9a. Projecao Estereografica Theory

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Graphical presentation of structural data sets
 (Stereonets, Rose diagrams)
2) Statistics of orientation data
3) Analysis of palaeostress data
4) Fold shape analysis
This programme was developed to be used for the following methods of structural geology:
5) Strain analysis
6) Structural maps
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Graphical Presentation
of Orientation Data
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Two-Dimensional Data: 
Rose Diagrams
(Data from the digital elevation model of the Bushveld)
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Rose Diagrams with overlapping Sectors from the Bushveld
(Digital elevation model)
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Methods of measuring structural data (planes and lineations)
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Tipos de notação
Americana
Em trama (mergulho ou dip-direction)
Mão direita
Mão esquerda
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1. StereographicProjection
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Equal Angle Projection
Characteristics:
This net is called
„WULFF NET“
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Lambert‘s Projection
a
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Equal Area Projection
Characteristics:
This net is called
„Schmidt Net“
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How to project planes
Stereographic projection
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Projection 
of a plane
Normal to the 
Plane:
aN = a + 180
jN = 90 - j 
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Presentation of Data in the Equal Area Projection
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Contouring Methods
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Data Set of 30 Linears
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Overlay of Counting Grid and Data 
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Counting
The number of data
in each calotte is counted
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Contour Lines
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The Distance Method 
(Scalar Product Method) 
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Contour Diagrams
Synopsis of bedding poles of the Bushveld
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Statistics of Orientation Data
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This is obviously wrong.
 Statistical measures of orientation data can only be found by application of vector algebra.
It is not possible to apply linear statistics to orientation data.
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What is the difference between orientation data and other structural data? 
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with v = 1 we get: 
 Vx = cos   cos 
 Vy = sin   cos 
 Vz = sin 
Transformation of a/j into Cartesian Coordinates (vx, vy, vz)
c= cosj
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Measures Derived from Addition of Vectors (Orientation Data):
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Centre of Gravity vs. Mode
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Vector sums of orientation data: 
If the data were real vectors with polarity, then they would show max. isotropy in a random distribution.
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Problems of axial data:
If the angle between two lineations is > 90°,
the reverse direction must be added.
Structural data, like fold axes or poles to planes, have no polarity.
They are not unipolar vectors, but bipolar axes.
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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)? 
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From these limits a measure for the
 
 Degree of Preferred 
 Orientation (R%) 
 
can be found:
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Distributions:
The Spherical Normal Distribution (unimodal distribution)
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Probability Measures:
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The Cone of Confidence
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Is there a geometric equivalent of the concentration parameter? 
Isotropic distribution in
a small circle with apical
angle w
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Statistical parameters:
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Examples for Spherical Aperture and Cone of Confidence
Yellow: Spherical aperture
Green: Cone of confidence
Confidence = 99%
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Common Distributions
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Axes of Inertia:
Amax, Amedium and Amin are the principal axes
of the inertia matrix (Lagrange). They are the
eigenvectors of this matrix. 
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Eigenvalues and Eigenvectors
Eigenvalue = 1
(no change of length
in horizontal direction)
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Calculation of Eigenvalues
Transformation-
 Matrix:
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Calculation of Eigenvectors
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The Orientation Matrix (Tensor) and it´s Eigenvalues:
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Orientation Ellipsoid
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Eigenvectors of a Cluster Distribution
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The Eigenvalues of Cluster-Distributions
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 Eigenvectors of a Great Circle Distribution
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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 from the Bushveld
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The Woodcock Diagram
Umgezeichnet nach Woodcock, 1977
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Woodcock Diagram
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Vollmer Diagram
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Woodcock Diagram of Bedding of Bushveld
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Vollmer Diagram of Bedding of Bushveld
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Significant Distributions
Redrawn after Woodcock & Naylor, 1983
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Non-Random Distribution
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Eigenvalues and –vectors of Typical Distributions
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Submaxima in a data set
Are they signifi-
cantly separated?
Crocodile River Dome F2
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Cluster Finding
test angle = 10° 
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Three Clusters Found
test angle = 20°, multiples of random = 2
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Crocodile R. Dome, F2
Multiple of random = 1
Multiple of random = 6
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Paleostress Data
The aim of paleostress analysis is to reconstruct 
the stress ellipsoid and it's orientation in space 
which existed during a mountain building process.
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Displacement Data
Displacement data:
1) azimuth and dip of
the fault
2) azimuth and dip of
the striation
3) polarity of striation
From the orientation of the slickenside
striations we receive the direction of the
maximum resolved shear stress (MRSS).
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Displacement systems
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Presentation of Displacement Data
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How can we find a palaeostress tensor from these data?
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Without further assumptions
only a "reduced stress tensor"
can be derived from field observations
The reduced stress tensor consists of 4 components:
The directions of the three principal stresses s1, s2, s3,
and a shape factor of the stress ellipsoid (R).
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Focal Plane Analysis of Sumatra-Earthquakes
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The Dihedra Method
The projection globe is 
divided in
 compressional and
extensional dihedra
Find the directions 
of principal stresses
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Combination of Data from 2 Faults
s1 lies in the 
intersection
of compressional
dihedra
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Computer Diagram of Paleostress
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Other Counting Methods
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Stress Fields in the Southern Bohemian Massiv
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Displacement Data from Pretoria
Tom Jenkings Drive
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Displacement Data in a Hoeppener Diagram
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The Shape Factor of the Stress Ellipsoid
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Relation between the Shape Factor (R) and the MRSS
Bott's formula:
l, m, n = direction cosines of the normal
to the slickenside plane
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Hoeppener Diagrams for Theoretical Striation Patterns 
These lines are the trajectories for MRSS directions
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Theoretical Striation Patterns on an Even Distribution of 315 Slickenside Planes
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Estimation of the Shape Factor (R)
0-10: eigenvalues of theoretical 
striations on the measured fault
planes.
Red dot: eigenvalues of 
the measured striations
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Plot of a Hoeppener Diagram
Poles to faults,
displacement directions
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Fold Shape Analysis
1) Isogons
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Construction of Isogons
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Dip Isogons
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Import of a Bitmap
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Draw Dip Isogons
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Fold Shape Analysis
2)The Ramsay-Classification of Folds
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Fold Shapes and Isogons
a
d/T0
Class 1A
Class 1B
Class 1C
Class2
Class 3
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1. Import of a Bitmap
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Rotating of the Bitmap
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3. Draw Isogons
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4. Plot the Graph
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Fold Shape Analysis
3) Fourier Analysis
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Comparison with Harmonic Waves
Classification with Fourier coefficients:
Assuming the fold is part of a harmonic wave,
this wave can be described by superimposed sinus- 
and
cosinus waves.
f (ab) = a0 + a1cosa + a2cos2a + a3cos3a +…
 + b1sina + b2sin2a + b3sin3a +…
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Reduction of Coefficients
1.) In natural folds we can always put the origin of the
 coordinate sytem at an inflection point of the folds,
 i.e. a0 = 0 
 and a1cosa + a2cos2a + a3cos3a +…. = 0
2.) For a complete discription of the shape we need only
 a quarter fold. That means all even coefficients
 become 0.
The remaining coefficients are:
 b1, b3, b5,……
b5 and higher are so small that they can be neglected.
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Addition of b1 and -b3
– b3 : chevron folds
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Addition b1 + b3
+b3 : box folds
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Determination of b1and b3 from a Quarter Fold
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b1/b3-Diagram
box
 folds
sinus
folds
chevron
 folds
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Strain Analysis
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Nearest Neighbours
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Fry Method
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Examples for the Fry Method
Photomicrograph of deformed ooids
From Ramsay & Huber 1987
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Fry Method (continued)
Mark the centres of the ooids
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Fry Method (Result)
Done with Fabric7 software (Wallbrecher 2006)
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Projection Method (Panozzo 1984)
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Example
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Drawing of Polygons and their Centres
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Evaluation
Ellipticity = 1.30
Q = 141°
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The Rf/Phi-Method (Lisle 1985)
Deformation analysis
with elliptical markers
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The finite ellipticity of each pebble (Rf) results from the initial ellipticity (Ri) and deformation 
The ellipticity of the strain ellipse
 is calulated.
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Evaluation
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Rf/Phi-Diagram
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The newest version of the program 
can be downloaded from
www.geolsoft.com
Thank you very much for your attention!
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