ize geological structures. A keyword search of
GEOREF, a geological reference data base in July
2003 using the exact phrase “differential geometry” uncovered only fourteen references from
1963 to 2003. Of these ten are applications to
other geoscience subjects such as gravity and
geodesy, seismology, hydrogeology, and paleontology. The four remaining consider the threedimensional geometry of folds, the deformation
of Earth’s surface due to neotectonic crustal
motion (Zakarevicius, 2000; Grachev et al., 2001),
and the normal curvature of geological surfaces
(Bergbauer and Pollard, 2003). A search on the
phrase “Gaussian curvature” again uncovered
fourteen references including three on the estimation of strain and the prediction of fractures
within folds (Lisle, 1994, 2000; Ozkaya, 2002).
This section includes examples that provide
insights into how differential geometry can be
applied to problems in structural geology. These
examples are works in progress and we expect
more details to emerge during on-going studies.
None-the-less we hope that these examples will
encourage others to apply differential geometry
to structural problems.
3.3.1 Characterizing the shapes of
lineations on discrete surfaces
Lineations are found on discrete geological surfaces such as faults and intrusive contacts (Fig.
3.3). These superficial lineations typically are
aligned on an exposure such that multiple measurements of the orientations of linear elements
would have a standard deviation of few degrees.
However, faults and intrusive contacts may pass
through rocks with different mechanical properties, they may be influenced mechanically by adjacent faults or intrusions, and they may be curved
surfaces. Thus one should expect superficial lineations to vary in orientation over the surfaces on
which they are found. On a fault, for example,
slickenlines should form a systematic pattern that
reflects the relative motion of the two surfaces
during frictional sliding. Of course the direction
of relative motion at a point may change as a fault
develops, leading to overprinting of slickenlines
with different orientations. Where overprinting is
not an issue, one should be able to define a set of
three-dimensional curves lying on the fault
surface that are everywhere parallel to the local
direction of relative motion, and then use the curvature and torsion to characterize the shapes of
these curves.
To illustrate the fact that slickenlines do vary
systematically with position on a fault, and to
appreciate some of the challenges inherent to the
investigation of superficial lineations we turn to a
data set from the Chimney Rock fault array (Fig.
2.29). The four sets of faults in this region are displayed on the structure contour map constructed
on the base of the Carmel Formation. Note, for
example, that individual contours on this map are
truncated by the Frenchman Fault. When traced
to the north across the fault the sense of step is
consistently to the east. However, the magnitude
of the step decreases toward both terminations of
this fault. This change in step magnitude suggests
that the magnitude of the slip decreases from the
mid-section of the Frenchman Fault toward the
terminations and, indeed, the slip must go to zero
at the terminations by definition.
The distribution of dip slip is plotted versus
position along the trace of the Blueberry Fault in
Fig. 2.31. Note how the magnitude of the dip slip
(gray boxes) increases from zero at the eastern termination to more than 30 m near the middle of
the fault. The slip distribution is not continuous,
but jumps abruptly where members of other fault
sets intersect the Blueberry Fault. The distribution of slickenline rake (black diamonds) is
plotted versus position along the trace of the
Blueberry Fault. The rakes are approximately 90Њ
(down dip) near the northeastern and southwestern terminations of the fault, but decrease more
or less systematically to about 70Њ as one
approaches the intersections with the La Sal and
Little Faults. Across these faults the rakes abruptly
increase. The rakes are systematically greater than
90Њ (inclined toward the east) between the intersections with the La Sal and Little Faults, and less
than 90Њ (inclined toward the west) on the distal
sides of these intersections. These changes reflect
the mechanical interaction of the faults and are
consistent with elastic models of this interaction
(Maerten, 2000).
Exposure of the faults at Chimney Rock are
adequate to document the lateral variation in
rake of the slickenlines over a distance of almost
3.3 APPLICATIONS TO STRUCTURAL GEOLOGY
115
GEOREF, a geological reference data base in July
2003 using the exact phrase “differential geometry” uncovered only fourteen references from
1963 to 2003. Of these ten are applications to
other geoscience subjects such as gravity and
geodesy, seismology, hydrogeology, and paleontology. The four remaining consider the threedimensional geometry of folds, the deformation
of Earth’s surface due to neotectonic crustal
motion (Zakarevicius, 2000; Grachev et al., 2001),
and the normal curvature of geological surfaces
(Bergbauer and Pollard, 2003). A search on the
phrase “Gaussian curvature” again uncovered
fourteen references including three on the estimation of strain and the prediction of fractures
within folds (Lisle, 1994, 2000; Ozkaya, 2002).
This section includes examples that provide
insights into how differential geometry can be
applied to problems in structural geology. These
examples are works in progress and we expect
more details to emerge during on-going studies.
None-the-less we hope that these examples will
encourage others to apply differential geometry
to structural problems.
3.3.1 Characterizing the shapes of
lineations on discrete surfaces
Lineations are found on discrete geological surfaces such as faults and intrusive contacts (Fig.
3.3). These superficial lineations typically are
aligned on an exposure such that multiple measurements of the orientations of linear elements
would have a standard deviation of few degrees.
However, faults and intrusive contacts may pass
through rocks with different mechanical properties, they may be influenced mechanically by adjacent faults or intrusions, and they may be curved
surfaces. Thus one should expect superficial lineations to vary in orientation over the surfaces on
which they are found. On a fault, for example,
slickenlines should form a systematic pattern that
reflects the relative motion of the two surfaces
during frictional sliding. Of course the direction
of relative motion at a point may change as a fault
develops, leading to overprinting of slickenlines
with different orientations. Where overprinting is
not an issue, one should be able to define a set of
three-dimensional curves lying on the fault
surface that are everywhere parallel to the local
direction of relative motion, and then use the curvature and torsion to characterize the shapes of
these curves.
To illustrate the fact that slickenlines do vary
systematically with position on a fault, and to
appreciate some of the challenges inherent to the
investigation of superficial lineations we turn to a
data set from the Chimney Rock fault array (Fig.
2.29). The four sets of faults in this region are displayed on the structure contour map constructed
on the base of the Carmel Formation. Note, for
example, that individual contours on this map are
truncated by the Frenchman Fault. When traced
to the north across the fault the sense of step is
consistently to the east. However, the magnitude
of the step decreases toward both terminations of
this fault. This change in step magnitude suggests
that the magnitude of the slip decreases from the
mid-section of the Frenchman Fault toward the
terminations and, indeed, the slip must go to zero
at the terminations by definition.
The distribution of dip slip is plotted versus
position along the trace of the Blueberry Fault in
Fig. 2.31. Note how the magnitude of the dip slip
(gray boxes) increases from zero at the eastern termination to more than 30 m near the middle of
the fault. The slip distribution is not continuous,
but jumps abruptly where members of other fault
sets intersect the Blueberry Fault. The distribution of slickenline rake (black diamonds) is
plotted versus position along the trace of the
Blueberry Fault. The rakes are approximately 90Њ
(down dip) near the northeastern and southwestern terminations of the fault, but decrease more
or less systematically to about 70Њ as one
approaches the intersections with the La Sal and
Little Faults. Across these faults the rakes abruptly
increase. The rakes are systematically greater than
90Њ (inclined toward the east) between the intersections with the La Sal and Little Faults, and less
than 90Њ (inclined toward the west) on the distal
sides of these intersections. These changes reflect
the mechanical interaction of the faults and are
consistent with elastic models of this interaction
(Maerten, 2000).
Exposure of the faults at Chimney Rock are
adequate to document the lateral variation in
rake of the slickenlines over a distance of almost
3.3 APPLICATIONS TO STRUCTURAL GEOLOGY
115