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This is to certify that the
dissertation entitled

Elastic Anisotropy in the Baraboo Quartzite,
Baraboo, Wisconsin
presented by

James Matthew Carty

has been accepted towards fulfillment
of the requirements for

Masters degree in Geology .

MQW

Major professor

Date jam I?) /?85 ”Hugh F. Bennett

 

MS U i: an Waive Action/Equal Opportunity Institution 0-12771

 

 

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RETURNING MATERIALS:
Place in book drop to
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ELASTIC ANISOTROPY lN THE BARABOO QUARTZITE;
BARABOO, WiSCONSlN

By

James Matthew Carty

A THESE
Submitted to
Micnigan State University
in partial fulfillment
for the degree of

MASTER OF SCIENCE

l 985

I!
\g ‘0 {.1

Abstract
ELASTIC ANlSOTROPY IN THE BARABOO OUARTZlTE;
BARABOO, WISCONSIN
BY

James Matthew Carty

A suite of rock samples from the Baraboo Ouartzite were tested for
velocity anisotropy in order to observe if any pattern of elastic
anisotropy exists in this teCtonical‘ey deformed region. it is suggested
that from such a study one car. infer the strain patterns present at the
time of metamorphism.

The concept of the 0 ellipsoid was used to determine the
anisotropic orientations for rock samples. The following conclusions are
made. llAll rocks tested cssess elastic anisotropy. 2) The elastic

anisotropy has a reg on al trend indicating a minimum strain directior
parallel to the hinge line of the syncline. ‘3) The observed anisotropy
correlates well with the preferred ailigrzment of elongate ouartz grains,
and may be related to tre relative number of grain hour-dries enco'c ntered
by a wave to" eachp , "C opatag at‘lon direct ion 4)- The orientati o" of elastic
ar‘sotropy ca". bet :seo’ to interpret stress and strain relationships in a

tec toni cally deformecr gion

ACKNOWLEDGEMENTS:

i would like to thank Dr. Hugh F. Bennett for his help and guidance
through out the study i would also WI sh to thank Dr F W Cambray and
Dr. Thomas A. Vogei for their time and assistance during this study.

l thank Jim Raab for providing me with his grain shape data and
analysis.

A special thanks goes out to my friends in and near office 236,; l-Ir.
Mike Takacs, Srgt. Steve Mraz, J Franz Carr, Jimmy Tolbert, Dale
RezbeK?, Tim Wilson, Kyle Walden, and Rich "Rocket" Roty for their
emetiona‘ support, and for taKing my mind off the books once in a while.

Otner friendsl would like to thanK in much the same light are Steve
DworKin, Keith Hi l,l Tim Bartlet t, John Nelson, Steve Ronr,Jerry
Grantham, and Bob "Champ“; Deooes.

I wish to thank my parents most of all. Thank you for everything, and
in particular, your er couragement during this project and throughout my

entire education.

TABI. E OF CONTENT S

LIST OF TABLES ....................................... v
LIST OF FIGURES ...................................... vi
TNRODUC I ION ........................................ i
CAUSES OF ELASTIC. ANESOTRODY ........................... 7
REG ONAL GEOLOGY, BARABOO, WI. SCOIN‘S N .................... l2
PREVIOUS WORK ...................................... l7
M: THODS USED TO DETERMINE THE ANISOT ROP'IC ORIENTATION ....... 2i
DATA ANALYSIS ...................................... 28
Major Axes ....................................... 28
MM Axes ....................................... 33
Intermediate Axes ................................. 35
Axial Ratio Data ................................... 3S
DISCO SSIOIN ......................................... 38
CONCLUSIONS ........................................ S4
ADDENDI l
“47:0de ii Elastic Anisotropy ........................... S7
ARRENDIX 2
Direction-Velocity Chart ............................. 67

APRENDIX 3
t? ellipsoid Axes (Aroitrary CoordInate System) ............. 7'I

TABLE OF CONTENTS cont...

RPENDIX 4

0 eiIipsoid Axes (FinaI Orientation) ..................... 74
APPENDIX S

Theoreticai Axiai Ratio From Crystaiiographic Effects On

Anisotrooic Orientation .............................. 76
BIBLIOGRAPHY ....................................... 77

3V

LIST OF TABLES

Taoie 'I. Precambrian Statigraony of the Baraboo District ........ I3

Taoie 2. 0 eiiiosoid Axes,- Dip/Dip Direction ................. 26

\I

Figure i.

'71
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Figure 6.

Figure 7.

LIST OF FIGURES

Pre-Paleozoic Outcrops in the area of Baraboo

VI

Wisconsin .................................... 4

Geologic Cross Sections of the Baraboo Syncline ........ IS

Time Measurement Schematic Diagram ............... 23

Diagram of the Transducer Assemblies ............... 24

Stereoplots of the Principle Axes of the 0 ellipsoids. . . . .29

Map of the Baraboo Syncline, and Stereoplots

illustrating Principle Axes of Elastic Anisotropy and

Grain Shape .................................. 30

Axial Ratio Graph of the 0 ellipsoids

and average Grain Shape ......................... 37

Stereoplot showing the Principle Axes Orientation of

the 0 ellipsoid calculated for the Slate sample

collected from the Cambrian Slate Belt of Wales ........ 4i

Stereoplot showing the correlation between the axes of

anisotropy and the average principle axes determined

petrographically (Tilmann and Bennett, I 973) .......... 4S

. StereOplot showing the correlation between the axes of

anisotropy and the average principle axes determined

petrograohicaily for sample OO32 .................. 4S
' .. Diagram showing the successive approximeation

method of ‘inding the principle axes of a

second-ram: tensor (Nye, I957. p. IES) ............... 62

INTRODUCTION

Many studies using elastic wave velocity as a tool for interpretting
stratigraphy, lithology, and other physical properties of rock assume
that all rocks behave as an isotropic medium. Tilmann and Bennett
(19732) measured elastic wave velocities in rocks. and established that
the elastic shear wave birefringence can be used as a indicator of elastic
anisotropy. The study reported that the rotation of the sample within a
polarized shear wave field allows for the identification and
measurement of the travel times of two orthogonally polarized shear
waves. Although the sample may be nearly isotropic for P-wave
velocities, the relative difference between the travel times of two
orthogonally polarized shear waves establishes the existence of
homogeneous anisotrooy. All samples tested in their study possessed
shear wave birefringence and therefore were anisotropic.

Tilmann and Bennett (I973a) interpret their results to indicate that
elastic anisotropy may be the rule rather than the exception in rocks.
The basis of this study stems from the probability that most rocks
possess elastic anisotropy. Based upon this premise, it is the goal of

I

2

this research to verify that there is a pattern of elastic anisotropy
existing in rocks of a tectonically deformed geologic region, and
furthermore to suggest the relationship between the observed anisotropy
and the stress and strain patterns which presumably created the elastic
anisotropy.

In order to establish the orientation of anisotropy, the concept of
the 0 ellipsoid has been developed by Bennett (l972). The 0 ellipsoid is
a simple elastic stiffness figure the can be used as a tool for detecting
and describing preferred crystallographic orientations. The 0 ellipsoid
is a theoretical surface whose value for any particular direction Is the
sum of the squares of the three seismic wave type phase velocities in
that direction (Appendix 1).

One of the advantages of the 0 ellipsoid technique for measuring
anisotropy in rocks is that the orientation of 0 is determined by the
gross make-up of the physical properties comprising the rock (Bennett,
I972). Any large physical feature or group of small features that are
part of the rock have an integrated effect on the overall patterns of
anisotropy observed. As in other strain studies, It is an approximation to
infer general tectonics of an area from the data received from very few
samples. However, the orientation of the elastic anisotropy observed as

the principle axes of the 0 ellipsoids may be helpful in the pursuit of a

3
complete understanding of the resultant stress-strain relationships
present in a tectonically altered region.

Other advantages of the elastic anisotropy technique result from
the ellipticity of the 0 ellipsoid. in a single quartz crystal the
ellipticity of the 0 ellipsoid (19%) is greater than the ellipticity of its
optical indicatrix (0.6%). Thus, it is probable that the effects of any
deformation in a rock body can be more easily measured through the
orientation of elastic anisotropy.

The rock type and structure chosen for this study was the Baraboo
Ouartzite which makes up the Baraboo Synciine in south central
Wisconsin (Figure I). This structure was chosen for a number of reasons.
i) Part of the Baraboo Ouartzite is a very homogeneous, nearly
monomineralic, rock consisting of more than 80% quartz grains. These
grains range in size from fine sand to small pebbles (Dalziel and Dott,
l970). 2) The structure is well exposed so that samples could be taken
from different locations around the entire fold region. 3) The Baraboo
Synciine is well mapped and studied thoroughly, therefore we have
numerous models with which to compare our results in order to explain
the causes of anisotropy and the anisotropic pattern observed. 4) A
contemporary grain shape study has been completed on a similar suite of

roots in the Baraboo area from identical or nearby outcrops (Raab. I 985).

Figure I. Pre-Paleozoic outcrops in the area of Baraboo, Wisconsin.
Inset map - regional setting. Sample and its respective collection
location is denoted by the sample number. (From Dalziel and Stirewalt,
I975; Dalziel and Dott, 1970.)

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The principle 0 ellipsoid axes directions are compared to principle
strain directions interpretted from the grain shape orientations.

The objectives of this study are; I) to establish the orientation of
anisotropy present in each of the samples collected from several
localities on the Baraboo structure, 2) to determine if any pattern of
anisotropy exists in the structure, 3) to compare the results of this
study to others in the area that concentrated on stress and strain axes

orientations, and 4) to evaluate possible causes of the observed

anisotropy.

CAUSES OF ELASTIC ANISOTROPY

Tilmann and Bennett (i973a) concluded that anisotropy existed in
all of the samples that they tested, and therefore anisotropy in rocks is
probably the rule and not the exception. In order to evaluate any pattern
of anisotropy that exists in this study, it is necessary to discuss and
evaluate the various causes of anisotropy.

Velocity anisotropy may occur when the individual (anisotropic)
crystals in a material have preferred orientations over a volume large
enough to affect the transmission of seismic waves (Bennett, l972). The
anisotropy observed in the upper mantle may be caused by preferred
orientation of of olivine and orthopyroxene crystals. These crystals have
a pronounced velocity anisotropy (Crampin, et. al., I984). In crustal
rocks anisotropy may be present in regions where recrystalization
occurred, such as metamorphic terraines, or regions where fractional
crystalization occurred producing a preferred crystallographic
orientation within the rock. The multi-crystalline material will possess
anisotropy if the individual crystals have sufficient velocity anisotropy

7

(Crampin, et. al., I984).

A velocity and petrographic study done on a single Baraboo
Ouartzite sample was used to relate preferred crystallographic
orientation to the 0 ellipsoid orientation (Tilmann and Bennett, i973b).
A plot of the c-axes orientations of the quartz crystals was scattered.
However, the c-axes orientations were used to determine a mean optical
indicatrix by summing all crystals with an associated optical indicatrix
tied to each c-axis orientation. The summation of ellipsoids yields an
ellipsoid, and the process used in their study weights each crystal
equally. The principle axes of the measured 0 ellipsoid closely
coincided with the principle axes of the mean optical indicatrix. The
major sonic axis correlated well with the optic intermediate, the minor
sonic axis correlated well with the major optic axis, and the
intermediate sonic axis correlated well with the optic minor axis. It
was concluded that the 0 ellipsoid method closely detects preferred
crystal orientations, and that the close coincidence between the sonic
and Optical surface axes in the nearly optically isotropic quartzite
sample is strong evidence that subtle fabric orientation is accurately
reflected by the sonic axes orientation.

Another cause of velocity anisotropy is due to lithologic anisotropy

(Crampin, et. al., I984). A sedimentary solid has lithologic anisotropy

9
when the individual grains, which may or may not be elastically
anisotropic, are elongated or flattened. These shapes are alligned by
gravity or fluid flow when the material is first deposited, or later by
homogeneous deformation. Velocity was studied in argillaceous and
carbonate rocks to a depth of 2 km (Brodov, et. al. l984). Anisotropy in
carbonate rocks was found to be a function of their lithologic
heterogeneity, the amount of ordered fracturing, and the amount of clay.
The highest anisotropy was found in argillaceous rocks irrespective of
their depth and degree of heterogeneity.

A study on the Grenville marble (Tilmann and Bennett, l973b)
showed that the anisotrooic orientation reflected structural layering and
crystal fabric. The minimum velocities were oriented normal to the
micaceous layering. An elastic wave travels slower in a direction
perpendicular to the micaceous foliation (Brodov et.al., I984), creating
the anisotropic effect of a layered medium (Postma, I955).

Effective seismic anisotropy can also occur in an otherwise
isotropic rock that contains a distribution of inclusions, such as dry
(vapor filled) or liquid filled cracks or pores which have preferred
orientations (Crampin, et. al., l984). Any preferred allignment of these
cracks is likely to be the result of a variety of stress-induced processes.

A study was completed by Babuska and Pros (I984) that related

lO
velocity anisotropy in quartzite to the distribution of microcracks. The
velocity anisotrooy in their study was calculated using only P-waves.
The orientations of 440 microcracks and 3 IS grain boundries were
determined, and it was discovered that the direction of high P-wave
velocities corresponded well to the directions where almost no
perpendiculars to cracks and grain boundries were found. The correlation
was slightly better for grain boundries than for microcracks. But more
importantly, it was revealed that the nature of microdiscontinuities in
rocks plays a very important role in the effects on the compressional
wave velocities.

In the study of a plastically deformed granite boulder (Tilmann and
Bennett, 1973b) it was found that the principle shape axes of the
ellipsoidal boulder and the 0 ellipsoid axes closely coincided. The
granite had been homogeneously deformed during a regional tectonic
event. It was concluded that the elastic anisotropic orientation
accurately reflects subtle fabric orientation, and can be a useful
technique in detecting and describing regional strain.

in the present study a suite of quartzite samples was collected
from various locations around the Baraboo Synciine. The measured
anisotropy present in these samples may reflect an internal fabric

caused by regional strain patterns. In order to interpret regional

i I
patterns of the anisotropy, it is necessary to look at the geology and

structural history of the Baraboo region.

REGIONAL GEOLOGY; BARABOO WISCONSIN

The following is a brief description of the geology of the Baraboo
region as indicated by Dalziel and Dott (l970). Precambrian rocks are
locally exposed along the axis of the Wisconsin Arch as inliers to the
flat-lying lower Paleozoic sediments. The largest and best known of
these inliers occurs in the Columbia and Sauk Counties, south-central
Wisconsin, where Precambrian rocks, mainly quartzite, form an elongate
ring of hills known as the Baraboo Ranges (Figure l, page 4).

The Baraboo Ranges consist of massive, pink, maroon, or purple
colored Baraboo Ouartzite. The Baraboo Ouartzite is the oldest and most
extensive member of the Baraboo Group. The Baraboo Group consists of
the Baraboo Ouartzite, the Seeley Slate, and the Freedom Formation. The
Baraboo Ouartzite is presumed to be more than 4000 feet thick, and
appears to rest straigraphically upon poorly exposed acidic igneous
rocks. The Precambrian succession in the Baraboo district inferred from

subsurface as well as surface data is shown in table I.

I2

l3
Table I. (Dalziel and Dott, I970)

 

PRECAMBRIAN STRATIGRAPHY OF THEBARABOO DISTRICT

......... Rowley Creek Slate (maximum known thickness I49 feet). . . . . . . . .
........... Dake Ouartzite (maximum known thicknessZMfeet). . . . . . . . . . .
...................... (Unoonformity?)......................

. .Freedom Formation (dolomite and ferruginous slate; maximum thickness .....
1000 feet)

........... Seeley Slate (maximum known thickness 370 feet). . . . . . . . . . .
............. Baraboo Ouartzite (thickness over 4000 feet). . . . . . . . . . . . .

...................... (Unconformity?)......................

.............. Rhyolitic "basement" (thickness unknown). . . . . . . . . . . . . .

 

Only samples of Baraboo Ouartzite were collected for this study. In
hand sample, and in thin section the quartzite appears isotropic with no
obvious anisotropic pattern such as a well developed foliation or
lineat‘on. The quartzite is massive, and consists of more than 80%
quartz grains that range in size from medium to coarse size sand grains
with sporadic rounded granuals and fine pebbles. in thin section the
grains are Observed to be heavily sutured. The pebbles are most
commonly white quartz, and rarely exceed one inch in diameter.

An argi’laceous rock is commonly found within the Baraboo
Ouartzite. Phyllite occurs as layers or lenses within massive quartzite

beds. Most phyllite layers are only a few inches thick, but some reach

14
several feet in thickness. These localities occur at the same high
stratigraphic level in the Baraboo Ouartzite and are roughly on strike
with one another.

Mineral assembleges in the phyllite determined by X-ray diffraction
indicate that the Baraboo Ouartzite reached the lower greenshist f acies
of regional metamorphism. The entire Precambrian section has been
folded into a complex doubly plunging syncline with an axial surface
striking approximately east - northeast and west-southwest, and dipping
steeply north-northwest. At the present level of erosion, the syncline is
about 25 miles long and has a maximum width of IO miles. The north
limb is nearly vertical and the south limb dips gently northwards,
generally about lS° (Figure 2).

The Precambrian rocks of the Baraboo district were probably
deposited and tectonically deformed in a Precambrian mobile belt. Most
of the structures observed can be attributed to the effects of one main
phase of deformation during which the Baraboo Syncline was formed.
Some structures observed only in the phyllitic layers are the results of
one or two later, but not unrelated deformation episodes. The various
structures record different stages in the strain history that resulted
from one regional stress system.

The axial plane is highly variable throughout the length of the fold. A

15

 

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l6
rough estimate of the axial plane may be based on measurements of the
quartzite cleavage at the two closures. The axial plane dips from
approximately 6S'/3SO‘ to 707350“. The plunge of the syncline at the
two closures is based on bedding-cleavage intersections. The pluge of
the eastern and western closure are 2S’/27S', and 2S'/07S'

respectively.

PREVIOUS WORK

Much of the early work on the petrology, structural geometry, and
structural relationships was done near the early part of this century.
Weidman (I904) correctly recorded the stuctural geometry of the
Baraboo Syncline and Baraboo Ouartzite. It is accepted that Van Hise
(I893) was first to appreciate the structural configuration of the
quartzite on the basis of bedding/cleavage relations. (Dalziel and Dott,
i970)

Other studies on the analysis of the structure have also been
completed. Riley (l947) did structural analysis on a microscopic scale.
His data was obtained in a study based on the structural relationships of
deformed quartz grains. Dalziel (I969) completed a study on the
structural relationships based on the orientation of quartz filled tention
gashes. Dalziel and Stirewalt (i975) also did work on the "Stress
history of fold development" of the area based on their interpretation of
deformed quartz and tension gash orientations, and on a collection of
data from Riley ( I947).

I7

18

The study by Riley (1947) and Dalziel and Stirewalt (1975)
determined stress directions in the Baraboo Syncline based on the
interpretation of preferred orientation of quartz subfabric (c-axes,
deformation lamellae, and microfractures) in the Baraboo Ouartzite. 0n
the scale of the entire syncline the direction of least stress was
interpreted to be nearly vertical and perpendicular to the hinge line of
the f old. In hinge zones, the axes of greatest and intermediate stress lie
in the bedding with the axis of greatest stress perpendicular to the
hinge. However, on the fold limbs the axis of greatest stress was
reported to be parallel to the hinge line (Dalziel and Stirewalt, 1975).

Within this same study, the principle stress directions for the
Baraboo Syncline was interpreted from tension-gash bands and extension
joints. The results using the second criteria showed that, at the time of
the formation of the nonslickensided extension fractures and the quartz
filled tension-gash bands, the axis of least stress was parallel to the
hinge line of the fold. The axis of greatest stress was close to being in
the bedding and perpendicular to the hinge line (Dalziel and Stirewalt,
1975). Two distinctly different orientations for the principle axes of
stress were interpretted from the two respective studies. Although the
orientations of the principle axes of stress derived from the two studies

are different, it is suggested that each strain feature represents a

19
different stage in the deformational history of the Baraboo Syncline. The
stress system deduced from the tension-gash study is believed to have
been in operation throughout a considerable part of the folding history.
The data was interpretted to indicate that the axis of greatest stress
basically has a north-south orientation, the axis of least stress has an
east-west orientation, and the axis of intermediate stress is nearly
vertical. it is believed that the quartz microfabric represents a stress
pattern characteristic of a stage in the shortening history late in the
fold development (Dalziel and Stirewalt, 1975).

Another study, in progress, attempts to describe several structural
relationships and the history of fold development of the Baraboo Syncline
based on the principle axes orientation of deformed sand grains and
pebbles that comprise Baraboo Ouartzite (Raab, 1985). Samples for that
study were collected at many of the same geographic locations as this
elastic anisotropy study. The method used to determine the preferred
orientation of grain elongation is based on the method developed by Fry
(1979). Grain deformation represents the strain response of the grains
witrin the quartzite to an induced stress. The long axis of the grains is
the 31 direction which represents the direction of least compressional
finite strain (Hobbs, et. al.,1976). The short axis is the %3 direction,

and the direction of maximum compressional finite strain.

20

The results of Raab's (1985) study indicate that the orientation of
13 is generally north-south, parallel to the axis of maximum regional
compression, and M is parallel to the hinge of the syncline. The
interpretation of the data indicates that the direction of least
compressional finite strain was parallel to the hinge line. The finite
strain directions inferred by the grain shape study are in general
agreement with the stress directions interpretted from the tension-gash

and extension joint study completed by Dalziel and Stirewalt (I975).

METHODS USED TO DETERMINE THE ANISOTROPIC ORIENTATION

All of the samples collected are Baraboo Ouartzite and contained no
observeable minerals other than quartz grains, which are observed
microscopically to be highly sutured. A suite of samples was collected
from various sites on the Baraboo syncline. These sites varied in
geographic location representing a vast area of the structure (Figure I).
The samples were collected from the north limb, where the beds are
nearly vertical and may be overturned, the south limb, where the beds dip
gently toward the north-northwest, and near both the western and
eastern closures of the syncline. All of the samples measured for
anISOtropy were oriented in the field with respect to the geographic
north and horizonal. Samples 0032 and UPN23 were collected and
oriented in the field by Raab (1985).

Once the samples were collected, they were cut into an
approximately 4 X 4 inch cube, with each cubic face labelled as to its
proper geographic orientation. Sample 4 was cut into two Cubes, 4A and
48. Further cutting of the cube edges and the corners produced a 26
Sided shape posessmg octrahedral and dodecahedral faces. The cubic

2i

22
faces were chosen as the arbitrary X, y, and 2 directions with the
apprOpriate Miller indicies. The dodecahedral and the octrahedral faces
were then oriented with their appropriate Miller indicies with respect to
the original cubic faces.

The measurement of the wave velocities is performed on a special
set-up devised by Bennett (1968) (Figure 3) patterned after Birch (1960)
and Jamison et. al. (1963). A lathe holds the sample, and measures the
distance between the transducers. The time taken by a wave to travel
througn the entire device, the transit time, is simultaneously displayed
on the oscilliscope (CRT), and is measured with respect to the excitation
pulse time. The transit time is equal to the time taken by the wave to
travel through the apparatus and the sample (Figure 4).

0n the CRT there are two signal traces (Figure 3). The first signal
displays wave transit times. In order to calculate the time taken to
travel through the sample only, the zero sample length time for the P
-wave and the shear wave are subtracted from the respective total
transit times. This procedure is done by simply joining the two wave
guides together without a sample and recording the apparatus transit
time. The other signal constantly displays the excitation pulse time to
which the transit time is referenced (Figure 3). Both the excitation

pulse time and the apparatus transit time are subtracted from the total

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25
time leaving only the time taken for each wave to travel through the
sample.

The travel times of the P-wave and the orthogonal shear wave set
are measured without removing the sample from the lathe. The first
shear wave is identified on the CRT and its time is measured. The
sample is rotated 90' on the devise so that the separation between the
two snear waves can be observed. The second orthogonal shear wave is
identified, and its time is recorded from the CRT. The P-wave is tnen
located, and its time is recorded from the CRT.

The lathe is designed to measure the distance between the two
wave guides. The length measured represents the distance that each
wave must travel for that direction. This distance is the length of the
sample. it is now a simple equation to determine the velocity of wave

ravelling through the sample; Distance/time = velocity. The velocities
measured are absolute between different samples (Appendix 2; Time,
distance, direction, and velocity).

The 0 ellipsoid for each sample is calculated as a matrix solution
using the methods discussed in the section on modelling the 0 ellipsoid
(Appendix 1). An ellipsoid can be defined using values that represent six
directioins. The 0 ellipsoids are defined using the 01' values in nine to

thirteen directions in order to least square fit the best 0 ellipsoid to

26

the data. This 0 ellipsoid is oriented with respect to an arbitrarily

chosen X, y, and z coordinate system (Appendix 3). The 0 ellipsoid is

then oriented with respect to the geographic x; y; and z'coordinates,

shown in Appendix 4, using methods previously discussed in Appendix 1.

The final adjustment to azimuth and hade of the principle axes are

calculated from the column vectors in the final geographically oriented 0

ellipsoid matrix. The azimuth and hade of the axes are then adjusted and

recorded as dip and dip-direction (Table 2).

 

4A

48

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0032

Q ELLIPSOID AXES:OLP/DIPDIRECTION

I;.O.;.
38’l087‘
18°/092‘
l6'/058°
21’/19l‘
49‘/156‘
53°/123°
2°/26S°
60'/OS4'

2'/083°

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4‘/352‘
72'/263°
74°l224°
22‘l092'
4l‘l326‘
35°/290°

4‘/355°
18‘/176'

25‘/171'

p
. . A
. -

SI'/255°
l‘/OOl‘
3°ll49‘
59'/322‘
l‘l240'
8°/023'
86°/160‘
23'/278°
65°/335‘

A
.L

 

The anisotropy observed in sample 4 is also involved in evaluating
the reproducibility of the results. Sample 4 was made into two testable
samples, 4A and 4B respectfully. The orientations of the two 0
ellipsoids derived from these two samples will be compared to each

other as well as the anisotropy present over the entire Baraboo

SLI‘UCLUFG.

DATA ANALYSIS

Shear wave birefringence was observed in all but one of the
samples tested in this study, and therefore therefore displayed velocity
anisotropy (Tilmann and Bennett, 1973a). Only one sample, sample 6, did
not allow elastic waves to be transmitted. It was highly fractured, and
therefore the velocities could not be measured. The anisotropic axes for
the remaining samples were determined (Table 2), and the major, minor,
and intermediate axes of the resulting 0 ellipsoids are plotted on
stereogram projections (Figure 5). The ellipticity of the anisotropy was
found from the magnitude of each principle axis of the calculated 0

ellipsoids (Appendix 3).

WES:

The 0 ellipsoid major axes for all samples, except 7 and 8, have
orientations that generally trend in an east-west direction (Figure 5a).
The major axes also tend to parallel the fold axis in samples 3, 4A, 4B,
9, i l, UPN23, and 0032 all of which have an east-west major axes trend
(Figures 1 and 6). Only two of the samples, 7 and 8, displayed a large

28

 

(b)

 

(C)

Figure 5. Stereoplots (equal-area, lower hemisphere projections)
of the principle axes of the Q ellipsoids. (a) Major axes;
(b) Minor axes; (c) Intermediate axes.

Figure 6 Stereoplots (equal-area, lower hemisphere projections)
illustrating principle axes of elastic velocity anisotropy deduced from
the O ellipsoid method. Also the principle axes of grain shape from data
provided by Raab (1985).

The solio lines on the stereoplots represent the bedding plane existing at
each outcrop. The dashed lines on the stere0plots represent the cleavage
plane existing at each outcrop.

0 ellipsoid axes Grain shape axes (4A, 48 only)
Major . Long ( A1) 0 A
Minor 1 Short (A3) D A
in: Int. (A2) {5’ A

 

* The 0 ellipsoid principle axes and the grain shape principle axes
orientations are‘determined from the same sample.

** The samples used for determination of the 0 ellipsoid and the grain
shape prinClpIe axes orientation were collected from the same outcrop.

31

 

Figure 6.

32
diviation from the basic trend. These two exceptions both have major
axes with a north-south trend.

In Figure 6 the 0 ellipsoid orientations are plotted along with the
orientations of grain shape data taken from a recent study (Rabb, I985).
The stereoplots combined the 0 ellipsoid and the grain shape data for
sets where samples were collected at the same or nearby outcrops. The
elastic anisotropy and grain shape orientations were calculated from the
same sample for locations 0032 and UPN23. These two samples show a
good correlation between the orientation of the major 0 axis and the
grain Shape long axis, A1.

The grain shape and 0 ellipsoid data plotted on stereoplots 9 and
1 1A (Figure 6), were determined from samples collected from the same
respective outcrops. There is a good correlation between the major 0
axis and the long axis, ‘AI, in stereoplot 1 1A, but not in stereoplot 9.

The 0 ellipsoid data plotted on stereoplots 3, 4A, 4B, 7, 8, and I 18
(Figure 6), was compared with grain shape data calculated from samples
collected from nearby outcrops. The grain shape data from sample UPN23
is also plotted on sample 3. The major 0 axis correlates well with the
long axis, M.

Grain shape data plotted with elastic anisotropy data from samples

4A and 48 were determined from samples collected at two different

33

outcrops on either side of a small anticline in the area. One sample was
collected from an outcrop about one half mile north of site 4 where the
beds dip to the north, and the other sample from an outcrop about one
half mile to the south where the beds dip to the south. There is a good
correlation between the major 0 axes in both 4A and 48. There is also a
good correlation between the major 0 axes and the long grain axes, Al,
between the four sample (Figure 6).

Samples 7 and 8 have superimposed grain shape data obtained from
a single sample collected about one mile east of sample location 8 near
the entrance to Devil's Lake State Park in Figure 6. The major 0 elastic
axes of either of the two samples does not correlate well with Al.

The major elastic axis of anisotropy in sample 11, stereoolot 1 IB,
is also compared to grain shape data from an outcrop at the nose of the
eastern closure. As in 1 1A, there is a good correlation between the

major elastic axis and the long axis, 9.1 (Figure 6).

W:

The minor axes of the 0 ellipsoid appear to lie in a plane
perpendicular to the major axes trend, as expected, (Figure 5a and 5b),
but the data displayed on Figure 6 indicates that there is some

relationship between the bedding plane and the orientation of the minor

34
axis of anisotropy. Samples 3, 1 i, UPN23, 0032, 4A, and 48 all have
minor axes orientations that are at a high angle to the bedding plane.
Samples UPN23 and l I collected from locations near the western and
eastern closures, and samples 3 and 0032 collected from outcrops
located on the north limb have nearly horizonal minor 0 axes orientations
dipping less than 25' and north-south azimuths (Figure 5b). The minor 0
axis orientation in samples 7, 8 and 9 has a different trend than the 0
ellipsoids calculated from the other samples. Samples 7, 8, and 9 seem
to have minor axes orientations nearly within their respective bedding
planes (Figure 6).

The minor 0 axis orientations are also plotted with the grain shape
results in Figure 6. There are two correlations that can be interpreted
from the stereoolot. The first is the relationship between the
orientation of the minor 0 axis and the orientation of the grain shape
short axis, A3. This correlation occurs in samples 3, 9, and UPN23
(Figure 6). The second correlation is between the minor 0 axis andthe
intermediate length, A2. Samples 4A, 4B, 1 1 (stereoplot 118), and 0032
show this correlation (Figure 6). These first two correlations
interpreted from the data on the stereoplots are best made when the
major 0 axis can be correlated with M.

A third possible correlation between the minor 0 axes and the grain

35
shape M axes is not significant. Although sample 7 indicates this
relationship, the result is interpreted as an exception because it exists
in orrly one sample. Samples 8 and l 1A have no correlation between the

minor axis of elastic anisotropy and any of the grain shape axes.

 

"ne intermediate axes of the 0 ellipsoid are highly variable
throughout the entire fold region. There appears to be an intermediate
axes trend in a plane perpendicular to the major axes trend. There is no
strnt‘e correlation between the intermediate 0 axes and the bedding
olar-e. ax‘al plane, or regional geography (Figure 5c). in most cases there
is a cooo correlation between the major 0 axis and M. it follows that
the other 0 and snape axes ten: to lie a plane defined by the major 0

8X95

X*’- T T:

:r-e magnitudes of the principle 0 axes were determined as shown in
Aster-pix 3. The shape of the 0 ellipsoid can be determined with axial
retire The 0 ellipsoid is prolate if the intermediate axis/minor axis
ratio "as less than the major axis/intermediate axis ratio. The opposite

relat'onsrzio yields an oblate ellipsoid. These 0 ellipsoid axial ratios

36
were blotted along with the axial ratios of grain snape (Raab, 1985) in
order to interpret the data (Figure 7). The calculated 0 ellipsoids were
mostly oblate in shape, and did not possess any Obvious relationship with

the grain shape ratios.

(X/N’.‘

2.0.

1.8..

1.6a

37

 

 

 

Prolate
O
b
l
a
t
e
e UPN23.0032
° 48
. 1.307 ‘9 .T4A
I I T T Y T I T T I
1 0 1 2 1 4 1.6 1 B 2 0

(W2)

Figure 7 Plot of the axial ratios of D ellipsmd and
quartz grain Shapes

X, V, and Z are the principle axes of the 0 ellipsmd and
quartz grain shapes where X > V :- 2.

Locations of grain shape data
North Limb I
South Limb .
Eastern Closure *
Western Closure®

 

DISCUSSION

A review of the equations governing wave velocity for both
P-waves and shear waves travelling through an elastic media reveals
that the velocity is controlled by elastic constants divided by the

density of the media.

v e (CU/[0W eq. (l)

in equation (1), V is the velocity, Cij represents a collection of
constants of elastic modulii, and p represents the density of the media
(Dobrin, 1976) (Garland, 1979). Density is direction independent for the
wave equation when the wave is travelling through one lithology.
Therefore the change in velocity for a given direction must be controlled
by directionally dependent elastic constants. Crampin (1984) wrote that
all classes of anisotropy are more or less directly oriented by an
existing or palaeo stress-field. A dependence on stress means that the
observed anisotropic orientation can be interpreted in terms of the

existing stress field at the time when the anisotropic allignments were

38

39
fixed. When a body of rock is subjected to a stress, the rock will strain,
or deform, in order to most efficiently alleviate the stress applied. It is
believed that the observed anisotropy in this study directly reflects
strain patterns within the rock that resulted from the regional stress
field which formed the Baraboo Syncline.

In order to more fully understand the relationships between
anisotropic orientation and strain, a slate sample collected from the
Cambrian Slate Belt of Wales was tested for elastic anisotropy. The
slate sample contained deformed reduction spots. The deformed
reduction spots in the sample tested were were flattened. Each spot
possesses one short axis and two long axes of nearly equal length.
Reduction spots can be thought of as a physical strain ellipsoids, and
their shape can be used to determine the direction of principle axes of
strain (Ramsey, 1967) (Tulis and Wood, 1975) (Meyer, i983). The short
axis of a reduction spot is equal to the direction of compressional strain,
and the long axis of the reduction spot is the direction of extensional
strain.

The sample was arbitraraily oriented with the short axis of the
redu tion spots being coincident with the (0,0,1) direction, and the long
axes being coincident with the (1,0,0) and (0, l ,0) directions. The minor

anisotrooic axis of the 0 ellipsoid for this sample closely coincided

40
with the (0,0, l ) direction of the sample, parallel to the direction of
greatest compressional strain (Figure 8). The major and intermediate
principle axes of the 0 ellipsoid were nearly equal in magnitude values
and within the plane perpendicular to the axis of greatest compressional
strain.

Although the orientation of the strain induced anisotropy is also
dependent on mineralogy of the sample, and the strain habits of that
particular mineral group, it can be expected that in general the minor
axis of the 0 ellipsoid is parallel to the direction of greatest strain.
The major axis of the 0 ellipsoid should therefore be alligned parallel
to the direction of least compressive strain in samples collected from
tectonicly altered regions.

A general overview of the geometry of the Baraboo Syncline
suggests that the regional compressional strain direction was generally
in a north-south direction, nearly perpendicular to the hinge line of the
syncline, but finite strain patterns existing at the time of deformation
were probably highly variable within the folding strata. The minor axes
of the 0 ellipsoids from the north limb and fold closures have a
north-south orientation (Figure 6). This trend does not exist in the
samples collected from locations on the south limb (Figure 6). According

to classic fold modelling, the orientation of least compressional strain

 

 

 

f

Figure 8. Stereoplot (equal-area, lower hemisphere projection)
showing the orientation of the principle axes of the Q ellipsoid

calculated for the slate sample collected from the Cambrian
Slate Belt of Wales.

Major I

Minor 0

Intermediate .

42
at the time of folding should have been vertical. in general, the major
axis orientation of the 0 ellipsoids are not vertical, but are oriented
nearly parallel to the hinge line of the fold where the orientation of the
intermediate strain axis is expected to be.

The regional stress pattern at the time of deformation is thought to
have been homogeneous, but the pattern of strain observed within the
deformed sedimentary layering is very complex. The complexity of the
observed strain pattern arises from the variations in lithology as well as
the changes in fold geometry. Throughout the developmental history of
the syncline, the orientation of the fold geometry changed relative to the
assumed homogeneous regional stress field giving rise to futhur
complexities in the observed strain patterns (Dalziel and Dott, l970).
Several studies have been completed in the Baraboo Syncline region that
discuss the orientation of the principle axes of stress and strain that
may have affected the area during the deformation.

Dalziel and Stirewalt (1975) studied the preferred orientations of
quartz c-axes, quartz lamellae, microfractures, tension-gashes, and
extension joints in order to locate the principle stress directions
existing at the time of syncline formation. The stress axes deduced from
the tension-gash study are in general agreement with the elastic

anisoropy data interpretations. Therefore, the ODSGWGG anisotropic

43
pattern probably reflects strain patterns induced during the same phase
of deformation that created the tension gashes.

The stress directions inferred by the grain shape study are in good
agreement with the results of the tension-gash and extension joint study
competed by Dalziel and Stirewalt (1975). in therory, 13, the direction
of greatest compressional strain, should be coincident with the direction
of maximum stress (Hobbs, et. al., 1976). in most of the cases, the major
axis, ‘ast direction, of the 0 ellipsoid possesses the same general trend
as the ‘Xl direction (Figure 6), but the A3 direction of grain shape
coincces with both minor and intermediate axes of the 0 ellipsoid
(Figure 6). The observed major Q axes orientation correlates w ll with
stress and strain directions interpreted from previous studies. in order
to mc'e fully understand the relationship between anisotropic
oriertations and regional deformation processes the causes of anisotropy
must as evaluated.

Structural layering due to the allignment of micaceous minerals
couic :e a contributing factor in the observed anisotropy. Micaceous
minerals tend to ailign in a plane perpendicular to an induced
compressional strain. The effects of structural layering in the Baraboo
Ouartzite may result from a micaceous foliation developed perpendicular

to the regional compressional strain that caused folding and

44
metamorphism of the Baraboo Syncline. However, the quartzite samples
collected are nearly pure quartz and possess no observeable micaceous
foliation. The effect of structural layering due to the allignment of
micaceous minerals could only be a minimal contributing factor in this
study, and is not considered important to the anisotropic pattern
observed.

Tre observed anisotropy could be a result of crystallographic effects
Within crustal rocks (Crampin et. al., 1984). The elastic anisotropy of a
single quartz crystal can be represented by a 0 ellipsoid of revolution
with an ellipticity of 19%. A previous studies indicate that the overall
pattern of c-axes orientations in the quartz grains comprising the
BaraCC-o Ouartzite appears to display a ramdom distribution when plotted
on a stereographic projection (Dalziel and Stirewalt, 1975) (Tilmann and
Bennett, 1973b) (Figure 9).

T’ne c-axes orientations in this study also appeared to show a
random distribution (Figure l0). The orientation of the c-axes of quartz
grains comprising sample 0032 were measured petrographically from an
oriented thin section. The average principle axes orientation was
calcc'ated by summing ellipsoids of revolution representing each c-axis
orier:ation. Eacr. ellipsoid was given a 19% eliipticity equal to the

elast‘c ellipticity of a single quartz crystal calculated from elastic

Figure 9. From Figure l (Tillmann and Bennett, 1973b) p. 6835.

Figure 10. Stereoplot (equal-area, lower hemisphere projection) showing
the correlation between the axes of anisotropy for sample 0032 and the
averaged principle axes of its axes determined optically.

0 observed. Major . Minor 0 intermediate 6)

0calculated:* Major I Minor E] Intermediate El

*From optically determinined orientations

46

Ian... Ouartzite

 

6‘ -ua aanie ”W

gen-rjm ' : 3
n .12. II

6. I A O

"mea

Figure 9

Fig. l. Equal-area projection of 200 quartz c
area for the bamboo quartzite. The direction of
the observed maximum P wave velocity, the 0p-
tical surface. and the sonic Q ellipsoid principal
axes are also plotted, M, m, and I being the el-
lipaoidal major, minor, and intermediate princi-
pal axea. respectively. Clou coincidence between
the axes of the two surfaces indicates that the
orientation of the Q ellipaoid describes the pre»
ferred crystallographic orientation

 

Figure l0

47
constants (Clark, 1966). The summed ellipsoids generated one final
QUBC’lC surface that represents the average c-axis orientation. These
average optic axes were compared to the opserved 0 ellipsoid axes for
sample 0032. There is a direct correlation between the axes of the
observed 0 ellipsoid and the average 0 ellipsoid determined from
optical orientations in sample 0032. The magnitude of the quadric
surfa:e axes was calculated in order to evaluate the amount of
anisc~tropy contributed by the crystallographic effect.

The eliipticity of the quadric surface generated by the
crystallographic - elastic relationship is 8%. The ellipticity of the
meass-red 0 ellipsoid is 23% (Appendix 5). The contribution of the
cryStailographic effect to the elastic anisotropy present in sample 0032
only atcounts for one third of the ppserveo elastic anisotropy. However,
the Ctrreiation between the 0 ellipsoid axes and the axes of the quadric
surf a:e generated by the averaged c-axes for 0032, differs from the
corre'ation reported by Tilmann and Bennett (19730) (Figure 9 and 10).
The s gnificance of crystallographic effects on elastic anisotropy in the
Barat-po Ouartzite therefore remains in tic-apt. in order to better evaluate
the relationship between the 0 ellipsoid and the average optical
orier-tation in a quartzite, a more detailec study is necessary. Other

Dnys":al features must be considered in order to more fully explain the

observed anisotrOpy.

There is undoubtably a direct correlation between the orientation of
the major axes of the 0 ellipsoids and the hi orientation of average
grain shape orientation (Raab, 1985). it therefore seems likely that the
preferred orientation of the long axes of the quartz grains is the major
factor influencing the orientation of the major axis of elastic anisotropy
in the Baraboo Ouartzite. The orientation of the observed anisotropy
resulting from a preferred allignment of elongate quartz grains seems
to reflect the relative number of grain boundaries a wave encounters
over a certain distance proportional to the direction of wave propagation.
The effect of grain boundaries on elastic wave velocity may be much the
same as preferrably oriented microcracks in a rock body. Any boundaries
encountered by a wave propagating through an elastic medium may
increase its compliance. An increase in compliance decreases the
elastic constant of the medium for that direction thereby decreasing the
velocity of wave propagation (Equation 1). A P-wave travelling through
the long axis of many grains will encounter a proportionally smaller
nu. per of grain boundaries and therefore travel faster than a P-wave
travelling througn the short axis of many grains. A study by Babuska and
Pros ( 1984) supports this theory.

in the data set of Figure 6 the orientations of the minor 0 axis and

49
the A3 directions do not always coincide as might be expected if the
elastic anisotropy is caused by triaxial ellipsoidal grain orientation. The
data from sample sites 0032, l 1, 4A, and 4B show the minor 0 axis has
an orientation close to the $2 direction, and the intermediate 0 axis has
an orientation close to the i3 direction. An explanation of this
coincidence may be that the shapes of the quartz grains measured in
these sections are nearly prolate. A prolate grain has two short axes
that are close to being equal in length. Therefore elastic wave
propagating through the sample in these two respective directions will
encomter close to the same number of grain boundaries, and their final
anisotropic orientation will be in a plane perpendicular to the long grain
and rrajor 0 axes.

The O ellipsoid and grain shape (Raab, 1985) axial ratios were
plotted on Figure 7. Prolate grain shapes are more abundant for samples
collected from the north limb and the eastern closure. This possibility
is somewhat less for samples collected from the south limb (Raab,
l985). However, there is no obvious relationship between the data of the
two studies. Therefore, the significance of the axial ratios of the 0
eléitspids is uncertain.

The Q orientations for locations 7 and 8 (Figure 6) are different

than. 35089 from the other samples, and thS 00 DOI reflect the same

50
strain pattern. Furthermore, the 0 ellipsoid orientations do not
correlate well with the grain shape data derived from sample collected
from a nearby location. The reason for the different pattern of at these
two locations is unknown, but it may be due to the abundance of phyllite
in the surrounding rock.

The location of both samples is on the south limb of the syncline on
US. i—tighway 12 (Figure l) in an area where the Baraboo Range has a
natural topographic low. Because micaceous minerals are much less
reslstant to weathering than quartz, the topographic low may have been
caused by weathering of a stratagraphic section where there was an
intensely interbedded phyllite within the quartzite. in an area about one
mile east of sample site 8, near the entrance to Devil's Lake State Park
(Figure l), the interbedded phyllite reaches thicknesses of up to ten feet
witr thinner interbedded quartzite layers (Dalziel and Dott, 1970).

The principle axes of strain affecting a quartzite in a region of
little or no phyllite may be different than those affecting a quartzite in
a region of intensively interbedded quartzite and phyllite. Phyllite is a
..ucr more ductile rock than a quartzite (Clark, 1966). Therefore the
phy'?‘-‘Ft9 will strain much more than the interbedded quartzite when the
FOCK unit is subjected to a regional stress. The resulting strain patterns

in tre ouart21te may be altered and reflected in the observed anisotropic

5i
orientation. Dalziel and Dott ( 1970) reported that the geometry of the
folded interbedded quartzite and phyllite zones on the south limb are
essentually concentric. Slickensides on the bedding plane indicate that
active bedding slip occurred during folding. Therefore an operating
"flexural slip" component may have resulted in the reorientation of the
princ‘ple stresses affecting the quartzite. Any reorientations of the
stress patterns affecting the quartzite beds will also alter the resultant
strafr patterns and elastic anisotropy.

Crenulation cleavage and kink bands are preserved in the phyllite
layers near the collection area for samples 7 and 8. These features are
evidence of a second phase of deformation in the Baraboo Syncline. Much
of the strain affecting this area during second phase deformation may
have peer. taken up by the phyllite. The interbedded quartzite layers may
have seen rotated during the second phase deformation. These beds
wou'ac not be strained as much as quartzites from other areas that have
little or no interbedded phyllite. The interbedded quartzite would
probaply retain the anisotropy induced during an earlier phase of
deformation.

The 0 ellipsoid from sample 9 does not correlate well with the
grair shape data (Figure 6). This sample did not transmit the elastic

waves as well as the other samples, and the signal received by the CRT

52
was weak and displayed a great deal of interference. in hand sample this
rock is observed to be more fractured than other rocks tested except
sample 6, which did not transmit any readable signal. Although these
fractures may be responsible for the poor transmission qualities as well
as tne noncorrelation with the data from the grain shape analysis, they
may nave been formed as the result of another physical strain response
to an induced stress.

Although the 0 ellipsoid technique does not consistantly reflect
identical strain patterns throughout the deformed region, this technique
does nave advantages over other strain measurements. Two of the
advantages are previously discussed. 1) The orientation of O is
determined by the gross make-up of the physical properties comprising
the rock. 2) in a single quartz crystal the ellipticity of the 0 ellipsoid
(19%: is greater than the ellipticity of its optical indicatrix (0.6%).
However, one of the most important advantages of the technique is its
potential for expansion. With further-understanding of anisotropic
behavior in rock stratigraphic units, such as the Baraboo Group, elastic
anisctropy may be measured in situ over large areas. Measuring tre
anist-tropy in situ would reduce the amount of error involved in keeping
the sample oriented throughout the testing period. Strain features

present in a stratigraphic rock unit vary in size and orientation. Tne

53
orientation of the observed elastic anisotropy may vary depending on
which of these physical features most greatly effects the elastic wave
tranmittion for each particular wavelength. The relationships between
those different orientations of elastric anisotropy might lead to a better
understanding of the deformational history of an area, and the regional

tectonics existing throughout the time of its formation.

CONCLUSlONS

Assuming that elastic anisotropy may be present in rocks which
have undergone metamorphic deformation, the basis of this study was to;
i) Prove the existence of velocity anisotropy, 2) to observe what
patterns of anisotropy exist in a tectonically deformed region, and 3)
determine the relationships between orientation of elastic anisotropy
and stress and strain. The orientation of the elastic anisotropy is
estabTished using the 0 ellipsoid technique (Bennett, 1972).

All of the samples tested display shear wave birefringence, and
therefore are anisotropic. The orientation of the principle 0 axes of the
0 ellipsoids for each sample established the principle axes of elastic
anisotropy. in 7 out of 9 samples tested there exists a major axis trend
that is nearly parallel to the hinge line of the syncline in an east-west
direCt‘ion. Samples collected from the north limb and the eastern closure
dispiay a minor axis trend nearly perpendicular to the hinge line in a
nortr-south direction.

=rom the interpretation of elastic anisotropy data derived from a
slate sample, the principle strain axes directions for the Baraboo

54

55

Syncline are inferred from the measured 0 elipsoid. The regional
pattern of elastic anisotropy reflects the strain directions present at
the time when the elastic anisotropy was induced. The direction of least
compressional finite strain is interpreted to be parallel to the major
axes :f elastic anisotropy. This result is not consistent with the fold
geometry or strain directions inferred by fold modelling. However, the
resuts are consistent with the stress directions deduced from measured
oriertations of quartz filled tension gashes by Dalziel and Stirewalt
(1975}.

The major 0 axes correlate well with the long grain shape axes in 7
of l-C- locations. The minor 0 axes correlate with both the short and

intermediate grain shape axes. The relationship of the the quartz grain

SllZC E‘

H

. to finite strain is assumed to be the same as finite strain
relat'onships in reduction spots.

A major portion of the observed anisotropy can be attributed to the
effe:t preferred orientation of elongate quartz grain (Raab, 1985).
Prev'ous work with elastic waves (Babuska and Pros, 1984) (Brodov,
1984‘- suggests that microdiscontinuities within a rock body effect
elastic wave propagation velocities. The relative velocity of an elastic
P-v. ave travelling through quartzite in a certain direction is thought to

be i' .iersely proportional to the number of grain boundries encountered.

56
Part of the measured elastic anisotropy can be attributed to a
preferred orientation of quartz crystals comprising the quartzite. The
calculated elastic anisotropy, generated by the preferred orientation, is
shown to be responsible for only one third of the observed elastic
nisOtropy.

Although, as in most studies, more work is needed to more fully
understand the characteristic behavior, causes, and relationships of
elast'c anisotropy, the results of this study lead to the conclusions that;
(l) a’.‘ of the rocks which could be tested tested possess elastic
anisotropy, (2) the elastic anisotropy has a regional trend that probably
results from strain patterns in the rock induced by a regional stress, (3)
the anisotropic orientation correlates well with the preferred
orientation of the grain elongations, and (4) the 0 ellipsoid method of
measuring anisotropic orientation can be a valid approach used to study
and understand the stress and strain relationships in a tectonically

altered region.

APPENDlCiES

APPENDEX l: MODELLlNG THE ANISOTROPIC ORIENTATION

in order to establish the orientation of anisotropy, the concept of
the 0 ellipsoid has been developed by Bennett (1972). The 0 ellipsoid is
a simple elastic stiffness figure the can be used as a tool for detecting
and describing preferred crystallographic orientations. The 0 ellipsoid
is a theoretical surface whose value for any particular direction is the
su .l of the squares of the three seismic wave type phase velocities in
that direction. The samples collected for this study, which are crystal
aggregates, are considered as elastic long wave equivalents to single
crystals.

The calculated value, 0/, of the 0 quadric for the 1th direction is

given by the equation:

Each 0/ value is the sum of the squares of the. three seismic wave

velocities for any one particular propaga ion direction. Where p is the

57

58

material density, V, is the P-wave velocity, and 1/2,

and V3 are the
velocities of two orthogonally polarized shear waves for the 1th
direction (Bennett, 1 972). A polarization plane is defined as the plane
that contains the propagation direction and shear wave particle motion
(Tilmann and Bennett, 1973b) There are two polarization planes for each
sample which are essentially orthogonal for any propagation direction.
Since density is direction independent, the term is incorporated into the
0/ term without affecting the shape or orientation of the 0 ellipsoid.
The locus of 01' values, in a minimum of six directions, is represented by
an ellipsoidal surface. Thus the calculated value of the 0 ellipsoid in
the 1th direction will be referred to as 0/'(Tilmann and Bennett, 1973b).
The principle axes of the ellipsoid represent the principle axes of elastic
wave velocity amsotrooy for that sample.

The 0 ellipsoid can be calculated in terms of elasticity

coefficients, 5,], (Bennett, 1972). The equation for the 0 ellipsoid is

7- 2 ,- _. .— 2! a
o-/((,,+L55 c66)+m((22+t44+r

[72(633‘ r443 (55% 2’77””24‘ 534+ r ‘+ Eq(Al.2)

t (55+ (435% 2/f77Cc‘,6+ [26+ (45)

59

When the propagation directions I, m, and n are referred to the
principle axes X, y, and z

[24* (54“ 556—0
(15+ [35+ [45:0 Eq.(Al.3)
[16 (25+ (450

The relationships in Equation Al.3 are true for all crystal systems

the crystallogr

excep' the triclinic and monoclinic. Thus the 0 e1 .ipsoid comcides with

In mnnnrl

' \J .\.-‘& l

aphic orthogonal axes and optical indicatrix axes except

inic and triclinic systems (Bennett, 1972). The 0 ellipsoid

C'y5t2500F33=

calcu‘ated from the different elastic wave velocities with respect to the

nic axes of Alpha ouartz has a 19% differenre between the

(F

major and minor velocity anisotropy axes based on elasticity data clark
196 -.

l A 19"? difference is quite large when compared to the 0.6%

difference between the optic "0 and "“E axes of Alpha quartz.

"i

I n

The "matrix method" is used for the construction of the O ellipsoid,
accordi g to Nye ( 1957), where the values of each 01' on a triclinic
crysta are measured. Each sample is cut into a cube. Because notr-‘ng is

krcv-r beforehand about the orientation of the principle axes of elastic

60
nisotropy in the crystal, arbitrary orthogonal axes x, V, and z, righthand
coordinate system, are chosen and labelled with (1,0,0), (0,1,0), and
(0,0,1) Miller indicies respectively. The edges of the cube are parallel to
these axes (Nye,l957).
The least squared value 0L5 of the 0 ellipsoid for the 1th

direction is given by

0:51 = #23,, + 07/2a22 + 019333 + Eq.(Al.4)

2(07/)(/7/)a2, + 2: m)( //)a,. + 2( //)( 07/)a.2

v."‘.e."e U; ml; and m; are directional cosines for the 1th direction
relative to an arbitrary set of orthogonal axes X, y, and z The a (alpha)
terms are the elements of a 3 X 3 symmetric ellipsoid matrix (Tilmann
and Bennett, 1973b). The elements of the a (alpha) matrix are the only
uni-'nowns They are determined by the least qua-res method outlined in
Nye ( ‘ 957). This procedure is based on the matrix equation relating the
measured 01' values to the directional cosme matrix 0 (theta) and the 3

(alpha) matrix by

0 = Ba Eq.(A1.5)

Tre 0 matrix elements are the measured 01' values from V1,

V2, and
1/5. The B (theta) matrix is con tructed b using the l, m, and n

directional cosine coefficients. The 3 (alpha) matrix is therefore

determmed by solving for 3 (alpha). (Tilmann and Bennett, 1973b)

a = (0t,8)"'°(0t)0 Eq. (A16)

To refer this matrix to its principle axes the method of "successive

approximation" is used. Figure 1 1 shows the principle of the method. An
arbitrary unit vector represented by a single column matrix 1, is chosen,
and Ice elastic stiffness matrix, u. , for unit change in elastic wave

veloc-ty is calculated from the ecuation (hye, 1957).

C
II

30,) Eq. (Al.7)

it is known from the property of the representation quadric that the

direCtzon of u, is that of the normal at poznt P, where i, cuts the

62

 

 

 

 

 

Figure l i The successwe apprommotion method of finding the
principle axes of a second-rank tensor. (Nge, 1957 p 165)

63
urface of the quadric. The new unit vector I2 is then taken parallel to

upa and the corresponding matrix u is computed A third unit ector l3

2
paraiiel to u2 is taken, and so the process is repeated. if the quacr ic is
an ellipsoid, the vectors converge on to the shortest, minor, axis

(Nye, ‘- 957 pp165, 166). By inverting the a (alpha) matrix the long.
major. axis is similarly located. The intermediate axis is the cross
product of the major and minor axes (Tilmann and Bennett, l9730).

The matrix thus caiculated represents the orientation of the
principle 0 axes of thea ellipsoid of one samp :e for an arbitrary set of
axes, 0. in order to interpret the data for this study, the principle 0
axes must be referred to the geographic coordinate system. This is done

througn matrix multiplication (Butkov, l968)

MO

0' Eq.(A‘i .8)

he cubic faces of the sampi e were oriented with respect toa
origir .a: face that was oriented in the field. Therefore the geograpric
ori entation for each of the formery .;}/, and 7direct ions is Known, and
the tree geographic directions are calculated and fit into a 3 X 3

directiona'. cosine row matrix, M. The final rotation is completed py

64
multipling the 3 X 3 directional cosine row matrix for the true
geographic directions, M, with the 0 ellipsoid principle axes column
matrix, 0, resulting in a directional cosine column matrix, 0', for the
true geographic principle 0 axes orientation of the 0 ellipsoid.

One of the biggest advantages of the 0 ellipsoid is that it provides
a simple mathematical surface that can be "least squared" to measured
data taken. in more than six directions (Nye, 1957). The mean squared
deviation from. this surface statistically tests for anisotropy in the
sample, and indicates the degree to which the anisotropy present in the
sample is homogeneous (Bennett, 1972).

Comparing the measured ellipsoidal values 01‘ with the calculated
ellipsoidal values (Piss provides a test for sample homogeneity. Three
different types of standard deviations are calculated and compared
(Tilrrann and Bennett, l973b). The standard deviation SM is thought of
as the deviation of the-measured ellipsoidal values from the best fit

sphere to the measured values.

5m. = i z ( and/22W” l”? cc. (Al.9)

til/is the measured lllpsoida‘: value in the /'th direction (EQ. All),

0777 is the arithmetic. mean ..easured value, and r: is the number of

65
propagation directions, I, measured.
The standard deviation 517“ is the deviation of the calculated

ellipsoidal values from the best fit sphere.

501w = i z (OLSi - anvil/n1”? Eq. (A1.l0)

(its. is the calculated ellipsoidal value (Ed. Al.4) in the 1th
direction.
The third standard deviation SDe is the deviation between the

measured and the calculated ellipsoidal values.
509 =1 2 (02' - OLSl)2/n]”2 Ed. (Al.l 1)
if the data tidints lie exactly on the. ellipsoidal surface then 50a =
O, and 50m and 50k are equal.
Samples acting as psuedosingle crystals With. the properties of

homogeneity and elastic behavior are indicated by the relationship

517k 2 50m > 5179.

FT}
F3

A
>
M
‘1

Sample heterogeneity is indicated by (Tilmann and Bennett, l973b)

66

50k) 50c) 5Dke. Eq.(Al.l3)

Heterogeneity is not considered to be consistent with the concept
of elastic behavior as a psuedosingle crystal. Heterogeneity is
cons‘dered to be the random form of variance of the physical properties
of the sample. A heterogeneous surface does not possess the property of
centrosymmetry. Centrosymmetry is an essential property of
homogeneous anisotropy.

The correlation coefficient is also calculated for the purpose of
testing how well the theoretical values of the 0 ellipsoid surface
corre‘ates with the measured values. Serf ct correlation between the
theore‘ca’ values and the measured values would yield a correlation.
coef‘fclent equal to lo. 1f the two sets of values are perfectly
independent and heterogeneous, the value of the correlation coefficient
would pproach zero. in general it is accepted that a good correlation

between the two sets of values is greater than. or equal to 0.9.

APPENDiX 2

 

 

 

 

 

 

San-1’: P. :g ‘5
Direot‘onljji) Iimeiusec) Distanoelccn) Yemeni/(ml ,3)
P 61 52 P 61 52
(1.0.0) 18.70 27.00 28.50 10.518 5.625 5.896 5.691
(0,1,0 18.20 25.75 26 .75 10.551 5.786 4.089 5.957
(0, ,1) 18.58 28.50 26.75 10.568 5.582 5.658 5.876
(1.0, , 19.61 28.25 29.50 11.082 5.655 5.925 5.757
{-1 0 20.08 50.88 29.00 11.567 5.762 5.747 5.989
(0,1,‘ ) 20.70 5025 51.75 11.712 5.658 5.871 5.689
{0,-1.1 19 .45 50.25 28.65 1 1.400 5.861 5.768 5.982
1.1.07 ’8 95 50.65 28.75 11.4‘2 6.022 5 727 5.970
(1:1 ,0) 18.95 29.56 50.25 1 1.288 5.957 5.845 5.751
(1 ’ " 20.55 51.25 50.25 11.105 5.807 5.777 5.902
(- . 20.20 51 .58 51 .75 12.055 5.958 5.856 5.790
(1,-1 19.70 50.75 50.00 11.615 5.896 5.777 5 872
{£212 20 .20 50.50 51.25 1 1.887 5.885 5.897 5.804
Samo‘e 4A
Bird; o" . Iimeiusec) Distance (cm) l’elorltt'i l'rri/seo)
L 61 82 P 5t 52
(1,0,0 2 .25 52.75 55.75 70.168 4.590 5.119 2.857
{0,1,0} 26.25 55.75 55.25 10.124 5.860 2 .854 2.874
(0,0,1: 25.75 54.00 57.75 10.55‘ 4.454 5 ”Ci-1 2.795
(1,0,1) 25.50 54.75 57.50 10.696 4.550 5.077 2.851
(-‘ .0,‘I- 25.25 55 .00 57.25 10.655 4.556 5.026 2.845
(0,1,1) 27.00 57.00 59.00 11.019 4.067 2.968 2.815
10,-? .' 28.00 59.75 59.25 11.656 4.157 2 92" 2.966
(1,1,0 ; 24.50 57 .25 54.75 10.516 4.266 2 .806 5 .008
11,-1.0‘ 26.75 58.25 56.50 10.617 5.957 2.767 2.900
(1,1,1) 26.50 58.50 41.25 11.557 4.517 2.971 2.775
(-'.- .1 27.75 41.50 40.00 11.252 4. '96 2.806 2.911
(l,-1 1; 26.50 57.75 59.50 11.717 4.221 2.965 2.852
1-' - 2525 57.25 58.75 11,155 553'? 2 971 2.85.6

 

67

68

 

 

 

 

 

 

 

mm R

P 51 52 P 51 52
(1.0.0) 21.50 51.25 55.50 10.411 4.852 5.558 5.114
(0.0.1 1 21 .25 50.50 55.00 10.556 4.989 5.476 5.029
(1,0,1) 25.00 54.75 57.00 11.155 4.855 5.215 5.017
(-1,0, 1 l 22.00 55.75 54.75 1 1.062 5.055 5.097 5.186
(0,1,' 3 26.25 56.00 59.25 1 1.476 4.402 5.210 2.944
(0,-1 ,1 '2 26.00 57.00 40.50 1 1.981 4.650 5.254 2 .975
(1,1 ,0) 25.50 55.50 55.00 1 1.128 4.745 5.528 5.186
( 1 ,-1 .0) 24.50 56.50 55.00 1 1 .005 4.509 5.027 5.156
(1,1 ,'. l 24.00 58.50 76.75 1 1.579 4.758 2.966 5.107
(~13, 1 25.00 57.00 57.75 11.580 4.642 5.156 5.074
(:1: 24.00 58.75 453.00 11.755 4.928 5.052 5 1;
Same Z
Direction. 1111 Timelusec) Distance (cm; Yelooitvlitrn/seo)

P SL 52 P j! 52
(1,00 9.55 29.68 28.55 10.084 5.211 5.598 5.552
(0,1,0; 19.55 29.65 50.65 10.561 5.458 5.562 5.446
10.0.”; 18.65 28.05 28.50 10.124 5.456 5.509 5.577
(1,0,1 1 21 .00 52.50 55.55 1 1.554 5.492 5.571 5.458
(~1.0.‘ ‘ 21.40 51.80 52.95 1 1.445 5.547 5.598 5.475
(0.1.1:- 1985 50.95 51.95 11.168 5.827 5.611 5.498
(0.-' ' 20.75 51.55 51.55 11.564 5.477 5.602 5.602
(1,1,c 20.15 51.55 50.55 11.125 5.527 5.526 5.641
(',-1 ’ 21.55 51.10 52.60 11.214 5.204 5.606 5.440
(1.1 l 20.58 50.95 52.45 11.594 5.595 5.685 5.514
(- 1, 20.58 52.55 51.50 1 1.044 5.420 5.595 5.529
(1,— 21.85 54.45 55.45 1 1.968 5.471 5.477 5.581
1-1 - 22 21-15 5135 52.41 11.455 5.4142 5.582 5528

nrn'. 0

Direogor’ .3. Iimetusecl Distance (cm) YelooitvtLrn/sao)

P j; 52 j) 51 52
(1 ,0.01 20.50 29.75 50.25 10.749 5.244 5.615 5.555
(0. 1.01 19.25 29.58 29.88 10.714 5.566 5.647 5.586
(0.0." 19.25 28.75 29.25 10.104 5.249 5.54 5.454
(1,0,1 2 19.95 50.00 51.00 10.904 5.475 5.655 5.517
(4.0," 21.00 32.50 33.25 11.525 5.392 3.4.54 5.406
(0,1,1) 20.75 51.75 51.50 11.505 5 447 5.560 5.586
10,-1 l 2‘. .00 50.75 52.25 11.196 5.551 5.64‘ 5 4'72
(1,1,w 21.20 52.50 51.75 1 1.257 5.500 5.457 5.559
11,-13 21.25 55.25 52.00 1 1.559 54.5 5.471 5.606
(1,: 1: 20.88 51.50 5 .00 11.148 5.541 5.559 5.484
i-l, ‘1 21.75 52.75 5‘. .00 ”1.656 5 550 5.555 3 475
11,-. l 21.50 52.00 52.50 11.415 5.509 5.5 17 5.512
1‘“: .- I— 2 1-75 54 7E 52.50 11,587 5 527 3 555 1:55

 

~5th

 

 

 

 

 

 

 

 

 

 

D'Lrectijnilm Timeiusec) Distance (cm) “10.11101 km/sec)

P 51 52 P 51 $2
(0,1,0) 19.15 50 44 29.44 10.460 5.469 5.456 5.555
(0.0.1) 16.60 29.44 29.94 10.272 6.188 5.489 5.451
(-1,0,1) 21.50 50.94 51.44 12.116 5.655 5.916 5.854
(0.1.1) 2': .95 51.94 52.69 11.955 5.458 5.757 5.651
(0,- 1 .13 20.58 50.69 52.44 1 1.689 5 .757 5.809 5605
(1.1.0) 22 .25 52.69 52.19 12.509 5.552 5.765 5.824
(1,4,0) 21.57 55.19 52.44 12.189 5.655 5.675 5.758
1-1,1,1} 22.25 52.44 51.94 12.105 5.565 5.819 5.850
(1,-1.' .‘ 21.75 52.69 51.94 12.595 5.699 5.792 5.881
01:15.; 2115 5169 51.44 12.105 5.565 5819 5.850
82.”: e ‘
011‘5311211191 Imejuseg) Distance (cm) Veloczivi yam/sec)

P 31 82 P 8L 52
(1,0,0‘ 18.75 28 .‘5 29.50 10.655 5.68“. 5.788 5.61‘.
(0,1,0, 18 .00 28.50 27.50 10.444 5.802 5.665 5 .798
(0.0.1) 18 .75 8 .0 28 .58 10 .444 5 .570 5.750 5 .681
(-1.0.“ 22 15 52.88 52.00 12.111 5.474 5.684 5.785
(0.1.1) 22.00 55.00 52.25 12.511 5.596 5.751 5.818
(1,1,0) 21.15 52.00 52.50 12.151 5.752 5.797 5.759
(1.- ‘ .C.‘ 20.88 51 .50 55.25 1 1.895 5.698 5.776 5.577
(1,1 11 22.25 54.25 55.00 12.794 5.750 5.755 5.656
(- ., 2.1? 22 88 54.25 55.50 12.581 5.500 5.675 5.756
t :1 '1 22.15 55.00 455.75 12.577 5.594 5751 5.667
Diresimm 1.) Timeiusec) 015183108 (cm) velociM lam/sec)

P SL 52 P 81 52
(0, .0) 2.58 51.00 54.00 10.262 4.586 5.510 5.018
(0.0.1) 21.75 51.75 54.00 10.574 4.862 5.550 5.110
(- ‘. ,0, .. 24.25 54.75 54.25 11.567 4.687 5.271 5.519
(0.1,‘1 25.75 57.65 56.50 1 1.824 4.978 5.145 5.240
104.12 25.00 57.50 58.50 12.105 4.841 5.217 5.144
(1 1,0) 25.25 54.00 52.75 11.559 4.498 5.541 5.470
(.,-'.,CL' 25.65 55.88 52.58 1 .599 4.407 2.902 5.216
{1,1,1 26.75 4‘. .58 45 .00 12.725 4.756 5.075 2.959
f-' 1,'.. 25.2 580" 58.75 11.956 4.755 5.146 5.086
11,-1.1) 24.25 58.25 56.75 11.687 4.819 5.055 5.180
(-‘ -‘ L. 25.75 58.25 $57.50 12509 4J80 5.218 5.28"

 

 

 

 

 

 

Will—1.11113; 11ng Distance (cm) Velocity/1’ Inna/sec.)
P 51 52 P 51 52

(1.0.0) 21 .00 29.25 29.75 10 .523 5 .01 1 3 .598 3 .537
(0.1.0) 18.13 26.50 27.75 10.391 5.733 3.921 3.744
(0.0.1) 18.50 27.88 26.50 10.132 5.477 3.635 3.823
(1.0.1) 21.63 31.38 33.50 1 1.471 5.305 3.656 3.424
(-1.0.11 22.13 30.88 32.75 1 1.570 5.229 3.748 3.533
(0.1.1) 18.00 28.50 27.75 10.317 5.732 3.620 3.718
(0.-1.‘) 18.88 29.63 28.75 10.584 5.607 3.573 3.681
( 1.1.0) 19.75 28.88 29.75 10.665 5.400 3.694 3.585
11,-1.0) 20.50 30.50 31.50 11.120 5.424 3.646 3.530
11.1.1) 20.38 30.75 31.50 11.397 5.594 3.706 3.618
1-1.1.1) 21.13 32.13 31.75 11.633 5.507 3.621 3.664
11,-1.1) 22.13 32.00 33.25 11.811 5.338 3.691 3.552
L—'.-:.:1 1.38 32.50 31.75 11.648 5.449 3.584 3.669
‘ + mn‘o

91(83282113} 11.851632) 313181129 12:) Yemfiw 111111332)

2 51 52 P 31 52

(1.0.01 17.03 26.71 35.81 10.305 6 .051 3.858 2.877
10.1.0: 15.50 26.05 36.06 10.246 6.611 3.931 2841
(0.0.1) 20.2 35.31 35.31 10.307 5.090 2.919 2.919
(1.0.1 I 21.25 35.31 34.06 11.506 5.415 3.255 3.378
(- 1 .0.1) 22- .00 36 .06 35.44 11.760 5.346 3.261 3.319
10..) 21.25 33.56 33.81 11.534 5.428 3.437 3.411
10,-1.1) 21.75 33 .81 34.81 1 1.628 5.346 3.439 3.340
("..C. 78.23 28.94 38.94 11.466 6.326 3.962 2.944
11,-1.0) 16.25 29.06 40.31 11.516 6.310 3.962 2.857
(1.1,?) 22.65 34.81 38.31 12.710 5.611 3.651 3.318
:-1.1,1 : 22 .25 36.96 35.69 12.789 5.748 3.460 3.583
(7,-13: 22.25 34.06 35.81 12.405 5.575 3.642 3.464
1.1: 1.1) 23.15 35 .81 35.81 12.842 5.547 3.586 3-586

 

APPENDIX 3

 

0 ELLIPSOID AXES (ARB1TRARY COORDWATE SYSTEM)

 

MAGMTUDE DiRE CTIONAL COSENES AXiS

 

Qarflr!}e 5

66.632 (0.094, 0.979, -0.'181) NAQCR

60.026 (0 .220 0 .157. C963) WNOR

62.890 (0.971. '0. ‘31, 201) 1NTER"1E01ATE
YAQCR/T".?NCR= T . ' T0 581.1097" "7".“ A1E= ‘ 059 ?.\'TERT":EDTATE/T"1?NCR= °' .048

“m 1

38.426 ( 0.983, 0.171, '0064) MAJOR

29.935 {-0165, 0.982, 0.089) MINOR

57.393 ( 0.078, -0.077, 0.994) !NTER.’“.E0§A.TE
.‘1A303..'/". .1‘.’3R=‘..254 NAJCRANTERT‘EDIATE:1.028 ENTERfiED; TE/H...0R=1.248

45.884 ( 0.782, 0.197, -0.59’.) 1'1. JCR
35.871 (-0.171, 0.980, 0.099) MeNOR
45.613 ‘ ( 0.599, 0.023. 0.80?) 114TERME01ATE

1"":A.f0R/1“31NOR=5.279 NMORNNTERMEDMTB1.052 INTERVED;A?E/M1NOR=1.217

5.2661313 2

57135 (0.299. 0.422, 0.856 12.135;

51.122 (3.925, - .350, 43.15:) 3:22.132

55 664 (0.236, 0. 36. 4.4951 NTEWED‘ATE
v.4..13=:/r<:113== : .1 1.1 VAJOR/f-‘QTERT‘fE’J'ATE: 1.027 11115982341175/81329 1.059

 

\l

72

 

0 ELLIPSOiD AXES (ARBITRARY COORDLNATE SYSTEM)

 

 

MAGNITUDE DIRECTIONAL COSENES AXIS
55.948 (-0.188. 0.975. 0119) MAJOR
51.690 (-O.539, -0.203, 0.817) MINOR
53.574 ( 0.821, 0.089, 0.564) 1NTERME01ATE

1":A..0?/1‘"1II\1‘0R=1.082

64.387
54.503
62.245

VIA 'CE/T‘GENORH

45.548
39.045
42.452

"7AJC:Z./;“.;'\.OR=‘; .15

U1 (n m
m o -

(‘0
010 u:
m (N N

LO;-

I'ifiuCR/f". 1 110R=

182

‘. 275'
1. b»,

MAJOR/INTERNEDiATEflO-"é‘é 1NTERMED1ATE/M1N0R=I.037

W

( 0.953, -o.oao, 0.291) MAJOR
( 0.031, 0.985. 0.170) MINOR
{-0.300, -0.153, 0.943) 1NTERMEDIATE

MAJOR/INTERMED1ATE=1.034 ENTERMEDEATE/M1N0R=1.142

"1 1 1?
(-O.109, 0.9.74, -C.201)~ MAJOR
{-0515, 0.1 '17, 0.849) 1111.021

( 0850,0196. 0.485.) FNTERVEDIATE
YAJC‘1R1.TEF.*1£:ATE=1.C:S :NTERMEC..‘.’.E/:~:1:.‘32=1.066

5?,m'71gg'19393
(-0.199, -0.154. 0.968)
( 0.893, 4.4.35, 0.115)
1 0.403, 0.887, 0.225)

1'1AJOR
WNOR
INTERNEDIATE
”AJO-’/1\"R".”‘ A .E='.

.073 1N’ER1‘1EDTATE/1‘41NOR=1.087

Canal-41E (“n-’1’)

(0 043, 0964, 0.263) ”TM-0R
(0. 998. -0. 054 0 034) MNOR
(0. 047, 0.261 ‘ =0. 9.6 ’) ENTERKEDIATE

1 1Aw-x/IN7ERI‘13 A7E=1. C67 £11‘TER1‘1E01A7E/1‘11N0R=1.135

 

73

 

0 ELLIPSOID AXES (ARBITRARY COORDINATE SYSTEM)

 

 

14501917005 DIRECTIONAL COSINES AXES
50155201012
55.529 (~0.004, 0999,0020) 11700::-
41 497 (-0.025, -0.020, 0999) 1111102.
50558 < .0999, 0.004, 0.025;: fk’ER‘EDEA’E

1"1/1.0‘;.Z.F.“"11N0R=‘1.626 1‘"1A1.I0r./111.’T ER. 1E01ATE= 1 9.: 115TE RVED1 ATE ""11. ‘ $R=1 460

 

APPENDIX 4

(.7 ELLIPSOID AXES

 

DIRECTIONAL COSINES WITH RESPECT TO GEOGRAPHIC NORTi-z

 

MAJOR MINOR INTERMEDIATE
50mm

( 0.063 -O.983 0.175)

I 0.752 0.130 0.609)

{-0.622 0.064 0.781)
Sam DIE £5

( 0.028 -0.035 0.999)

{-0.95' -0.3’.O 0.079:

{-0.307 0.95I 0018)
Sam. Q‘ge QB

{-0.528 -0.076 0.846}

{ 082': 0.228 0.516)

I 0.277 -O.959 0.053}
- .0? Z

(-O.9‘:3 —0.040 0406‘.

( 0.775 -0.936 0.305)

I 0.355- 037: 0.853)

. ,qI

( 0.51.6- 0.630 0.498)

{-0.303 4.393 0.857)

075' -0.6~60 0.02'1‘;
92mph O

{-0.323 '0244 0.914.)

I 0.504 0.764 0.403)

1 0794 4:567 0.131)
3?an 1e 1 ':

~'. 00,4 -0.993 0.077)

00.995 -0.035 0.042)

I 0.037 0.055 0.997)

 

7S
0 ELLIPSOID AXES

 

 

DEREC. IONAL COSINES WITH RESPECT TO GEOGRAPHIC NORTH
MAJOR MINOR INTERMEDIATE
m 1 D '
( 0.311 -0.944 0.108)
{-0.416 -0.006 0.909)
1 0.855 ' 0.311 0.394)
mm} '1'!
(~0158 0.929 0.336)
1 0.995- 0082 0.059)
(-0.037 -0.428 0.903)

 

APPENDIX 5

.ENTA TION (DIRECTIO" AI. COSINES)
AXES ORIENT/A ION OFT H LOUADRIC SIP: CE (0032)

MAJOR ’I.I NOR INTERMEDIATE

 

a= 100* 3‘: 'e6 -.8979 .3104
c=‘ 79* .9494 .2823 — 1375
.0159 .3110 .9422
I‘viagrsétuoe 55 6495 51.5169 55 I454
DID/DID 0.750000 27055 307145" 60" /"'}.-’I°
EI1ICIIIcItII=Nam/0000? = 55.6496/535369 = 1.0502

 

EINAI. E'..PSO IT ORIENTAT ION (DIRECT . II I L COSINES)
MEASURED C? EEQRSOID PRINC 13-75 AXES CIT- E’T‘ATI ONI OO32

ll ..‘“ ',o-l Ch ..‘-TE'T)"TTT‘ ..'...
‘T I I I h 0'.- ‘I: h
. INI .' .t- A:

_

 

34 033 964
Wag“ 1.3:» 2 39643 7.9429 2 205
130/300 1‘ "00000 2'/C53° 25V .7.‘ 6 /33:>

 

If!

* Each. ouartz c—ax': 0201150 Is reoresenteo‘ by a LineoretIca‘I elast‘c
e ”@5003 of revqutéorI, III III and major axes "a" a-“rc? "c * "
a 9300': e cuartz c“, 555‘. wIt-r: an e‘IastIic eIIéotIcéz‘. 07? I992 (C1302, 19

76

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77

I l l l . I l I Qua 1- III I | J u ‘1