entations over the entire net. In other words this
population has no preferred orientation. As the
counter is randomly positioned on the net the
number of points within the counter, n, varies.
The distribution of n is a binomial distribution
because the counting circle divides the population into two mutually exclusive sets: those inside
the counter and those outside. On average the
number of points in the counter will be fN
because the area of the counter is a fraction f of
the area of the net. The mean, m, and standard
deviation, s, of this binomial distribution are
(Krumbein and Graybill, 1965, p. 102):
(2.94)
Recall that the standard deviation is a measure of
the spread of the distribution about the mean. As
the counter gets very small, f → 0, m → 0, and s →
( fN)
1/2 . As the counter approaches the size of the
net, f → 1, m → N, and s → 0.
It is recommended (Kamb, 1959a) that the
radius of the counter be chosen such that m ϭ 3s.
In other words, for the population with no preferred orientation, the number of points within
the counter, on average, would be three times the
standard deviation. The fabric diagrams can be
contoured at values of 0, 2s, 4s, 6s, etc. Contours
drawn from counts using this prescription are
very smooth. If such large counts cluster in one
region of the fabric diagram and produce closed
contours with values greater than 3s, one can
interpret the population as having a preferred orientation. Substituting the expressions for the
mean and standard deviation from (2.94) into the
condition m ϭ 3s and solving for f, we find:
(2.95)
Note that a 1% counter area, sometimes chosen
arbitrarily for the contouring of fabric diagrams,
corresponds to N ϭ 891. This is an unusually large
number of points for fabric studies. Implementations using a 1% area and fewer than 891 points
are likely to produce irregular contours that have
no statistical significance because the counter
area is too small.
Substituting the ratio of areas for f in the previous equation, and solving for the radius of the
counter, r, we have:
f ϭ
9
N ϩ 9
m ϭ f N, s ϭ [ fN(1 Ϫ f )] 1ր2
(2.96)
Given the radius of the Schmidt net, R, and the
number of points, N, in the population (2.96) provides the radius of the counter and the distance
between the intersection points of the square grid
that overlays the net. Using the counter and grid
so defined one can construct the orientation
density diagram to display graphically the statistical significance of a data set containing the orientations of a linear fabric or the normals to a
planar fabric.
2.3.4 Field and model angles and
analytical orientation statistics
We turn now to the relationships between geographic angles and coordinates, and the angles and
coordinates used in data analysis and model construction. Recall that the orientations of planar
and linear elements that approximate geological
structures are measured in the field using two geographic angles, the azimuth, ␣, and the inclination, . Different terms are associated with the
azimuth of strike (␣ s ), dip (␣ d ), plunge (␣ p ), and
normal (␣ n ), and appropriate subscripts distinguish these. Similarly, different subscripts for the
inclination angle, , distinguish the dip ( d ),
plunge ( p ), and plunge of the normal ( n ). For any
planar element the azimuth and plunge of the
normal line (pole) may be used to specify the orientation, so in fact we only need to consider how line
segments are oriented in three-dimensional space
to account for the orientations of all linear and
planar elements that approximate geological structures. For data analysis and model computations it
is convenient to describe the orientation of any line
segment using three direction angles (␣ x , ␣ y , ␣ z ) that
relate the line to a Cartesian coordinate system,
rather than the geographic system. Here we introduce the relationships that transform field data in
the geographic coordinate system to a Cartesian
system using these direction angles. The textbook
by Groshong (1999) provides additional discussion
of these and other techniques of three-dimensional
geometry as used in structural geology.
Consider the orientation of the line segment OP
relative to the orthogonal geographic coordinate
system composed of the axes east, north, and up
(Fig. 2.26a). Regardless of their specific geological
r ϭ
3R
√N ϩ 9
2.3 ORIENTATIONS OF STRUCTURAL ELEMENTS
67
population has no preferred orientation. As the
counter is randomly positioned on the net the
number of points within the counter, n, varies.
The distribution of n is a binomial distribution
because the counting circle divides the population into two mutually exclusive sets: those inside
the counter and those outside. On average the
number of points in the counter will be fN
because the area of the counter is a fraction f of
the area of the net. The mean, m, and standard
deviation, s, of this binomial distribution are
(Krumbein and Graybill, 1965, p. 102):
(2.94)
Recall that the standard deviation is a measure of
the spread of the distribution about the mean. As
the counter gets very small, f → 0, m → 0, and s →
( fN)
1/2 . As the counter approaches the size of the
net, f → 1, m → N, and s → 0.
It is recommended (Kamb, 1959a) that the
radius of the counter be chosen such that m ϭ 3s.
In other words, for the population with no preferred orientation, the number of points within
the counter, on average, would be three times the
standard deviation. The fabric diagrams can be
contoured at values of 0, 2s, 4s, 6s, etc. Contours
drawn from counts using this prescription are
very smooth. If such large counts cluster in one
region of the fabric diagram and produce closed
contours with values greater than 3s, one can
interpret the population as having a preferred orientation. Substituting the expressions for the
mean and standard deviation from (2.94) into the
condition m ϭ 3s and solving for f, we find:
(2.95)
Note that a 1% counter area, sometimes chosen
arbitrarily for the contouring of fabric diagrams,
corresponds to N ϭ 891. This is an unusually large
number of points for fabric studies. Implementations using a 1% area and fewer than 891 points
are likely to produce irregular contours that have
no statistical significance because the counter
area is too small.
Substituting the ratio of areas for f in the previous equation, and solving for the radius of the
counter, r, we have:
f ϭ
9
N ϩ 9
m ϭ f N, s ϭ [ fN(1 Ϫ f )] 1ր2
(2.96)
Given the radius of the Schmidt net, R, and the
number of points, N, in the population (2.96) provides the radius of the counter and the distance
between the intersection points of the square grid
that overlays the net. Using the counter and grid
so defined one can construct the orientation
density diagram to display graphically the statistical significance of a data set containing the orientations of a linear fabric or the normals to a
planar fabric.
2.3.4 Field and model angles and
analytical orientation statistics
We turn now to the relationships between geographic angles and coordinates, and the angles and
coordinates used in data analysis and model construction. Recall that the orientations of planar
and linear elements that approximate geological
structures are measured in the field using two geographic angles, the azimuth, ␣, and the inclination, . Different terms are associated with the
azimuth of strike (␣ s ), dip (␣ d ), plunge (␣ p ), and
normal (␣ n ), and appropriate subscripts distinguish these. Similarly, different subscripts for the
inclination angle, , distinguish the dip ( d ),
plunge ( p ), and plunge of the normal ( n ). For any
planar element the azimuth and plunge of the
normal line (pole) may be used to specify the orientation, so in fact we only need to consider how line
segments are oriented in three-dimensional space
to account for the orientations of all linear and
planar elements that approximate geological structures. For data analysis and model computations it
is convenient to describe the orientation of any line
segment using three direction angles (␣ x , ␣ y , ␣ z ) that
relate the line to a Cartesian coordinate system,
rather than the geographic system. Here we introduce the relationships that transform field data in
the geographic coordinate system to a Cartesian
system using these direction angles. The textbook
by Groshong (1999) provides additional discussion
of these and other techniques of three-dimensional
geometry as used in structural geology.
Consider the orientation of the line segment OP
relative to the orthogonal geographic coordinate
system composed of the axes east, north, and up
(Fig. 2.26a). Regardless of their specific geological
r ϭ
3R
√N ϩ 9
2.3 ORIENTATIONS OF STRUCTURAL ELEMENTS
67