MW\\\\\\\\‘\\\T\\\"\\T\T\\‘~; mu \ujox'ni‘fmlk‘uin

311E315 3 1293 0170

LIBRARY

Michigan State
University

 

This is to certify that the
thesis entitled
COMPARISON OF THE QLSi ELLIPSOID AND ELLIPSOIDAL
REDUCTION SPOTS USED TO DETERMINE FINITE STRAIN IN

THE PRECAMBRIAN KONA SLATE MEMBER:

MARQUETTE COUNTY, MICHIGAN
presented by

Erick C. Nefe

has been accepted towards fulfillment
of the requirements for

 

M. S 0 degree in GeOlogy

 

77L“ka <7" [4; ML; {16/

/Major professor

 

Date November 24, 1980

 

0-7639

m:

25¢ per day per item
RETURNING LIBRARY MATERIALS:
N

v. - Place in book return to remove
$3315!” ‘4' charge from circutation records

I "i\\\\ I.
‘ {

. I II‘ .,
Jim
‘1

V

 

 

 

 

 

© 1980

ERICK CLEMENS NEFE

All Rights Reserved

COMPARISON OF THE QLS ELLIPSOID AND ELLIPSOIDAL REDUCTION
. i
SPOTS USED TO DETERMINE FINITE STRAIN IN THE PRECAMBRIAN

KONA SLATE MEMBER; MARQUETTE, MICHIGAN

BY

Erick C. Nefe

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Geology

1980

ABSTRACT

COMPARISON OF THE QLS. ELLIPSOID AND ELLIPSOIDAL REDUCTION SPOTS USED
TO DETERMINE FINITE STRAIN IN THE PRECAMBRIAN KONA SLATE MEMBER:
MARQUETTE COUNTY, MICHIGAN

By

Erick C. Nefe

It has been suggested by Bennett (1972), Tilmann and Bennett (1973 a,b)
and Anderson (1977) that the Q ellipsoid method may be a valuable tool for
describing regional tectonic forces.

Kona slate in the Marquette synclinor ium containing reduction spots
has been examined by Wes tjohn (1978). Westjohn (1978) concluded that the
reduction Spots demonstrated h5% flattening in the Z axis, with extensions
of 60% and 15% in the X and Y axes, respectively.

The QLS. ellipsoid (constructed by anisotropic velocity measurements)
of the Kona Slate exhibited similar axial orientations and similar axial
ratios. The average deviation (resultant vector) of the axes of QLSi
ellipsoid from the known reduction spot was (1.80, 1.10, 1. ho ) for the
major axis, (2.60, 6.80, 6.30) for the intermediate axis and (1.60, 6.60 ,
6.80) for the minor axis using an orthogonal set of (X, Y, Z) axes and a
deviation technique described by Fisher (1953) and McElhinny (1973).

Comparison of the mean axial ratios for the QLSi ellipsoid demon-
strated a h8% flattening in the Z axis, with extensions of 62% and 21%
in the X and Y axes, respectively. It is concluded, from the close
agreement between the QLS. ellipsoid and the reduction spot ellipsoid

orientations, that the QLS ellipsoid is a valuable tool for describing

regional finite strain in an area.

ACKNOWLEDGMENTS

I would like to thank both Dr. Cambray and Dr. Long
for suggestions during the laboratory work and during the

early stages of writing.
I would also like to thank Dr. Hugh Bennett for helping
me throughout the laboratory work and the final stages of

writing of this thesis.

ii

IOD'U'U
5951.4.

:0 O
m

R It n H
BI 01 BI
(D

0

LIST OF SYMBOLS

element of 3 x 3 matrix

circle of 95% confidence around the resultant vector (R)
intermediate axis

an estimate of the precision parameter

directional cosines for a selected set of orthogonal
axes X, Y and Z, respectfully

minor axis

major axis

number of measurements

number of measurements in the ith direction
directional cosine matrix

probability (for this paper p = 0.05)
density

calculated ellipsoidal value

least square ellipsoidal value

average data mean

average trace element

resultant vector

distance from origin to reference Q ellipsoid
diviation of Qi from Q5

deviation of Qi from QS

deviation of Q o from -
Lsi Qm

dev1ation of QLS. from Qs

iii

dl

>4 <1 < < <1 rt

deviation of Qi from lei
time

phase velocity

P wave phase velocity

SV wave phase velocity

SH wave phase velocity

distance

iv

TABLE OF CONTENTS

Introduction . . . . . . . . . . . . . . . . .
Geologic Setting . . . . . . . . . . . . . . .

Laboratory Procedures. . . . . . . . .
Sample Preparation. . . . . . . .
Thin Section Preparation. . . . .
Operation of Ultrasonic Apparatus . .
Sample Measurement and Calculations . .

Q-Ellipsoid Method . . . . . . . . . . . . . .
Error Analysis. . . . . . . . . . . . .
Cutting and Reduction Spot Variation
Apparatus. . . . . . . . . . . . . .

Discussion and Results . . . . . . . . . . . .
Thin Sections . . . . . . . . . . . . . .
Velocities and Q Ellipsoids . . . . . . .

conCIuSion O O O O O O O O O O O O O O O O O 0

Appendix A . . . . . . . . . . . . . . . . . .
Tables I-VI

Appendix B . . . . . . . . . . . . . . . . . .
Computer Program used for Calculations

Bibliography . . . . . . . . . . . . . . . . .

11
11
12
15
24

25
31
31
31
33
33
33
48

49

82

89

1.

2.

LIST OF FIGURES

Northwestern Upper Penninsula of
Michigan Showing the Study Area. . . . . .

Sample Location Sites. . . . . . . . . . .

3a-3e. Sample Preparation . . . . . . . . . .

4.

ll.

12.

l3.
14.

15.

Schematic Diagram of Ultrasonic Apparatus.
Photograph of Ultrasonic Apparatus . . . .
Photograph of Lathe Assembly . . . . . . .
Photograph of Lathe Assembly . . . . . . .
Schematic Diagram of Transducer Assembly .
Photograph of Transducer Assembly. . . . .

Photograph of Transducer Assembly with
Sample in Measuring Position . . . . . . .

Photograph of a Signal on the Cathode-Ray
Tube Display of the Oscilloscope . . . .

Equal Area Stereonet Plot of the Field
Orientations of the X axes of the
Reduction Spots and the Respective

Plots of the X axes of the QLS

Ellipsoids . . . . . . . . . .i. . . . . .

Plot of Westjohn's (1973) Axial Ratios . .

Plot of QLS Axial Ratios. . . . . . . . .
1

Plot of the Relationship Between

Westjohn's (1973) Data and the Q
I 0 LS.
Ax1al Ratios . . . . . . . . . . .1. . . .

vi

_13

16
16
18
18
20

20

22

22

39
42

44

46

LIST OF TABLES

Ia. Propagation Directions, Directional
Cosines and Sampling Distances . . .

Ib. Propagation Directions, Directional
Cosines and V1, V2 and V3 Mean
Velocities . . . . . . . . . . . . .

II. Qi Ellipsoid Values and QLS Values

0 o G _ _
w1th Assoc1ated Qm’ 05' m’ me’

<sse and Percent Sample Uniformity .

III. Directional Cosines of the Principle
Axes of the QLS Ellipsoids. . . . .
i

IV. Degree of Deviation Between the
Known Axes of the Reduction Spots
and the Computer Generated Axes

VI.

of the QLS.‘ . . . .
1

Lo 2S-and lo E1Q

gY gYLSi

Thin Section Data. .

vii

Data.

76

78

80

I NTRODUCTION

Early earthquake seismologists considered that anisotropy
was possible, and used it to help explain anomolies in seismic
observations Rudski (1911); Neuman (1930); and Byerly (1934).
These early earthquake seismologists led the way and as seis-
mology and ultrasonic equipment became more advanced, many
surface rocks were also found to be anisotropic on a small
scale McCollum and Snell (1932); Weatherby, Born and Harding
(1934; Ricker (1953); White and Sengbush (1953); Cholet and
Richards (1954); Uhrig and Von Melle (1955); White, Heaps and
Lawrence (1956); Jolly (1956); Dunoyer and Laherrere (1959);
Shimozura (1960); Duda (1960); Macpherson (1960); Brace (1960);
Anderson (1961); Backus (1962); Gassman.(1964); Schmidt (1964);
Crampin (1970); Nur (1971); Cerveny (1972); Cerveny and Psencik
(1972); Anderson, Minister and Cole (1974); Crampin (1975, 1977);
Meissner (1977); Schlue (1977); Levin (1978); Berrman (1979);
and Crampin and Kirkwood (1979). Anisotropy in rock samples
has also been indicated at the ultrasonic level Tocher (1957);
Motveyeu and Martyanou (1958); Musgrave (1959); Balakrishna
(1959); Kopf and Wawryik (1961); Birch (1960, 1961); Klima
and Babuska (1968); Thill (1968, 1969); Tilmann and Bennett
(1973 a,b) and Anderson (1977). According to the above re-
search, velocity anisotropy may be caused by preferred crystal

orientation, grain orientation, stress fields, structural

1

2
layering, microfracture and macrofracture orientation, direc-
tional porosity and/or permeability. Bennett (1972) further
states that velocity anisotropy can be used as an indicator
of these structural and petrofabric patterns.

Bennett (1972) developed a simple seismic model for deter-
mining principle aniSotropic directions within crystal aggre-
gates. In general his model uses an elastic stiffness figure
referred to as the Q ellipsoid. With this model three different
body waves with orthogonal particle motion can propogate in
anisotropic media in any prescribed direction. Thus, the Q,
which determines the surface of the Q ellipsoid, is the sum
of the squares of the three phase velocities for a given di-
rection, multiplied by the density. The principle axes of
the Q ellipsoid are identical to the orthogonal crystallo-
graphic axes of a single crystal Bennett (1972).

Tilmann and Bennett (1973b) applied the Q ellipsoid con-
cept to three rock types; a quartzite, a marble and a plas-
tically deformed granitic boulder. For all three samples the
elastic velocities were measured. The quartizite and the marble
were compared to their respective optical petrofabric analyses,
while the results of the granitic boulder were compared to its
shape axes. It was observed that each rock sample behaved as a
homogeneous pseudosingle crystal. Conclusions made were that
the orientations of the Q ellipsoids were controlled by crystal
orientations and structural effects, and that the shape axes of
the plastically deformed granitic boulder closely coincided with
the Q ellipsoid axes. Thus, the Q ellipsoid method may be use-

ful in regional tectonic studies.

3

It has been suggested by Bennett (1972), Tilmann and
Bennett (1973b) and Anderson (1977) that the Q ellipsoid may
prove useful in describing regional tectonic forces. Thus it
is the purpose of this research to test that hypothesis, re-
gionally.

A study area was selected where in situ finite strain
indicators are present. Westjohn (1978) used ellipsoidal re-
duction spots in the Kona slate member of the Marquette syn-
clinorium (Figures 1 and 2) as one means of determining finite
strain for that member. The purpose of this research is to
select similar samples from similar sites and determine the
relationship between the in situ ellipsoidal reduction spots

and the theoretical Q ellipsoid.

GEOLOGIC SETTING

The area of study is generally south-west of Marquette in
the Upper Peninsula of Michigan (Figures 1 and 2). Van Hise
and Bayley (1897) were the initial investigators in this area
and they have written extensive geologic reports interpreting
the Proterozoic history of the Upper Peninsula of Michigan.
Other geologic study was done by Gair and Thadden (1968) in
the Marquette and Sands guadrangles and Puffet (1974) studied
the Negaunee guadrangle. Taylor (1973) did extensive mapping
of the Kona dolomite formation and lithologically described
each member in detail.

The Kona dolomite formation is part of the Chocolay Group
from the Middle Precambrian metasediments. Gair and Thadden
(1970) felt a need to clarify the terminology of the Middle
Precambrian sediments of this area. They proposed that the
name "Marquette Range Supergroup" replace the term Animikie
Series for the Middle Precambrian strata. For more detail
regarding the lithology of the area the reader can consult
any of the following references.

The major structural feature in the area is the Marquette
Synclinorium, which is a west trending trough of deformed
Middle Precambrian metasediments (Figure l). The trough it-
self is approximately six miles wide in the north-south direc-
tion and thirty miles long in the east-west direction. The
Middle Precambrian trough is surrounded by Lower Precambrian
rocks to the north and south. Cannon (1973) and Van Schmus

(1976) believe the Penokean Oregany, which occurred approximately

Figure 1. Map of the Northwestern Upper Peninsula of
Michigan showing the study area location.

 

   
 

 

LAKE

Upper Precambrian roc ks

ERIOR
47° SUP

 

.....

46°

 

80 mllu

85°

‘-

-.. . .. . . _..—. r. -.—_ -_ —.—— - , .-._ - .-. - ""-'—'T'——_ -—

  
    
        
   
 

wj ..__ —.._ ——-.

0- ~
“ " ‘ .‘ 61 950300!“

elem nohdnoauq "qu

H 0 I H 3 q U 3 “301 nohdnoeuq mam

chow nhdmoanq iowoJ

 

83JIM OE

 

 

.-- _ .l _ ‘
'65- '-.-i '
;.
.r l.
. 5 - lt-
- \
r \
‘ i
a . -_ \
r .‘ \ .
I I'. 1" \I ‘
I ‘I -
.' _\
\
l . K I
. '.
~.. . .15 l‘
' _ I
. . - - - .
‘ n
, . | a
. ' | ~ 9
Pl I
. ’_ .,
I
_ . 3
I .
‘ D
I . l -
o I .
V. \
'K
I‘ §_. .-- f,
.\. -_ _.
C. h
a .“
u ' I
I
'-.
I
- I
l I
. .
.
-' .-- .
'-

Figure 2.

Map of the study area showing Westjohn's eight
site locations where reduction spots "occur".
Westjohn's site locations are listed above

each point and the samples collected for this
study are numbered below each point. Site la
and 7a were added by this author. Westjohn's
sites 2 and 3 were found to be in error because
no Kona argillite was present in either area.
This is supported by Taylor (1973).

/l\

42' L'

 

 

 

 

 

 

 

out,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. . . mi
N . o muxn<a
3...: y
a.“
'67.
91
, / 1 AK
(J on a
s
A OOOOOV Av
mzez «\ivr
I
. ~ 35
m 55
o. \
0 I
:2. . .2631 u e no t mtm umza< m2
c. .utm c. 3.6 w 2..» WM.
2 E. a.
z 2. p
n 2:. e _ 3:...
o. a v u-
m mtm n 2:. N. .1 .p
/ Wm.
V \VI)\
when-RSI:
:. .9:

 

 

 

 

 

9

1.85-1.95 B.y. ago was responsible for the deformation of
the trough. The tectonic event included deformation, meto-
morphism, extrusive and intrusive igneous activity.

Cannon (1973) proposed that the Middle Precambrian meta-
sediments were deposited on a peneplaned Archean basement com-
plex and folded by two processes. First, regional gravity
sliding produced gentle folding and then vertical faulting
in the Archean basement rocks formed the folded Marquette
supergroup into the Marquette trough. Klasner (1978) proposed
a similar model but his model consists of four phases of de—
formation. With Klasner's four stage model continuing meta-
morphism took place during the first three stages. Phase I,
consiated of gravity sliding of soft sediment off an ancestral
Penokean range located in central Wisconsin. Phase II, con-
tinued regional deformation took place. Phase III, deforma-
tion.was due to uplift of the lower Precambrian basement as
rigid blocks. Metamorphism peaked in this phase. Phase IV,
produced continued uplift of basement rocks and this uplift
produced grabens such as the Marquette trough.

Westjohn (1978) suggested that more evidence was needed
to support or reject Cannon's and Klasner's model. Thus he
selected the Middle Precambrian Kona delomite formation for
his research to determine finite strain for the area. He
selected a slate (argillite) member of the Kona formation for
several reasons. The Kona slate member is well exposed through-
out the trough, it has a well developed secondary fabric, and
it contains reduction spots and deformed veins. Westjohn used
reduction spots (ellipsoidal green to yellow bodies, which were

believed to be spherical prior to deformation) and deformed

10
veins to conclude that the Marquette trough was shortened
approximately 45% normal to the trough.

The major stimulus for this research is that the slaty
units of the Kona formation have a well developed secondary
fabric according to Westjohn (1978). The purpose of this re-
search is to answer the questions, will this well developed
secondary fabric be detected by anisotrOpic ultrasonic phase
velocity measurements and, if so, how is the theoretical Q
ellipsoid related to the in situ ellipsoidal reduction spots

for a regional study area?

ll

LABORATORY PROCEDURES

Sample Preparation:

The Kona slate samples with reduction Spots which were
collected from the Marquette synclinorium (Figure 2) were
large (,.100 lbs.). All ten were marked with field orientation
measurements. Due to a well formed cleavage, most samples were
less than four inches thick (perpendicular to cleavage). After
trimming of the large odd shaped samples, the samples were cut
into four inch "cubes". It was critical during all cutting
that opposite sides be parallel so that there would be a good
coupling between the samples and the wave guides of the ultra-
sonic equipment (Figure 3d). Two cubes were cut from each sample,
if size and condition of the sample permitted, to assure that
measurements were representative of each sample. The samples
were marked as l, l', 2, 2', etc. All samples contained re-
duction spots and the reduction spots determined the cutting
of each cube. Figure 3a-e shows the cutting process and orien-
tation of the reduction spots with respect to the cubes. For
all samples the cleavage was in the plane (XY plane) of the
major and intermediate axis of the ellipsoidal reduction spots.
An observed axes variation of 100 was noted between reduction
spots of the same sample. After both sets of cubes were com-
pleted, the corners of the cubes which were perpendicular to

the cleavage were cut at 45°, forming octagons (Figure 3b).

12

Tilmann and Bennett (1973b) suggest that since the P
and S wave velocity surfaces may be quite complex in shape,
the maximum value chosen from just a few measurements may not
be the true surface maximum. So to insure that the true maxi-
mum value was chosen, the octagons were cut in such a way as
to permit measurement in nine directions (Figure 3c).

Each samples nine propagation directions were marked using
an orthogonal set of X, Y and z axes (Figure 3e). Directional
cosines were used to determine direction of the signal through
the sample (Table I). Directional cosines are simply the angle
between the signal propagation direction and the respective X,
Y and Z axes (Figure 3e).

Measurements were then taken in nine directions for each
sample.

Attenuation of the signal in the Z direction (0.0, 0.0,
1.0) made it necessary to cut off an approximate 2.0 cm plate
from each sample. This smaller sampling distance caused less
attenuation and signal time picks could be measured more
accurately.

Thin Section Preparation:

 

Thin sections were prepared for all samples in the XY plane.
Due to the very fine fabric of the slate, the grain orientations
were measured at lOOX or 200x. All thin sections displayed a
definite lineation of ellipsoidal pyrite grains in the same
direction as the ellipsoidal reduction spots. The average
deviation from the x axis for 50 pyrite grains per sample are
listed in Table VI and the results are discussed in the result

and discussion section of this research.

Figure 3a.

Figure 3b
and 3c.

Figure 3d.

Figure 3e.

13

Shown is a typical field sample. The cleavage
of the sample is in the plane of the paper.
The dotted lines represent cut directions. Two

Isamples were cut from each large field sample

if size and shape permitted.

Cutting was done so that nine propagation
directions could be measured.

It was very important that all opposite sides
were parallel for two reasons. First, so good
coupling between the wave guides and the sample
could be accomplished and second so that the
wave path (dotted line) would be a true repre—
sentation for the propagation direction.

Labeling of the directional cosines with respect
to the XYZ axes is shown in these two examples.

14

2:36- .06 ..:.os.o.

::.o...o...:.os.o 6.8

on um

3.0.0.. 6.0:

.o.o..:.o...o..:.o....o¢ 3.0.2356. .:.o...o.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>. x AO-Oa°.°u°.-vx x -N....-..
. N .
2.3.8 ..:.o...o .od. 3......
. 2.53.0 .o.o._:.o...o. :63 >
3.. .o.o.o.o.~
o m n n a n
>
t /
\\ x
X j.
x x
N 5°
1 \
/ \
I? \

 

 

 

 

 

ZO_F<m<mmmm m4m2<w

‘O

15

Operation of Ultrasonic Apparatus

The apparatus is shown in Figures 4-11. The apparatus con-
sists of two pulse generators, a lathe assembly, an amplifier,
a high-low pass filter and an oscilloscope. An electrical wave
form is sent out by the pulse generator. One pulse is sent di-
rectly to the oscilloscope (line B) where it is used to deter-
mine time = 0. The other pulse (A) is sent through the sample
by a P—S conversion technique (Figures 8 and 9), the signal is
then amplified, filtered and the visual response is shown on
the oscilloscope (A). The change in time from t = 0 (B pulse)
and the signal (A pulse) is determined. This is the "total
transit time" it took for the Signal to get through the sample
and the transducer assemblies (Figures 8 and 9). The amount
of "sample transit time" is found by first measuring the "time
delay", which is calculated when the transmitting and receiving
transducer assemblies are placed in direct contact. For this
research the transducer assembly "time delay" for the P—wave
was 45/«sec.and 60/nsec.for the S-waves. Thus the zero sample
length times or the "time delay" substracted from the "total
transit times", yields the "sample transit times". For a more
detailed explanation of the electronics behind the apparatus
operation consult Bennett (1968).

A similar process was used to determine the sample length.
The lathe was calibrated in such a way so that one 3600 clock-
wise rotation of handle A (Figure 6) brought the wave guides
0.2539 cm closer. Each 3600 degree rotation was marked on the
lathe bed platform. The shaft of the handle was further cali-

brated (Figure 6) to allow the measurement accuracy of 1 0.0025 cm.

16

Figure 4. Schematic diagram of the ultrasonic apparatus.

Figure 5. Photograph of the ultrasonic apparatus used
for this research.

l7

ULTRASONIC APPARATUS

PULSE GENERATORS

 

 

 

 

 

 

 

OSCILLOSCOPE
AMPLIFIER
SAMPLE HI/LO PASS A
41"
E B
PM" It FILTER
out —A
sync. out 3 sync. In

 

 

 

 

 

 

 

Figure 6.

Figure 7.

18

Handle A is located on the far left of the
lathe assembly and it is used to measure
sample length to an accuracy of -0.0025 cm.

This gauge “is located on the far right of
the lathe assembly and used to insure that
the same amount of pressure was exerted on
each sample by the wave guides.

l9

 

20

Figure 8. A schematic diagram of the transducer assembly.

The P-S conversion was done by cutting the

prisms at‘z 48 or the critical angle for the
shear wave.

Figure 9. Photograph of the transducer assembly.

2'1

TRANSDUCER ASSEMBLIES

 

 

 

 

 

 

 

Figure 10.

Figure 11.

22

Transducer assembly with sample in measuring
position.

Photograph of the cathode-ray tube display of
P-S conversion transducer wave form arrival
after passing through a sample of the Kona
slate. The sweep speed of the oscilloscope
was 20 jAsec.per division. The bottom trace
is the impulse trigger or t = 0. The top
trace is the wave form arrivals. The P wave
can be seen having a "total transit time" of
approximately 62 jksec. The 5 wave can be
seen having a "total transit time" of approxi-
mately 88 /usec.

23

 

 

24
Overall based on repeated measurements it was found measurements
were accurate to :0.0005 cm. The total estimated error for the
distance measurement is :0.0075 cm.

To insure that the pressure exerted by the closed trans-
ducer assemblies, around the sample, was the same for all samples
a gauge measuring inch lbs. was used (Figure 7).

Sample Measurement and Calculations:

Apparatus operation having been explained, one sample cal-
culation will be shown.

Sample #10', prOpagation direction 1

Distance: 19.0492 cm.(distance with wave guides closed)
- 9.8802 cm.(distance with sample)

= 9.1690 cm.(sample length)

Time: 59.2/M.sec.(total transit time)
-45.0‘/~sec.(transducer assembly transit
time for P-wave)

Velocity: 9.1690 cm/l4.2 x 105 sec = 6.457 km/sec

Qi and QLS-' the accompanying standard deviations, and

the determination of the principle axes were calculated
using a computer program shown at the appendix of this

report.

The resulting velocities are listed in Table Ib and the

sampling distances are listed in Table Ia.

Q-ELLIPSOID METHOD

Q Ellipsoid:

 

Bennett (1972) developed the Q ellipsoid as a tool for
specifically detecting preferred crystallographic orientations.

As stated in the introduction:

2
2

Q = Qé = (Vi + V + vi) (1)

where r) is density and can be considered constant, V1 is the P
wave phase velocity in the ith_direction and V2 and V3 are the
phase velocities Of the two orthogonally polarized shear waves
for the iEh direction. The principle axes of the Q ellipsoid
always coincides with the optical indicatrix axes for a single
crystal in the cubic through orthrombic system Bennett (1972).
Therefore, for a cubic crystal the Q ellipsoid would reduce

to a sphere. For uniaxial and biaxial crystals the Q surface
would become an ellipsoid of revolution and a triaxial ellip-
soid, respectfully. When considering crystal aggregates one
can consider them as an elastic long-wave equivalent to a single
crystal, so the locus of the values will be represented by an
ellipsoidal surface. Tilmann and Bennett (1973b) have set
three criteria if the Q surface is ellipsoidal. First, the
material is homogeneous and anisotropic; second, the principle
anisotrOpic directions are described by the principle axes of
the ellipsoid; and third, the percent difference between the
axes of the ellipsoid is a measune of the degree of elastic
anisotropy, which is controlled by anisotropic crystal orien-

tation and structural effects.

25

26

Equations of the elastic wave theory in anisotropic media
were used to derive Equation (1), Bennett (1972). Equation (1)
or Qi will be treated as the calculated value of the Q ellipsoid
in the ith direction. The calculated Qi values can be least

squares fit by using the equation:

_ 2 ex
QLSi ’ 1i “11 + mi"“22 + “i“33 + 21mi"i"‘23 + 2nili 31 + 21imi°"12
where QLS is the least square value in the ith direction, (limini)
i

are the measurement directional cosines.for an arbitrarily chosen
set of orthogonal axes X, Y and Z and the o<'s are the elements
of a symmetric 3 x 3 matrix.

Equation (2) by means of referring to the principle axes

becomes:
_ 2 2 2 _ 2 2 2
QLsi"l"‘11‘“‘“°‘22+n"‘33‘1QI‘”““92"n93 (3)
= . . 0‘ . . .
wherecx ij 0, 1 ¢ 3. Thus, the (°<1l’°‘22' 33) c01nc1de Wlth

the major, intermediate and minor axes.

By setting:

_ 1
l r

where r is the distance from the origin to the reference Q ellip-

soid, and by substitution of:

1:5
r
:1

m r (5)
Z
n:-
r

one finds that

1-952 22 22
r2 — (r) Oi + (r) O2 + (r) O3

27

_ 2 2 2
1 — x Ql + y Q2 + 2 Q3 . (6)

Equation (6) is now in the form of an ellipsoid whose
principle axes have lengths of (Ql)-%, (Qz)-k, and (Q3)-%, which
are the major, intermediate and minor axes, respectfully.

The elements of thecx matrix in Equation (2) are determined
by the least square method outlined by Nye (1957). The deter-
mination of the elements of thecx matrix is based on the matrix

equation:
Q = eex' (7)

where the Qi values are related to the directional cosine matrix
9 and thec< matrix. The Q matrix elements are the measured Qi
values from equation (1). The a matrix is constructed by using
the directional cosines which are the l, m and n coefficients of
Equation (2). Thecx matrix is determined by solving Equation (7)
for

_ -1
ac - (Ste) GtQ (8)

which yields the computational form for determination of the
matrix. For a more detailed mathematical explanation consult
Nye (1957, p. 164-165).

The principle axes of the Q ellipsoid can then be calcu-
lated, from the best fit «:matrix. By this method described by
Nye (1957) unit vectors which are normal to the least squared
Q surface are successively calculated until they converge onto
the major axis. The minor axis is found by inversion of the
matrix and repeating the process. The intermediate axis is

then determined by the cross-product of the major and minor

28

axes. By using the directional cosines of the major, inter-
mediate, and minor axes from Equation (2), the magnitude of
these axes can be determined.

After the QLS. surface has been determined it can be com-
pared to the measured Qi values and this comparison provides
a statistical test for homogeneous anisotropy. In an aniso-
tropic media the measured Qi values will vary with direction
and the values should approximate an ellipsoid. Five standard
mean-square deviations can be calculated to check the accuracy

of the Qi values. The equations are as follows:

 

(I n 2‘ a
G R = a 12:1 (0595.) J (9)
_ '1 n 21 s5
6 s — a 12:1 (Qi-QS) (10)
_ _ (1' n _ _ 2 as
6me — E 1: (QLSi'Qm) J (11)
k n
- .1.
6se _ 11 1= (QLS.-Qs)i % (12)
.. 1
F1 11
6e = I.) 1%. (Qi-QLS.)2J 15 (13)
.. l

 

where n is the number of measurements; Qi is the calculated

ellipsoid value in the ith direction; Q is the best fit Q

LS.
1
value in the ith direction; QR is the mean Q value from the

data,
-_1“Q
gm _ n 5:1 i

and QS is the average trace element of the symmetric matrix

 

29

where
QLSl = QLS along direction (1.0, 0.0, 0.0)
QL82 = QLS along direction (0.0, 1.0, 0.0)
QLS3 = QLS along direction (0.0, 0.0, 1.0)

QS serves as a radius of a best fit sphere for the data.
The standard deviations (Gm, 63) measure the deviation
Of the Qi values from the mean and best fit sphere, respectfully.

The standard deviations (5-

me' see) measure the deviations of

the calculated QLS values from the mean and best fit sphere,

respectfully. The standard deviation 6e measures the devia-

tion of the measured Qi from the best fit Q surface. If all
data points fall exactly on the ellipsoidal surface «Ge = O and
G.— = 63-.
ms m
The uniformity of sampling will in part determine the re-

liability of the least squared Q surface. The following equa-

tion will indicate the uniformity of sampling:

QR _ Q5

Q8

x 100 = percent uniformity of sampling (l4)

30

where once again Qfi is the mean of the data and Q8 is the
average trace element. When the percent ratio is small, the
sampling is uniform. As the sampling becomes less uniform, the
percent ratio will increase. (Table lb).

Homoqeneous anisotropy and behavior as a pseudocrystal can
be determined by comparison of the standard mean-square devia-
tions. If 6e = 6a and 6e = as the data is homogeneous iso-
tropic. Therefore, Ge < Gse and Ge< Gme occur in order for
homogeneous anisotropy to be exhibited. Homogeneous anisotropy
is demonstrated by:

61$ a £356 > a;

e (15)

6s a 6se 7 6e (16)

where 61-?! Will approach Gs and 6 me Will approach Gse whenever
Equation (14) is very small or equal to zero; Ge is the best
measurement of data scatter which includes sample inhomogeneity
and errors in measurement.

Inhomogeneous anisotropy is demonstrated by the relationship:

65 > 6e>6me (l7)
and
GS>6e >Gse (18)

Inhomogeneity will be indicated if there is a variance of pre-
ferred crystal orientation, irregular compositional or structural
differences Within the sample and if errors in measurements exceed
the degree of anisotropy.

For the folloWing study Equations (l)-(18) were applied to

seventeen rock samples and the results are shown in Tables I-VI.

31

Error Analysis:

 

Cutting and reduction spot variation.

 

The angles of the rocks were cut as accurately as possible,
but due to clamping and other equipment used in cutting, the
angles of each rock sample showed an accuracy of i2.00.

The samples were always cut with the Z axis perpendicular
to cleavage and the XY plane being the cleavage plane. By this
method both Wood (1974) and Westjohn (1978) suggest that the re-
duction spots major and intermediate axes are defined. However,
a 100 variation was observed between the cleavage plane and the
true major and intermediate axes of the reduction spots. This
variation is further supported by Westjohn (1978). Thus, the
true maximum and intermediate axes of the ellipsoidal reduction
spots of these samples are accurate to 112.00.

Apparatus.

 

Bennett (1968) estimated the time measurement accuracy, the
distance accuracy and the veolcity accuracy for the ultrasonic
equipment as 1'0.7%, 10.33% and 1.00%, respectively. However,
the sampling length in this study was shorter, so the time and
distance accuracy was recalculated and the resulting velocity
accuracy was also recalculated.

Due to attenuation in the Z (0.0, 0.0, 1.0) direction the
sampling distance was much smaller in this direction than other
propagation directions. The approximate sampling distance was
1.27 cm. By using the lathe setup, measurements were good to
1'0.0075 cm. Thus, the accuracy of the distance is good to i0.59%
for this study. The time accuracy was recalculated by taking 10
measurements of the same sample and finding the percent error.

It was found to be 0.69% or ==O.7%.

32

By using:

= 1.30% (19)

 

AV AX 6t AV
' V

where A‘V is the percent velocity error, 4.x is the percent
V X

distance error and 1L2 is the percent time error. This smaller
sampling distance wiIl indeed cause greater error, although it
is possible to compensate, in part, for this by increasing the
sweep speed of the oscilloscope. A reasonable estimate for the
total error of the velocity measurements is 12.0%.

Using an example of how a velocity error of :2.0% will

affect the Qi values is given below:

_ 2 2 2
Qi/—V1+V2+V3

if V = 6.00 km/sec, V = 3.00 km/sec and V = 2.50 km/sec,

l 2 3
then
Qi = 51.520 mz/sec2
Now if V1 = 6.00 + (6.00 x 0.02) km/sec, V2 = 3.00 + (3.00 x
0.002) km/sec and V3 = 2.50 + (2.50 x 0.02) km/sec,
Qi = 53.865

Then the Qi values are good to:

53. - . , . _
853.8651 250 x 100 $3 5.0% (max1mum pOSSIble error)

 

DISCUSSION AND RESULTS

 

Thin Sections:

 

A definite lineation of opaque pyrite crystals were noted
for all samples in the XY plane. The crystals formed small
ellipsoids Whose major axis corresponded, within the amount of
experimental error, to the major axis of the ellipsoidal reduc-
tion spots (Table VI).

Velocities and the Q Ellipsoids

 

Of the eighteen samples, seventeen showed excellent results.
One sample, #3, was severely fractured causing very poor mea—
surements in all directions. To define the elastic ellipsoid
it is necessary to measure the three phase velocities in a mini-
mum of six noncoplanar directions Bennett (1972).

All samples were measured in nine directions except sample
#7, which was measured in seven directions due to fracturing and
poor signal transmission.

The measured phase velocities showed a common relationship
for all samples. V1, V2 and V3 were the fastest, in propagation
direction 1 for most samples. For all samples the slowest
velocities were observed in propagation direction 3, which is
most probably related to air spaces in between the cleavage
surfaces. These two observations are quite obvious from the
velocity data (Table Ib). Thus, a relationship can be seen
between the "fast" propagation direction and the major axis of

the ellipsoidal reduction spots. The slowest velocity direction,

likewise corresponds to the minor axis of the reduction spots.

33

34
The resulting Qi from propagation direction 1 is always the
largest of the nine propagation directions and the Qi for prOpa-
gation direction 3 is always the smallest (Table II).

From Table I and Table II an obvious trend is shown. Sta-
tistical calculations from Table II prove that the fit of the Qi
values and the theoretical QLS. values conform to the proper
statistical tests mentioned in the methods section of this paper.
As a second test the Qi and QLS. values correlation coefficients
were compared by using a linear regression program incorporated
in a Texas Instruments SR-56. The nine Qi values were entered
as the X coordinates and the respective QLS. were entered as the
Y data. The r values were as follows:

% probability it is a valid
correlation coefficient for

9 measurements (taken from
M. Lamont, L. Douglas, R.

 

Sample ‘3 Oliva, 1977)
#1 0.964596 99.9
#2 0.910346 99.9
#2' 0.976061 99.9
#4 0.952303 99.9
#4' 0.948849 99.9
#5 0.817783 99.0-99.9
#5' 0.774328 95.0-99.0
#6 0.996321 99.9
#6' 0.880862 99.0-99.9
#7 0.978584 99.9
#7' 0.940235 99.9
#8 0.881884 99.0-99.9
#8' 0.872623 99.0-99.9

35

% probability it is a valid
correlation coefficient for
9 measurements (taken from
M. Lamont, L. Douglas, R.

 

Sample E Oliva, 1977)
#9 0.804959 99.0-99.9
#9' 0.784228 95.0-99.0
#10 0.856649 99.0-99.9
#10' 0.967885 99.9

By the statistical tests of Table II and the above correlation
coefficient calculations, one can assume their is a definite
relationship between Qi and QLS.’

The percent uniformity (from Table II) ranged from 3.9%
for sample #9 to 19.9% for sample #4', which indicates a uni-
form sampling.

Sample homogeneity and elastic behavior as a pseudocrystal

is exhibited by all samples due to the fact that:

 

6 - Z G-— > 6
m me e

>
65 "' Gse >Ge

is true for every sample.

Sample homogeneity and elastic behavior as a pseudocrystal
has been proven: Next the data was used to determine the princi-
ple axes magnitudes and directions.

The theoretical QLSi ellipsoidal axes direction (Table III)
are compared to the known axial alignments of the ellipsoidal
reduction spots in Table IV. The major and intermediate axes
are assumed to lie precisely in the XY plane (in reality 112,0
variation is possible, see error analysis). The QLS. axes

directions (Table IV) show excellent correspondence in relation

36

to the reduction spot axes and when one considers a 112.00
variation for the reduction spots and a 15.0% maximum error
associated with the Qi values, it can be concluded that the
axes directions of the theoretical QLS. ellipsoid and the re-
duction spots are very closely related. The calculated means
and standard deviations of Table IV could be questioned, due
to the fact that these are linear functions, while the data is
in three dimensional coordinates. Fisher (1953) mathematically
devised a way to check data dispersed on a sphere by using
directional cosines.

To test the accuracy of a group of measurements, Fisher
(1953) has shown that the true mean direction of a population
of N directions lies within a circular cone about a resultant
vector R, with semi-anglecx}, at the probability level (l-P),

for k > 3 where:

cos-<(l_p) = 1 - -Nl;—R<%)1/N‘l - 1. (20)

All of the variables in Equation (20) can be calculated so one
can solve for °¢ .
The resultant vector length R is calculated by using the

known directional cosines,
2 2
R2 = (21:) + (2mi) + (Zni) (21)

Next, the direction of the resultant vector is found by using

the following equations:

1N

xR = R i=1 1i (22)
1%

YR =‘R i=1 mi (23)

37

N
_ 1
ZR" 1";

R 1 n. (24)

1

Fisher (1953) gives an estimate of the precision parameter k as:
k = ——— (25)

if kris large clustering in one area of the sphere will occur.
McElhinny (1973) used equations (20)-(25) and an assumed pro-
bability level of p = 0.05 to derive:

,< = 140

(26)
95 (RN)

where °‘95 is the circle of 95% confidence around the resultant
vector. For a more detailed mathematical explanation, the
reader can consult Fisher (1953) and McElhinny (1973). The
results of using equations (20)-(26) on the axial orientations

of the QLS ellipsoids (Table IV) were as follows:
i

X axis

Resultant vector orientation directional cosines (0.9995,

0,0199, 0.0247)
- degrees (1.80, 88.90, 88.60)

°<95 = 4.8580 - 95% of the data is within 4.8580
of the resultant vector

k = 51.903

Y axis

Resultant vector orientation directional cosines (0.0451,

0.9930, 0.109)
- degrees (87.40, 6.80, 83.70)

°‘95 = 6.1550 - 95% of the data is within 6.1550
of the resultant vector

k = 32.328

38
Z axis

Resultant vector orientation - directional cosine (0.0280,
0.1154, 0.992)

- degrees (88.40, 83.40, 6.80)

-‘95 = 3.6370 -95% of the data is within 3.6370
of the resultant vector

k = 92.592

These results show excellent correlation between the resultant
vector orientations and the "known" reduction spot axial

orientations. From the small °< values it can be concluded

95
that there is a very small amount of scatter for the seventeen
QLS. axial orientations.

1 One last method can be used to compare the relationship
between the axial orientations. An equal area stereonet plot

comparing the X axis of the Q ellipsoid orientation and

LS.

1
the field measured X axis orientation of the reduction spots
is shown in Figure 12. This method however has an added 1'2.0
error, due to the :2.00 accuracy of field measurements. The

Q and reduction spots deviate by a mean value of 8.50 which

LS.
is :gain well within experimental error. Sample #7 is the
worst fit of all the data. This may be due to the fact that
Sample #7 was measured in only seven propagation directions due
to fracturing. These fractures may have caused other propaga-
tion directions to be measured inaccurately.

Similar orientations of the axes has been shown by three
independent methods, so next the relationship between the axial
ratios will be examined. The magnitudes of the axes for the
QLS. ellipsoids are listed in Table III. The values were used

1

to tabulate the data in Table V using the equations:

39

Figure 12. Equal area stereonet plot of the field orienta-
tion of the X axes of the reduction spots (iR)
and the respective plots of the X axes of the
QLS. ellipsoids (iQ), taken from Table IV.

1

The degrees of difference between the iR and iQ
were measured directly from the stereonet and
are listed below:

 

 

Sample Number Deviation of iQ from iR
10 6°
20 5°
2'Q 12°
4Q 4°
4 'Q 6°
so 1°
5 'Q 8°
6Q 11°
6'Q 0.5°
7Q 22°
7'Q 4°
so °
8'Q 2°
90 15°
9'0 16°

100 11°
lO'Q 16°

Mean 8.3

’
/
/
I
’

IORO‘:‘-"IOQ

40

.050
SR
\
\ I
\5 0
2'!
/"—_°20
/
(I
2'0
9R
P‘x‘
I ‘s‘
/ ‘90
IR an/ 00
Reg
’\
so , '0
o, [070
9G /
/
/
/
/
/
/
/
of..7'o

41

log (26)

MIX
’ ll
3’

NH“

log = B (27)

where X is the magnitude of the major axis, Y is the magnitude
of the intermediate axis and Z is the magnitude of the minor
axis. This allowed comparison of the QLS. axes ratios to
Westjohn's (1978) axes ratio data. Figur: 13 shows Westjohn's
values plotted from all samples throughout the study area.
Figure 14 shows the QLS. axes ratios after using Equations
(20) and (21). Figure 15 is a combination of QLS. axes ratios
and Westjohn's reduction spot axial ratios. A definite simi-
larity between both sets of data can be observed. Westjohn
did extensive work in the Harvey syncline (site 8) and the
Negaunee outcrop (site 1). Sample #l's (from the Harvey syn-
cline) QLS. axial ratio shows a direct relationship with
Westjohn's reduction spots axial ratios. Sample #2 and 2'

(from the Negaunee outcrop) QL axial ratios show a poorer

Si

fit. Sample #2' fits fairly well into Westjohn's twenty-three
plotted points. Sample #2 however, does not fit as well into
Westjohn's Negaunee area data. Trying to correlate the QLSi
axial ratios and the reduction spots axial ratios on a one

to one basis is impossible due to scatter in both sets of data.
Thus a plot of the average values would be of more importance.
The mean values for the axial ratios of the reduction Spots

for the Harvey syncline and the Negaunee outcrop are both very

close to the QLS mean axial ratio (Figure 15).

Figure 13.

42

Plot of 46 deformation ellipsoidal reduction
spots showing the variation of data from the
site 8 or the Harvey syncline (each 0 repre-
sents one ellipsoid from the Harvey syncline)
and the variation between site 1 or the
Negaunee outcrop (each 0 represents one ellip-
soid from the Negaunee outcrop), Taken from
Westjohn (1978). Means are plotted with large
symbols.

- -._..__——— ____..-

43

ON.-

OO.N

_n.N

0..»

 

>\N @O .._

 

60‘.

 

#6..
r 4 r d t r -
O
O
O
O
O
O
J
I00
0’
4 .A.
60/0 1'
o $
Aw
I
.1
OGW

o¢.o

10*

I

10*

AA AA

AA A A

I ‘ - A A A A A A A

 

«.0

«.6 >\x 00..

¢.O

0.0

 

pl
. . L Q‘ K
, A
\
._
I bl
. t n A
t .
x
u
\A
\ ,I
\/ ( (\VA 1 ..1. <1
x
2‘ \
\x
\ \\
r \
I
x x
a \

s p N

 

 

 

 

.. x... u... 0 ...

44

Figure 14. Plot of 17 QLS. surfaces to show variation of
the data throughout the study area. Each O
represents one QLS. ellipsoid. Correlation
between Westjohn's sites and the numbered QLS.

1

ellipsoids is given below:

 

 

Westjohn's Sites Numbered Samples Taken
Site 1 (Negaunee) 2 and 2'

Site la (Negaunee) 8, 8', 9 and 9'

Site 4 10 and 10'

Site 5 7, 7'

Site 6 4, 4', 5, 5'

Site 7a 6 and 6'

Site 8 (Harvey) l

45

>\N 00..

 

 

~20: 0.0a 0.0: ¢.Oo 0.0.. N0. ..0:
0‘ .
.O- N 1
h A
o .0 no, .
”N — I °_‘ .0 A _.o
o .w Ln
\ C \ O .
0 . 1 _
0 $ .
Aw.
_ . I
or... s s w.‘ «d
A.
A
V i .
cod T .3 >\x 00..
fit O/V * a
200 00 W0
6
4. 0.2.... x .
.9 1
.T A
J w
0/ #00 ohwx 1 A
.N t .o o .s. A
.0 4 w t 0
a? I
4
[Var/O 1’ ”Oh\
. s
O
b
I 0.00’\ A
0..» t A. we 4 1 0.0
O °~ . .
0e.
20 .
4' .
a... U
4

 

 

 

0N0 0N0 «0.0 0v.0 00.0 00.0 05.0

.010; .o u. s. 0.1.. a O L . s ..D .. n a . . . .1.... . n . n . 0,. u t... . , v p. .

46

Figure 15. Plot of the relationship between Westjohn's
reduction spot data (0 Negaunee area, 0
Harvey area) and the QLS ellipsoid data (0).
1

Means are plotted in large symbols.

 

 

 

47

0«..

00..

00.«

.0.«

0..»

 

>\N 004

 

 

 

5.0-. 0.0: 0.0: v .0- 0.0- «.0- ..0-
1- --J-- I - 14-4444- -44-- 4- dlri- 4- i)<.-
A
a
.0. .
N
o u .
n .
O. . .o i
I \ O o .
N O . O
C‘ . ‘ n N o 0‘ .
o o
.v 00 n
o 0 our % .
O _ O O...
o I .
A
. ..x o A
Oh: .
v C 1
C _
.
O ‘ A
C
C C .
C
V a
an. 00/ x. .
_° C “O .
0 0x
4 a.
o« .
0 .
Ax .
ooo ms 1
0/ It Wx .
OF 60 $0 I
‘ A
4- * A .
A! O A
O \
9/ 1v 00.3 1
.1v 1‘-
o .
n a
.00
4
.. o 4 1.
«so
no \ I
0?
.o .
Jr .
.
s
00
.o
n P b p p
0«.0 0«.0 «0.0 00.0 00.0 00.0 05.0

 

..e

«.0

no (X 00-.

0.0

-.Oh'r-—

 

 

 

 

rl.-.‘|-JI.II"I|I' N -.1‘..t‘llllo I‘tlxil.l. A. f..u| - ..t - it‘ll .I... - 4"! l. v... - ..i ‘1‘ Die-1!: (, Illitfl-Iu-o'l! s .DJIDI -.D Jill». i-.'ll|l¢l.l'.o.l|l'.$
4| . . t .
. . .. . I .
I . r \ s
.
\.
.
..
I A
I 1/ w
I s
, . .
r . .

Ht.-.— a,

.
a.

\ . . .
- s \t

. .. .
-..-(- st. . v-(lllf'pi 1.39.11»... .. nut. iilbhu slit-.01.-. vt.( « . :ttlll.‘ . ..

.
I

.
-..-I.1l .1.-

r“-

48

Conclusions:

 

Statistically the Q ellipsoid has been shown to be
very similar to the reduction spots, in both axes direction
and axial ratios. Thus, the Q ellipsoid method can be used,
as reduction spots can be used, to determine finite strain,

regionally, for this area.

 

APPENDIX A

49

50

Table Ia

Propagation direction, directional cosines and sampling distance.

PrOpagation

Direction

\OmxlO‘Ul-waI-J \OCDxlO‘U‘lIbb-JNH \DmdeI-waI-J

\OQQO‘U‘l-waH

51

Directional
Cosines

 

(0.0, 0.0
(0.70711, 0.70911, 0.0)
(-0.70711, 0.70711, 0.0)
(0.97030, 0.0, 0.24192)
(0.97030, 040, -O.24l92)
(0.0, 0.97030, 0.24192)
(0.0, 0.97030, -0.24l92)

#2

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.89490, 0.0, 0.44620)
(0.89490, 0.0, -0.44620)
(0.0, 0.89490, 0.44620)
(0.0, 0.89490, -0.44620)

#2'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.93969, 0.0, 0.34202)
(0.93969, 0.0, -0.34202)
(0.0, 0.93969, 0.34202)
(0.0, 0.93969, -0.34202)

#4

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.95630, 0.0, 0.29237)
(0.95630, 0.0, -0.29237)
(0.0, 0.95630, 0.29237)
(0.0, 0.95630, -0.29237)

Sampling
Distance
(Meters)

0.1029038
0.0978221
0.0203201
0.1158244
0.1117604
0.0980443
0.0988063
0.0939803
0.0932183

0.1018543
0.1092204
0.0609602
0.1181104
0.1186184
0.0985523
0.0977903
0.0999493
0.1037593

0.1140464
0.1102364
0.0165101
0.1203964
0.1150624
0.1087124
0.1099824
0.1085854
0.1092204

0.1069344
0.1028703
0.0200661
0.1193804
0.1168404
0.1036323
0.1033783
0.0980443
0.0990603

Propagation
Direction

52

Directional
Cosines

 

 

\OGDQO‘W-bWNH \OQOU'lobLQNE-J \OmxlO‘IU'l-QWNH

\qummbwwl‘

#4'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.97815, 0.0, 0.20790)
(0.97815, 0.0, -0.20790)
(0.0, 0.97815, 0.20790)
(0.0, 0.97815, -0.20790)

#5

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.95882, 0.0, 0.28401)
(0.95882, 0.0, -0.28401)
(0.0, 0.95882, 0.28401)
(0.0, 0.95882, -0.28401)

#5'

(1.0, 0.0, 0.0)
(0.0, 1.0, 0.0)
(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.92387, 0.0, 0.38268)
(0.92387, 0.0, -0.38268)
(0.0, 0.92387, 0.38268)
(0.0, 0.92387, -0.38268)

#6

(1.0, 0.0, 0.0)
(0.0, 1.0, 0.0)
(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
{-0.70711, 0.70711, 0.0)
(0.87036, 0.0, 0.49242)
(0.87036, 0.0, -0.49242)
(0.0, 0.87036, 0.49242)
(0.0, 0.87036, -0.49242)

Sampling
Distance
(Meters)

 

0.0930913
0.1023623
0.0157481
0.1113794
0.1113794
0.0947423
0.0989003
0.1018543
0.0972823

0.0972823
0.0985523
0.0180341
0.1066804
0.1092204
0.0946153
0.0960123
0.0930913
0.0962663

0.0980443
0.0993143
0.0170181
0.1051563
0.1031243
0.0923293
0.0939803
0.0982983
0.0982983

0.1046483
0.1059183
0.0182881
0.1193804
0.1168404
0.1021083
0.1028703
0.1035053
0.1037593

Propagation

Direction

 

\DCDQO‘U’IDDJNH \OCDQQU‘Irfi-DJNH \DG)\IO‘W:>U)Nl—'

\DCDQO‘UIubUJNH

53

Directional
Cosines

 

#6'

(1.0, 0.0, 0.0)
(0.0, 1.0, 0.0)
(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.93041, 0.0, 0.36650)
(0.93041, 0.0, -0.36650)
(0.0, 0.93041, 0.36650)
(0.0, 0.93041, -0.36650)

#7

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.85717, 0.0, 0.51504)
(0.85717, 0.0, -0.51504)
(0.0, 0.85717, 0.51504)
(0.0, 0.85717, -0.51504)

#7'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.84805, 0.0, 0.52992)
(0.84805, 0.0, -0.52992)
(0.0, 0.84805, 0.52992)
(0.0, 0.84805, -0.52992)

#8

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.89101, 0.0, 0.45399)
(0.89101, 0.0, -0.45399)
(0.0, 0.89101, 0.45399)
(0.0, 0.89101, -0.45399)

Sampling
Distance
(Meters)

 

0.1028703
0.0970283
0.0173991
0.1106174
0.0988063
0.0991873
0.0990603
0.0913133
0.0908053

0.0990603
0.0992491
0.0203201
0.1092204
0.1054921
0.1008383
0.1013463
0.0990603
0.0993143

0.1031243
0.0974093
0.0157481
0.1122684
0.1109984
0.0957583
0.0962663
0.0973492
0.0965232

0.1021083
0.0960123
0.0210821
0.1096014
0.1099824
0.990603

0.990603

0.0919483
0.0920753

Propagation

Direction

 

“DwQO‘LnubWNH \Omflmtfl-hWNH koooqmmc-wwld

\DCDQGLNDLUNI"

54

Directional
Cosines

 

#8'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.86603, 0.0, 0.50000)
(0.86603, 0.0, -0.50000)
(0.0, 0.86603, 0.50000)
(0.0, 0.86603, -0.50000)

#9

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.81915, 0.0, 0.57358)
(0.81915, 0.0, -0.57358)
(0.0, 081915, 0.57358)
(0.0, 0.81915, -0.57358)

#9'

(1.0, 0.0, 0.0)
(0.0, 1.0, 0.0)
(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.85717, 0.0, 0.51504)
(0.85717, 0.0, -0.51504)
(0.0, 0.85717, 0.51504)
(0.0, 0.85717, -0.51504)

#10

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.91706, 0.0, 0.39875)
(0.91706, 0.0, -0.39875)
(0.0, 0.91706, 0.39875)
(0.0, 0.91706, -0.39875)

Sampling
Distance
(Meters)

 

0.1013463
0.0972823
0.0152401
0.1104904
0.1089664
0.0986793
0.0993143
0.0960123
0.0961393

0.1057913
0.0970283
0.0195581
0.1130304
0.1089664
0.1041403
0.1043943
0.0904243
0.0967743

0.1003303
0.0967743
0.0170181
0.1073154
0.1092204
0.1007113
0.1002033
0.0919483
0.0947423

0.1089664
0.1028703
0.0193041
0.1186184
0.1193804
0.1054103
0.1046483
0.0980443
0.0990603

Propagation
Direction

55

Directional
Cosines

 

 

\OQQGU'IuwaI"

#10'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)

(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.87882, 0.0, 0.47716)
(0.878882, 0.0, -0.47716)
(0.0, 0.87882, 0.47716)
(0.0, 0.87882, -0.47716)

Sampling
Distance
(Meters)

 

0.0906783
0.0929643
0.0157481
0.0968378
0.0988063
0.0906783
0.0901703
0.0908053
0.0906783

56

Table Ib

Propagation directions, directional cosines and V1, V2 and V3

mean velocities.

Propagation
Direction

57

Directional
Cosines

 

 

KDCDQO‘U'IDUJNH \DmQO‘UIhWNl-J \DmNO‘U‘IQWNH

\DmQOSUIanNH

#1

(1.0, 0.0, 0.0)
(0.0, 1.0, 0.0)
(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.97030, 0.0, 0.24192)

(0.097030, 0.0, -0.24192)

(0.0, 0.97030, 0.24192)
(000' 0097030, -0024192)

#2

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.89490, 0.0, 0.44620)
(0.89490, 0.0, -0.44620)
(0.0, 0.89490, 0.44620)
(0.0, 0.89490, -0.44620)

#2'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(“0.70711, 0.7011, 0.0)
(0.93969, 0.0, 0.34202)
(0.93969, 0.0, -0.34202)
(0.0, 0.93969, 0.34202)
(0.0, 0.93969, -0.34202)

#4

(1.0, 0.0,
(0.0, 1.0,
(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.95630, 0.0, 0.29237
(0.95630, 0.0, -0.29237)
(0.0, 0.95630, 0.29237)
(0.0, 0.95630, -0.29237)

0.0)
0.0)

Velocities (km/sec)

 

V1

 

6.899
4.996
4.515
6.436
5.882
6.587
6.587
5.221
5.122

6.701
6.277
3.332
5.320
5.815
6.009
6.189
5.152
5.188

6.307
5.575
3.370
6.258
5.922
6.025
5.968
4.760
5.176

6.520
5.715
3.541
6.079
6.350
5.956
6.006
5.160
4.717

v2

 

3.499
2.385
1.992
2.597
2.517
3.083
2.559
2.349
3.329

3.223
3.309
2.622
3.374
3.057
3.159
3.134
3.163
3.183

3.050
2.966
1.437
2.864
2.831
2.465
2.747
2.582
2.552

3.073
3.025
1.705
3.061
2.465
2.528
2.901
2.935
2.984

V3

 

2.780
1.932
1.494
1.631
2.192
2.159
2.196
1.880
2.589

2.380
2.061
2.420
2.461
1.990
2.722
2.686
2.603
2.688

2.816
2.212
1.425
2.863
2.626
2.259.
2.382
2.433
2.212

2.700
2.246
1.674
2.676
1.866
2.115
2.034
2.113
2.006

Propagation
Direction

 

\oooqmmwaI—I \Dmflamofi-WNH \DmflmU'lnwaH

\OCDQO‘UInbUJNI—i

58

Directional
CoSines

 

#4'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0
(-0.70711, 0.70711, 0.0)
(0.97815, 0.0, 0.20790)
(0.97815, 0.0, -0.20790)
(0.0, 0.97815, 0.20790)
(0.0, 0.97815, -0.20790)

#5

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.95882, 0.0, 0.28401)
(0.95882, 0.0, -0.28401)
(0.0, 0.95882, 0.28401)
(0.0, 0.95882, -0.28401)

#5'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.92387, 0.0, 0.38268)
(0.92387, 0.0, -0.38268)
(0.0, 0.92387, 0.38268)
(0.0, 0.92387, -0.38268)

#6

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.87036, 0.0, 0.49242)
(0.87036, 0.0, -0.49242)
(0.0, 0.087036, 0.49242)
(0.0, 0.87036, -0.49242)

Velocities (km/sec)

 

V1

7.165
6.146
3.228
6.471
6.914
6.672
6.537
4.897
5.154

6.856
5.120
4.031
5.569
5.553
5.151
5.255
4.638
4.439

6.528
5.518
4.612
6.114
6.288
5.108
5.026
4.693
5.149

6.708
5.695
4.156
5.527
6.215
6.303
6.197
5.335
5.136

V2

 

3.003
3.002
1.549
3.027
3.060
2.979
2.886
2.927
3.047

3.196
2.857
1.932
3.055
3.092
2.902
2.321
2.633
2.839

3.242
2.623
2.086
3.039
3.033
2.971
2.990
2.933
3.027

3.133
2.878
1.345
3.450
3.192
2.503
2.611
2.828
2.556

V3

 

2.135
2.497
1.519
2.916
2.175
2.369
2.483
2.186
2.212

2.766
2.730
1.892
2.487
2.629
2.351
2.321
2.024
2.382

2.967
2.192
2.038
2.577
2.672
2.239
2.487
2.152
2.429

1.938
1.629
1.270
2.003
2.101
1.919
2.125
2.193
2.256

ProPagation
Direction

59

Directional
Cosines

 

 

mmxlONUIhUJNH @QOUIIbWNP-J \DCDQO‘Mowat-J

\omqmmwai-I

#6'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.93041, 0.0, 0.36650
(0.93041, 0.0, -0.36650)
(0.0, 0.93041, 0.36650)
(0.0, 0.93041, -0.36650)

#7

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.85717, 0.0, 0.51504)
(0.85717, 0.0, -0.51504)
(0.0, 0.85717, 0.51504)
(0.0, 0.85717, -0.51504)

#7'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.84805, 0.0, 0.52992)
(0.84805, 0.0, -0.52992)
(0.0, 0.84805, 0.52992)
(0.0, 0.84805, -0.52992)

#8

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.89101, 0.0, 0.45399)
(0.89101, 0.0, -0.45399)
(0.0, 0.89101, 0.45399)
(0.0, 0.89101, -0.45399)

Velocities (km/sec)

 

 

 

 

Vi V2 V3

6.429 3.164 2.091
5.880 3.109 1.702
4.579 1.540 1.487
6.012 3.225 1.941
6.199 2.943 1.659
5.166 2.867 2.740
5.054 3.076 2.580
4.659 2.801 1.776
4.935 3.089 1.949
6.126 2.975 2.197
3.492 1.917 1.814
5.985 2.395 2.304
5.042 2.083 1.867
4.826 3.147 2.572
4.233 2.514 2.117
3.706 2.719 2.078
6.366 3.105 2.515
5.477 1.917 1.517
3.099 1.549 1.520
5.197 3.153 2.389
5.477 2.891 1.894
5.094 2.616 2.176
5.477 2.395 1.981
5.170 2.518 1.999
5.170 2.518 1.999
7.091 3.283 2.745
5.121 3.085 2.574
3.482 1.770 1.741
6.563 3.321 2.609
6.625 3.293 2.391
5.214 2.966 2.144
5.106 2.948 2.476
4.839 2.736 1.973
4.651 2.970 2.423

Propagation
} Direction,

 

\DQQO‘Wnwal-J \DmflmU‘lbWNl" \OWQO‘U‘IbWNH

\OCDQO‘U‘IIwaI-J

60

Directional
Cosines

 

#8'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.86603, 0.0, 0.50000)
(0.86603, 0.0, -0.50000)
(0.0, 0.86603, 0.50000)
(0.0, 0.86603, -0.50000)

#9

(1.0, 0.0,
(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)

0.0)

(0.81915, 0.0, 0.57358)
(0.81915, 0.0, -0.57358)
(0.0, 0.81915, 0.57358)
(0.0, 0.81915, -0.57358)
#9'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)

(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)

(0.85717, 0.0, 0.51504)

(0.85717, 0.0, -0.51504)
(0.0, 0.85717, 0.51504)

(0.0, 0.85717, -0.51504)
#10

(1.0, 000' 000)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)

(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.91706, 0.0, 0.39875)
(0.91706, 0.0, -0.39875
(0.0, 0.91706, 0.39875)
(0.0, 0.91706, -0.39875)

Velocities (km/sec)

 

V1

 

6.756
6.196
5.080
6.424
6.410
6.017
6.017
5.053
4.492

7.006
6.383
5.752
5.976
6.233
4.866
5.931
4.861
5.692

7.166
5.752
5.673
6.426
6.425
4.706
5.330
4.421
5.206

6.564
5.779
3.530
6.590
6.218
5.667
4.757
4.180
4.503

V2

 

3.197
3.189
1.438
3.269
3.263
2.274
2.586
2.017
2.641

3.296
3.234
1.686
3.078
3.287
2.590
2.451
1.966
2.968

3.216
3.349
1.605
3.272
3.290
3.033
3.191
2.179
3.267

3.061
2.991
1.697
3.089
3.109
2.252
2.844
2.200
2.814

V3

 

2.791
1.938
1.373
2.361
2.369
1.701
2.197
1.861
2.334

2.784
2.494
1.657
2.239
2.698
2.170
2.139
1.706
2.601

2.818
2.794
1.519
2.567
2.411
2.189
2.753
1.940
2.533

2.471
1.994
1.424
2.188
2.287
2.233
2.265
1.865
2.293

Propagation
Direction

 

\oooqoxmwaI-J

61

Directional
Cosines

 

#10'

(1.0, 0.0, 0.0)

(0.0, 1.0, 0.0)

(0.0, 0.0, 1.0)
(0.70711, 0.70711, 0.0)
(-0.70711, 0.70711, 0.0)
(0.87882, 0.0, 0.47716)
(0.87882, 0.0, -0.47716)
(0.0, 0.87882, 0.47716)
(0.0, 0.87882, -0.47716)

Velocities (km/sec)

 

V1

 

6.457
5.738
3.262
6.093
6.025
5.038
4.901
4.681
5.334

V2

 

3.119
3.099
1.594
3.027
2.994
2.963
2.892
2.987
3.084

V3

 

2.830
2.626
1.495
2.645
2.341
2.699
2.390
2.377
2.591

62

Table II

Qi ellipsoid values and Q values with associated Qm’ Qs’

5,, and percent sample uniformity.

G;-
HV 6 6se’ e

s’ GI-I-le'

63

 

 

 

 

Sample #1

Propagation Direction Qi QLSi
1 67.568 60.984
2 34.381 38.155
3 26.585 26.571
4 50.827 52.114
5 45.738 47.026
6 57.555 60.368
7 54.759 57.522
8 36.311 33.623
9 44.020 41.332
6 — = _ = =

m 12.051 0111 46.416 03 41.903
6 s = 12.868
6 - = 11.622

me

- 12 468 Qm-Qs lO 7%
G‘se — . Qs .
6e = 3.183
Sample #2
1 60.956 58.395
2 54.598 48.158
3 23.833 24.999
4 45.743 52.589
5 47.120 53.965
6 53.497 50.821
7 55.340 52.665
8 43.323 43.070
9 44.272 44.019
6 a = 10.110 Qm = 47.631 QS = 43.8507
6 s = 10.794
6 file = 9.204
Qm-QS

G = 9.9502 x 100 = 8.6%

se Q3
6

e = 4.184

64

 

 

 

Sample #2'

Propagation Direction

1

2

3

4

5

6

7

8

9

<3 - =
m 12.034

6 s = 13.4472
- = 11.746
me

Gse = 13.1899

Ge = 2.618

Sample #4

1

2

3

4

5

6

7

8

9

G - = 11.309
m

6.3 = 12.7514

6 - = 10.770
me

6 se = 12.2755

6»

= 3.451
e

57.180
44.771
15.452
55.562
49.980
47.480
48.837
35.254
38.197

Qm = 43.634

 

59.244
46.856
18.248
53.485
49.881
46.338
48.625
39.705
35.178

Qm = 44.173

Qm-Qs

Qs

 

Q
LSi

55.754
43.012
14.133
52.174
46.593
50.206
51.565
38.158
41.110

Qs = 37.6330

54.694
43.188
16.967
50.743
47.139
50.325
52.612
43.209
38.683

QS = 38.2830

65

 

 

 

 

 

Sample #4'
Propagation Direction Q Q
i LS.
i
1 64.913 63.747
2 53.020 46.648
3 15.127 14.287
4 59.540 54.019
5 61.898 56.377
6 59.002 62.497
7 57.227 60.721
8 37.326 43.542
9 40.741 46.956
6 i = 15.128 Qm = 49.866 QS = 41.5607
6 s = 17.2578
6 me = 14.354
Qm-Qs

6 se = 16.5837 05
G = 4.776

e

Sample #5

1 64.880 51.908
2 41.830 37.687
3 23.561 21.688
4 46.532 44.410
5 47.308 45.186
6 40.482 48.686
7 42.050 50.254
8 32.540 35.947
9 33.439 36.846
C- 1.11 = 10.922 QIn = 41.401 Qs = 37.0943
6 = 11.7433

3
6 me = 8.932

Qm-Qs

G - = 9.9167 x 100 = 11.6%

se Q8
6 = 6.285

66

Sample #5'

 

Propagation Direction

 

= 9.798

Q10.3882

O 0) ‘\ \oarqchUMbgguaH

- = 7.587

8.3348

n 0
n

6.200

Sample #6

 

- - 10.289
m.
= 10.8307
8
- =10.251
me
= 10.7947
e

as 6) 0 fl 5) U3m~JOHJGIWbOPI

=0.882
8

61.928
42.133
29.775
53.257
55.878
39.932
40.386
35.258
41.575

0111 = 44.458

Qm-Qs
Qs

58.569
43.369
20.694
46.462
53.229
49.675
49.736
41.269
38.001

x 100 =

Qm =44.556

 

Q

52.962
43.695
26.364
47.019
49.639
48.839
49.293
37.997
44.315

03 =41.007o

58.112
43.931
21.481
47.639
54.406
49.200
49.260
40.122
36.854

05 = 41.1747

67

 

 

 

Sample #6'

Propagation Direction
1
2
3
4
5
6
7
8
9
«6 - = 8.933

m.

= 9.7234

6's;
G.- = 7.7541

me

= 8.6032

G'se
c = 4.2318

e

Sample #7

1 (1,0,0)
2 (0.1.0)
3 (0,0,1)

4 0(1/ 2' 1/ 210)
(‘l/ 2: 1/ 2:0)
(0.857,0,0.515)
(0.857,0,-0.515)
(0,0.857,0.515)
(0,0.857,-0.515)

a mmqmm

m

G = 11.056
8
me

6 = 10.836
se

6 =2.216
e

55.722
47.137
25.550
50.312
49.841
42.414
41.661
32.706
37.695

Q = 42.560

m

meos
Qs

51.205
19.160
46.865
33.246
39.809
28.720
25.445

x 100

Qm =34.921

 

9.9%

Q

50.682
42.509
22.970
46.832
46.360
47.336
45.582
37.390
42.378

Q =38.7203

47.117

17.683
46.865
36.028
42.591
28.720
25.445

Qs = 32.400

 

68

 

 

 

Sample #7'

Propagation Direction
1

2

3

4

5

6

7

8

9

G .. = 10.270

m
6 fl =‘- 10.665
G.- = 9.656

me
6 = 10.075

se
6 = 3.497
e
Sample #8
1
2
3
4
5
6
7
8
9
<3 _ -

m 14.849
6 s = 15.7413
6 - =

me 13.095
5 se = 14.0986
6 =-- 7.001

e

Qi

56.498
35.974
14.314
42.658
41.943
37.527
39.658
37.065
37.065

Qm = 38.078

meos

Qs

 

68.595
42.367
18.288
60.909
60.451
40.580
40.892
34.794
36.324

Q .= 44.800

m

 

x 100 =

Q
LSi

50.626
39.458
15.523
45.399
44.685
39.703
41.834
32.737
32.737

Qs =

60.526
45.287
12.916
53.136
52.678
50.557
50.870
37.851
39.380

35.202

Q =39.5763

S

13.2%

69

 

 

 

Sample #8'

Propagation Direction
1
2
3
4
S
6
7
8
9
G - = 11.680

111

-=-.- 12.13 5
Gs . l
c,- = 10.192

me

= 10.7070
Gse
= 5.704
Ge
Sample #9

1
2
3
4
5
6
7
8
9
G a - 11.201
6 s = 11.3392
6 .— =

me 9.018
Gse = 9.1922
6 =6.644

e

63.654
52.316
29.760
57.528
57.347
44.269
47.718
33.064
32.600

Qm = 46.473

Qmugs

Qs

 

67.698
57.425
38.674
50.120
56.934
35.095
45.759
30.405
47.973

x 100 =

Qm =47.796

 

7.6%

Q
LSi

60.344
45.718
23.519
53.122
52.941
49.413
52.863
40.401
39.937

56.972
50.691
30.375
50.465
57.199
42.890
53.554
35.223
52.791

05 = 46.0127

Q =43.1937

70

 

 

 

Sample #9'

Propagation Direction
1

2

3

4

5

6

7

8

9

G - = 12.341

In _

G s = 12.6178
me
6 se = 10.0287
= 7.658
G e
Sample #10
l
2
3
4
5
6
7
8
9
G - -

m 13.529
6 s = 14.5357
G fie = 11.590
6 se = 12.7503
6 =6.979

(D

69.635
59.108
37.066
58.589
57.918
36.137
46.170
28.057
44.192

m

 

58.562
46.313
17.411
57.757
53.560
42.173
35.848
25.791
33.453

Qm =41.207

 

Q = 47.764

Q
LSi

59.520
47.538
28.353
53.865
53.194
46.236
56.269
34.382
50.517

Q 3 45.1370

8

53.387
41.774
12.520
49.680
45.482
50.052
43.727
33.291
40.954

05 = 35.8937

14.8%

Propagation Direction

 

0 a (x 6) & 'mcn~umtnou»uaw

ii = 12.060
3 512.9140

iie = 11.673
8 = 12.5528

3.032

71

Sample #10'

59.430
49.424
15.416
53.283
50.745
41.445
38.096
36.484
44.676

111

Qm-Qs

Qs

 

Q = 43.222

x 100 = 11.9%

Q

53.316
49.324
13.176
52.590
50.051
45.852
42.502
36.998
45.190

‘ Q _= 38.6053

5

72

Table III

Directional cosines of the principle axes of the QL ellipsoid;

S.
i

with the associated cosine between major (Mo) and minoe (Mo)

axes to show axes fit using Nye's approximation technique.

73

 

 

 

Magnitude (m2[sec2) Directional Cosines
Sample #1

61.396 (0.994,0.0930,0.061)

21.942 (-00097'00461'00882)

42.371 (0.054,-0.883,0.467)

0.000034 = cosine between Mo and mo axes

 

Sample #2
58.478 (0.998’*00063'-00033)
24.943 (0.035,0.026,0.999)
48.131 (-00060p-00998'00029)

-0.000007 = cosine between MO and mo axes

 

Sample #2'
56.392 (0.977,0.212,-0.036)
13.935 (0.020,0.077,0.997)
42.572 (0.215,-0.974,0.071)

-0.000000 = cosine between Mo and mo axes

 

 

 

Sample #4
55.023 (00989'0014GI-00037)
16.220 (0.059,-0.152,0.987)
43.606 (0.138,-0.978,-0.159)
-0.000000 = cosine between Mo and mo axes
Sample #4'
63.952 (0.996,-0.078,0.050)
13.670 (-00040’00125'00991)
47.059 (-0.084,-0.989,0.121)
0.00000 = cosine between Mo and mo axes
'Sample #5
51.984 (0.999’-00227'-00469)
21.576 (0.048,0.052,0.997)
37.723 (-00020'—00998'00053)

-0.00007 = cosine between MO and mO axes

H-
0 0303

H 5 3 H0802 OHOBOE: 0H0

000

H33

00

74

 

 

 

Magnitude (mg/secz) Directional Cosines
Sample #5'
53.147 (0.990,-0.l38,0.011)
25.267 (0.022,0.236,0.971)
44.607 (-0.136,-0.962,0.237)

0.000003 = cosine between Mo and mo axes

 

Sample #6
58.884 (0.976,-0.218,-0.012)
21.319 (-0.007,-0.085,0.996)
43.322 (-00219'_00972'00084)

-0.000000 = cosine between Mo and mo axes

 

Sample #6'
50.696 (0.999,0.027,0.016)
22.296 (-0.021,0.l78,0.984)
43.169 "(0.024,-0.984,0.178)

0.000003 = cosine between Mo and mo axes

 

Sample #7
50.618 (0.927,0.366,-0.084)
16.550 (0.176p-00229'00957)
28.108 (0.331,-0.902,-0.277)

0.000012 = cosine between MO and mO axes

 

Sample #7'
50.677 (0.999'0003gy-00034)
15.482 (0.337’-000005'00999)
39.447 (00038p-00999'-00002)

0.000000 = cosine between Mo and mo axes

 

Sample #8
60.530 (0.999,0.018,-0.004)
12.887 (0.004,0.029,0.999)
45.311 (0.018p-00999'00029)

0.000000 = cosine between Mo and mo axes

Sxmbol

H
O 0303

#43:: H H H
00 00 0030;;

000

H 3
0 030

75

 

 

 

Magpitude (mz/secz) Directional Cosines
Sample #8'
60.451 (0.998100009'-00054)
23.408 (0.054'-00013'00998)

-0.000004 = cosine between Mo and mo axes

 

Sample #9
58.573 (0.956,-0.274,-0.103)
25.385 , (0.202,0.363,0.910)
54.080 (-0.212,-0.891,0.402)

-0.000029 = cosine between Mo and mo axes

 

Sample #9'
60.859 (0.958,0.182,-0.219)
23.948 (0.144,0.355,0.924)
50.604 (0.246'-00917'00314)

-0.000023 = cosine between Mo and mo axes

 

Sample #10
54.045 (0.987,0.137,0.085)
11.105 (-0.109,0.175,0.979)
42.532 (0.119,-0.975,0.187)

0.000000 a cosine between Mo and mO axes

 

Sample #10'
53.693 (0.959,0.282,0.013)
12.415 (-0.052,0.133,0.990)
49.708 (0.277,-0.950,0.142)

H
05

H33
000

H
00

H 3
O 030

76

Table IV

Degree of deviation between the known axes of the reduction

spots and the computer generated axes of the QLS , which was

calculated from the measured velocity data..

 

mush/N GHMHmquHmu.CH HO COHHUGHHD

 

mwxfi HOCHZ mo COHUUDHHQ

 

Mde HOan MO GOfluomuflQ

vmwmm.aa wmvmo.m ammov.m «mmbmom wmwwh.oa mvmmmov mmvmo.v memmm.m mmmmm.w .>mo .m

mamma.vm mmmmm.~HI Hmom¢.hm mmmvm.oa mmmmo.vm mmbmm.mm mmmv¢.Hm ovmwmobm Hmmmm.m sum:
«Humm.am Nmmma.mHI mv¢om.mb mmoHN.m HMObm.~m wmmoo.mm H¢mmm.mm mnmmm.mn mmmmm.wH .0H*
whaam.mh haoam.NHI Nvmha.mm omahm.HH ~mv¢m.mh mmmvm.wm mbwoaomm vmmva.mm scorn.m oa*
mmonm.ah haomm.mml mvamh.mh hummm.mm mmmha.mm wmadb.am Hrmmm.moa mmmmm.mh mmvmm.ma .m*
vmmmm.mm ommmo.hNI mwmwm.moa mmomm.vm mmmm>.mm mm¢¢m.mh mamomomm NmHmm.moH wmmvo.ha mw
mmmmb.om mwhmm.o;I oneam.mm mmmma.m mvmmm.om mmaam.mm anamo.mm Hmmh¢.mm mammaom .mw
mmmmm.mm bmmmm.H:I bmmmm.mm omomm.a vsomm.mm ovmh>.mm bmvmm.om mwmmm.mm ommmo.a m¢
mmmoa.om movmma.m I mneom.hm ummwmm.a mmmmo.om mammo.mm Hmmmm.am momom.hm vaNm.N .hw
vamo.moa mmvwm.mml momhm.o> mmmmb.ma mmmHN.moa mmmmm.mb mmbmh.vm memm.mm mommo.mm hw
mmmah.mh mmabm.oal wmmmw.mm NO¢NM.OH H>m¢h.mb mmHmH.Hm mmomo.mm Nmm¢¢.mm mmaamxa. .mw
mmovm.vm omaom.mal ommmm.moa maomm.v manhm.vm mvmmm.om mmmmw.om mmmmm.moH mhmvo.ma m¢
_/ oammm.mb hmOFm.mHI mommm.bm bmwmb.ma mwwmm.mb mamab.mm Nmmmm.mm mmmdm.hm mmawm.h .m*
7 maomm.mm mmam~.m I mvbmH.Hm bmomo.v ormao.hm ham¢m.bm mmhmm.mm maoom.am Nmmma.m m¢
vamwo.mm 8mmm¢.m I mmmmb.vm mhmmm.b wmvmm.~m ommom.~m oamaa.bm mmahv.vm mvamm.m .v¢
wmmva.mm mwva.NHI mmmmo.~m mmbmm.m mmv¢h.mm vwmwm.mm ommma.mm NHmmm.Hm vahm.m v¢
mbH¢m.mm mmmmo.mHI wmvmm.>h Nammm.v omwmm.mw whamm.mm hmamo.mm ommmh.>h mmHN¢.NH .~*
mmmmm.mm Harmm.m I mmmm¢.mm mmmmm.mw bvhb¢.mm mmmmm.hm owoam.am wmmam.mm mmmoo.v ~*
NHNvH.Nm mmmmo.mmI oomam.mm mvmaa.mm mma¢m.mm mmmmm.mm mmmmv.mm wommm.vm mvmmm.m aw

Ammusm

Isoo an 040

ummnv sowumu

Iswwuo quo

Acowuowm

OomIN OOIM Oomlx OoIN Oomlw Oomux OomIN Oomuw 0o" MMMHMMMMMMW

soflumucmfluo

pomm coma

Inapmu s3ocx

Amwmummav Ammmummav Ammmummov

Table V

WOOd'S (1974) technique of plotting log % and log é-was used
for the QLsi
Z is the magnitudetof the minor axis and Y is the magnitude of

data, where X is the magnitude of the major axis,

the intermediate axis. The ratios were then used to compare

to Westjohn's plots, Figures 12, 13 and 14.

#1
#2
#2'
#4
#4'
#6
#6'
#7
#7'
#8
#8'
#9
#9'
#10
#10'
#5
#5'

Mean = 0.119
S. Dev. = .061

Variance = .004

79

KIN

log

 

.161
.085
.122
.232
.133
.133
.070
.256
.109
.126
.121
.035
.080
.104
.034
.139

.076

H
.0
a

Nun

 

-.286
-.286
-.485
-.429
-.536
-.308
-2.87
-.230
-.406
-.546
-.291
-.329
-.325
-.583
-.602
-.243

-0247

Mean = 0.376
S. Dev. = 0.132

Variance = 0.017

80

Table VI

Thin sections were done for all samples. Ellipsoidal pyrite
grains were observed at 100-200X. The average orientation
direction of the major axis of 50 pyrite grains is compared
to the major axis of the ellipsoidal reduction spot.

81

Sample # Deviation (clockwise rotation
of the microscope stage are
positive)

1 - 0.5°
2 +10.0°
3 + 0.55"
4 - 0.50
5 + 0.50
6 + 9.50
7 + 0.5°
8 - 3.5°
9 0.0°

10 - 600

APPENDIX B
Computer Program used to Tabulate Tables I-IV,

written by Tilmann and Bennett (1973).

82

823

.11.:3.ss

PAGE

03/17/00

FIN B .0 0800

7"! 175 0"! 1

q
1

PROGRAM SONI

urur,ourvurl

FROG?! M $08.10 (

s
.5
01
«1‘ I
no It
'45L 3
09.5 1
51" 0
PLO 6
IL... 1|
LET vl
L ru 0
[‘2‘ ol
09‘ E
u I H
0590 0|
I c 0
NO I:
7". 3
1c 1
3.10 9
F... 6
T. .L (
SUN 2
ES. 0
3. 0‘
SE E
0 F N
TEE cl
ER I
55 \o
E 0 6
ISL 0
T7,: 6|:
I N (3
co: 0. V
Cr." ‘3
LNR cl!
El... El.
VH7 hV
R.‘ Cl 0
DEC I)
Ev! )6
.KEO 6 0
U05 IO
5 l. 3“
LEI: 1V 9
Snow. (6”)
IA...» ‘13
.K YT...
ESA EA‘
C“ HUF—
JAIAI. v: I I
2A 9.."
€7.10 )3 0
R05 3 .0
. KY 13 I
LV. . 0'.
LEL 10‘
7.0L DP
“ I. ll. all.
0 0"
"RF 77°
00° S‘N
‘0 N00
nfius EVI-
DAI MPH
.I—IX .1an
95‘ 00c
I
bufvb b

I

wheat 0 1s oxnectzou or MEASUREMENTS.

RE‘JZNG IN 0 HAIRIX,

 

_
.
u '
I
’
z
N
In
D
x 1
i a
I 1
u 0 .
_ V ..
H ,.
0;
I .
HA .
o
1
,
x
H
IA
x
o
I.
D
I
I
O
o 0
I I:
'0 x
C )2
\l E 3,
3 R Is,
0 v. 1:
1 D :0
a H J1
J o It?
I c. ’3
‘1’ l J‘
Jo x '0
1 o 9.x
..0 1 (5
(0 I C!
02 1 (o
”(F “NI”
'00Q01’11...
11‘ 0(10‘U

.. ”TN-Col .- Z'IN

 

 

 

 

 

__ E
_. v M
. ‘ _
V a
5“ 2 1
TN S .
N0 . N .
flu; . 7 _
o0 _ .
.8 , n u 1
RE _ 1 1
UR . E 1
sm u .
A .u i
E _ I .
NE 1 .
H u 1 .
at _ m 1
.IO _ 7
T _ X
F . 2
on a 0 _
56 V _
UN A .
LI I _ _
A0 a
VN P
cc H
ED. 6
Hi i 0
.IEI X
R... 0
SRT 1
7.00 I
«ca y
I))
00 E ) X 312
EET I I I 9 I 9
RT T X .K III
E I 0 r- ((.l
H... C )1 .u 000
"IV. ‘a 0 3 3 Mn 0 O i
0. 3 L 5 )))
than I E 1 c A 222)))
XSC 1 V :5 T o u u 231
I C ._ H J1 E 4 u y 1 2 ..
1.9).» J 0 Ir H )))v.!.. .
Til I 1 )3 .I 123(((
.AU )0: .. Jl. l J 000
NH" Jo x 9 I E III. C c
I 11 0 IX H (((222
V5 .4 u 1 (5 T 30L(((
E0 (0 .. .V .. __ = .. = = z
NU‘ V2 1 (0 G ))))))
.r-L. N(F H N )H N 123.956
..uL. W l 13:51 .21... 7.
5V 12(0):. 10(U T
N X Nan-IN; :7.le A
1;! 1110.350 fffit. .1
053m VIIDHTVH iiflol E
0.5.. N1ARH‘P.)Z. QKN l
EHA ceicuficiofio a
R!" CDPFCPF/DPF G
O
0
0
o 2 5 2 .
0 01 0 _
I. 11 Z
_

..v 9010....

1, 10X,12HY'IETA HETRXX/ll

 

 

g
'NG '4- narnix

“III-H

N i I! I I l...b1 01E
1?... 3.1....1U01‘11‘U

6
o
z

I
T
f
I
T
E
1
~I
2
NI
.1
Y
GENEQAT

, NMERE THETAI. HATKX IS THETA MATRIX TIMES

I:
)
I
n-
I
_ I I:
, a
z z u
0 F E
I I s
6' x o
3 n P
9 1 s
(
u p M
V I R
O I cl
2 I X
o X 1 I
O 0.. n \I T
I: v.
2 m A u. E
I I H J .l
u H II .H
v . u I s
I
o H T E .u
2 0 E H X
0 x “E vl o
J 0 T5 3 4‘
iv 1 (v .0. l
1 AI. 69. J 1
I ’h "5 bid 6. H
Near. #1 :N 01: 0.1
I‘U0( T.» 11‘ 3‘
ALVNZ'I ‘R a 2 TNZV.

,U

.1. -._-.. .

FROG!!! 50410

N
.00
p

220

2
J
I.
‘2
L

{-f (4’

21
0

CIR-(v. 0‘

u
p

N
O
U

668‘-

r ((K'II'K' f.t.C:
’ I

'rouLouznc ozrznu:uss ELLIPSOIOflL Ax:s."'”

£34

74/175 opt-1 FTN 6.0..96

00401-1 6 A A
Pnzurzsl (THETAttl ),J=1 NI

Vehiat(l60,5¥.(9F10.5.7x))

figurxnu-

5:31,; ~

Dch'i ,6

éggigl N

“67:11;JJIerEIATII.xI-rnaraIK,JI 7 _
unasuno#H£T&1(x,J

ONTlNY‘

HEYII ..JI:sun ,

CNYINUL 1. ...... -.i --__i ___..-
Pazwrzzo

C5157 (1H1110X11 3117045711 N‘YRIX/I)

0301:} 6

n:vrz 1( HETA1(I J). 4:1 _. .. __ .._ ,_,___.
ng:if(180,5x,(GE12.5 'exI)”

SWIJUTZNE 1’0 GENERATE DIVERSE .. .. _ ..- .._.. -._ 1.. ._

n 6
ciLL auvnAIIn.1u6141.cnvxuv.ner

GENERATING THETAZ n4. RIX. wuss: TNEIAZ "Aszx IS xuvznse IIHES"

'hEl’l TF. ANtPOS

03/17/00 .11. 33.53

...-.__.....a -._. _

'pacs

 

 

 

 

 

0011.1, . 5. . .---.. __...,_ ._.__.....__.._-i___
004J=1fi
8d
Harizi. J)=CATINV(I,K)'THETIT(K,J) ..7 m ._-i_1 i-_i__i__i__.,_i
su~:sunofM€TAZII.JI
gcyrzuug
H:!AZ( ,J)=sun
crurxnuz -. - __-_ .._i --_.
GENERATING ALPHA MATRIX
ocsz=1,6 ._-_ _ --___-_
SUH=0
08K=1,N
LP4£(.)=TFETA3(I,K)‘A(K)
SUM:SUmALPm.(.) --11 _
ccwtruu:
AEPHA(11=$UH
g_~txuu;
k117222 5 ,..__ _.-__.__
rckqar(1HI,10X,17H1~VERS£ or THETAIIID
ognx=16 _ _
P.1Nf223.(CAT.uv(I JI.J=1
FCKHLI(1H0,5X,\6E15.5,10X)? .._ . ._-.._ 3 ,-___.,.,
conrzuus
sznrzos ALPHA
FCENIT(16110X12HLLPHA nAtnIxIIIIoX.Ezn.Io/II

 

2

 

TOLLOVING [ETERNINES STANDARC DEVIAYICNS

CALL STDDEV

 

CALL nus
DNA OE'K V THE FOLLnuING 03054 »
iii’g °?Eé S‘¥°EJ"%1“SEE°S126‘?f“E:§“ I§sz n: or .E.su.E,,N,$
341‘ EL. .05 11 ‘me 19 A»(E VELSC 71E 50? 0 .
$109 .

 

1

03/17/00 .17.33.53 ~no:

FTN 8.00690

85

75/175 O’T‘1
1” S 0 V
CUHVON L M 6).)(13,3).EHEAS(13).E(13).N

500230? E T
P (

SUOROUTZNE STDOEV

. z
. m . m . _ _ “ m . v “E .
. . . u H M . _ 1 _ 1 M G _
H _ w . _ . _ fi 1 , n n
_ . n . _ . * . _ h _ _
1 . . .
_ . H . . _ H m _ . m . w
. u . u _ _ . M H _ w
.. w . - “ u m . a . . ..
_ u _ _ . _ .
. . 1 .
. . . . i n I 3
m _ . _ _ _ 3 u . 1 w _ H 1 m an .
_ . _ _ . . _ _ _ m . . . . . .. _
. . w 1 .. _ _ . . _ . 2 . 3 .
, _. . . — . _ _ . .
. . _ . 1 _ _ _ _ . i m _ _ 7 n
. . 1 . . _ . n _ _ _ _ _ H _ 1 1
_ _ _ h . _ _ _ m . . . _ m I u M _ . ._
. .
1 _ 1 . 1 i . . . . m o .
. — — . . _ — ~ — N g
_ . a . I
_ . 1 . . 1 _ 7
. . w _ . . . . fl _ . 1 .
M . _ 1 . _ 1 i _ _ . l
I . . _ i . . . 3
L! H I w I _ _ a P o c .R
AA . _ . u . _ I Di
0 H . _ . u L T PIN
)P . 1 H H . b . _ . g L .1 1.0
.ZL _ . . . S , . h . _ _ . E c. .051
O 4h . . N _ E . . . _ _ N T
o. H m h , U u _ . _ . . _ . M T. mun!
\l, H . __ L . _ . s 0.x
32 H 1 . 0 h _ u . . . _ c E SFV
Io . . V . . . r _ 1 . I 0 .u H.£
I, . . _ a _ u u u m _. ..I 9 u “.1.—D
1' . u D I . . u . . . E I‘ O 2‘.-
01 .2 0 . . . . R I 1. I 0
I. I R 5 i . . . _ 0 0 I XOR
1.1 09:. . . i E o X 6H..-
01 0c¥t1 . . H h 5 113
)0 EAOL . i u m I I N
)0 1 . ackaL ” . i _ _ m N i 5 .INA
2) . UMPE M _ _ . . 0 0 OCT
.1 H1 ETLT . x F 045
Q P}. H I. m 0 112
1. L0 .._. E E EV I
an. S SOLOL I 5 1L!
ca 0‘ w .QTCTFU _ _ _ s P. .E06
E )0 U R R . . _ E I 0
U 20 L LEIEI U L 05/
L O I. I ‘LCLC L L HR!
0 0 .l V VC C a 0 E R/u I
V )5 RT RT V 05
2‘ 0 L111: _ 0 SN”.
L II) E ICFCF L T SCAI
A To”, on“ D . ‘ CIVIC.
0 (F6 10 ITTTT _ . D E HTS:
O (00 SA SFEFE 0 5 C5 I155
.8 (o H PE 9. B 0 S o R .50.?
z )2 L L LSSSS 2 . I 1 CIHQX
L )0 0 L0 LEUEU I. o I L F 1 FR 6R
It 1). T 2.8N5N 2 O . l. 0 T 0 ID 0
:. .()) .1 I O ) 2 F. x v591R/N
‘32 0E SSHSfl O S 2 o 0 FTP. u/O
L H .0 TL "U U ) E O O 0 )h. “00 .9.
a P...) W N". N .- .I H O ) . E I 0. STLHHVJT
L...) F 314....n , I C ) C 2 A P. (5 E8884 ..u
.._. ”"02 L1 8 NHJFC I. R S R u W .p. o 1 "EV I70...
T I fix» . S I. . v1 10 3 AUG “231V
5 )3. . w. m... 1301.3 A C u... _ I. 3 I. I p. )1 T : HDUIFF
“ 22‘ 70.. .I 3 23:36»: E o I. “M 1 . .I a 1 5 1r ” TGFH 5 ID
0 (a ..V. I H N 1R9). F H1 ) )1 1 1 E )1 1 I. E I. 0. 035.42..
r. 0 7.0 F (c... I UUIIJI E: E I:- E E V. :9— E T1! V N) 3x 0 1U I 9 D
H )() E SEC 1 SSLSL .V V (V V , CV & (V V CV A 75 no CXSLXOK
T 101 IT 119 E hALAL )A 1 51 N4 .4 NS c NA .1 N5 2E E2 3 €54.5HA)
.1. I F5 .01 NV {CECE 1... N... .uS IS )5 .41. .5. .IS )5 4‘ )U M. I 9. it .609
5 7.1... E HOE [A "H H. (O [:0 E0 1!. It ET E. El. 10 ET 1... £0 1 («0"an N o
N ((1 TV N)C6V ES 5 5 NEE ET NHE VT "(E V? NHL VT N‘E Vk HANIN M I“ IHNLO
I N003... 5.. Iv... JAEV: 557...... :4.va . ..EVE an... 6VC30. ..EVEAF. IEVEAQiiv .21 31.0.. 1....I1
T 0!.)10U Es 0 1(SASUAS 0 o 0 . I 1(Abnu 01((AUSQ 01‘ Anus: 01(AUSCO 1(IflU53000 10‘ 7 01.7013. .SF
0 1‘)..." BE 3085255.."5503 E E :85NSSVU:8SN:SV::25N::08:aSN:SaV::SN:sZTL.—ZT VZT ZTTH’N
a 33311 a! 1:131:113H31 EHHSS 11:1:s3111211:zillzllExllisll:1:11:I1ETI:1T4 TT‘FTRJhEQ
.0 T)I\OT .1.. FEBC. affoXv-tr. 5EETEEth...bE-CT:.H...E/E.TE1~CEO£ET:.S€:.9EETESNHO0NH TVNLVHIZSJ
N le)N RR VV1........VVN.r.F.VV mmmow 1VVNVOVV1VVNV VV1VVNVOVVIVVNVOVVlVVHVOIv... .21.... NIKC.AP.V I Tm
E O‘N)o Y: ‘ahCIIAJO. 998‘ EAOAHA‘OA‘O‘ AAOAAP.AH.AAO. AC.nhAA0InACAH..HCCOL..C CECE HCFXOTI
0 OEWSC cc 55060538035 [0.89. DgCSRSSDch-RSS.OgsphssofiscSRSSQQ-SgPPFWOPF U CTFHLFFMUWOHHC
O
. O
o . o
1 a. 2 T .a H
b . 0 r. .0 . 9 0 0 S 0 0
HUB 1.9.9 0.0 1 .930 b 9.90 H , 1 1 1 . 1 Z . Z 3 Z Z
_ , . . . . _ _ m
. 1 . I
m _ _ 1
. . .. . u
. , i .c
I I _ R
8
i . U
. .s .
s I u z a 3 u w w. n n _ m 5.. r .._. . I m

£36

"‘3!”

 

 

 

 

 

... ..1‘.

’ suaaouvzuc IXIS 70/175 007-1 rru 0.0.090 03/17/00 .17.:3.53 '“5102 1
. 1 s- noutzur 0x13 . A . -
ggggg'ohP215001'3991302?;i?u:%71 01711 01711 01711 0113 31 HU(71
:111733 7317!? .61711:30171I.c:¢)l1 ' ’ ’ ’ ’
s ¢ususzvfi txccpis 30 . ~ - .—
0-050310" “190115!
0 1.11:01051 1
1 1,2)=ALEHAib£
- 1p3)8‘L)‘M 5 - 11 - - _.
10 0 2.11:010n016)
a 2,2)8ALPHA’2
‘ 3.3)3‘LFH5 ‘0
. a .112019H115 _- - 1-1. - -- _..__.. 1.11.1.1.- 1
A 3.21::Lph1<~1
15 A 3.3)IALPHA(3)
U .
'3 'A’ na7n1x coauzsrcuo 70 ALPnA uttRIX or azoucco ELLIPSOIDAL narnxx ~—-———-1 ~——-
9&103200 ‘
70 200 gogggl(1H1.10x,23H00100107£n ALPoa n1731x1
' 02303251 (111 3:1,31 ' ' ' “’
231 F3§‘§L¢i‘°m 3F50.101
zs Buhoims - 1 1 1.....1... 1-1-1---- 1.1-1.1-... -111- -
xz1120.57735
, 7 11:0.57735
. 3 x v 7 IS r:Psr 37:007zvs oxnecrxsn - - . 1-.111111111111111111.11.11.111.--_111_1_
30 g Ai.51.01 ALE use: 70 cuecx 2100 as 73011 AXIAL V010:
" a1-u1,1;ou2.21uu,31
1 - {'111511;553I2’.“2, .210013,311111.110013,31-¢0¢1,z10'21— (012.31'¢ -..__1111_1_1,_1 - 11111-- 1.
35 g :01 31": ~ 0013 3 o: 21.02 ,3 .. 11-111 1 o 0 z 3 '0
“fig-(3% .29011?§.110‘211-701§101151’2; 5311 ( ’ ’ ( ‘ ’ ’
‘"" “ ”'Paznrzoz ' ‘ '""”
502 roaqar11n1.20x,3:u0£7£nuxuattou'07 003001911 axxs.///1
.0 agagi1,10 -
"" ""‘"““" "u uo11=u1n '01:.11ox1u10111.2107(n1'011,31
x NolgsH‘N v .1 9X:v)‘a(2p2)OY(N)'AtZ,3;
‘5 0 v No1 an n '1 5.1 ox "10013.2101101'113,3
3 0,0.c ARE NCHHALIZEO u.x.v ”"‘””""”“'"""’”*"”"”' ”" ”' ’“"
37101". 1-vzox«uo11-¢2ov1no11-°21
- - 4§7N1u§n 11/sa.5 . .111-11111111111111111111
s0 u1=x¢u+1 (SONG
c u1s711o1 ISORO?
H N01 3 iN
. X ".1 89 fl - .. .._“..- -.-..--. .- -._..1_.-.._1.1.1.. -._—.1-
’5 9 Y 7461):: m _
10 7 Min 0 AXIS r LIP o ”A mru r n 1 1x H06
1 E 115. S8. 205 A36 new 0331 fiku 013' roan R5220 vécgéa. 351§UALLI
00 6 6:3‘flf T
1- A an," - -1 - -11. 1--1-1.--1111111_._1_111_11 1__
I0 CNHNUEG
‘5 b203 rd 00310.0 .60 LPHA10151.5NBETA1.13XEHGAHHA1.I.F15.S,2F20.5)
q ostcanznc HAGnItuoe or tars 0x13 ‘ ‘ ‘ ”"“ ‘ “‘““
b
3501 1 ooA'-0¢1 211c00-111 3
5:3160535’ 11o00§011212100 00-012:31 1 -- 11.1- 1- -1
70 2019-113 11o035;0¢3‘21oc 100113.31
a» e ;:s a:
US$331."
'5' .200 :cu00711ho,17u000N1700£ no. 1 . ,r11.s1 " "“”“ ”‘ ‘
b
0 cutcx 2: no
0. , - - — - -_. 1-11 — .._..- -.

d

If

2

03117100 .17.33.53 PAGE

FIN 50 . .59.

£37

"3-(‘1'1HJG’WH ) O‘Bl'fll‘lfl'tfl

7&1175 O’Yi1

SUBROUYINE AXIS

——
u-
a...” ..

.. a _

-
.. ——-. .——. —.—.-. .

 

 

 

'05: u 1» HAJOR AX or
7v, naéuzr 'c or Aifs

._---—.._- . --..‘.._ . .--.
-._—_——.. _

. -o “.._.-4--- ..-- - ..——-..... -n-

.. .. .._-~.. .._—.._-..

 

A2913!,6HGIHHAZ.’9F1§.5.2FZO.5)

,”
. 333 .
’ i .
1.23 1
ls“ c
.3 II
ham». h
. o o 0 I 0
X NNN’ .l
I . g“ ‘ ’6
R. VVJIO . .6‘
T YYYN . .fl
I 000‘ . 10
N ‘1’" . 11
E 222' F8
5 O 9 9 9. 9‘
u ‘ (.0232 I), 9
O (‘1... E333 8 I:
R “‘O a 9 9 9’ ‘1
.t F 8353! HQEEJZ 22‘
ol 0 O O O 0.. 5“l\0 0 lo
‘ ‘1’” O’HA‘. o. 56
L E NNNN X. 0 I F 00 o
s (I‘l‘l\ 5“.“ CF N“ g
a a a XX“ five. vl 1.4
O E 3 X x!!! 96092 EC F
F V. 7. D 1; ca 011.. 9:1,); .UA £0
N M 1 R u ’11)? 52220 UH G
X I a T. 111. H a 9 :E T.( I...
J o J A A 0990777 P123E I. 0
fl .0 .A .v H I. «#:333200 :3!((o.un1 as.
U M I 8 ” fl (((1WER “MAAIa GA 01
x O ISA-O 3...... an... R.
H a F A I M a. & 063N3wm lanai. 3a.. ET
9.. ’ I P I m O O O I)” “ ’dUBDHHs 26
I“ u w m M AL“ ‘ ,’)W))’\l’"~ 8x0 9 O 0.17. 2H 0
"Pub” 111", 8"],(64LO «‘6’
m 8 T B 9 G ‘(l‘l‘O 6 § ((1! .Q 31.11....“ ’0... ll.
l. \a E ‘ C to E N HHHTNNNGBC GO 9 I )KHOHZO
I J N H3 I N K 0 f. U HRhFE‘CEC 0H123Q ’41... DH”
33 i I V 81 v. 33‘ o o o r- 1 s .. .. BXY x x 3)) {6111(5715311
G a! T N10 V 1.... 001 A o 1))SHXY)))NNMUD(AAA=0(A1((
N 11‘! U 332" E 111N333 L 1 111s:s:111(((N2foooOZtHZYS
1 3:3... 3 If... x ..IJI),) U 810007)...)Ooocesf3llb QBHTI 371.5
m. XJET .I 1.9"... 2‘ .._.J 97-111 c JONVNONNNVNNGAXaf81533fNNZNHI
1 77am m “trim fl ooumuwi m MNHXHMWMCHIN .wWJMWfihmmmfiuflfl
S Wmac s ”on”: R DDAORXVN Dmuxv smut? MCGHFDEFHFFZPFI
. . a 1. .5
. 6 . n 1
. 9. .6 1 .5 I. 9. 2
.b b b 76.90 9.5.96 .0 500 Z

"’—6ac£

-..-_.-..--_H ..‘..

 

03117150”.17.33.53

-- —-. -._-n— _-

 

_..—.—. ..-. .._- .

T 0F 7ND FOUND ‘X13

0
rtu 5.00593 '

31.61

XX.

was?a33:..sez_§5#:;2~9;usg°§§3:22:35
2,

GOING
I
N

53.13x.6nc:nnA3./,rzs.s.zrzo;s)
.6 I
éfivuio

F
C

03‘
3'02»):

HMGN DE N
'3- uiI‘a’mu8
SJ'! 0F CLLIPSCIDAL “15.68.80.“
CHECKXNG 0LTHOGO MLIT" OF NAJOR ”.0 MINOR 1X15
(nb‘flAGDOCC«’CAG)

2 8.0 3
Z

1

(sz

:23"

0.5!.ssucosxuc actuzeu HAJOR AND nxuon axxs.zx.rxo.e)

W

9.1103” ZOEB".29F3002,

75/115 opt-r
s
QHIGNO 3
123(65‘016

gfignrzzo cosa’°
rev-1“"
' ‘
(N5

Suanoutxus Axxs
:
55 a
220

7.

8E3

 

 

 

 

 

 

 

‘
" suannurzuz BNNHAt rurxrs opts: FYN b.09q98 03/17/00 .17.33.53 éisé'
. 1 suaaourzut Ouvutftu a,ax.ocra n.“ -
‘ ozncusxou atu,u».eziu,ut
3 rzuas nv ‘ auJB DETERH INA vs at navn can u av PIVOYAL conneusxuc
a s 3 u=cxn.‘u Agfi a. alsxuve “as or a. 0:1 =a£r£ zn§r£ or a , r ~ ,h- —— -
a A=1.a
-1. g ccucaars xoautztv «Atnxx - ._n - .mm-w-..
b
01:13:,"
~ 912.}: g,"
. .15 a J =0.o .- -mw -- -_----- .~ —-- - .—_.___ _- -
oozof=1.u
15 go 81:1,!)s1.o
3 . BEGIN PIVOTAL couocu3A7zcu .171 --A-u m - _ -_ - -_ _~____.-..H--wm--.f..__.n.-
. 2 J Is szorzuo nan
20 o ”-1 n
.. - .. - 388(J’ J, 4‘... _. - Wm..._~.... . ,W, ..................1 .._...” .... J... ....... m...“
‘ ACA'P
z, 5 ozvxas szorrxu; now av DIABONAL ELENEN!
h . . ..V _. - --_... _J_.... .___-_..___ 1.- . , -._—1...- -. -. .
tJ’ x229 J,x)/P
'0“, g.’ ”(5.x)a (J,x)/P
3. "g 1 1s tut new to as aenuzco " ‘""""‘“""“’"'“““"“"‘“‘”"”' ‘—““‘”'“""
V b
225ft'1'"
_- so.J)ss.so -- .. - - - _.. _ -.m__.--"- _,. - "a- __.- __ ___u____u._.
5.888(1 J)
35 s§f34=1 K
ni=L ' M)~a(g uw'sava .
_u - - 53 '1! N)=8 1:.uz- in (J.ny‘sava ._,J,. - 11.- -._. -- ."1.___,“"_.._. - -._..-"_-
55 cEnrinue
so o~r1~uz
no er:
. TUiN ..--_ - -- _ -_ -- .._..- .._ -..~___-- -._ -._—.._ __
:‘o

1

BIBLIOGRAPHY

89

BIBLIOGRAPHY

Anderson, C. W., 1977. Large Scale Velocity Anisotropy and
the Q-ellipsoid Method, M.S. Thesis, Michigan
State University, East Lansing, Michigan.

Anderson, D. L., 1961. Elastic Wave Propagation in Layered
Anisotropic Media, J. Geophys.'Res., V. 66,
N0. 9' pp. 2953-2963.

 

Anderson, D. L., Minister, B. and Cole, 0., 1974. The Effect
of Oriented Cracks on Seismic Velocities, J;
Geophys. Res., V. 79, pp. 4011-4015.

 

Backus, G. E., 1962. Long-wave Elastic Anisotropy Produced
by Horizontal Layering, J. Geophys. Res., V.
67' NO. 11' pp. 4427-4440.

 

Balakrishna, S., 1959. Effect of Dimensional Orientation
on Ultrasonic Velocities in Some Rocks, Indian
Acad. Sci. Proc., Sec. A, V. 49, No. 6, pp.
318-321.

 

Bennett, H. E., 1968. An Investigation into Velocity Aniso-
tropy through Measurements of Ultrasonic Wave
Veldcities in Snow and Ice Cores from Greenland
and Antartica, Ph.D. Thesis, University of
Wisconsin.

Bennett, H. E., 1972. A Simple Seismic Model for Determining
Principle Anisotropy Direction, J. Geophys. Res.,
V. 77' pp. 3078-3080.

 

Berryman, J. G., 1979. Long-wave Anisotropy in Traversely
Isotropic Media. Geophysics, V. 44, No. 5,

 

Birch, F., 1960. The Velocity of Compressional Waves in Rocks
to 10 Kilobars, Part 1; J. Geophys. Res., V. 65,
NO. 4' p. 1083-1102.

 

Birch, F., 1961. The Velocity of Compressional Waves in Rocks
to 10 Kilobars, Part 2, J. Geophys. Res., V. 66,
NO. 7' pp. 2199-2224.

 

90

91

Brace, W. F., 1960. Orientation of Anisotropic Minerals in
a Stress Field, Discussion, Chap. 2, G.S.A.
MemOir' NO. 79' pp. 9-20.

Byerly, P., 1934. The Texas Earthquake of August 16, 1931,
Ball. Seismological Soc. Am., V. 24, pp. 81-99.

Cannon, W. P., 1973. The Penokean Orogeny in Northern Michigan
in Huronion Stratigraphy and Sedimentation (ed.
G. M. Young), Geol. Assoc. Canada, Spec. Paper
12, pp. 251-271.

 

Cannon, W. F. and Gair, J. E., 1970. A Revision of Strati-
graphic Nomenclature for Middle Precambrian
Rocks in Northern Michigan, Geol. Soc. Am. Ball.,
V. 81, pp. 2843-2846.

 

Crampin, S., 1970. The Dispersion of Surface Waves in Multi-
layered Anisotropic Media, Geophys. J., V. 21,

- 1975. Distinctive Particle Motion of Surface
Waves as a Diagnostic of Anisotropic Layering,
Geophys. J., V. 40, pp. 177-186.

 

- 1977. A Review of the Effects of Anisotropic
Layering on the Propagation of Seismic Waves,
Geophys. J., Roy. Astr. Soc., V. 49, pp. 9-27.

Crampin, S. and Kirkwood, S. C., 1979. Approximations to the
Velocity Variations in Systems of Anisotropic
Symmetry, Geophys. J., V. 57, pp. 697-703.

 

Cerveny, V., 1972. Seismic Ray Trajectory Intensities in In-
homogeneous Anisotropic Media, Geophys. J.R.
AStr. Soc., V. 29' pp. 1'13.

 

 

Cerveny, V., and Psencik, I., 1972. Rays and Travel Time Curves
in Inhomogeneous Anisotropic Media, Z. Geophysik,
V. 30, pp. 565-577.

 

Cholet, J. and Richards, H., 1954. A Test on Elastic Anisotropy
Measurements at Berriane (North Sahara), Geophys.

PI‘OSE. ' V. 2' pp. 232-246.

Duda, S. J., 1960. Elastic Waves in Anisotropic Medium Accord-
ing to a Macroscopic Measurement Underground.
Geophy;.Prosp., V. 8, No. 3, p. 429-444.

 

Dunoyer, De Segonzac and Laherrere, J., 1959. Application of
the Continuous Velocity Log to Anisotropic
Measurements in Northern Sahara; Results and
Consequences, Geophy. Prosp., V. 7, No. 2, pp.
207-217.

 

92

Fisher, R. A., 1953. Dispersion on a Sphere, Proc. Roy. Soc.
London, A217, pp. 295-305.

 

Gair, J. E., 1968. Preliminary Geologic Map of the Palmer 7%
Minute Guadrangle, Marquette County, Michigan,
U.S.G.S., open file.

Gair, J. E. and Thadden, R. E., 1968. Geology of the Marquette
and Sands Quandrangle, Marquette County, Michigan,
U.S.G.S., V. 197.

Gair, J. E., Thadden, R. E. and Jones, B. P., 1961. Folds and
Faults in the Eastern Part of the Marquette Iron
Range, Michigan, U.S.G.S., Research Article 178,
pp. 76-78.

Gassman, F., 1964. Introduction to Seismic Travel Time Methods
in Anisotropic Media, Pure Applied Geophy., V.
58' pp. 53-112.

 

Hagedoorn, J. G., 1954. A practical example of an Anisotropic
Velocity Layer, Geophys. Prosp., V. 2, pp. 52-60.

Irving, E., 1964. Paleomagnetism and its Application to Geolog-
ical and Geophysical Problems, Wiley, New York.

Jamieson, J. D. and Haskin, H., 1963. The Measurenent of Shear
Wave Velocities in Solids using Axially Polar-
ized Ceramic Transducers, Geophysics, V. 28,
pp. 82-90.

 

Jolly, R. N., 1956. Investigation of Shear Waves, Geophysics,
V. 21, No. 4, pp. 905-938.

 

Klasner, J. S., 1978. Penokean Deformation and Associated
Metamorphism in the Western Marquette Range,
Northern Michigan, Geol. Soc. Am. Ball., V. 89,
p. 711-722.

 

Klima, K. and Babuska, V., 1968. A comparison of Measured and
Calculated Anisotropies of Marble, Studia Geophys.
Geodaet, V. 12, pp. 377-384.

 

Kopf, M. and Wawryik, M., 1961. Acoustic Velocity and Susceph
tibility Measurements on Rocks of the Triassic
and Zechstein from the West Thurington Basin,
Geologic, V. 10, No. 2, pp. 214-230.

Kraut, E. A., 1963. Advances in the Theory of Anisotropic
Elastic Wave Propagation, Rev. Geophys., V. 1,
NO. 3, pp. 401-448.

 

Lamont, M. 0., Douglas, L. L., and Oliva, R. A., 1977. Calcula-
tor Decision-making Sourcebook, (Texas Instruments

93

Levin, F. K., 1978. The Reflection, Refraction and Differention
of Waves in Media with an Elliptical Velocity
Dependence, Geophysics, V. 43, pp. 528-537.

 

- 1979. Seismic Velocities in Traversely Iso-
tropic Media, Geophysics, V. 44, pp. 920-928.

 

Macpherson, J. D., 1960. Anisotropy of the Robk in the Approaches
to Halifax Harbor, Nova Scotia, Seismol. Soc. Am.
Ball., V. 50, NO. 4' pp. 575-579.

 

Matveyer, A. K. and Martyanor, Ye, G., The Dependence of Ultr-
sonic Velocity in Coals on their Degree of Meta-
morphism, Afiad. Nauk. SSSR Do klady, V. 122,
No. 3, pp. 459-461.

 

McCollum, B. and Snell, F. A., 1932. Assymmetry of Sound Vel-
ocity in Stratified Formations. Early Geophys.
Papers, Soc. Exploration Geophysicists, pp. 216-
227, 1947, Reprinted from Physics, 1932.

 

McElhinny, M. W., 1973. Paleomagnetism and Plate Tectonics,
Cambridge University Press, pp. 77-83.

 

M eissner, R. and Fakhimi, M., 1977. Seismic Anisotropy as
Measured under High Pressure, High Temperature
Conditions, Geophys. J., V. 49, No. 1, pp. 133-
143.

 

Musgrave, M. J. P., 1959. The Propagation of Elastic Waves in
Crystals and other Anisotropic Media, Report
Progr. Phys. ' V. 22' pp. 74-96.

 

Neuman, F., 1930. An analysés of the S-wave, Ball. Seismo-
logical Soc. Am., V. 20, pp. 15-30.

 

 

Nur, A., 1971. Effect of Stress on Velocity in Rocks with

 

Nur, A. and Simmons, C., 1969. Stress Induced Velocity Aniso-
tropy in Rock: Experimental Study, J. Geophys.
Res., V. 74' pp. 6667-6674.

 

Nye, J. R., 1957. Physical Properties of Crystals, London:
Oxford Clarendon Press.

 

Postma, G. W., 1955. Wave Propagation in Stratified Medium,
Geophysics, V. 20, pp. 780-806.

 

Puffet, W. P., 1974. Geology of the Negaunee Quadrangle,
Marquette County, Michigan, U.S.G.S., Profes-
sional Paper, 788.

Ricker, N., 1953. The Form and Laws of Propagation of Seismic
Wavlets, Geophysics, V. 18, pp. 10-40.

 

94

Rudski, M. P., 1911. Parametrische Darstellung der Elasti-
schen We11en in Anisotropen Median, Acad. Sci.
Cracovic Ball., V. 8a, pp. 503-536.

 

 

Sato, R. and Lapwood, E. R., 1968. SH Waves in a Transversely
Isotropic Medium-1, Geophys. J. Res. Ast. Soc.,
V. 14' pp. 463-470.

Schlue, J. W., 1977. A Physical Model for Surface Wave Azimuthal
Anisotropy, Ball. Seis. Soc. Am., V. 67, pp.
1515-1519.

Schmidt, E., 1964. A Case of Anisotropy of Seismic Velocity,
Geoexploration, V. 2, pp. 28-35.

 

Schimozura, D., 1960. Elasticity of Rocks and some Related
Geophysical Problems, Japanese Jour. Geophysics,
V. 2, No. 3, 88 pp.

Talbot, C. J., 1970. The Minimum Strain Ellipsoid using De-
formed Quart Veins, Tectonophysics, V. 9, pp.
47-76.

 

Taylor, G. L., 1973. Stratigraphy, Sedimentalogy, and Sulfide
Mineralization of the Kona Dolomite, Ph.D.
Thesis, Michigan Technological University,
Houghton, Michigan.

Thill, R. E., 1969. Correlation of Longitudal Velocity Vari-
ation with Rock Fabric, J. Geophys. Res., V. 74,

Thill, R. E., McWilliams, J. R., and Bur, T. R., 1968. An
Accoustical Bench for an Ultrasonic Pulse
System, Bu Mines Rept. Invest., V. 7164, 22 pp.

Tilmann, S. E. and Bennett, H. F., 1973a. Ultrasonic Shear
Wave Birefringence as a Test of Homogeneous
Elastic Anisotropy, J. Geophy. Res., V. 78,
pp. 7623-7629.

- 1973b. A Sonic Method for Petrofabric
Analysis, J. Geophys. Res., V. 78, pp. 8463-
8469.

Tocher, D., 1967. Anisotropy in Rocks under Simple Compression,
Transactions American Geophysical Union, V. 38,
NO. 1' pp. 89-94.

Tullis, T. E. and Wood, D. S., 1975. Correlation of Finite
Strain from both Reduction Bodies and Preferred
Orientation of Mica in Slate from Wales, Ball.
Geol. Soc. Am., V. 86, p. 632-638.

95

Uhrig, L. F. and Van Melle, F. A., 1955. Velocity Anisotropy
in Stratified Media, Geophysics, V. 20, No. 4,
pp. 774-779.

Van Schmus, W. R., 1976. Early and Middle Proterozoic History
of the Great Lakes Area, North America, Phil.
Trans. R. Soc. London, A, 280: pp. 605-628.

 

Van Hise, C. R. and Bayley, W. S., 1895. The Marquette Iron
Bearing District of Michigan, U.S.G.S., 15th
Ann. Rep.’ pp. 485-650.

Van Hise, C. R. and Leith, C. K., 1911. The Geology of the
Lake Superior Region, U.S.G.S., Mon. 52, 614 pp.

Weatherby, B. B., Born, W. T. and Harding, R. L, 1934. Granite
and Limestone Velocity Determinations in the
Arbuckel Mountains, Oklahoma, Ball. A.A.P.G.,
V. 18, pp. 106-118.

 

Westjohn, D. B., 1978. Finite Strain in the Precambrian Kona
Formation, Marquette County, Michigan, M.S.
Thesis, Michigan State University, East Lansing,
Michigan.

White, J. E., Heaps, S. N. and Lawrence, P. L., 1956. Seismic
Waves from a Horizontal Force, Geophysics, V.
21, No. 3, pp. 715-723.

White, J. E. and Sengbush, R. L., 1953. Velocity Measurements
in near Surface Formations, Geophysics, V. 17,
pp. 54-70.

 

Wood, D. S., 1974. Current Views of the Development of Slaty
Cleavage, Ann. Rev. Earth Sci., V. 2, pp. 1-35.

 

"‘Wll’llllflflll’llflr