Statistical Analysis of Directional Features - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 276

Keywords Directed and undirected lines • Unit vector fields • Bjorne formation
paleodelta • San Stefano quartzphyllites • TRANSALP profile • Ductile extrusion
model • Pustertal tectonites • Italian dolomites • Adria microplate • Defereggen
“Schlinge”
8.1 Directed and Undirected Lines
Various geological attributes may be approximated by lines or planes. Measuring
them results in “angular data” consisting of azimuths for lines in the horizontal
plane or azimuths and dips for lines in 3-D space. Although a plane usually is
represented by its strike and dip, it is fully determined by the line perpendicular to
it, and statistical analyses of data sets for lines and planes are analogous. Examples
of angular data are strike and dip of bedding, banding and planes of schistosity,
cleavage, fractures or faults. Azimuth readings with or without dip are widely used
for sedimentary features such as axes of elongated pebbles, ripple marks, foresets of
cross-bedding and indicators of turbidity flow directions (sole markings). Then
there are the B-lineations in tectonites; problems at the microscopic level include
that of finding the preferred orientation of crystals in a matrix (e.g., quartz axes in
petrofabrics). Another example is the direction of magnetization in rocks. Statistical
theory for the treatment of unit vectors in 2-D (Fisher 1993 ) and 3-D (Fisher
et al. 1987 ) is well developed.
8.1.1 Doubling the Angle
It is useful to plot angular data sets under study in a diagram before statistical
analysis is attempted. Azimuth readings may be plotted on various types of rose
diagrams (Potter and Pettijohn 1963 ). If the lines are directed, the azimuths can be
plotted from the center of a circle and data within the same class intervals may be
aggregated. If the lines are undirected, a rose diagram with axial symmetry may be
used such as the one shown in Fig. 8.1 . The method of doubling the angle then is
used to estimate the mean azimuth (Krumbein 1939 ). The average is taken after
doubling the angles and dividing the resulting mean angle by two. In Agterberg
( 1974 ) problems of this type and their solutions are discussed in more detail.
Krumbein's solution is identical to fitting a major axis to the points of intersection
of the original measurements with a circle. Axial symmetry if preserved when
the major axis is constructed. This method can be extended to the situation of
undirected lines in 3-D space. Then the first principal component or dominant
eigenvector is computed using the coordinates of the intersection points of
the lines, which pass through the center of a sphere, and the surface of the sphere
( cf . Sect. 1.4.1 ).