2. 1 Evaluation of Fabric Data and Statistics of Orientation Data

3. 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 Orientation Remote Sensing Data Measures of Orientation Data are: azimuth and dip angle [  /  ]

4. 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:

5. 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 = 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. (n = number of data)

6. 5 What is the difference between orientation data and other structural data? 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:

7. 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 

8. 7 Vector sums of orientation data: if the data are real vectors with polarity (palaeomagnetic data) we have max. isotropy in a random distribution and max. anisotropy in a parallel orientation:

9. 8 Measures derived from addition of vectors (orientation data): The Resultant Length Vector: The Vector Sum: The Normalized Vector Sum: Azimuth and Dip of the Centre of Gravity: The Centre of Gravity:

10. 9 Problems of axial data: If the angle between two lineations is > 90°, the reverse direction must be added.

11. 10 Flow diagram for the vector addition of axial data:

12. 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: we conclude that the vector sum of any axial data must be in the limits:

13. 12 From these limits a measure for the Degree of Preferred Orientation (R%) can be found:

14. 13 Distributions: The Spherical Normal Distribution (unimodal distribution) Fisher Distribution (Fisher, 1953) Concentration-Parameter (k): Watson, 1966 For axial data: Wallbrecher, 1978

15. 14 Density Function: Probability Measures: The Cone of Confidence: 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

16. 15 The Cone of Confidence

17. 16 Geometric equivalent of the concentration parameter: From this we derive the spheric aperture: For large numbers of data: Isotropic distribution in a small circle with apical angle 

18. 17 Examples for Spherical Aperture and Cone of Confidence Fold axes Rio Marina (Elba Italy Fold axes Minucciano Tuscany Yellow: Spherical aperture Green: Cone of confidence Confidence = 99%

19. 18 Spherical Normal Distribution Aus Wallbrecher, 1979

20. 19 Significant Distributions Umgezeichnet nach Woodcock & Naylor, 1983

21. 20 The moment of Inertia (M) Rotation axis is. Length ofis undefined: is the radius of the globe: m = 1all masses m are: Moment of Inertia: For the entire Globe:

22. 21 Axes of inertia: Cluster Distribution: Great circle distribution : Partial Great circle:

23. 22 The Orientation Matrix

24. 23 Eigenvalues: normalized: Eigenvectors: The Orientation Matrix and it´s Eigenvalues: Orientation Tensor

25. 24 The Eigenvalues of Cluster-Distributions

26. 25 Eigenvectors of a Cluster Distribution Spherical Aperture Cone of Confidence Eigenvectors (length indicates size of eigenvalues. Sum equals the radius of the diagram.) Foliation Psarà Island Greece

27. 26 Eigenvectors of a Great Circle Distribution Eigenvectors (length indicates size of eigenvalues. Sum equals the radius of the diagram. Campo Cecina Alpe Apuane Italy

28. 27 the circular aperture (  ): From this we derive a measure for the length of a partial great circle. We call this measure Eigenvalues of Partial Great Circles

29. 28 Examples for Partial Great Circles Punta Bianca Gronda Ponte Stazzemese Forno Alpe Apuane, Italy Punta Bianca Gronda heavy lines = circular aperture

30. 29 2-Cluster-Distributions

31. 30 Eigenvalues and –vectors of typical distributions

32. 31 The Woodcock-Diagram Girdle : 0 < m < 1 Girdle : 0 < m < 1 Cluster: 1 < m < 8 Umgezeichnet nach Woodcock, 1977