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Evaluation of Fabric Dataand
Statistics of Orientation Data
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Which types of data are most common in structural geology?
1) Deformation Data:
Elongation [%]Shear strain []Strain rate [d /dt]
2) (Paleo-) Stress Data [Mpa]:
Stress Tensor(Stress Ellipsoid)Deviatoric Stress
3) Orientation Data:Field Measures (compass)Bedding, Schistosity, Lineation, etc.Lattice Preferred OrientationRemote Sensing Data
Measures of Orientation Data are: azimuth and dip angle [/]
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Classical Methods of Evaluation of Orientation Data:
2) Data distributed in 3 dimensions: Equal area projections (Schmidt, 1925)
1) Data distributed in 2 dimensions Rose diagrams:
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It is not possible to apply linear statistics to orientation data. Example:The mean direction of the directions 340°, 20°, 60° is 20° The arithmetic mean is: (340 + 20 + 60) / 3 = 140this is obviously nonsense. Statistical masures of orientation data can only be found by application of vector algebra.
The mean direction can be derived from the vector sum of all data.
n
iiv
1
(n = number of data)
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What is the difference between orientation data and other structural data?
1iv
1) They have no magnitudes, i.e. they are unit vectors:
2) Most of them (bedding, schistosity, lineations) have no polarity!
This type of orientation data can be described as bipolar vectors or axes:
v�
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How can we convert measures of orientation data
(/)into vectors of the form (Vx, Vy, Vz) ?
with v = 1 we receive: Vx = cos cos Vy = sin cos Vz = sin
with v = 1 we receive: Vx = cos cos Vy = sin cos Vz = sin
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Vector sums of orientation data: if the data are real vectors with polarity (palaeomagnetic data) we have max. isotropy in a random distribution
01
n
iiv
and max. anisotropy in a parallel orientation:
nvn
ii
1
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Measures derived from addition of vectors (orientation data):
The Resultant Length Vector: n
ivR1
The Vector Sum:
2
1
2
1
2
11
nn
i
n
i
n
zyxvR
The Normalized Vector Sum:
n
RR
Azimuth and Dip of the Centre of Gravity:
n
iR xR
x1
1
n
iR yR
y1
1
n
iR zR
z1
1
R
RR x
yA arctan
Rzarcsin
R
RS
The Centre of Gravity:
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Problems of axial data:
If the angle between two lineations is > 90°,the reverse direction must be added.
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Flow diagram for the vector addition of axial data:
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What is the vector sum of axial data?
In case of max. anisotropy (parallel orientation) the sum will equal to the number of data, but what is the minimum (max. isotropy)?
It can be shown that the vector sum of a random distribution of axial data is:
21
nv
n
ii
we conclude that the vector sum of any axial data must be in the limits:
nRn
2
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From these limits a measure for the Degree of Preferred Orientation (R%)
can be found:
1002
%n
nRR
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Distributions:The Spherical Normal Distribution (unimodal distribution)
Fisher Distribution (Fisher, 1953) kzyxF ,,, 000
Concentration-Parameter (k):
Rn
nk
1ˆ k̂0 Watson, 1966
For axial data: k̂2 Wallbrecher, 1978
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Density Function: cos
sinh4),( ke
k
kf
Probability Measures:
The Cone of Confidence:
11
1arccos1
1
n
PR
Rn
P is the level of error (0.01, 0.05 or 0.1 are common levels,they equal 1%, 5% or 10% of error)
Fisher Distribution
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The Cone of Confidence
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Geometric equivalent of the concentration parameter:
From this we derive the spheric aperture: kn
ˆ
11
2arcsin
For large numbers of data:k̂
2arcsin
2% cosR
Isotropic distribution ina small circle with apicalangle
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Examples for Spherical Aperture and Cone of Confidence
Fold axesRio Marina (Elba
Italy
Fold axesMinucciano
Tuscany
Yellow: Spherical apertureGreen: Cone of confidence
Confidence = 99%
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Spherical Normal Distribution
Aus Wallbrecher, 1979
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Significant Distributions
Umgezeichnet nach Woodcock & Naylor, 1983
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The moment of Inertia (M)
vuvu
vu
cos
22
cos1
sin
a
a
zzyyxx vuvuvu cos
Rotation axis is u .1u
Length of u is undefined:
1vv is the radius of the globe:m = 1all masses m are:
2amM 2aM Moment of Inertia:
n
i iKugel aM1
2For the entire Globe:
2cos1M 21 zyyxx vuvuvuM
)222(1 22222zzyyzzxxyyxxzzyyxx vuvuvuvuvuvuvuvuvuM
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Axes of inertia:
Cluster Distribution:
Great circle distribution:
Partial Great circle:
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The Orientation Matrix
2
2
2
1
zyzxz
zyyxy
zxyxx
vvvvv
vvvvv
vvvvv
uM
2
2
2
iiiii
iii
i
Kugel
zzyzx
yyx
x
unM
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Eigenvalues: n 321
normalized: 1321
Eigenvectors:321
The Orientation Matrix and it´s Eigenvalues:
2
111
2
11
2
1
n
i
n
ii
n
ii
n
i
n
ii
n
i
zzyzx
yyx
x
LOrientation Tensor
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The Eigenvalues of Cluster-Distributions
23
221
sin3
21
sin3
1
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Eigenvectors of a Cluster Distribution
Spherical Aperture
Cone of Confidence
Eigenvectors(length indicatessize of eigenvalues.Sum equals the radiusof the diagram.)
FoliationPsarà IslandGreece
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Eigenvectors of a Great Circle Distribution
Eigenvectors(length indicatessize of eigenvalues.Sum equals the radiusof the diagram.
Campo CecinaAlpe ApuaneItaly
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2sin
2
1 22
the circular aperture ():
22arcsin2
From this we derive a measure
for the length of a partial great
circle. We call this measure
Eigenvalues of Partial Great Circles
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Examples for Partial Great Circles
PuntaBianca Gronda
PonteStazzemese Forno
Alpe Apuane, Italy
PuntaBianca Gronda
04.01
2.02
76.03
04.01
3.02
66.03
03.01 26.02
71.03
02.01
21.02 77.03
heavy lines =circular aperture
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2-Cluster-Distributions
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1
9 0
8 0
7 0
6 0
4 0
3 0
2 0
1 0
0
}
2cos
2sin
0
23
22
1
2cos
2sin
0
23
22
1
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Eigenvalues and –vectors of typical distributions
S p h e re3
1321 n o t
d e f in e d
S tre tch e dro ta tio n a le llip so id(c ig a r)
321 3 in ce n tre o f
th e c lu s te ran d 21
n o t d e f in e d
G ird leD is trib u tio n
F la tro ta tio n a le llip so id(D isk )
01
2
132 32
1 is th eB -ax is
an d n o t
d e f in e d
2 -C lu s te rD is trib u tio n
T h re e ax ia le llip so id 321
32
1
an dth e g rea t c irc leth ro u g h b o th c lu s te rs ; isth e p o le
o n
iso tro p ,R an d o mD is trib u tio n
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The Woodcock-Diagram
)/ln(
)/ln(
12
23
m)/ln(
)/ln(
12
23
m
Girdle:
0 < m < 1Girdle:
0 < m < 1
Cluster:1 < m < 8
Umgezeichnet nach Woodcock, 1977 )/ln(
)/ln(arctan][
23
12%
GonG )/ln(
)/ln(arctan][
23
12%
GonG