Quantitative Analysis and Visualization of Nonplanar Fault Surfaces Using Terrestrial Laser Scanning (LIDAR)—The Arkitsa Fault, Central Greece, As a Case Study

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 variation in surface orientation, with spreads fault reactivation, fault seal, and fl uid fl ow in of ~±20°–25° in strike and ±10° in dip. Most of faulted reservoirs and aquifers. Despite this, most Many fault surfaces are noticeably nonpla- the variation in orientation refl ects decimeter- modeling of fault dynamics assumes idealized nar, often containing irregular asperities and and meter-scale corrugations and longer planar geometries, and no failure criteria have yet more regular corrugations and open warping. wavelength warps of the fault panels. Average been developed for curved faults. Terrestrial laser scanning (light detection and wavelengths of corrugations measured on two One of the main limitations in incorporating ranging, LIDAR) is a powerful and versatile of the panels are 4.04 m and 4.43 m. Whereas realistic fault geometries into dynamic fault mod- tool that is highly suitable for acquisition of the orientations of the three fault surfaces els is the paucity of quantitative studies of three- very detailed, precise measurements of slip- show signifi cant variations, the orientations dimensional geometry of natural fault zones. In surface geometry from well-exposed faults. of fault striae are very similar between the this paper we present detailed data acquired using Quantitative analysis of the LIDAR data, panels, and are tightly clustered within each terrestrial laser scanning (TLS), also known as combined with three-dimensional visualiza- panel. Fault-slip analysis from each panel ground-based LIDAR (light detection and rang- tion software, allows the spatial variation in shows that the local stress fi eld is consistent ing), from well-exposed fault surfaces in an various geometrical properties across the fault with the regional velocity fi eld derived from area of active tectonism, the Aegean. The main surface to be clearly shown. Plotting the varia- global positioning system data. The oblique purpose of this paper is to describe methods to tion in distance of points from the mean fault slip lineations observed on the fault panels process, visualize, and analyze these kinds of plane is an effective way to identify culmina- represent a combination of two contempora- large geospatial data set, including techniques to tions and depressions on the fault surface. neous strain components: north-northeast– highlight spatial variation in different geometric Plots showing the spatial variation of surface south-southwest extension across the Gulf of parameters across a fault surface. orientation are useful in highlighting corru- Evia and sinistral strike slip along the west- gations and warps of different wavelengths, northwest–east- southeast Arkitsa fault zone. REGIONAL SETTING as well as cross faults and fault bifurcations. Analysis of different curvature properties, INTRODUCTION The study site for this work is the area of spec- including normal and Gaussian curvature, tacularly exposed fault surfaces of the Arkitsa provides the best plots for quantitative mea- Many fault surfaces commonly observed in fault zone, reported by Jackson and McKenzie surements of the geometry of corrugations outcrop are markedly nonplanar, and often display (1999), located on the southern side of the north- and folds. However, curvature analysis is corrugations, asperities, and longer wavelength ern Gulf of Evia in central Greece (Fig. 1A). highly scale dependent, so requires careful fi l- low-amplitude curvature. Such irregularities can The area is currently undergoing active north- tering and smoothing of the data to be able to have major effects on the mechanical behavior of south to northeast-southwest regional extension, analyze structures at a given wavelength. the fault during slip, and a better understanding with rates on the order of 1–2 mm/yr (Clarke at Three well-exposed fault panels from the of the interplay between irregular geometry, fault al. 1998), in response to the complex tectonic Arkitsa fault zone in central Greece were kinematics, and dynamics is an important step in interplay between subduction beneath the Hel- scanned and analyzed in detail. Each panel improving our knowledge of a variety of geologi- lenic Arc, backarc extension in the Aegean, and is markedly nonplanar, and shows signifi cant cal processes, including earthquake nucleation, the westward movement of the Anatolian plate

*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 ﬁ gures; 1 table; 3 animations.

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 Jones et al. height A (meters asl) 2500

1000

500

300

39°N 200

Almyr A os-Sperchios e graben g e a 100 Fig.1B n KVF N AK . F ARFZ Gu KF lf 50 Z of E v AT ia F 0

projection: WGS84 S. Gulf of E via Gu lf o f C ori nth

38°N

Corinth Athens

22°E 23°E 24°E B 22°55' 23°00' 23°05' Logos

Fig. 2 NORTHNORTH 38°45' A rkit Arkitsa sa F ault Zone G u lf o f Livanates A ta Livanates la n Fault ti

Megaplatanos

38°40' At Zeli alan Atalanti ti F ault Zone

Tragana

10 km

466 Geosphere, December 2009

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 LIDAR analysis and visualization of nonplanar faults at Arkitsa, Greece

well constrained. Some historical earthquakes Atalanti fault show similar values, on the order Figure 1. (A) Satellite topography data have been recorded in the area, but none can be of 0.27–0.4 mm/yr (Ganas et al., 1998). In addi- (Shuttle Radar Topography Mission) directly associated with the Arkitsa fault zone, tion to the overview given in Jackson and McK- from central Greece, showing location of or with any other fault with certainty, other enzie (1999), further description was given by the study area on the southern side of the than the 1894 rupture events of Atalanti fault Kokkalas et al. (2007) and Papanikolaou and Gulf of Evia. KVF—Kamena Vourla fault; (Ambraseys and Jackson, 1990). Royden (2007). AKF—Agios Konstantinos fault; ARFZ— The Arkitsa fault zone together with the Agios There is evidence of Holocene seismic activ- Arkitsa fault zone; KFZ—Kallidromo fault Konstantinos and Kamena Vourla faults form a ity along the fault, based on the geomorpho- zone; ATF—Atalanti fault; asl—above sea west-northwest–east-southeast, left-stepping, logical expression of the Arkitsa region, as well level. (B) Tectonic map of the area around north-dipping fault margin that links with the as the back-tilted terraces on the hanging-wall Arkitsa. Fault picks are based on detailed southern margin of the Almyros-Sperchios block of the Arkitsa fault, and the exposure of an aerial photo analysis and ground-truthing. graben (Fig. 1). The Arkitsa fault zone, with ~1-m-high scarp of unweathered fault surface a length of ~10 km, separates Late Triassic– that existed prior to quarrying. In addition, the Jurassic platform carbonates in the footwall coastline in the region shows evidence of Holo- (e.g., Dewey et al., 1986; Doutsos and Kok- from lower Pliocene–Quaternary sediments in cene uplift and subsidence that is controlled by kalas, 2001; Kokkalas et al., 2006). In central the hanging wall. The amount of throw appar- the motion and location of the faults (Roberts Greece, much of the extensional strain is local- ently varies along strike, typical of highly seg- and Jackson, 1991; Stiros et al., 1992). ized into a number of west-northwest–east- mented fault zones. The minimum throw is esti- The Arkitsa fault is best exposed in an out- southeast–trending grabens, which are mostly mated to be ~500–600 m, based on the thickness crop 450 m south of the main Athens-Lamia characterized by complex geometries in map of Neogene–Holocene sediments in the hanging national road (Fig. 2), where recent quarry- view (Fig. 1B), and a high degree of segmen- wall, plus the scarp height and topographic relief ing activity during the past 20 yr has removed tation along strike (Doutsos and Poulimenos, of the footwall block. This allows a slip rate of hanging-wall colluvium to reveal clean, fresh 1992; Ganas et al., 1998; Kokkalas et al., 2006). 0.2–0.3 mm/yr to be estimated for the Arkitsa exposures of the upper 65–70 m of three large Most of the major normal faults in the region are fault zone, given that activity on faults of the fault panels. These fault panels form the north- thought to have been active in the Pleistocene, Evia rift zone started within the past 2–3 m.y. ern scarp edge of a footwall block that rises to although the relative time of their activity is not (Ganas et al., 1998). Slip rates for the adjacent 255 m above sea level (a.s.l.). The hanging-wall

Evia n Gulf of Nor ther

to L amia Athens oad to National R

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 467

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 Jones et al.

sediments form the low-lying plain at 70 m a.s.l. sequences. This is a straightforward option in ther examples). There are a number of different We have recorded the detailed geometry of the the RiScan Pro software that is used to drive the solutions available to capture additional color fault panels using terrestrial laser scanning, and LMS-Z420i, and in our experience can increase information. Several scanner manufacturers use this paper presents a description of the acquisi- longitudinal precision to 10 mm or better. on-board video circuitry mounted inside the tion, visualization, and preliminary analysis of The fault zone was scanned using a standard scanner. The Riegl LMS scanners use a separate the laser-scan data. acquisition strategy of taking multiple overlap- off-the-shelf digital single lens refl ex camera ping scans from a number of different tripod (dSLR), which is tightly calibrated so that the DATA ACQUISITION positions (scan stations) along the length of the optical properties of the camera-lens combina- outcrop. This aims to minimize any large gaps tions are known. The camera is attached to the A detailed survey of the exposed fault planes in the data (data shadows) caused by the irregu- top of the scanner using a precision-engineered and surrounding footwall was carried out in early lar topography of the outcrop surface. In prac- mount (Fig. 3). In this way, the pixels in the 2D December 2005 using terrestrial laser scanning tice, there will always be some data shadows, images taken with the camera can be accurately (TLS), also commonly termed ground-based but careful planning of the data acquisition can mapped to the 3D point data from the scanner. LIDAR (light detection and ranging). TLS is a help to keep these to a minimum. In this study, Although using a separate dSLR requires more very rapid surveying method in which the pre- 10 scan stations were used along the outcrop time and effort during data acquisition, the main cise geometry of the outcrop can be measured length, with 9 stations sited <75 m from the advantage is that this solution provides very in detail by recording the three-dimensional outcrop, and 1 station ~500 m away to provide high resolution and faithful color rendition. (3D) position of many millions of points across extra coverage of the footwall hillside above the surface of an exposure. Since the method is the fault scarp (Fig. 2). Over a period of 2 days, Merging and Georeferencing the Scan Data nonpenetrative, the raw output is a surface 3D 25 scans with a total of 75 × 106 points were dataset (Dogan Seber, 2007, personal commun.; acquired from these 10 stations. Scans taken from different scan stations are Jones et al., 2008), containing no subsurface Data from each of the scan stations need to merged together based on common refl ector data from within the outcrop. The high reso- be merged together into a single model. This is targets and dGPS, as described above. We used lution provided by TLS allows outcrops to be achieved using two methods in combination. dGPS to measure 16 tie-point positions—12 analyzed quantitatively in great detail, leading First, a set of highly refl ective cylindrical tar- of which gave subcentimeter spatial precision to increased use in geosciences (e.g., Ahlgren et gets is placed carefully in prominent positions following post-processing. These 12 positions al., 2002; Jones et al., 2004; Bellian et al., 2005; spread around the survey area. Then from each were used for georeferencing (10 refl ector posi- McCaffrey et al., 2005; Pringle et al., 2006; and scan station, all the refl ectors that are visible tions plus 2 dGPS base stations). many others. See also Wawrzyniec et al., 2007, from that station are scanned at high resolu- Semiautomatic merge functionality is and associated papers in the Geosphere themed tion. By matching the same refl ector from dif- included in the RiScan Pro software, which iter- issue “Unlocking 3D earth systems—Harness- ferent scan stations, all the different scans can atively minimizes the error involved in matching new digital technologies to revolutionize be stitched together. In addition, a differential ing scan stations to each other. For this project multi-scale geological models”). global positioning system (dGPS) was used the mean of the average errors (i.e., the mean of The equipment we used at Arkitsa was a Riegl to locate the positions of the most prominent matching all stations to each other) was 5 mm. LMS-Z420i (Fig. 3) (LMS—laser measurement refl ectors, so that they can be georeferenced to Once the colored scans from different scan system), which is a long-range scanner with an a global coordinate system. For this we used an stations have been merged, then the data can effective range of up to 1000 m, a maximum reso- Ashtech Promark 2 in survey mode, capable of be rendered as a virtual outcrop, either as a lution (i.e., angular acuity) of 0.004° (equivalent subcentimeter precision if the satellite confi gu- large model covering the entire area scanned to 7 mm at 100 m), and a specifi ed longitudinal ration is suitable (McCaffrey et al., 2005). (Fig. 5A) and/or as higher resolution models for precision of 5 cm. The long range of the scanner detailed analysis of more localized parts of the was essential for enabling us to capture the mor- DATA PROCESSING outcrop (Fig. 5B; see also Animations 1 and 2 phology of the entire footwall block along the associated with Figs. 5A and 5B, respectively). length of the hillside, and together with its high- A number of standard processing steps are Normally, at this stage of our specifi c work fl ow, resolution color capability and very rapid speed needed to combine the raw scan data into a vir- the data are still in the form of a point cloud of data acquisition (12,000 points/s), this made tual copy of the real outcrop. The virtual outcrop (Fig. 4B; i.e., the outcrop surface has not been the LMS-Z420i an ideal instrument for this is then suitable for further geospatial and geo- meshed), as this is usually adequate to give a project. However, this model of scanner may be logical analysis. good impression of the nature of the outcrop less suitable for some other kinds of close-range prior to further processing. project, because data acquired when the scanner Adding High-Defi nition Color Information is used in normal mode generally contain sig- Cleaning the Data nifi cantly more noise than shorter range scan- The main functionality of a terrestrial laser ners such as the Trimble GS100/GS200/GX200 scanner measures only the distance traveled by The next step in the processing work fl ow or Leica HDS3000 (Lichti and Jamtsho 2006; the laser pulse, plus the intensity of the beam generally depends upon the overall purpose of Renard et al., 2006; Geospatial Research Ltd., that is returned. This provides a grayscale image the project. If the main priority is to recreate unpublished tests). Therefore, when using Riegl of the outcrop (Fig. 4A). In our experience, the a virtual copy of the outcrop that is as true to LMS scanners (e.g., Z360, Z390, Z420i) for capability to superimpose high-resolution color life as possible, then the point cloud data can be detailed studies of small-scale surface morphol- information onto the raw point cloud (Fig. 4B) fi ltered, meshed, and draped with bitmap poly- ogy (e.g., Renard et al., 2006; Sagy et al., 2007), can be extremely valuable during subsequent gons taken from the scan photos (Figs. 4D–4F). it is important to improve precision by changing geological interpretation based on the virtual However, for the work presented in this paper, the mode of operation to acquire multiple scan outcrop (see White and Jones, 2008, for fur- the aim has been to use the TLS data to derive

468 Geosphere, December 2009

Fault panel B (see text)

Nikon D70 dSLR (with 14 mm, 50 mm & 85 mm lenses)

precision scanner camera window mount

USB cable (camera to PC)

Riegl LMS-Z420i scanner tilt mount

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 469

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 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 refl ected laser beam. Red scale bars are 10 m and vertical. (B) High-defi nition RGB (red-green- blue) 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 fi ltered (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

PaPPanelnel A PanelPaanenel B PanelPaPannel C

oloolivevee ggrogrovesves 70m707 m aasls sesecsectiontion between arrows is ca. 900m long

pylppylonsons are cca.a.. 3503350mm apart

pypylpylony onon pylpylonon

B 66767.4m7.4.4m (d((down-dipdowown-n ddip mammarksarkks frffromromo wwateratater rrun-off)unun-oofff )

opoopenenen ttensioneennssiiono ccrcrescenticresescecentn icc frfracturesracactureres ffrfracturesraacctuturees

mammainainn ssliplip vevvectorectc oorr

small thrust faults

Figure 5 (continued on following page). Annotated screenshots from video ﬂ y-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. Sepa- rate colored scans have been merged and georeferenced, but the data have not yet been ﬁ ltered, 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 471

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 Jones et al.

quantitative information about the geometry and More critically, it is very important that the data cifi c directions (e.g., Brown and Scholz, 1985; kinematics of the main fault surfaces, and so the are smoothed in order to eliminate any irregu- Power et al., 1987), or as a 2D property of the main priority was to extract the scan points from larities (noise) on scales smaller than the preci- surface as a whole (Renard et al., 2006). In this the large fault panels and isolate them from the sion of the scanner. In practice, both these aims paper we present additional methods of analysis background data. This meant that much effort can be met by using RiScan Pro’s octree fi lter- that allow other geometrical properties of fault was needed to clean the scan data by removing ing functionality (e.g., Foley et al., 1997), typi- planes to be studied. This approach shows that all traces of vegetation, talus, colluvium, weath- cally using an edge threshold of 10–15 cm. The different types of analyses tend to highlight dif- ered parts of the outcrop, and points refl ected fi ltered result is a heavily decimated point cloud ferent geometrical properties of the faults, and off airborne dust particles (Fig. 4C). This can (Fig. 4D) that can then be meshed (Fig. 4E), and therefore a range of methods should be used in be a very laborious process, and automated if necessary also image draped (Fig. 4F). This combination when analyzing the 3D geometry approaches to data cleaning have the potential produces a processed data set that is ready for of fault surfaces. to save considerable time and effort (Erik Mon- more specifi c visualization and geometrical and Unlike roughness, which is typically expressed sen, Schlumberger, 2006–2008, personal com- geological analysis. as an overall bulk property of a fault surface, muns.). attributes such as orientation, curvature, and Geospatial Analysis and Visualization distance from the mean plane are more usefully Filtering and Smoothing analyzed in terms of their spatial variation across Detailed analysis of fault slip surfaces can the surface of the fault. After such geometrical Before the geometry of the cleaned fault sur- lead to better understanding of fault zone pro- properties are derived for each point on the surfaces can be analyzed, it is necessary to fi lter cesses and fault dynamics. TLS has already face, the calculated values can then be mapped the data to reduce the amount of points down proven to be an effective tool for quantify- back onto the fault surface by recoloring each to a more realistic size ready for computational ing surface roughness of faults (Renard et al., corresponding point in the virtual outcrop data. processing. At this stage, each of the three fault 2006; Sagy et al., 2007). Roughness is a use- Displaying the colored fault surfaces using 3D panels is represented by between 10 × 106 and ful parameter to characterize the overall surface visualization software provides a powerful tool 16 × 106 points, and for computational effi - irregularity of rock (or other materials), and can to highlight spatial variation in fault geometry ciency it is ideal to reduce this to <200,000. be measured along 1D sampling lines in spe- (e.g., Fig. 5C).

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.

472 Geosphere, December 2009

Spatial Variation in Perpendicular Distance variations, several cycles of the color ramp can that point. Therefore, plotting all the vectors on from the Mean Plane be used (Fig. 6B). a stereonet shows the amount of variation in the In essence, this method of analysis is com- orientation of a single fault panel (Fig. 7B). In This is a simple method to highlight the parable to a contour plot of the fault surface. practice, not all stereonet software is capable deviation of a fault surface from a plane, in this It is therefore particularly suitable for high- of handling tens of thousands of points, but for case the mean plane through all measured points lighting culminations and depressions on the panel B analyzed here, a random 10% subsam- (nodes) on the fault. This method was used by surface, and is capable of revealing relatively ple of the entire data set is suffi cient to show Renard et al. (2006) and Sagy et al. (2007) to subtle features of fault surface topography. In that the orientation of the panel varies by at least visualize fault irregularities. There are a num- contrast, the method is less suitable for high- 40°–50° in strike and 20°–30° in dip. ber of different ways to calculate the orienta- lighting the precise orientation and wavelength To show how this variation in orientation is tion of the mean plane, including 3D regression of corrugations and striae, for which methods distributed spatially across the fault panel, each methods and analysis of the moment of inertia involving variation in orientation and curvature point in the fi ltered point cloud is allocated a (Fernández, 2005). A useful alternative is to are more appropriate. color according to its orientation relative to use vector calculus to derive the orientation of the mean orientation of the panel as a whole. the mean of all node normals. This approach is Spatial Variation in Fault Surface This entire process is implemented in software effi cient, since we need to calculate node nor- Orientation and can be rapidly applied to the TLS data in mals as an intermediate step for other types of a single operation, but in order to explain and analysis (see following for more details about This section describes a method for visualiza- illustrate the underlying concept of the method, the calculation of node normals and the mean tion of the spatial variation in the orientation of it is described here as a series of separate steps. plane). Once the orientation of the mean plane is a fault surface across the outcrop. Variation in The mean orientation of the fault panel is known, 3D trigonometry can be used to measure orientation is calculated relative to the mean ori- taken as the vector mean of all the normals to the the perpendicular distance of each point from entation of the fault panel, as described below. tangent planes for all the nodes. Strictly speak- the mean plane. The values for perpendicular Once the scans from a fault panel have been ing, each normal should be weighted according distance are then projected back onto the fault merged, cleaned, fi ltered, and meshed (Fig. 4), to the area of the polygons surrounding the node, surface and shown using a standard color ramp the panel geometry is represented by tens of but the difference is extremely small unless the (Fig. 6A). To highlight more subtle geometrical thousands of points (nodes), connected to form meshing process has produced a large range in a mesh of small triangular polygons (Figs. 4E polygon sizes. The angle between each of the and 7A). For each node in the mesh, the 3D normals and the mean orientation of the fault positions of adjacent nodes are used to defi ne panel is also calculated at this stage (this angle the node’s tangent plane (this is a standard pro- is the scalar product of the cosines of the two cedure in 3D computer graphics; e.g., Foley et vectors). The strength of the color assigned to al., 1997), and the vector normal of the tangent each node is dependant on the magnitude of this plane represents the pole to the fault plane at 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 Animation 1. Overview of the Arkitsa fault strength of the color (saturation) is proportional zone, showing fault panels A, B, and C (see to the angular distance (plunge) of the point from Figure 5 for further information). The ani- the center of the net, with gray used for values mation shows low resolution LIDAR data at the center, and stronger colors with increasing that includes the prominent ridge form- angle outward. The color saturation is ramped ing the hanging wall behind the exposed to reach 100% at 4 standard deviations from the fault panels. The viewpoint in the anima- mean (the small circle in Figs. 7C, 7D), so that tion travels along the fault scarp, then rises Animation 2. Medium resolution LIDAR data all outlying points are allocated full saturation. up to look down-dip along the fault panels from panel B (see Figure 5 for further infor- Using 4σ is an arbitrary amount, but using a from above. From this vantage point the mation). At this resolution the prominent slip- threshold in this way (rather than the entire net fault segmentation and the curvature and parallel striations and corrugations are clearly or the full range of angular differences) helps to corrugation of the fault panels are evident. visible, and the morphology of the fault plane increase the sensitivity of the method. For panel To view the avi animation, use software can be studied in detail. To view the avi ani- B, 97.6% of the data are within 4σ of the mean. such as QuickTime or Microsoft Media mation, use software such as QuickTime or Once a color value is calculated for every Player. If you are viewing the PDF of this Microsoft Media Player. If you are viewing the point, the color is allocated to the correspond- paper or reading it offl ine, please visit PDF of this paper or reading it offl ine, please ing node in the meshed view of the fault panel http://dx.doi.org/10.1130/GES00216.SA1 visit http://dx.doi.org/10.1130/GES00216.SA2 (cf. Figs. 7D, 7E). Our software implementa- or the full-text article on www.gsapubs.org or the full-text article on www.gsapubs.org to tion outputs the resultant colored mesh in either to view the animation. view the animation. VRML (virtual reality modeling language;

Geosphere, December 2009 473

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 Jones et al.

www.web3d.org/x3d/specifications/vrml/) lengths) appear as thin, bright bands of con- tures and thrust faults that cut the main surface or VTK (Visualization Toolkit) format (www trasting color, while fault bends representing of the fault panel. .vtk.org), suitable for display in many standard larger scale changes in orientation (~10–50 m graphics packages, including Kitware’s open- wavelengths) appear as broad belts of overall Curvature Analysis source program ParaView (www.paraview color variation. For example, Figures 7D and 7E .org). Meshed surfaces colored in this way help show wide areas of predominantly northwest- The curvature of a surface is a measure of its to highlight surface irregularities of different southeast strike in red colors, separated by a departure from planarity. There are a number sizes, particularly corrugations, fault bends, region of west-northwest–east-southeast strike of different ways to express curvature, includ- and secondary crosscutting faults (Figs. 7E and (shown in cyan). Thin traces shown in green and ing normal, Gaussian, and mean curvatures (see 8). Small-scale corrugations (~0.5–5 m wave- yellow correspond to small-scale tension frac- Appendix). These were described in relation to

A secondary fault 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 fi gures (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 fi gures (B) use fi ve cycles. See Figure 5A for the relative position of the three panels.

474 Geosphere, December 2009

B C D

Equal area lower hemisphere projection n = 5726 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 ﬁ ltered to 57,261 points (using a 15 cm octree ﬁ lter), and meshed to form a continuous surface com- posed 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 475

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 Jones et al.

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), Berg- is no longer stretched to include small areas of face. Plots of Gaussian curvature from panels A bauer and Pollard (2003a, 2003b), and Pearce anomalously high curvature). This method is and B are shown in Figures 9E and 9F. These et al. (2006). In mathematical terms, curvature an effective way to highlight the corrugations plots show that there is negligible Gaussian cur- is a second-order derivative of the surface, and on the surface, and is the best plotting method vature across the fault panels, suggesting that hence curvature values are independent of the to use for precise measurements of geometrical there are no noticeable culminations on the pan- surface’s orientation (i.e., if a surface is rotated, aspects such as wavelength of the corrugations. els (at least at the scale at which the surfaces are the orientation of the surface at a given point Based on peak-to-peak measurements from smoothed). This contrasts with the plots shown will generally change, but curvature properties these plots, the average wavelength of meso- in Figure 6, which clearly show that domes and of the surface at that point will not). For folded scale corrugations on panel A is 4.04 m, and on basins are present. This apparent contradic- or corrugated surfaces such as the fault panels panel B is 4.43 m. tion arises because of the scale dependence of at Arkitsa, curvature values vary across the sur- Fault-plane corrugations are believed to form the curvature analysis: the domes and basins face, and analyzing the spatial variation of dif- from overlapping faults by two main mecha- in Figure 6 have wavelengths of ~101 m, while ferent curvature parameters is a useful step in nisms, including lateral propagation of curved the smoothing of the data used in Figure 9 was understanding the three-dimensional geometry fault tips and linkage by connecting faults (Fer- chosen to highlight the corrugations (with typi- of the surface. rill et al., 1999). Such mechanisms operating on cal wavelengths ~100 m). In order for the curva- Measurement of curvature is highly scale initially en echelon fault arrays might explain ture analysis to detect and quantify larger scale dependant, and analysis will emphasize the the meso-scale corrugations seen in each of the structures, it is necessary to smooth the data shortest wavelength structures of a surface. Arkitsa fault panels. Large-scale corrugations more coarsely, to iron out small-scale features. Hence, it is important to remove any high- may also form by the progressive breakthrough The slight mottling effect seen in parts of the amplitude, short-wavelength noise from the of originally segmented fault systems (e.g., plots represents very small variations in curva- data, as this will mask genuine features of sur- Childs et al., 1995), which is probably the most ture values. These are caused by minor residual face geometry (e.g., Bergbauer and Pollard, likely cause of the decametric-scale warping at noise in the smoothed surface, and refl ect the 2003a). In this study we have minimized noise Arkitsa. Other mechanisms for the formation of limits of precision of the laser scanner equip- by resampling and smoothing the cleaned fault corrugated fault surfaces include reactivation of ment. In these specifi c examples, increasing the panels shown in Figures 4, 7, and 8. preexisting faults and fractures (e.g., Donath, sensitivity of the plotting method only empha- The results of the curvature analysis are 1962), and later folding of a preexisting fault sizes the small amounts of noise in the data, shown in Figure 9 for fault panels A and B. surface (e.g., Holm et al., 1994; Krabbendam without highlighting any other aspects of sur- Displaying the magnitude of normal curvature and Dewey, 1998). face geometry. shows the location of the larger corrugations and Gaussian curvature can be a useful param- The various curvature plots shown in Fig- cross faults (Figs. 9A, 9B; Animation 3), but is eter to highlight domes, basins, and saddles on ure 9 help to emphasize that the output of not particularly effective in showing more subtle the fault surface. It is defi ned as the product of curvature analysis can be heavily dependant geometrical features. Sensitivity of the plots can the principal curvatures, such that it is nonzero on the processing steps involved (particularly be increased (Figs. 9C, 9D) by plotting the abso- if there is curvature along both of the principal the way that the raw data are cleaned and lute values of normal curvature, and by using directions. Thus Gaussian curvature provides a smoothed), and the visualization parameters an upper threshold (such that the color scale quantitative measure of culminations on the sur- used for plotting the results. Quantifi cation

476 Geosphere, December 2009

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

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 Jones et al.

of curvature parameters is very useful, but Variation in Slip Vector Orientation measurements of slip lineations along the length because the curvature of fault surfaces is gen- of the fully exposed fault panels allows us to test erally quite subtle (e.g., compared with the cur- A number of previous studies have shown whether systematic variations in slip direction vature seen in folded bedding), more time and that the orientations of slip lineations can show are also present in this part of the Arkitsa fault. effort is needed to ﬁ ne-tune the variables used signiﬁ cant systematic variation along the strike At Arkitsa, fault striae that are subparallel to for analysis and visualization. For this reason it of a fault, particularly in parts of the fault that millimeter- to meter-scale corrugations are very may often be ineffective to use geological soft- are close to the tip zones or in relay zones (Rob- prominent on the surface of the fault panels in ware packages that have black box curvature erts, 1996; Maerten, 2000; Maniatis and Ham- outcrop and are also clearly visible in the TLS functionality that cannot be customized on a pel, 2008). The use of laser scanning to make point clouds (Figs. 5–9). In addition, there are case by case basis. a large number of precise, spatially referenced also other orientations of fault striae locally

A Panel AB Panel B

+0.338 +0.338

0.000 0.000

-0.338 -0.338

normal curvature normal curvature C D 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 Geosphere, December 2009

≥ ≥ preserved; however, these are less well devel- ordered such that E1 E2 E3, with E1 being the oped, and are below the centimeter resolution of primary orientation of slip vectors. The normal-

our TLS data. Further work is needed to relate ized eigenvalues (S1, S2, and S3) reﬂ ect the vari- these secondary slip directions to fault kinemat- ance of the vector data. For example, panel A A Panel A ics and regional tectonics, and in the following with an S1 value of 0.9981 shows that 99% of analysis we focus only on the most prominent the total data are consistent with the speciﬁ ed

set of slip lineations. orientation of E1, within statistical signiﬁ cance. Although the orientation of the fault surfaces Similar values of 0.9952 and 0.9971 have been shows signiﬁ cant variation within a single panel derived for panels B and C, respectively. The (Figs. 7 and 10), the orientation of the slip vec- relative size of each S value indicates the shape

tors on each panel is remarkably consistent of the distribution of the data. Data having S1 >> ≈ (Fig. 10). Along the polished fault panel sur- S2 S3 suggest a highly clustered distribution, as faces, the well-defi ned striations concentrate is the case for all three examined fault panels. along a north-northwest–south-southeast azi- In addition, the fabric shape parameter K can muth (mean azimuths 336°–344°), plunging be used to test for girdle distributions (K < 1.0), toward the north-northwest (mean plunge values clustered patterns (K > 1.0), or a uniformly ran- 51°–57°). This confi guration is consistent with dom distribution (K ≈ 1.0), while parameter C slightly oblique normal dip-slip motion with a determines the strength of the shape parameter n = 124 component of left-lateral displacement. Within K (Woodcock, 1977; Woodcock and Naylor, equal angle lower hemisphere each fault panel, typical variation from the aver- 1983). If C > 3.0, a preferred orientation and age plunge values is ~4°–9°, while variation shape are strongly indicated, and if C < 3.0, there B Panel B from mean slip azimuth is ~7°–15°. are weak orientation and shape fabric of the data. Our analysis is based on 902 slip vectors Values of K and C parameters for all three pan- picked from the digital surface (124 measure- els show marked clustering with strong preferred ments for panel A, 678 measurements for panel orientation and consistent fabric shape. B, and 100 for panel C). This is a comparable The statistical analysis of populations of though more extensive study to that presented in minor faults, usually based on measurements Kokkalas et al. (2007), for which 121 slip vec- of fault orientation, slip direction, and sense tor measurements were used. Statistical proper- of slip, is commonly used for areas of brittle ties for the entire data set are given in Table 1. deformation. Results from this type of fault slip The eigenvector analysis we applied is a non- analysis are typically used to make inferences parametric method that is particularly suitable about the stress fi eld at the time of deformation for analyzing directional data (Fisher et al., (assuming that the applied stress is homoge- 1987; Mardia and Jupp, 2000). Eigenvectors are neous). By defi nition, fault-slip data provide more direct information about kinematics than n = 678 the mechanics of faulting, with slickenlines equal angle lower hemisphere used as direct indicators of displacement directions on fault planes. In general, this analysis C Panel C 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 Animation 3. Enhanced virtual outcrop in fault slip data from each panel suggests that the which false color, based on the curvature Arkitsa fault zone is slipping in response to an properties of the surface, is projected onto n = 100 extensional stress fi eld with a north-south ori- the surface of fault panel B. See Figure 5 equal angle lower hemisphere entation of minimum principal stress (σ ), with and text for further details. To view the 3 a small deviation toward the north-northeast Figure 10. Equal-angle stereonets of slip avi animation, use software such as Quick- orientation calculated only for panel B (for fur- vector orientations for each of the fault pan- Time or Microsoft Media Player. If you are ther details, see Kokkalas et al., 2007). els A, B, and C. See Figure 5A for relative viewing the PDF of this paper or reading it Although the stereonet plots (Fig. 10) and position of the panels. offl ine, please visit http://dx.doi.org/10.1130/ statistical analysis (Table 1) show that the slip GES00216.SA3 or the full-text article on data are very tightly clustered, we have also ana- www.gsapubs.org to view the animation. lyzed the data to try to detect small, statistically

Geosphere, December 2009 479

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 Jones et al.

TABLE 1. ORIENTATION OF EIGENVECTORS (E1–E3) AND MAGNITUDE OF of the faults contains surface irregularities, cor- NORMALIZED EIGENVALUES (S1–S3) OF THE SLIP-VECTOR DATA FOR EACH OF rugations, fault bends, and crosscutting minor THE MAIN FAULT PANELS AT ARKITSA, CENTRAL GREECE faults, and deviates signifi cantly from planarity. E1 E2 E3 S1 S2 S3 K C Each panel shows a spread of at least ±20° in the Panel A 344/51 165/39 75/0 0.9981 0.0013 0.0007 10.62 7.26 orientation of the surface perpendicular to the 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 main slip vector. Normal curvatures delineate Note: The ratio of the eigenvalues indicates the shape of the distribution of data: K is clearly the location of corrugations on the sur- the fabric shape parameter, while parameter C determines the strength of parameter K faces of the panels. Gaussian curvature is very (Woodcock 1977; Woodcock and Naylor, 1983). The values for K and C indicate very low at this scale of analysis. Corrugations have 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 axes orientated parallel to the main slip direc- the LIDAR (light detection and ranging) data allow us to quantify the orientation of the tion, and have average wavelengths of ~4 m. slip vector across the whole of the exposed fault panels with high precision. The orientations of the slip vectors are tightly clustered, plunging ~55° north-northwest. This is consistent with the present-day regional signifi cant spatial variations in the orientation ing that the stress/strain tensor has not changed velocity fi eld derived from GPS measurements. of slip vectors across the fault panels. This is signifi cantly in orientation since the last major These results can be compared with measure- because, as noted by Jackson and McKenzie slip event recorded on the Arkitsa fault surface. ments given by Resor and Meer (2006). (1999, p. 5), some striae appear to curve toward Despite the close match between our fault data The high resolution of the LIDAR data helps the top of the exposed panels. Although we con- and regional strains inferred from GPS, we still to illustrate that the detailed geometry and cur with their fi eld observation (some parts of cannot be certain whether the precise orientation kinematics of the fault zone are highly com- panel B do show gradual variation in orientation of oblique slip we have measured at Arkitsa is plex, and that different complexities exist at of ~1°–2° between the top and bottom of the entirely indicative of regional strain, or whether different scales. It is not yet clear the degree outcrop), this variation appears to be of local- there are also additional, more localized effects to which this part of the Arkitsa fault zone ized extent, and apparently does not refl ect the close to the eastern tip of the Arkitsa–Kamena is typical of faults in general, or whether the whole of panel B, or the other exposed panels. Vourla fault zone (cf. Roberts and Ganas, 2000). high structural complexity is related to the Note, however, that we have not yet been able to regionally complicated rotational strains seen make quantitative measurements from the TLS CONCLUSIONS throughout the Aegean. In either case, much data that are precise enough (±0.5°) to refi ne the further work is needed to unravel these com- estimated variations in slip vector orientation Terrestrial laser scanning (i.e., LIDAR) is an plexities on the outcrop scale, and to relate the made by Jackson and McKenzie (1999). Further effective tool for acquiring detailed geometri- spatial variation in fault geometry to larger work is needed to optimize the processing of the cal data from exposed outcrop surfaces, and is scale fault kinematics and dynamics. TLS data prior to repicking of the fault striae. therefore ideal for studying surface irregularities and curvature of nonplanar faults. Field acquisi- ACKNOWLEDGEMENTS Comparison of Arkitsa Slip Vectors and tion of LIDAR data is rapid, though the millions We thank Phil Clegg and Tim Wright for assistance Regional Velocity Field of data points gathered will subsequently need to during fi eld work, Jonny Imber for discussion on be merged, cleaned, and fi ltered prior to further faulting, Mark Pearce for collaboration on curvature, The present-day regional stress fi eld, derived geological analysis. Various methods of analysis and Steve Waggott and Ruth Wightman for collabora- from fault-slip data, focal mechanisms of recent can be used to highlight different geometrical tion related to laser scanning. Thomas Dewez kindly earthquakes, and several GPS campaigns, rep- properties of the scanned surfaces. Roughness commented on the manuscript, and Emily Brodsky, Ioannis Papanikolaou, and Ramon Arrowsmith pro- resents north-south to north-northeast–south- provides a bulk measure of the overall surface vided useful reviews and editorial comment. southwest extension of central Greece (Roberts irregularity of a fault, while other methods allow and Jackson, 1991; Armijo et al., 1996; Rob- the spatial variability of fault geometry to be APPENDIX—MATHEMATICAL erts and Ganas, 2000; Hollenstein et al., 2008). studied. These methods include measuring the DEFINITIONS OF CURVATURE Recent GPS velocities and strain rate tensor cal- perpendicular distance of points from the mean culations, derived from the velocity fi eld, indi- plane, plotting variation in orientation relative to Curvature in Two Dimensions cate that velocity differences observed across the mean plane, and curvature analysis (includ- For a folded surface given by curve z = f(x), a the North Evia Gulf are small (2–4 mm/yr; ing normal, mean, and Gaussian curvature). typical defi nition of curvature (e.g., Ramsay, 1967, Hollenstein et al., 2008). This is consistent with Studying curvature allows quantitative measure- p. 346–347) is given by the low seismic activity of this rift structure. ments of fault bends and corrugations, although Regional strain rates can be resolved into nor- the measurements are highly scale sensitive and dz2 mal and shear components relative to the west- require care during data preparation and analy- 2 = dx northwest–trending Arkitsa–Kamena Vourla sis. Spatial variation in the various geometrical curvature, C 3 . (1) fault zone (Figs. 12 and 13 of Hollenstein et al., properties can be highlighted by projecting the ⎡ ⎛ dz ⎞ 2 ⎤ 2 ⎢ + ⎥ 2008). Normal strain rates represent low exten- resultant analytical values back onto the virtual 1 ⎜ ⎠⎟ ⎣⎢ ⎝ dx ⎦⎥ sional strain values in a north-northeast orienta- fault planes (Animation 3). 3D visualization tion, while shear strains indicate sinistral motion software is particularly effective in making it along the Arkitsa–Kamena Vourla fault zone. easy to compare the derived parameters with the In two dimensions (2D), fold hinges are defi ned as areas where C is a maximum or minimum. Crests and These geodetic calculations fi t closely with the underlying original fault geometry. troughs on the folded surface occur where the fi rst- north-south orientation of the stress tensor we Analysis of three well-exposed fault panels order derivative of the curve (i.e., the gradient, dz/dx) derive from the Arkitsa fault slip data, indicat- from Arkitsa (central Greece) shows that each is zero.

480 Geosphere, December 2009

Curvature in Three Dimensions Pearce et al. (2006) and Mynatt et al. (2007) pre- Childs, C., Watterson, J., and Walsh, J.J., 1995, Fault overlap sented a more complete derivation of the curvature zones within developing normal fault systems: Geo- equations given here. logical Society of London Journal, v. 152, p. 535–549, For the study of fault plane geometry, three- doi: 10.1144/gsjgs.152.3.0535. dimensional analysis of curvature is preferable to Clarke, P.J., and 13 others, 1998, Crustal strain in central 2D. For each point on a surface, r, given by Curvature Parameters Greece from repeated GPS measurements in the interval 1989–1997: Geophysical Journal Interna- The normal curvature is the magnitude of curva- tional, v. 135, p. 195–214, doi: 10.1046/j.1365-246X ture corresponding to the maximum and minimum .1998.00633.x. ⎡ x ⎤ Dewey, J.F., Hempton, M.R., Kidd, W.S.F., Saroglu, F., principal curvature directions, k and k . These values = ⎢ ⎥ 1 2 and Sengor, A.M.C., 1986, Shortening of continental r ⎢ y ⎥ , (2) help to deﬁ ne the 3D traces of hinge lines along folds lithosphere: The neotectonics of Eastern Anatolia—A or corrugations. It can sometimes be useful to visual- ⎣⎢zxy(,)⎦⎥ young collision zone, in Coward, M.P., and Ries, A.C., ize absolute (i.e. modulus) values of normal curva- eds., Collision tectonics: Geological Society of Lon- don Special Publication 19, p. 3–36. tures (|kn|), as shown in Figures 9C and 9D. Donath, F.A., 1962, Analysis of basin-range struc- there are two directions in which the amount of nor- Gaussian curvature, G, is the product of the two principal curvature values: ture, south-central Oregon: Geological Soci- mal curvature is a maximum or minimum. These ety of America Bulletin, v. 73, p. 1–16, doi: are the maximum and minimum principal curvature 10.1130/0016-7606(1962)73[1:AOBSSO]2.0.CO;2.

directions, k1 and k2. The magnitude and orientation Doutsos, T., and Kokkalas, S., 2001, Stress and deforma- Gkk= . (9) tion in the Aegean region: Journal of Structural Geol- of k1 and k2 are given by the characteristic roots of the 12 shape operator L: ogy, v. 23, p. 455–472, doi: 10.1016/S0191-8141 Analysis of Gaussian curvature indicates whether a (00)00119-X. Doutsos, T., and Poulimenos, G., 1992, Geometry and kine- surface is developable (Lisle, 1992, 1994), and hence LIII= / , (3) matics of active faults and their seismotectonic sig- can be useful to reveal domes, basins and saddles. nifi cance in the western Corinth-Patras rift (Greece): Mean curvature, M, is the average of the two prin- Journal of Structural Geology, v. 14, p. 689–699, doi: where I and II are the two fundamental forms of cipal curvature values (Roberts, 2001; Bergbauer and 10.1016/0191-8141(92)90126-H. the surface (Bergbauer and Pollard, 2003a, 2003b; Pollard, 2003a, 2003b): Fernández, O., 2005, Obtaining a best fi tting plane through Pearce et al., 2006; Mynatt et al., 2007). The fi rst 3D georeferenced data: Journal of Structural Geology, fundamental form is given by v. 27, p. 855–858, doi: 10.1016/j.jsg.2004.12.004. =+ Mkk()/122 . (10) Ferrill, D.A., Stamatakos, J.A., and Sims, D., 1999, Normal fault corrugation: Implications for growth and seismicity of active normal faults: Journal of Structural Geol- ⎡ ⋅⋅⎤ ogy, v. 21, p. 1027–1038, doi: 10.1016/S0191-8141 rrxx rr xy I = ⎢ ⎥ , (4) Curvature Properties (99)00017-6. ⋅⋅ ⎣⎢rrxy rr yy⎦⎥ Fisher, N.I., Lewis, T., and Embleton, B.J.J., 1987, Statisti- cal analysis of spherical data: Cambridge, Cambridge where Since curvature is a function of the second derivative (equations 1, 3, and 7), it is an invariant prop- University Press, 344 p. ⎡ ⎤ Foley, J.D., van Dam, A., Feiner, S.K., and Hughes, J.F., 1997, erty of the curve. This means that curvature values Computer graphics: Principles and practice (second edi- ⎢ 1 ⎥ are independent of the dip of the fault surface. In ∂ ⎢ ⎥ tion): Boston, Massachusetts, Addison-Wesley, 1175 p. = r = contrast, the positions of crests and troughs are not Ganas, A., Roberts, G., and Memou, P., 1998, Segment rx ⎢ 0 ⎥ , (5) ∂x ⎢ ⎥ invariant features, but are closely dependent on the boundaries, the 1894 ruptures and strain patterns ∂z orientation of the fault. For this reason, the method of along the Atalanti fault, central Greece: Journal of ⎢ ⎥ Geodynamics, v. 26, p. 461–486, doi: 10.1016/S0264 ⎣∂x ⎦ analysis of spatial variation of fault surface orientation shown in Figures 7 and 8 is done relative to the -3707(97)00066-5. mean fault plane. Healy, D., Jones, R.R., and Holdsworth, R.E., 2006, New insights into the development of brittle shear fractures and from a 3-D numerical model of microcrack interaction: ⎡ ⎤ REFERENCES CITED Earth and Planetary Science Letters, v. 249, p. 14–28, ⎢ ⎥ 0 doi: 10.1016/j.epsl.2006.06.041. ∂r ⎢ ⎥ Ahlgren, S., Holmlund, J., Griffi ths, P., and Smallshire, R., Hollenstein, C., Muller, M.D., Geiger, A., and Kahle, H.-G., = = ⎢ ⎥ 2002, Fracture model analysis is simple: American 2008, Crustal motion and deformation in Greece from ry 1 . (6) ∂y ⎢ ⎥ Association of Petroleum Geologists Explorer, Sep- a decade of GPS measurements, 1993–2003: Tec- ⎢ ∂z ⎥ tember, 2002. tonophysics, v. 449, p. 17–40, doi: 10.1016/j.tecto ⎣⎢∂y ⎦⎥ Ambraseys, N., and Jackson, J., 1990, Seismicity and associ- .2007.12.006. ated strain of central Greece between 1890 and 1988: Holm, D.K., Fleck, R.J., and Lux, D.R., 1994, The Death Geophysical Journal International, v. 101, p. 663–708, Valley turtlebacks reinterpreted as Miocene-Pliocene folds of a major detachment surface: Journal of Geol- The second fundamental form is doi: 10.1111/j.1365-246X.1990.tb05577.x. Armijo, R., Meyer, B., King, G., Rigo, A., and Papanasta- ogy, v. 102, p. 718–727. siou, D., 1996, Quaternary evolution of the Corinth rift Jackson, J., and McKenzie, D., 1999, A hectare of fresh stri- and its implications for the late Cenozoic evolution of ations on the Arkitsa fault, central Greece: Journal of ⎡ ∂2 ∧∧∂2 ⎤ the Aegean: Geophysical Journal International, v. 126, Structural Geology, v. 21, p. 1–6, doi: 10.1016/S0191 r ⋅ r ⋅ p. 11–53, doi: 10.1111/j.1365-246X.1996.tb05264.x. -8141(98)00091-1. ⎢ 2 n n⎥ ⎢ ∂xxy∂∂ ⎥ Bellian, J.A., Kerans, C., and Jennette, D.C., 2005, Digital Jones, R.R., McCaffrey, K.J.W., Wilson, R.W., and Hold- II = , (7) outcrop models: Applications of terrestrial scanning sworth, R.E., 2004, Digital fi eld data acquisition: ⎢ 22∧∧⎥ ∂ r ∂ r lidar technology in stratigraphic modelling: Jour- Towards increased quantifi cation of uncertainty during ⎢ ⋅ n ⋅ n ⎥ nal of Sedimentary Research, v. 75, p. 166–176, doi: geological mapping, in Curtis, A., and Wood, R. eds., ⎣⎢∂∂xy∂ y2 ⎦⎥ 10.2110/jsr.2005.013. Geological prior information: Geological Society of Bergbauer, S., and Pollard, D.D., 2003a, How to calculate London Special Publication 239, p. 43–56. normal curvatures of sampled geological surfaces: Jones, R.R., Wawrzyniec, T.F., Holliman, N.S., McCaffrey, where the unit normal nˆ is given by Journal of Structural Geology, v. 25, p. 277–289, doi: K.J.W., Imber, J., and Holdsworth, R.E., 2008, Describ- 10.1016/S0191-8141(02)00019-6. ing the dimensionality of geospatial data in the Earth sci- Bergbauer, S., and Pollard, D.D., 2003b, How to calculate ences—Recommendations for nomenclature: Geosphere, ∂rr∂ normal curvatures of sampled geological surfaces: v. 4, p. 354–359, doi: 10.1130/GES00158.1. × Erratum: Journal of Structural Geology, v. 25, p. 2167, Kokkalas, S., Xypolias, P., Koukouvelas, I., and Doutsos, ∧ ∂xy∂ doi: 10.1016/S0191-8141(03)00100-7. T., 2006, Postcollisional contractional and exten- n = . (8) ∂ ∂ Brown, S.R., and Scholz, C.H., 1985, Broad bandwidth sional deformation in the Aegean region, in Dilek, Y., rr× study of the topography of natural rock surfaces: Jour- and Pavlides, S., eds., Postcollisional tectonics and ∂xy∂ nal of Geophysical Research, v. 90, p. 2575–2582. magmatism in the Mediterranean region and Asia:

Geosphere, December 2009 481

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021 Jones et al.

Geological Society of America Special Paper 409, p. nal of Structural Geology, v. 29, p. 1256–1266, doi: physical Research, v. 105, p. 23,443–23,462, doi: 97–123, doi: 10.1130/2006.2409(06). 10.1016/j.jsg.2007.02.006. 10.1029/1999JB900440. Kokkalas, S., Jones, R.R., McVaffrey, K.J.W., and Clegg, Papanikolaou, D.J., and Royden, L.H., 2007, Disruption of Roberts, S., and Jackson, J., 1991, Active normal faults in P., 2007, Quantitative fault analysis at Arkitsa, central the Hellenic arc: Late Miocene extensional detach- central Greece: An overview, in Holdsworth, R.E., and Greece, using terrestrial laser-scanning (LiDAR): Geo- ment faults and steep Pliocene-Quaternary normal Turner, J.P., compilers, Extensional tectonics: Faulting logical Society of Greece Bulletin, v. 40, p. 1959–1972. faults—Or what happened at Corinth?: Tectonics, and related processes: Key Issues in Earth Sciences, Krabbendam, M., and Dewey, J.F., 1998, Exhumation of v. 26, TC5003, doi: 10.1029/2006TC002007. Volume 2: London, Geological Society of London, UHP rocks by transtension in the Western Gneiss Pearce, M.A., Jones, R.R., Smith, S.A.F., McCaffrey, p. 151–168. Region, Scandinavian Caledonides, in Holdsworth, K.J.W., and Clegg, P., 2006, Numerical analysis of fold Sagy, A., Brodsky, E.E., and Axen, G.J., 2007, Evolution R.E., et al., eds., Continental transpressional and trans- curvature using data acquired by high-precision GPS: of fault-surface roughness with slip: Geology, v. 35, tensional tectonics: Geological Society of London Spe- Journal of Structural Geology, v. 28, p. 1640–1646, p. 283–286, doi: 10.1130/G23235A.1. cial Publication 135, p. 159–181. doi: 10.1016/j.jsg.2006.05.010. Stiros, S.C., Arnold, M., Pirazzoli, P.A., Laborel, J., Laborel, Lichti, D., and Jamtsho, S., 2006, Angular resolution of Power, W.L., Tullis, T.E., Brown, S.R., Boitnott, G.N., and F., and Papageorgiou, S., 1992, Historical coseismic terrestrial laser scanners: Photogrammetric Record, Scholz, C.H., 1987, Roughness of natural fault sur- uplift on Euboea Island, Greece: Earth and Planetary v. 21, no. 114, p. 141–160, doi: 10.1111/j.1477-9730 faces: Geophysical Research Letters, v. 14, p. 29–32, Science Letters, v. 108, p. 109–117, doi: 10.1016/0012 .2006.00367.x. doi: 10.1029/GL014i001p00029. -821X(92)90063-2. Lisle, R.J., 1992, Constant bed-length folding: Three-dimen- Pringle, J.K., Howell, J.A., Hodgetts, D., Westerman, A.R., Wawrzyniec, T.F., Jones, R.R., McCaffrey, K.J.W., Imber, sional geometrical implications: Journal of Structural and Hodgson, D.M., 2006, Virtual outcrop models of J., Holliman, N.S., and Holdsworth, R.E., 2007, Intro- Geology, v. 14, p. 245–252, doi: 10.1016/0191-8141 petroleum reservoir analogues: A review of the current duction: Unlocking 3D Earth systems—Harnessing (92)90061-Z. state-of-the-art: First Break, v. 24, p. 33–42. new digital technologies to revolutionize multi-scale Lisle, R.J., 1994, Detection of zones of abnormal strains in Ramsay, J.G., 1967, Folding and fracturing of rocks: New geological models: Geosphere, v. 3, p. 406–407, doi: structures using Gaussian curvature analysis: Ameri- York, McGraw-Hill, 568 p. 10.1130/GES00156.1. can Association of Petroleum Geologists Bulletin, Renard, F., Voisin, C., Marsan, D., and Schmittbuhl, J., White, P.D., and Jones, R.R., 2008, A cost-effi cient solution v. 78, p. 1811–1819. 2006, High resolution 3D laser scanner measurements to true color terrestrial laser scanning: Geosphere, v. 4, Maerten, L., 2000, Variation in slip on intersecting normal of a strike-slip fault quantify its morphological anisot- p. 564–575, doi: 10.1130/GES00155.1. faults: Implications for paleostress inversion: Journal ropy at all scales: Geophysical Research Letters, v. 33, Woodcock, N.H., 1977, Specifi cation of fabric shapes of Geophysical Research, v. 270, p. 197–206. L04305, doi: 10.1029/2005GL025038. using an eigenvalue method: Geological Soci- Maniatis, G., and Hampel, A., 2008, Along-strike varia- Resor, P.G., and Meer, V.E. 2006, Characterizing geometry, ety of America Bulletin, v. 88, p. 1231–1236, doi: tions of the slip direction on normal faults: Insights kinematics, and fracturing of a wavy fault surface, 10.1130/0016-7606(1977)88<1231:SOFSUA>2.0.CO;2. from three-dimensional fi nite-element models: Journal Arkitsa, Greece: Eos (Transactions, American Geo- Woodcock, N.H., and Naylor, M.A., 1983, Random- of Structural Geology, v. 30, p. 21–28, doi: 10.1016/j physical Union), v. 87, abs. T21C–0440. ness testing in three-dimensional orientation data: .jsg.2007.10.002. Roberts, G.P., 1996, Variation in fault-slip directions along Journal of Structural Geology, v. 5, p. 539–548, doi: Mardia, K.V., and Jupp, P.E., 2000, Directional statistics: active and segmented normal fault systems: Journal of 10.1016/0191-8141(83)90058-5. Chichester, Wiley, 429 p. Structural Geology, v. 18, p. 835–845, doi: 10.1016/ McCaffrey, K.J.W., Jones, R.R., Holdsworth, R.E., Wilson, S0191-8141(96)80016-2. R.W., Clegg, P., Imber, J., Holliman, N., and Trinks, Roberts, A., 2001, Curvature attributes and their applica- I., 2005, Unlocking the spatial dimension: digital tech- tion to 3D interpreted horizons: First Break, v. 19, nologies and the future of geoscience fi eldwork: Geo- p. 85–100, doi: 10.1046/j.0263-5046.2001.00142.x. logical Society of London Journal, v. 162, p. 927–938, Roberts, G.P., and Ganas, A., 2000, Fault-slip directions doi: 10.1144/0016-764905-017. in central and southern Greece measured from stri- MANUSCRIPT RECEIVED 27 OCTOBER 2008 Mynatt, I., Bergbauer, S., and Pollard, D.D., 2007, Using ated and corrugated fault planes: Comparison with REVISED MANUSCRIPT RECEIVED 11 MAY 2009 differential geometry to describe 3-D folds: Jour- focal mechanism and geodetic data: Journal of Geo- MANUSCRIPT ACCEPTED 11 MAY 2009

482 Geosphere, December 2009

Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/5/6/465/3338067/i1553-040X-5-6-465.pdf by guest on 28 September 2021