Evaluation of Fabric Data
and
Statistics of Orientation
Data
1
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 Orientation
Remote Sensing Data
Measures of Orientation Data are: azimuth and dip angle [/]
2
Classical Methods of Evaluation of Orientation Data:
1) Data distributed in 2
dimensions
Rose diagrams:
2) Data distributed in 3
dimensions: Equal
area projections
(Schmidt, 1925)
3
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 = 140
this 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.

 vi
n
i 1
(n = number of data)
4
What is the difference between orientation data and other structural data?
1) They have no magnitudes, i.e. they are unit vectors:

vi  1
2) Most of them (bedding, schistosity, lineations) have no polarity!
This type of orientation data can be described as bipolar vectors or
axes:

v
5
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 
6
Vector sums of orientation data:
if the data are real vectors with polarity (palaeomagnetic data) we have
max. isotropy in a random distribution

 vi  0
n
i 1
and max. anisotropy in a parallel orientation:

 vi  n
n
i 1
7
Measures derived from addition of vectors (orientation data):
n


The Resultant Length Vector: R   vi
1
The Vector Sum:

R 
2
2
 n   n   n 
1 v   1 xi    1 yi    1 z 
n
The Normalized Vector Sum:
R 
2
The Centre of Gravity:

 R
S 
R

R
n
Azimuth and Dip of the Centre of Gravity:
1
xR  
R
n
 xi
1
1
yR  
R
n
y
1
i
1
zR  
R
n
z
1
i
yR
AR  arctan
xR
  arcsin z R 8
Problems of axial data:
If the angle between two lineations is > 90°,
the reverse direction must be added.
9
Flow diagram for the vector addition of axial
data:
10
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:
n

n
vi 

2
i 1
we conclude that the vector sum of any axial data must be in the limits:
n 
 R n
2
11
From these limits a measure for the
Degree of Preferred
Orientation (R%)
can be found:
R% 

2R n
n
100
12
Distributions:
The Spherical Normal Distribution (unimodal distribution)
Fisher Distribution (Fisher, 1953)
F x0 , y 0 , z 0 , k 
Concentration-Parameter (k):
n 1
ˆ
k

0  k̂   Watson, 1966
n R
For axial data:
2  k̂  
Wallbrecher, 1978
13
Fisher Distribution
k
k cos 
Density Function: f (,  ) 
e
4 sinh k
Probability Measures:
The Cone of Confidence:

1
 n R 


n 1
1






  arccos 1      1 


P


R





P is the level of error
(0.01, 0.05 or 0.1 are common levels,
they equal 1%, 5% or 10% of error)
14
The Cone of Confidence
15
Geometric equivalent of the concentration parameter:
Isotropic distribution in
a small circle with apical
angle 
R%  cos 
2
1
1
n
From this we derive the spheric aperture:   arcsin 2
kˆ
For large numbers of data:
  arcsin
2
k̂
16
Examples for Spherical Aperture and Cone of
Confidence
Fold axes
Minucciano
Tuscany
Fold axes
Rio Marina (Elba
Italy
Confidence = 99%
Yellow: Spherical aperture
Green: Cone of confidence
17
Spherical Normal Distribution
18
Aus Wallbrecher, 1979
Significant Distributions
Umgezeichnet nach Woodcock & Naylor, 1983
19
The moment of Inertia (M)
Rotation axis isu .

Length of u is undefined: u  1
v is the radius of the globe:v  1
all masses m are:
m=1
Moment of Inertia:M  m  a 2
For

a  sin 
2
a  1  cos 2 
M  a2
the entire Globe: M Kugel  i 1 ai
n
 
 
u v
cos      u  v
u v
2
cos   u x v x  u y v y  u z v z
M  1  u x vx  u y v y  u vz 
M  1 cos 2 
2
M  1  (u x vx  u y v y  u z vz  2u x vxu y v y  2u x vxu z vz  2u y v y u z20vz )
2
2
2
2
2
Axes of inertia:
Cluster Distribution:
Great circle distribution:
Partial Great circle:
21
The Orientation Matrix
2
vx

M  1  u  v y vx
vx v y
2
vy
vz vx
vx vz
v y vz
vz v y
vz
2
x


 nu  x y  y
x z y z z
2
i
M Kugel
2
i
i
i i
i
i i
2
i
22
The Orientation Matrix and it´s Eigenvalues:
Orientation Tensor
 n  2
  xi 
 1 
 n
L    xi y i
 1
 n
  xi z i
 1
 n

  yi 
 1

2
n
y z
i
i
1
Eigenvalues:
1  2  3  n
normalized:
1   2  3  1
Eigenvectors:








2
 n  
  zi  
 1  

12 3
23
The Eigenvalues of Cluster-Distributions
1 2
1  2  sin 
3
2 2
3  1  sin 
3
24
Eigenvectors of a Cluster Distribution
Foliation
Psarà Island
Greece
Spherical Aperture
Eigenvectors
(length indicates
size of eigenvalues.
Sum equals the radius
of the diagram.)
Cone of Confidence
25
Eigenvectors of a Great Circle Distribution
Campo Cecina
Alpe Apuane
Italy
Eigenvectors
(length indicates
size of eigenvalues.
Sum equals the radius
of the diagram.
26
Eigenvalues of Partial Great Circles
1 2
2  sin
2
2
From this we derive a measure
for the length of a partial great
circle. We call this measure
the circular aperture ():
  2 arcsin 22
27
Examples for Partial Great Circles
Alpe Apuane,
Italy
3  0.66
3  0.76
2  0.2
2  0.3
1  0.04
Punta
Punta
Bianca
Bianca
1  0.03
1  0.04
Gronda
Gronda
heavy lines =
circular aperture
2  0.26
Ponte
Stazzemese
3  0.71
1  0.02
Forno
2  0.21
3  0.77
28
2-Cluster-Distributions
90
80
70
60

1  0
 2  sin



2
3  cos
2
2
40
20
10
0
}
30


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

29

2
Eigenvalues and –vectors of typical distributions
isotrop,
Random
Distribution
S
p
h
e
r
e
Stretched
rotational
ellipsoid
(cigar)
Girdle
Distribution
2-Cluster
Distribution
Flat
rotational
ellipsoid
(Disk)
T
h
re
ea
x
ia
l
e
llip
so
id
1
1  2  3 
3
1  2  3
1  0
1
2  3 
2
1  2  3
not
defined
3 incentre of
the cluster
1 and 2
not defined
1 is the
B-axis
2 and 3not
defined
2 and 3 on
the great circle
through both
clusters;1 is 30
the pole
The Woodcock-Diagram
Cluster:
1<m<
8
ln( 3 / 2 )
m
ln( 2 / 1 )
Girdle:
0<m<1
Umgezeichnet nach Woodcock, 1977
ln( 2 / 1 )
G%   [Gon]  arctan
ln( 313 / 2 )