes orientation of the shape ellimoid was mea-
sured to an estimated accuracy of 110°.

It should be pointed out that for single
crystals of quartz and calcite the representative
optic and sonic surfaces are opposite in sign.
Therefore in crystal aggregates a maximum
optic axis might reasonably correspond to a
minimum sonic axis.

CONCLUsIONs

For all samples investigated the calculated Qt
values for each direction display an F value
signiﬁcant at the 0.01 conﬁdence level (Table
2). This indicates that the quartzite, marble,

8467

and granite are seismically anisotropic. The sum
of the squares of the three seismic wave type
velocities over the nine measured directions
describes an ellipsoidal surface and satisﬁes the
conditions of equation 8 (Table 5). Thus the
seismic anisotropy of the samples observed in
Table 2 results from the behavior of the poly-
crystalline material as an elastic long-wave
equivalent to a single crystal.

For the baraboo quartzite the Q ellipsoid
axes and the optical indicatrix type surface
axes closely coincide. The maximum angular
separation between the principal axes of the two
surfaces is 11° or less (Table 6). The optical
surface is nearly spherical, the major and minor
axes differing in magnitude by only 0.4%, com-
pared with a difference of 2.2% in a single

 

-a.us " ‘ .
a. - 1.028 I A

Fig. 2. Equal-area projection of 100 calcite
c axes for the Grenville marble contoured by the
Mellis method at intervals of l, 2, 3, and 4% of
the axes per 1% area. Stipled areas denote 3%
concentrations. Hatchured areas denote greater
than 4% concentrations per 1% area. Micaeeous
layering is in the N-S vertical plane. The ob-
served maximum P wave velocity, optical surface,
and Q ellipsoid principal axes are also plotted,
M, m, and I being the major, minor, and inter-
mediate axes, respectively, of the ellipsoidal sur-
faces. Location of the sonic minor axis is the
nesult of interaction between the structural fabric
and the crystallographic fabric.

8468

Boulder

Granite

 

asaonm
o: shape
X a VP I“

Fig. 3. Projection of the Q ellipsoid and shape
ellipsoid principal axes for the plastically de-
formed granitic boulder. The observed maximum
P wave velocity is also shown. Close coincidence
between the ellipsoidal axes indicates that the Q
ellipsoid technique may be useful in describing
regional tectonic forces.

quartz crystal. The near sphericity indicates a
weak preferred crystallographic orientation.
Statistical analysis of the scatter diagram of
Figure 1 yields a correlation coefficient of r =
0.103 [Chayes, 1949]. Therefore the degree of
orientation, as displayed in Figure l, is not
signiﬁcant at the 0.05 conﬁdence level (rm :
0.200). It is interesting to note that the calcu-
lated correlation coefﬁcient is signiﬁcant at the
0.10 conﬁdence level (r.no = 0.100). The magni-
tudes between the major and the minor sonic
Q ellipsoid axes differ by 10%, compared with
a difference of 18% for a single quartz crystal.
We conclude from the study on this sample
that the Q ellipsoid method readily detects pre-
ferred crystal orientations. The close coincidence
between the sonic and the optical surface axes
in this nearly Optically isotropic sample is strong
evidence that the sonic orientation accurately
reﬂects subtle fabric orientation.

The Grenville marble produces a Q ellipsoid
whose principal axes describe the effects of
structural and crystal fabric. To a ﬁrst approxi-

TILMANN AND BENNE'I'I‘I SONIC PETROGRAPHIC ANALYSIS

mation this sample behaves as a layered mate-
rial, the minimum velocities being normal to
the micaceous layering [Postma, 1955]. This
structural fabric would tend to locate the minor
sonic axis at the west pole of Figure 2. The
major optical surface axis corresponds to the
minor sonic axis (Table 6). Thus the crystal
fabric would tend to locate the minor sonic
axis coincident with the major Optical surface
axis. Interaction of these two fabrics would
place the minor sonic axis between the west
pole and the major optical surface axis. There-
fore the observed location of the minor Q ellip-
soid axis is as expected (Figure 2). The Q
ellipsoid axes differ by 27%, compared with
39% for a single calcite crystal, a rather strongly
anisotropic material being indicated.

The principal shape axes and the Q ellipsoid
axes of the granite boulder closely coincide
(Figure 3). The multiphase granite behaves
homogeneously as a pseudosingle crystal oriented
in response to plastic deformation. Thus the Q
ellipsoid technique may be useful in detecting
and describing regional tectonic forces.

For quartzite, marble, and granite the angles
between the Q ellipsoid maximum axis and the
observed maximum P wave velocity are 36°, 9°,
and 45°, respectively (Table 3 and Table 4).
For quartzite and marble the P wave velocity
measured in the direction nearly coincident
with the maximum Q ellipsoid axis was not the
observed maximum P wave velocity. Thus the
direction of the observed V, .m has not been a
reliable indicator of the Q ellipsoid major prin—
cipal axis. Indeed it is conceivable that the P
wave could indicate sample isotropy, whereas
the Q ellipsoid indicates seismic anisotropy
[Bennett, 1972]. Therefore the use of the P
wave velocity surface to determine preferred

 

 

TABLE 6. Optical and Sonic Axes Separation

Axes Separation
Quartst'te

msiMo 11°

8:mo 3°
Marble

)4on 21°

I81m° 4°

 

'I‘ILMANN AND BENNETT: SONIC PETROGRAPHIC ANALYSIS

crystallographic orientation must be made cau-
tiously.

The results of this study lead us to conclude
that (1) the elastic behavior of these rock
materials is testable and is shown to be that of
a homogeneous pseudosingle crystal, (2) in the
samples studied the. orientation of the sonic Q
ellipsoid is controlled by preferred crystallo-
graphic orientations of the materials and by
structural effects, (3) weakly preferred orienta-
tions are readily observable with the Q ellipsoid
method, (4) the sonic Q ellipsoid technique is
equally valid for single-phase or multiphase
materials. and (5) the P wave velocity surface
is not necessarily a reliable indicator of prin-
cipal anisotropic directions.

Acknowledgment. The authors wish to thank
Thomas Vogel for his assistance throughout the
course of this study.

8469
REFERENCES

Bennnett, H. F., A simple seismic model for de-
termining principal anisotropic direction, J. Geo-
phys. Res, 77, 3078—3080, 1972.

Chayes, F.. Statistical analysis of three-dimen-
sional petrofabric diagrams, in Structural Pe-
trology of Deformed Rocks, edited by H. W.
Fairbairn, p. 317, Addison-Wesley, Reading,
Mass, 1949.

Nye, J. F., Physical Properties of Crystals, p. 164,
Oxford at the Clarendon Press, London, 1957.
Postma, G. W., Wave prOpagation in a stratified

medium, Geophysics, 20, 780—806, 1955.

Tilmann, S. E., and H. F. Bennett, Ultrasonic
shear wave birefringence as a test of homo-
geneous elastic anisotropy, J. GeOphys. Res,
78, 7623-7629, 1973.

Turner, F. J., and L. E. Weiss, Structural Analysis
of Metamorphic Tectonites, p. 62, McGraw-Hill,
New York, 1963.

(Received April 13, 1973;
revised August 6, 1973.)

8

 

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