1

Evaluation of Fabric Dataand

Statistics of Orientation Data

2

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 [/]

3

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:

4

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)

5

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�

6

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

7

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

8

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:

9

Problems of axial data:

If the angle between two lineations is > 90°,the reverse direction must be added.

10

Flow diagram for the vector addition of axial data:

11

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

12

From these limits a measure for the  Degree of Preferred Orientation (R%)  

can be found:

1002

%n

nRR

13

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

14

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

15

The Cone of Confidence

16

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

17

Examples for Spherical Aperture and Cone of Confidence

Fold axesRio Marina (Elba

Italy

Fold axesMinucciano

Tuscany

Yellow: Spherical apertureGreen: Cone of confidence

Confidence = 99%

18

Spherical Normal Distribution

Aus Wallbrecher, 1979

19

Significant Distributions

Umgezeichnet nach Woodcock & Naylor, 1983

20

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

21

Axes of inertia:

Cluster Distribution:

Great circle distribution:

Partial Great circle:

22

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

23

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

24

The Eigenvalues of Cluster-Distributions

23

221

sin3

21

sin3

1

25

Eigenvectors of a Cluster Distribution

Spherical Aperture

Cone of Confidence

Eigenvectors(length indicatessize of eigenvalues.Sum equals the radiusof the diagram.)

FoliationPsarà IslandGreece

26

Eigenvectors of a Great Circle Distribution

Eigenvectors(length indicatessize of eigenvalues.Sum equals the radiusof the diagram.

Campo CecinaAlpe ApuaneItaly

27

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

28

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

29

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

30

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

31

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