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Geosphere
Quantitative analysis and visualization of nonplanar fault surfaces using
terrestrial laser scanning (LIDAR) −−The Arkitsa fault, central Greece, as a
case study
R.R. Jones, S. Kokkalas and K.J.W. McCaffrey
Geosphere 2009;5;465-482
doi: 10.1130/GES00216.1
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© 2009 Geological Society of America
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Quantitative analysis and visualization of nonplanar fault surfaces
using terrestrial laser scanning (LIDAR)—The Arkitsa fault,
central Greece, as a case study
R.R. Jones*
Geospatial Research Ltd., Department of Earth Sciences, University of Durham, Durham DH1 3LE, UK
Reactivation Research Group, Department of Earth Sciences, University of Durham, Durham DH1 3LE, UK
S. Kokkalas*
University of Patras, Department of Geology, Laboratory of Structural Geology, 26500 Patras, Greece
K.J.W. McCaffrey*
Reactivation Research Group, Department of Earth Sciences, University of Durham, Durham DH1 3LE, UK
ABSTRACT
Many fault surfaces are noticeably nonplanar, often containing irregular asperities and
more regular corrugations and open warping.
Terrestrial laser scanning (light detection and
ranging, LIDAR) is a powerful and versatile
tool that is highly suitable for acquisition of
very detailed, precise measurements of slipsurface geometry from well-exposed faults.
Quantitative analysis of the LIDAR data,
combined with three-dimensional visualization software, allows the spatial variation in
various geometrical properties across the fault
surface to be clearly shown. Plotting the variation in distance of points from the mean fault
plane is an effective way to identify culminations and depressions on the fault surface.
Plots showing the spatial variation of surface
orientation are useful in highlighting corrugations and warps of different wavelengths,
as well as cross faults and fault bifurcations.
Analysis of different curvature properties,
including normal and Gaussian curvature,
provides the best plots for quantitative measurements of the geometry of corrugations
and folds. However, curvature analysis is
highly scale dependent, so requires careful filtering and smoothing of the data to be able to
analyze structures at a given wavelength.
Three well-exposed fault panels from the
Arkitsa fault zone in central Greece were
scanned and analyzed in detail. Each panel
is markedly nonplanar, and shows significant
variation in surface orientation, with spreads
of ~±20°–25° in strike and ±10° in dip. Most of
the variation in orientation reflects decimeterand meter-scale corrugations and longer
wavelength warps of the fault panels. Average
wavelengths of corrugations measured on two
of the panels are 4.04 m and 4.43 m. Whereas
the orientations of the three fault surfaces
show significant variations, the orientations
of fault striae are very similar between the
panels, and are tightly clustered within each
panel. Fault-slip analysis from each panel
shows that the local stress field is consistent
with the regional velocity field derived from
global positioning system data. The oblique
slip lineations observed on the fault panels
represent a combination of two contemporaneous strain components: north-northeast–
south-southwest extension across the Gulf of
Evia and sinistral strike slip along the westnorthwest–east-southeast Arkitsa fault zone.
fault reactivation, fault seal, and fluid flow in
faulted reservoirs and aquifers. Despite this, most
modeling of fault dynamics assumes idealized
planar geometries, and no failure criteria have yet
been developed for curved faults.
One of the main limitations in incorporating
realistic fault geometries into dynamic fault models is the paucity of quantitative studies of threedimensional geometry of natural fault zones. In
this paper we present detailed data acquired using
terrestrial laser scanning (TLS), also known as
ground-based LIDAR (light detection and ranging), from well-exposed fault surfaces in an
area of active tectonism, the Aegean. The main
purpose of this paper is to describe methods to
process, visualize, and analyze these kinds of
large geospatial data set, including techniques to
highlight spatial variation in different geometric
parameters across a fault surface.
INTRODUCTION
The study site for this work is the area of spectacularly exposed fault surfaces of the Arkitsa
fault zone, reported by Jackson and McKenzie
(1999), located on the southern side of the northern Gulf of Evia in central Greece (Fig. 1A).
The area is currently undergoing active northsouth to northeast-southwest regional extension,
with rates on the order of 1–2 mm/yr (Clarke at
al. 1998), in response to the complex tectonic
interplay between subduction beneath the Hellenic Arc, backarc extension in the Aegean, and
the westward movement of the Anatolian plate
Many fault surfaces commonly observed in
outcrop are markedly nonplanar, and often display
corrugations, asperities, and longer wavelength
low-amplitude curvature. Such irregularities can
have major effects on the mechanical behavior of
the fault during slip, and a better understanding
of the interplay between irregular geometry, fault
kinematics, and dynamics is an important step in
improving our knowledge of a variety of geological processes, including earthquake nucleation,
REGIONAL SETTING
*E-mails: Jones: [email protected]; Kokkalas: [email protected]; McCaffrey: [email protected].
Geosphere; December 2009; v. 5; no. 6; p. 465–482; doi: 10.1130/GES00216.1; 10 figures; 1 table; 3 animations.
For permission to copy, contact [email protected]
© 2009 Geological Society of America
465
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Jones et al.
A
height
(meters asl)
2500
1000
500
300
39°N
200
Ae
Almyros-Spe
rchios graben
KVF
Fig.1B
AK
KFZ
F
N.
ARFZ
Gu
lf
of
50
Ev
i
0
S. G u
Gu
lf o
f
Co
rin
100
a
ATF
gean
projection:
WGS84
lf of
Ev
ia
th
38°N
Athens
Corinth
23°E
22°E
B
22°55'
24°E
23°00'
23°05'
Logos
Fig. 2
Arkit
sa F
ault Zone
38°45'
NORTH
Arkitsa
Gu
lf
of
Livanates
At
es
nat
Liva ault
F
Megaplatanos
al
an
ti
38°40'
Zeli
Atalanti
Ata
lant
i
Fault Zon
e
Tragana
10 km
466
Geosphere, December 2009
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LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece
Figure 1. (A) Satellite topography data
(Shuttle Radar Topography Mission)
from central Greece, showing location of
the study area on the southern side of the
Gulf of Evia. KVF—Kamena Vourla fault;
AKF—Agios Konstantinos fault; ARFZ—
Arkitsa fault zone; KFZ—Kallidromo fault
zone; ATF—Atalanti fault; asl—above sea
level. (B) Tectonic map of the area around
Arkitsa. Fault picks are based on detailed
aerial photo analysis and ground-truthing.
(e.g., Dewey et al., 1986; Doutsos and Kokkalas, 2001; Kokkalas et al., 2006). In central
Greece, much of the extensional strain is localized into a number of west-northwest–eastsoutheast–trending grabens, which are mostly
characterized by complex geometries in map
view (Fig. 1B), and a high degree of segmentation along strike (Doutsos and Poulimenos,
1992; Ganas et al., 1998; Kokkalas et al., 2006).
Most of the major normal faults in the region are
thought to have been active in the Pleistocene,
although the relative time of their activity is not
well constrained. Some historical earthquakes
have been recorded in the area, but none can be
directly associated with the Arkitsa fault zone,
or with any other fault with certainty, other
than the 1894 rupture events of Atalanti fault
(Ambraseys and Jackson, 1990).
The Arkitsa fault zone together with the Agios
Konstantinos and Kamena Vourla faults form a
west-northwest–east-southeast, left-stepping,
north-dipping fault margin that links with the
southern margin of the Almyros-Sperchios
graben (Fig. 1). The Arkitsa fault zone, with
a length of ~10 km, separates Late Triassic–
Jurassic platform carbonates in the footwall
from lower Pliocene–Quaternary sediments in
the hanging wall. The amount of throw apparently varies along strike, typical of highly segmented fault zones. The minimum throw is estimated to be ~500–600 m, based on the thickness
of Neogene–Holocene sediments in the hanging
wall, plus the scarp height and topographic relief
of the footwall block. This allows a slip rate of
0.2–0.3 mm/yr to be estimated for the Arkitsa
fault zone, given that activity on faults of the
Evia rift zone started within the past 2–3 m.y.
(Ganas et al., 1998). Slip rates for the adjacent
Atalanti fault show similar values, on the order
of 0.27–0.4 mm/yr (Ganas et al., 1998). In addition to the overview given in Jackson and McKenzie (1999), further description was given by
Kokkalas et al. (2007) and Papanikolaou and
Royden (2007).
There is evidence of Holocene seismic activity along the fault, based on the geomorphological expression of the Arkitsa region, as well
as the back-tilted terraces on the hanging-wall
block of the Arkitsa fault, and the exposure of an
~1-m-high scarp of unweathered fault surface
that existed prior to quarrying. In addition, the
coastline in the region shows evidence of Holocene uplift and subsidence that is controlled by
the motion and location of the faults (Roberts
and Jackson, 1991; Stiros et al., 1992).
The Arkitsa fault is best exposed in an outcrop 450 m south of the main Athens-Lamia
national road (Fig. 2), where recent quarrying activity during the past 20 yr has removed
hanging-wall colluvium to reveal clean, fresh
exposures of the upper 65–70 m of three large
fault panels. These fault panels form the northern scarp edge of a footwall block that rises to
255 m above sea level (a.s.l.). The hanging-wall
via
Gulf of E
n
r
e
h
t
r
No
to L
amia
oad
National R
to Athens
500 m
Figure 2. Extent of the terrestrial laser scan data coverage shown in pink, superimposed on an aerial photograph. Yellow circles
mark the position of the 10 scan stations used in this study.
Geosphere, December 2009
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Jones et al.
sediments form the low-lying plain at 70 m a.s.l.
We have recorded the detailed geometry of the
fault panels using terrestrial laser scanning, and
this paper presents a description of the acquisition, visualization, and preliminary analysis of
the laser-scan data.
DATA ACQUISITION
A detailed survey of the exposed fault planes
and surrounding footwall was carried out in early
December 2005 using terrestrial laser scanning
(TLS), also commonly termed ground-based
LIDAR (light detection and ranging). TLS is a
very rapid surveying method in which the precise geometry of the outcrop can be measured
in detail by recording the three-dimensional
(3D) position of many millions of points across
the surface of an exposure. Since the method is
nonpenetrative, the raw output is a surface 3D
dataset (Dogan Seber, 2007, personal commun.;
Jones et al., 2008), containing no subsurface
data from within the outcrop. The high resolution provided by TLS allows outcrops to be
analyzed quantitatively in great detail, leading
to increased use in geosciences (e.g., Ahlgren et
al., 2002; Jones et al., 2004; Bellian et al., 2005;
McCaffrey et al., 2005; Pringle et al., 2006; and
many others. See also Wawrzyniec et al., 2007,
and associated papers in the Geosphere themed
issue “Unlocking 3D earth systems—Harnessing new digital technologies to revolutionize
multi-scale geological models”).
The equipment we used at Arkitsa was a Riegl
LMS-Z420i (Fig. 3) (LMS—laser measurement
system), which is a long-range scanner with an
effective range of up to 1000 m, a maximum resolution (i.e., angular acuity) of 0.004° (equivalent
to 7 mm at 100 m), and a specified longitudinal
precision of 5 cm. The long range of the scanner
was essential for enabling us to capture the morphology of the entire footwall block along the
length of the hillside, and together with its highresolution color capability and very rapid speed
of data acquisition (12,000 points/s), this made
the LMS-Z420i an ideal instrument for this
project. However, this model of scanner may be
less suitable for some other kinds of close-range
project, because data acquired when the scanner
is used in normal mode generally contain significantly more noise than shorter range scanners such as the Trimble GS100/GS200/GX200
or Leica HDS3000 (Lichti and Jamtsho 2006;
Renard et al., 2006; Geospatial Research Ltd.,
unpublished tests). Therefore, when using Riegl
LMS scanners (e.g., Z360, Z390, Z420i) for
detailed studies of small-scale surface morphology (e.g., Renard et al., 2006; Sagy et al., 2007),
it is important to improve precision by changing
the mode of operation to acquire multiple scan
468
sequences. This is a straightforward option in
the RiScan Pro software that is used to drive the
LMS-Z420i, and in our experience can increase
longitudinal precision to 10 mm or better.
The fault zone was scanned using a standard
acquisition strategy of taking multiple overlapping scans from a number of different tripod
positions (scan stations) along the length of the
outcrop. This aims to minimize any large gaps
in the data (data shadows) caused by the irregular topography of the outcrop surface. In practice, there will always be some data shadows,
but careful planning of the data acquisition can
help to keep these to a minimum. In this study,
10 scan stations were used along the outcrop
length, with 9 stations sited <75 m from the
outcrop, and 1 station ~500 m away to provide
extra coverage of the footwall hillside above
the fault scarp (Fig. 2). Over a period of 2 days,
25 scans with a total of 75 × 106 points were
acquired from these 10 stations.
Data from each of the scan stations need to
be merged together into a single model. This is
achieved using two methods in combination.
First, a set of highly reflective cylindrical targets is placed carefully in prominent positions
spread around the survey area. Then from each
scan station, all the reflectors that are visible
from that station are scanned at high resolution. By matching the same reflector from different scan stations, all the different scans can
be stitched together. In addition, a differential
global positioning system (dGPS) was used
to locate the positions of the most prominent
reflectors, so that they can be georeferenced to
a global coordinate system. For this we used an
Ashtech Promark 2 in survey mode, capable of
subcentimeter precision if the satellite configuration is suitable (McCaffrey et al., 2005).
DATA PROCESSING
A number of standard processing steps are
needed to combine the raw scan data into a virtual copy of the real outcrop. The virtual outcrop
is then suitable for further geospatial and geological analysis.
ther examples). There are a number of different
solutions available to capture additional color
information. Several scanner manufacturers use
on-board video circuitry mounted inside the
scanner. The Riegl LMS scanners use a separate
off-the-shelf digital single lens reflex camera
(dSLR), which is tightly calibrated so that the
optical properties of the camera-lens combinations are known. The camera is attached to the
top of the scanner using a precision-engineered
mount (Fig. 3). In this way, the pixels in the 2D
images taken with the camera can be accurately
mapped to the 3D point data from the scanner.
Although using a separate dSLR requires more
time and effort during data acquisition, the main
advantage is that this solution provides very
high resolution and faithful color rendition.
Merging and Georeferencing the Scan Data
Scans taken from different scan stations are
merged together based on common reflector
targets and dGPS, as described above. We used
dGPS to measure 16 tie-point positions—12
of which gave subcentimeter spatial precision
following post-processing. These 12 positions
were used for georeferencing (10 reflector positions plus 2 dGPS base stations).
Semiautomatic merge functionality is
included in the RiScan Pro software, which iteratively minimizes the error involved in matching scan stations to each other. For this project
the mean of the average errors (i.e., the mean of
matching all stations to each other) was 5 mm.
Once the colored scans from different scan
stations have been merged, then the data can
be rendered as a virtual outcrop, either as a
large model covering the entire area scanned
(Fig. 5A) and/or as higher resolution models for
detailed analysis of more localized parts of the
outcrop (Fig. 5B; see also Animations 1 and 2
associated with Figs. 5A and 5B, respectively).
Normally, at this stage of our specific work flow,
the data are still in the form of a point cloud
(Fig. 4B; i.e., the outcrop surface has not been
meshed), as this is usually adequate to give a
good impression of the nature of the outcrop
prior to further processing.
Adding High-Definition Color Information
Cleaning the Data
The main functionality of a terrestrial laser
scanner measures only the distance traveled by
the laser pulse, plus the intensity of the beam
that is returned. This provides a grayscale image
of the outcrop (Fig. 4A). In our experience, the
capability to superimpose high-resolution color
information onto the raw point cloud (Fig. 4B)
can be extremely valuable during subsequent
geological interpretation based on the virtual
outcrop (see White and Jones, 2008, for fur-
Geosphere, December 2009
The next step in the processing work flow
generally depends upon the overall purpose of
the project. If the main priority is to recreate
a virtual copy of the outcrop that is as true to
life as possible, then the point cloud data can be
filtered, meshed, and draped with bitmap polygons taken from the scan photos (Figs. 4D–4F).
However, for the work presented in this paper,
the aim has been to use the TLS data to derive
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LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece
Fault panel B (see text)
Nikon D70 dSLR
(with 14 mm, 50 mm & 85 mm lenses)
scanner
window
precision
camera
mount
USB cable
(camera to PC)
tilt mount
Riegl LMS-Z420i
scanner
tripod
24V
battery
ethernet
cable
ruggedized laptop
(Panasonic Toughbook)
protective
carry case
Figure 3. Riegl LMS-Z420i laser scanner at station 1, in action scanning fault panel B (see text).
Geosphere, December 2009
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Jones et al.
A
B
C
D
up
east
north
E
F
Figure 4. Steps in processing of the terrestrial laser scan data. (A) An example area of raw point data, from fault panel B, showing only the intensity of the reflected laser beam. Red scale bars are 10 m and vertical. (B) High-definition RGB (red-greenblue) color applied to the points. (C) Extraneous breccia and vegetation have been removed from the data (this and subsequent
steps are usually best carried out after separate scans have been merged; see text). (D) Data are filtered (decimated) to reduce
the size of the data set and to smooth small wavelength noise in the data. (E) Data are meshed (triangulated) to form a continuous surface. (F) Full RGB color texture from photos draped onto the mesh.
470
Geosphere, December 2009
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LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece
A
top of ridge
g 255m asl
Pa
P
Panel
a ne
nell A
Panel
Pa
a ne
nel B
Panel
Pa
P
a ne
nel C
section
between arrows is
se
ca. 900m long
o ve
ol
e groves
g ro ves 70m
gro
70
7 m asl
as
olive
pylons
p ons are ca.. 350m
pyl
3 m apar t
350
pyl
py
yo
on
n
pylon
pyl
pylon
B
67
6
67.4m
7 .4m
.4m
.4
((down-dip
(d
d ow
own
n-- d i p m
ma
a
ark
rk
ks
marks
ffrom
fr
r om
om w
a ter
at
er rrun-off)
u n -o
un
o ff
f )
water
ope
open
op
en
n te
ttension
e n si
en
sio
on
n
frr a
acc tu
ture
rre
es
es
fractures
crescentic
ccrr e
es
s ce
c ent
e nt
n ic
c
fractures
f r ac
fr
fra
a c tu
t u re
es
m a in
ma
n slip
s lip
sli
p
main
vect
ve
c t or
ct
or
vector
small thrust
faults
Figure 5 (continued on following page). Annotated screenshots from video fly-throughs of the terrestrial laser scan data from Arkitsa (see
Animations 1, 2, and 3). (A) Low-resolution overview, showing the three faults panels A, B, and C that are the main focus of this study. Separate colored scans have been merged and georeferenced, but the data have not yet been filtered, and the fault surfaces are not yet isolated
and stripped of breccia and vegetation. Number of points in view is ~2.5 × 106. Looking south; asl—above sea level. (B) Medium-resolution
data from panel B. The downdip height of the panel is ~60.2 m (distance measured along the slip vector, shown with yellow arrows, is 67.4 m).
Geosphere, December 2009
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Jones et al.
quantitative information about the geometry and
kinematics of the main fault surfaces, and so the
main priority was to extract the scan points from
the large fault panels and isolate them from the
background data. This meant that much effort
was needed to clean the scan data by removing
all traces of vegetation, talus, colluvium, weathered parts of the outcrop, and points reflected
off airborne dust particles (Fig. 4C). This can
be a very laborious process, and automated
approaches to data cleaning have the potential
to save considerable time and effort (Erik Monsen, Schlumberger, 2006–2008, personal communs.).
More critically, it is very important that the data
are smoothed in order to eliminate any irregularities (noise) on scales smaller than the precision of the scanner. In practice, both these aims
can be met by using RiScan Pro’s octree filtering functionality (e.g., Foley et al., 1997), typically using an edge threshold of 10–15 cm. The
filtered result is a heavily decimated point cloud
(Fig. 4D) that can then be meshed (Fig. 4E), and
if necessary also image draped (Fig. 4F). This
produces a processed data set that is ready for
more specific visualization and geometrical and
geological analysis.
Geospatial Analysis and Visualization
Filtering and Smoothing
Before the geometry of the cleaned fault surfaces can be analyzed, it is necessary to filter
the data to reduce the amount of points down
to a more realistic size ready for computational
processing. At this stage, each of the three fault
panels is represented by between 10 × 106 and
16 × 106 points, and for computational efficiency it is ideal to reduce this to <200,000.
Detailed analysis of fault slip surfaces can
lead to better understanding of fault zone processes and fault dynamics. TLS has already
proven to be an effective tool for quantifying surface roughness of faults (Renard et al.,
2006; Sagy et al., 2007). Roughness is a useful parameter to characterize the overall surface
irregularity of rock (or other materials), and can
be measured along 1D sampling lines in spe-
cific directions (e.g., Brown and Scholz, 1985;
Power et al., 1987), or as a 2D property of the
surface as a whole (Renard et al., 2006). In this
paper we present additional methods of analysis
that allow other geometrical properties of fault
planes to be studied. This approach shows that
different types of analyses tend to highlight different geometrical properties of the faults, and
therefore a range of methods should be used in
combination when analyzing the 3D geometry
of fault surfaces.
Unlike roughness, which is typically expressed
as an overall bulk property of a fault surface,
attributes such as orientation, curvature, and
distance from the mean plane are more usefully
analyzed in terms of their spatial variation across
the surface of the fault. After such geometrical
properties are derived for each point on the surface, the calculated values can then be mapped
back onto the fault surface by recoloring each
corresponding point in the virtual outcrop data.
Displaying the colored fault surfaces using 3D
visualization software provides a powerful tool
to highlight spatial variation in fault geometry
(e.g., Fig. 5C).
C
Figure 5 (continued). (C) Enhanced virtual outcrop in which the results of curvature analysis for panel B are projected back onto
the LIDAR (light detection and ranging) data. The colors highlight spatial variation in the magnitude of maximum normal curvature (this is directly comparable to the data in Fig. 9B, but uses different color mapping). See text and Appendix for derivation of
curvature values.
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LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece
Spatial Variation in Perpendicular Distance
from the Mean Plane
This is a simple method to highlight the
deviation of a fault surface from a plane, in this
case the mean plane through all measured points
(nodes) on the fault. This method was used by
Renard et al. (2006) and Sagy et al. (2007) to
visualize fault irregularities. There are a number of different ways to calculate the orientation of the mean plane, including 3D regression
methods and analysis of the moment of inertia
(Fernández, 2005). A useful alternative is to
use vector calculus to derive the orientation of
the mean of all node normals. This approach is
efficient, since we need to calculate node normals as an intermediate step for other types of
analysis (see following for more details about
the calculation of node normals and the mean
plane). Once the orientation of the mean plane is
known, 3D trigonometry can be used to measure
the perpendicular distance of each point from
the mean plane. The values for perpendicular
distance are then projected back onto the fault
surface and shown using a standard color ramp
(Fig. 6A). To highlight more subtle geometrical
Animation 1. Overview of the Arkitsa fault
zone, showing fault panels A, B, and C (see
Figure 5 for further information). The animation shows low resolution LIDAR data
that includes the prominent ridge forming the hanging wall behind the exposed
fault panels. The viewpoint in the animation travels along the fault scarp, then rises
up to look down-dip along the fault panels
from above. From this vantage point the
fault segmentation and the curvature and
corrugation of the fault panels are evident.
To view the avi animation, use software
such as QuickTime or Microsoft Media
Player. If you are viewing the PDF of this
paper or reading it offline, please visit
http://dx.doi.org/10.1130/GES00216.SA1
or the full-text article on www.gsapubs.org
to view the animation.
variations, several cycles of the color ramp can
be used (Fig. 6B).
In essence, this method of analysis is comparable to a contour plot of the fault surface.
It is therefore particularly suitable for highlighting culminations and depressions on the
surface, and is capable of revealing relatively
subtle features of fault surface topography. In
contrast, the method is less suitable for highlighting the precise orientation and wavelength
of corrugations and striae, for which methods
involving variation in orientation and curvature
are more appropriate.
Spatial Variation in Fault Surface
Orientation
This section describes a method for visualization of the spatial variation in the orientation of
a fault surface across the outcrop. Variation in
orientation is calculated relative to the mean orientation of the fault panel, as described below.
Once the scans from a fault panel have been
merged, cleaned, filtered, and meshed (Fig. 4),
the panel geometry is represented by tens of
thousands of points (nodes), connected to form
a mesh of small triangular polygons (Figs. 4E
and 7A). For each node in the mesh, the 3D
positions of adjacent nodes are used to define
the node’s tangent plane (this is a standard procedure in 3D computer graphics; e.g., Foley et
al., 1997), and the vector normal of the tangent
plane represents the pole to the fault plane at
Animation 2. Medium resolution LIDAR data
from panel B (see Figure 5 for further information). At this resolution the prominent slipparallel striations and corrugations are clearly
visible, and the morphology of the fault plane
can be studied in detail. To view the avi animation, use software such as QuickTime or
Microsoft Media Player. If you are viewing the
PDF of this paper or reading it offline, please
visit http://dx.doi.org/10.1130/GES00216.SA2
or the full-text article on www.gsapubs.org to
view the animation.
Geosphere, December 2009
that point. Therefore, plotting all the vectors on
a stereonet shows the amount of variation in the
orientation of a single fault panel (Fig. 7B). In
practice, not all stereonet software is capable
of handling tens of thousands of points, but for
panel B analyzed here, a random 10% subsample of the entire data set is sufficient to show
that the orientation of the panel varies by at least
40°–50° in strike and 20°–30° in dip.
To show how this variation in orientation is
distributed spatially across the fault panel, each
point in the filtered point cloud is allocated a
color according to its orientation relative to
the mean orientation of the panel as a whole.
This entire process is implemented in software
and can be rapidly applied to the TLS data in
a single operation, but in order to explain and
illustrate the underlying concept of the method,
it is described here as a series of separate steps.
The mean orientation of the fault panel is
taken as the vector mean of all the normals to the
tangent planes for all the nodes. Strictly speaking, each normal should be weighted according
to the area of the polygons surrounding the node,
but the difference is extremely small unless the
meshing process has produced a large range in
polygon sizes. The angle between each of the
normals and the mean orientation of the fault
panel is also calculated at this stage (this angle
is the scalar product of the cosines of the two
vectors). The strength of the color assigned to
each node is dependant on the magnitude of this
angle. This is easier to conceptualize if the data
are rotated so that the mean orientation of the
panel is at the center of the stereonet (Fig. 7C).
For each point, the shade (hue) of the color
applied depends on the relative azimuth of the
point, with red for values to the west, cyan to the
east, yellows and greens in the southwest and
southeast quadrants, and magentas and blues
in the northwest and northeast quadrants. The
strength of the color (saturation) is proportional
to the angular distance (plunge) of the point from
the center of the net, with gray used for values
at the center, and stronger colors with increasing
angle outward. The color saturation is ramped
to reach 100% at 4 standard deviations from the
mean (the small circle in Figs. 7C, 7D), so that
all outlying points are allocated full saturation.
Using 4σ is an arbitrary amount, but using a
threshold in this way (rather than the entire net
or the full range of angular differences) helps to
increase the sensitivity of the method. For panel
B, 97.6% of the data are within 4σ of the mean.
Once a color value is calculated for every
point, the color is allocated to the corresponding node in the meshed view of the fault panel
(cf. Figs. 7D, 7E). Our software implementation outputs the resultant colored mesh in either
VRML (virtual reality modeling language;
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Jones et al.
www.web3d.org/x3d/specifications/vrml/)
or VTK (Visualization Toolkit) format (www
.vtk.org), suitable for display in many standard
graphics packages, including Kitware’s opensource program ParaView (www.paraview
.org). Meshed surfaces colored in this way help
to highlight surface irregularities of different
sizes, particularly corrugations, fault bends,
and secondary crosscutting faults (Figs. 7E and
8). Small-scale corrugations (~0.5–5 m wave-
A
lengths) appear as thin, bright bands of contrasting color, while fault bends representing
larger scale changes in orientation (~10–50 m
wavelengths) appear as broad belts of overall
color variation. For example, Figures 7D and 7E
show wide areas of predominantly northwestsoutheast strike in red colors, separated by a
region of west-northwest–east-southeast strike
(shown in cyan). Thin traces shown in green and
yellow correspond to small-scale tension frac-
secondary fault
tures and thrust faults that cut the main surface
of the fault panel.
Curvature Analysis
The curvature of a surface is a measure of its
departure from planarity. There are a number
of different ways to express curvature, including normal, Gaussian, and mean curvatures (see
Appendix). These were described in relation to
B
Panel A
Panel B
+
Panel C
_
+
_
Figure 6. Visualization method highlighting deviation from planarity, shown in terms of the spatial variation in perpendicular distance from the mean plane of the fault. This method is very effective for identifying asperities (bulges and depressions) on the fault
surface (Sagy et al., 2007). Left-side figures (A) use a single cycle of the color ramp, and are best for showing the overall topography
of the surface. Panel A shows a clear footwall depression (red) and bulge (blue, to the right of the depression). The sharp boundary
between depression and bulge is a secondary fault that cuts the main slip surface. Finer detail can be seen more easily using multiple
cycles of the color ramp; the right-side figures (B) use five cycles. See Figure 5A for the relative position of the three panels.
474
Geosphere, December 2009
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LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece
A
B
Equal area lower hemisphere projection
C
n = 5726
D
Equal area lower hemisphere projection
n = 5726
Equal area lower hemisphere projection
n = 5726
Figure 7 (continued on following page). Visualization method to show spatial variation in fault panel orientation. (A) Fault panel
B, looking nearly south. 7.8 × 106 points in the raw point cloud were stripped of surrounding wall rock and vegetation to leave
4 × 106 points; these were filtered to 57,261 points (using a 15 cm octree filter), and meshed to form a continuous surface composed of 116,008 triangles. Each triangle in the mesh can be allocated a color according to the relative orientation of its vector
normal compared with the orientation of the mean vector for the entire fault panel (see text). In practice, this is done in a single
operation programmatically using vector manipulation, but is also shown here schematically for a subset of the data to illustrate
the logical steps of the method. (B) Since the vector normal of each node in the mesh represents the pole to the fault plane at that
point, the data can be plotted on a stereonet to highlight the spread in orientations. The 26° small circle represents 4 standard
deviations and contains 97.6% of the data (cf. Healy et al., 2006). (C) The data are rotated so that the mean vector normal for
the entire panel is vertical (this is usually done only as an intermediate step within our software code, but is also illustrated here
on the stereonet). Each of the poles is colored according to its orientation relative to the mean, so that poles close to the mean are
given gray tones, and poles further from the mean are given bright tones. The azimuth of the pole determines its hue (e.g., here
red is W, cyan is E). Points plotted with circle symbols are from vertical and overhanging parts of the outcrop. (D) The colored
data are rotated back to their original orientation.
Geosphere, December 2009
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Jones et al.
E
50 m
Figure 7 (continued). (E) Color is allocated to all 116,008 triangles in fault panel B. This visualization method helps to highlight
the curvature and corrugations at different wavelengths, as well as the effect of crosscutting fractures.
geological surfaces by Lisle (1992, 1994), Bergbauer and Pollard (2003a, 2003b), and Pearce
et al. (2006). In mathematical terms, curvature
is a second-order derivative of the surface, and
hence curvature values are independent of the
surface’s orientation (i.e., if a surface is rotated,
the orientation of the surface at a given point
will generally change, but curvature properties
of the surface at that point will not). For folded
or corrugated surfaces such as the fault panels
at Arkitsa, curvature values vary across the surface, and analyzing the spatial variation of different curvature parameters is a useful step in
understanding the three-dimensional geometry
of the surface.
Measurement of curvature is highly scale
dependant, and analysis will emphasize the
shortest wavelength structures of a surface.
Hence, it is important to remove any highamplitude, short-wavelength noise from the
data, as this will mask genuine features of surface geometry (e.g., Bergbauer and Pollard,
2003a). In this study we have minimized noise
by resampling and smoothing the cleaned fault
panels shown in Figures 4, 7, and 8.
The results of the curvature analysis are
shown in Figure 9 for fault panels A and B.
Displaying the magnitude of normal curvature
shows the location of the larger corrugations and
cross faults (Figs. 9A, 9B; Animation 3), but is
not particularly effective in showing more subtle
geometrical features. Sensitivity of the plots can
be increased (Figs. 9C, 9D) by plotting the absolute values of normal curvature, and by using
an upper threshold (such that the color scale
476
is no longer stretched to include small areas of
anomalously high curvature). This method is
an effective way to highlight the corrugations
on the surface, and is the best plotting method
to use for precise measurements of geometrical
aspects such as wavelength of the corrugations.
Based on peak-to-peak measurements from
these plots, the average wavelength of mesoscale corrugations on panel A is 4.04 m, and on
panel B is 4.43 m.
Fault-plane corrugations are believed to form
from overlapping faults by two main mechanisms, including lateral propagation of curved
fault tips and linkage by connecting faults (Ferrill et al., 1999). Such mechanisms operating on
initially en echelon fault arrays might explain
the meso-scale corrugations seen in each of the
Arkitsa fault panels. Large-scale corrugations
may also form by the progressive breakthrough
of originally segmented fault systems (e.g.,
Childs et al., 1995), which is probably the most
likely cause of the decametric-scale warping at
Arkitsa. Other mechanisms for the formation of
corrugated fault surfaces include reactivation of
preexisting faults and fractures (e.g., Donath,
1962), and later folding of a preexisting fault
surface (e.g., Holm et al., 1994; Krabbendam
and Dewey, 1998).
Gaussian curvature can be a useful parameter to highlight domes, basins, and saddles on
the fault surface. It is defined as the product of
the principal curvatures, such that it is nonzero
if there is curvature along both of the principal
directions. Thus Gaussian curvature provides a
quantitative measure of culminations on the sur-
Geosphere, December 2009
face. Plots of Gaussian curvature from panels A
and B are shown in Figures 9E and 9F. These
plots show that there is negligible Gaussian curvature across the fault panels, suggesting that
there are no noticeable culminations on the panels (at least at the scale at which the surfaces are
smoothed). This contrasts with the plots shown
in Figure 6, which clearly show that domes and
basins are present. This apparent contradiction arises because of the scale dependence of
the curvature analysis: the domes and basins
in Figure 6 have wavelengths of ~101 m, while
the smoothing of the data used in Figure 9 was
chosen to highlight the corrugations (with typical wavelengths ~100 m). In order for the curvature analysis to detect and quantify larger scale
structures, it is necessary to smooth the data
more coarsely, to iron out small-scale features.
The slight mottling effect seen in parts of the
plots represents very small variations in curvature values. These are caused by minor residual
noise in the smoothed surface, and reflect the
limits of precision of the laser scanner equipment. In these specific examples, increasing the
sensitivity of the plotting method only emphasizes the small amounts of noise in the data,
without highlighting any other aspects of surface geometry.
The various curvature plots shown in Figure 9 help to emphasize that the output of
curvature analysis can be heavily dependant
on the processing steps involved (particularly
the way that the raw data are cleaned and
smoothed), and the visualization parameters
used for plotting the results. Quantification
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Panel A
LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece
n = 5927
equal area lower hemisphere projection
Panel C
n = 5843
equal area lower hemisphere projection
Figure 8. Spatial variation in fault panel orientation for panels A and C. See text and Figure 7 for description of the visualization method. Note that each of the panels are colored separately, relative to the mean orientation of vector normals for
that panel only.
Geosphere, December 2009
477
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Jones et al.
of curvature parameters is very useful, but
because the curvature of fault surfaces is generally quite subtle (e.g., compared with the curvature seen in folded bedding), more time and
effort is needed to fine-tune the variables used
for analysis and visualization. For this reason it
may often be ineffective to use geological software packages that have black box curvature
functionality that cannot be customized on a
case by case basis.
A
Variation in Slip Vector Orientation
A number of previous studies have shown
that the orientations of slip lineations can show
significant systematic variation along the strike
of a fault, particularly in parts of the fault that
are close to the tip zones or in relay zones (Roberts, 1996; Maerten, 2000; Maniatis and Hampel, 2008). The use of laser scanning to make
a large number of precise, spatially referenced
Panel A
B
measurements of slip lineations along the length
of the fully exposed fault panels allows us to test
whether systematic variations in slip direction
are also present in this part of the Arkitsa fault.
At Arkitsa, fault striae that are subparallel to
millimeter- to meter-scale corrugations are very
prominent on the surface of the fault panels in
outcrop and are also clearly visible in the TLS
point clouds (Figs. 5–9). In addition, there are
also other orientations of fault striae locally
Panel B
+0.338
+0.338
0.000
0.000
-0.338
-0.338
normal curvature
normal curvature
D
C
0.338
0.338
0.250
0.250
0.125
0.125
0.000
0.000
absolute normal curvature
absolute normal curvature
E
F
+0.047
+0.047
0.000
0.000
-0.026
-0.026
Gaussian curvature
Gaussian curvature
Figure 9. Curvature of main fault panels. (A, B) Magnitude of normal curvature. The main corrugations and cross faults are
visible, but small curvature variations are not highlighted. (C, D) Plots of absolute normal curvature. Values >0.25 are given the
same color: this increases the sensitivity of the variation in normal curvature. (E, F) Magnitude of Gaussian curvature. The plot
shows very low values of Gaussian curvature across the fault plane.
478
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LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece
preserved; however, these are less well developed, and are below the centimeter resolution of
our TLS data. Further work is needed to relate
these secondary slip directions to fault kinematics and regional tectonics, and in the following
analysis we focus only on the most prominent
set of slip lineations.
Although the orientation of the fault surfaces
shows significant variation within a single panel
(Figs. 7 and 10), the orientation of the slip vectors on each panel is remarkably consistent
(Fig. 10). Along the polished fault panel surfaces, the well-defined striations concentrate
along a north-northwest–south-southeast azimuth (mean azimuths 336°–344°), plunging
toward the north-northwest (mean plunge values
51°–57°). This configuration is consistent with
slightly oblique normal dip-slip motion with a
component of left-lateral displacement. Within
each fault panel, typical variation from the average plunge values is ~4°–9°, while variation
from mean slip azimuth is ~7°–15°.
Our analysis is based on 902 slip vectors
picked from the digital surface (124 measurements for panel A, 678 measurements for panel
B, and 100 for panel C). This is a comparable
though more extensive study to that presented in
Kokkalas et al. (2007), for which 121 slip vector measurements were used. Statistical properties for the entire data set are given in Table 1.
The eigenvector analysis we applied is a nonparametric method that is particularly suitable
for analyzing directional data (Fisher et al.,
1987; Mardia and Jupp, 2000). Eigenvectors are
Animation 3. Enhanced virtual outcrop in
which false color, based on the curvature
properties of the surface, is projected onto
the surface of fault panel B. See Figure 5
and text for further details. To view the
avi animation, use software such as QuickTime or Microsoft Media Player. If you are
viewing the PDF of this paper or reading it
offline, please visit http://dx.doi.org/10.1130/
GES00216.SA3 or the full-text article on
www.gsapubs.org to view the animation.
ordered such that E1 ≥ E2≥ E3, with E1 being the
primary orientation of slip vectors. The normalized eigenvalues (S1, S2, and S3) reflect the variance of the vector data. For example, panel A
with an S1 value of 0.9981 shows that 99% of
the total data are consistent with the specified
orientation of E1, within statistical significance.
Similar values of 0.9952 and 0.9971 have been
derived for panels B and C, respectively. The
relative size of each S value indicates the shape
of the distribution of the data. Data having S1 >>
S2 ≈ S3 suggest a highly clustered distribution, as
is the case for all three examined fault panels.
In addition, the fabric shape parameter K can
be used to test for girdle distributions (K < 1.0),
clustered patterns (K > 1.0), or a uniformly random distribution (K ≈ 1.0), while parameter C
determines the strength of the shape parameter
K (Woodcock, 1977; Woodcock and Naylor,
1983). If C > 3.0, a preferred orientation and
shape are strongly indicated, and if C < 3.0, there
are weak orientation and shape fabric of the data.
Values of K and C parameters for all three panels show marked clustering with strong preferred
orientation and consistent fabric shape.
The statistical analysis of populations of
minor faults, usually based on measurements
of fault orientation, slip direction, and sense
of slip, is commonly used for areas of brittle
deformation. Results from this type of fault slip
analysis are typically used to make inferences
about the stress field at the time of deformation
(assuming that the applied stress is homogeneous). By definition, fault-slip data provide
more direct information about kinematics than
the mechanics of faulting, with slickenlines
used as direct indicators of displacement directions on fault planes. In general, this analysis
is applied using individual fault-slip measurements from a large population of striated fault
surfaces, rather than many measurements from
a single nonplanar fault surface. Using TLS
data to make precise measurements of large
numbers of slip lineations and fault orientations
from a single fault surface also gives a unique
opportunity to determine the local stress (strain)
tensor for each fault panel, and hence to test the
consistency of slip direction between the different panels. Stress analysis using 100–150
fault slip data from each panel suggests that the
Arkitsa fault zone is slipping in response to an
extensional stress field with a north-south orientation of minimum principal stress (σ3), with
a small deviation toward the north-northeast
orientation calculated only for panel B (for further details, see Kokkalas et al., 2007).
Although the stereonet plots (Fig. 10) and
statistical analysis (Table 1) show that the slip
data are very tightly clustered, we have also analyzed the data to try to detect small, statistically
Geosphere, December 2009
A
Panel A
n = 124
equal angle lower hemisphere
B
Panel B
n = 678
equal angle lower hemisphere
C
Panel C
n = 100
equal angle lower hemisphere
Figure 10. Equal-angle stereonets of slip
vector orientations for each of the fault panels A, B, and C. See Figure 5A for relative
position of the panels.
479
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Jones et al.
TABLE 1. ORIENTATION OF EIGENVECTORS (E1–E3) AND MAGNITUDE OF
NORMALIZED EIGENVALUES (S1–S3) OF THE SLIP-VECTOR DATA FOR EACH OF
THE MAIN FAULT PANELS AT ARKITSA, CENTRAL GREECE
E1
E2
E3
S1
S2
S3
K
C
Panel A
344/51 165/39 75/0
0.9981 0.0013 0.0007 10.62
7.26
Panel B
336/57 90/14
188/70 0.9952 0.0026 0.0021 29.12
6.14
Panel C
338/54 235/10 138/34 0.9971 0.0012 0.0009 31.02
6.96
Note: The ratio of the eigenvalues indicates the shape of the distribution of data: K is
the fabric shape parameter, while parameter C determines the strength of parameter K
(Woodcock 1977; Woodcock and Naylor, 1983). The values for K and C indicate very
tight clustering of the data, with confidence level for all panels >>99% (representing the
probability that the distribution is not random). These statistical parameters show that
the LIDAR (light detection and ranging) data allow us to quantify the orientation of the
slip vector across the whole of the exposed fault panels with high precision.
significant spatial variations in the orientation
of slip vectors across the fault panels. This is
because, as noted by Jackson and McKenzie
(1999, p. 5), some striae appear to curve toward
the top of the exposed panels. Although we concur with their field observation (some parts of
panel B do show gradual variation in orientation
of ~1°–2° between the top and bottom of the
outcrop), this variation appears to be of localized extent, and apparently does not reflect the
whole of panel B, or the other exposed panels.
Note, however, that we have not yet been able to
make quantitative measurements from the TLS
data that are precise enough (±0.5°) to refine the
estimated variations in slip vector orientation
made by Jackson and McKenzie (1999). Further
work is needed to optimize the processing of the
TLS data prior to repicking of the fault striae.
Comparison of Arkitsa Slip Vectors and
Regional Velocity Field
The present-day regional stress field, derived
from fault-slip data, focal mechanisms of recent
earthquakes, and several GPS campaigns, represents north-south to north-northeast–southsouthwest extension of central Greece (Roberts
and Jackson, 1991; Armijo et al., 1996; Roberts and Ganas, 2000; Hollenstein et al., 2008).
Recent GPS velocities and strain rate tensor calculations, derived from the velocity field, indicate that velocity differences observed across
the North Evia Gulf are small (2–4 mm/yr;
Hollenstein et al., 2008). This is consistent with
the low seismic activity of this rift structure.
Regional strain rates can be resolved into normal and shear components relative to the westnorthwest–trending Arkitsa–Kamena Vourla
fault zone (Figs. 12 and 13 of Hollenstein et al.,
2008). Normal strain rates represent low extensional strain values in a north-northeast orientation, while shear strains indicate sinistral motion
along the Arkitsa–Kamena Vourla fault zone.
These geodetic calculations fit closely with the
north-south orientation of the stress tensor we
derive from the Arkitsa fault slip data, indicat-
480
ing that the stress/strain tensor has not changed
significantly in orientation since the last major
slip event recorded on the Arkitsa fault surface.
Despite the close match between our fault data
and regional strains inferred from GPS, we still
cannot be certain whether the precise orientation
of oblique slip we have measured at Arkitsa is
entirely indicative of regional strain, or whether
there are also additional, more localized effects
close to the eastern tip of the Arkitsa–Kamena
Vourla fault zone (cf. Roberts and Ganas, 2000).
CONCLUSIONS
Terrestrial laser scanning (i.e., LIDAR) is an
effective tool for acquiring detailed geometrical data from exposed outcrop surfaces, and is
therefore ideal for studying surface irregularities
and curvature of nonplanar faults. Field acquisition of LIDAR data is rapid, though the millions
of data points gathered will subsequently need to
be merged, cleaned, and filtered prior to further
geological analysis. Various methods of analysis
can be used to highlight different geometrical
properties of the scanned surfaces. Roughness
provides a bulk measure of the overall surface
irregularity of a fault, while other methods allow
the spatial variability of fault geometry to be
studied. These methods include measuring the
perpendicular distance of points from the mean
plane, plotting variation in orientation relative to
the mean plane, and curvature analysis (including normal, mean, and Gaussian curvature).
Studying curvature allows quantitative measurements of fault bends and corrugations, although
the measurements are highly scale sensitive and
require care during data preparation and analysis. Spatial variation in the various geometrical
properties can be highlighted by projecting the
resultant analytical values back onto the virtual
fault planes (Animation 3). 3D visualization
software is particularly effective in making it
easy to compare the derived parameters with the
underlying original fault geometry.
Analysis of three well-exposed fault panels
from Arkitsa (central Greece) shows that each
Geosphere, December 2009
of the faults contains surface irregularities, corrugations, fault bends, and crosscutting minor
faults, and deviates significantly from planarity.
Each panel shows a spread of at least ±20° in the
orientation of the surface perpendicular to the
main slip vector. Normal curvatures delineate
clearly the location of corrugations on the surfaces of the panels. Gaussian curvature is very
low at this scale of analysis. Corrugations have
axes orientated parallel to the main slip direction, and have average wavelengths of ~4 m.
The orientations of the slip vectors are tightly
clustered, plunging ~55° north-northwest. This
is consistent with the present-day regional
velocity field derived from GPS measurements.
These results can be compared with measurements given by Resor and Meer (2006).
The high resolution of the LIDAR data helps
to illustrate that the detailed geometry and
kinematics of the fault zone are highly complex, and that different complexities exist at
different scales. It is not yet clear the degree
to which this part of the Arkitsa fault zone
is typical of faults in general, or whether the
high structural complexity is related to the
regionally complicated rotational strains seen
throughout the Aegean. In either case, much
further work is needed to unravel these complexities on the outcrop scale, and to relate the
spatial variation in fault geometry to larger
scale fault kinematics and dynamics.
ACKNOWLEDGEMENTS
We thank Phil Clegg and Tim Wright for assistance
during field work, Jonny Imber for discussion on
faulting, Mark Pearce for collaboration on curvature,
and Steve Waggott and Ruth Wightman for collaboration related to laser scanning. Thomas Dewez kindly
commented on the manuscript, and Emily Brodsky,
Ioannis Papanikolaou, and Ramon Arrowsmith provided useful reviews and editorial comment.
APPENDIX—MATHEMATICAL
DEFINITIONS OF CURVATURE
Curvature in Two Dimensions
For a folded surface given by curve z = f(x), a
typical definition of curvature (e.g., Ramsay, 1967,
p. 346–347) is given by
curvature, C =
d2z
dx 2
3
.
(1)
⎡ ⎛ dz ⎞ 2 ⎤ 2
⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ dx ⎠ ⎥⎦
In two dimensions (2D), fold hinges are defined as
areas where C is a maximum or minimum. Crests and
troughs on the folded surface occur where the firstorder derivative of the curve (i.e., the gradient, dz/dx)
is zero.
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LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece
Curvature in Three Dimensions
For the study of fault plane geometry, threedimensional analysis of curvature is preferable to
2D. For each point on a surface, r, given by
⎡ x ⎤
r = ⎢⎢ y ⎥⎥ ,
⎢⎣ z( x, y) ⎥⎦
(2)
there are two directions in which the amount of normal curvature is a maximum or minimum. These
are the maximum and minimum principal curvature
directions, k1 and k2. The magnitude and orientation
of k1 and k2 are given by the characteristic roots of the
shape operator L:
L = II / I ,
(3)
where I and II are the two fundamental forms of
the surface (Bergbauer and Pollard, 2003a, 2003b;
Pearce et al., 2006; Mynatt et al., 2007). The first
fundamental form is given by
where
⎡rx ⋅ rx
I=⎢
⎢⎣rx ⋅ ry
rx ⋅ ry ⎤
⎥,
ry ⋅ ry ⎥⎦
⎡ ⎤
⎢1⎥
∂r ⎢ ⎥
=⎢ 0 ⎥,
rx =
∂x ⎢ ⎥
∂z
⎢ ⎥
⎣ ∂x ⎦
Pearce et al. (2006) and Mynatt et al. (2007) presented a more complete derivation of the curvature
equations given here.
Curvature Parameters
The normal curvature is the magnitude of curvature corresponding to the maximum and minimum
principal curvature directions, k1 and k2. These values
help to define the 3D traces of hinge lines along folds
or corrugations. It can sometimes be useful to visualize absolute (i.e. modulus) values of normal curvatures (|kn|), as shown in Figures 9C and 9D.
Gaussian curvature, G, is the product of the two
principal curvature values:
G = k1k2 .
(9)
Analysis of Gaussian curvature indicates whether a
surface is developable (Lisle, 1992, 1994), and hence
can be useful to reveal domes, basins and saddles.
Mean curvature, M, is the average of the two principal curvature values (Roberts, 2001; Bergbauer and
Pollard, 2003a, 2003b):
M = (k1 + k2 ) / 2 .
(10)
(4)
Curvature Properties
(5)
Since curvature is a function of the second derivative (equations 1, 3, and 7), it is an invariant property of the curve. This means that curvature values
are independent of the dip of the fault surface. In
contrast, the positions of crests and troughs are not
invariant features, but are closely dependent on the
orientation of the fault. For this reason, the method of
analysis of spatial variation of fault surface orientation shown in Figures 7 and 8 is done relative to the
mean fault plane.
and
⎡ ⎤
⎢0⎥
∂r ⎢ ⎥
=⎢ 1 ⎥.
ry =
∂y ⎢ ⎥
⎢ ∂z ⎥
⎢⎣ ∂y ⎥⎦
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(6)
The second fundamental form is
⎡ ∂2r ∧
⎢ 2 ⋅n
∂x
II = ⎢ 2
⎢ ∂r ∧
⋅n
⎢
⎢⎣ ∂x∂y
∂2r ∧ ⎤
⋅ n⎥
∂x ∂y ⎥
,
∂2r ∧ ⎥
⋅n ⎥
∂y 2
⎥⎦
(7)
where the unit normal n̂ is given by
∂r ∂r
×
∂x ∂y
.
n=
∂r ∂r
×
∂x ∂y
∧
(8)
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REVISED MANUSCRIPT RECEIVED 11 MAY 2009
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