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Roughness of fault surfaces over nine decades of length scales 1
Thibault Candela 1, François Renard
1,2, Yann Klinger
3, Karen Mair
2, Jean Schmittbuhl
4, and Emily 2
E. Brodsky 5 3
4
1ISTerre, University of Grenoble I, CNRS, OSUG, BP 53, 38041 Grenoble, France. 5
E-mail: [email protected]; [email protected] 6
2Physics of Geological Processes, University of Oslo, Oslo, Norway. E-mail: [email protected] 7
3Institut de Physique du Globe de Paris, Sorbonne Paris cité, Univ. Paris Diderot, UMR 7154 CNRS, 8
1 rue Jussieu, F-75005 Paris, France. E-mail: [email protected] 9
4UMR 7516, Institut de Physique du Globe de Strasbourg, Strasbourg, France. E-mail: 10
5Department of Earth and Planetary Sciences, University of California–Santa Cruz, Santa Cruz, 12
California 95064, USA. E-mail: [email protected] 13
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Abstract 15
We report on the roughness measurements of five exhumed faults and eight surface ruptures over a 16
large range of scales: from 50 micrometers to 50 km. We used three scanner devices (LiDAR, laser 17
profilometer, white light interferometer), spanning complementary scale ranges from 50 micrometers 18
to 10 m, to measure the 3-D topography of the same objects, i.e. five exhumed slip surfaces (Vuache-19
Sillingy, Bolu, Corona Heights, Dixie Valley, Magnola). A consistent geometrical property emerges 20
as the morphology of the slip surfaces shows a straight line covering five decades of length-scales in a 21
log-log plot where axes are fault roughness and spatial length scale. The observed fault roughness is 22
scale dependent, with a common self-affine behavior described by four parameters: two power-law 23
exponents H, constant among all the faults studied and slightly anisotropic ( 07.058.0// H in the 24
slip direction and 04.081.0 H perpendicular to it), and two pre-factors showing a quite large 25
variability. For larger scales between 200 m and 50 km, we have analyzed the 2-D roughness of the 26
surface rupture of eight major continental earthquakes. These ruptures also show self-affine behavior 27
( 1.08.0 RH ), which is consistent with the slip-perpendicular behavior of the smaller-scale 28
measurements. We show that small degrees of non-alignment between the slip orientation and the 29
exposed trace result in sampling the slip-perpendicular geometry. Although a data gap exists between 30
the scanned fault scarps and rupture traces, the measurements are consistent within the error bars with 31
a single self-affine scaling exponent in the slip-perpendicular direction, i.e. consistent dimensionality, 32
over nine decades of length scales. 33
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1. Introduction 40
Faults appear most commonly as more or less continuous linear breaks at the surface of the 41
Earth, but their traces show wavy irregularities at all scales (Brown and Scholz, 1985). These 42
irregularities, which we call roughness, control the geometry (Power et al., 1987), mechanics, and 43
transport properties of fault zones (Power and Durham, 1997) and contribute to their 3D architecture 44
(Bistacchi et al., 2010; Faulkner et al., 2010). Fault roughness may control several faulting processes 45
and parameters such as the total resistance to slip, the aseismic versus seismic behavior, the alteration 46
of shear resistance during sliding, the magnitude of stress concentration and heterogeneity in the fault 47
zone (Chester and Fletcher, 1997; Chester and Chester, 2000), and the deformation and damage of the 48
rock on either side of the fault (Arvid and Fletcher, 1994; Dieterich and Smith, 2008; Griffith et al., 49
2010). 50
The morphology of fault planes controls the dynamics of faulting and slip. Based on the results 51
of kinematic source inversion models that reconstitute the spatio-temporal evolution of slip during an 52
earthquake, several studies show that both coseismic slip and stress appear to be very heterogeneous 53
along the fault plane (Bouchon, 1997; Mai and Beroza, 2002). A possible explanation is that the fault 54
plane is rough and asperities concentrate stress and slip heterogeneities at various spatial scales 55
(Schmittbuhl et al., 2006; Candela et al., 2011a, 2011b). Earthquake activity and aseismic creep can 56
create and destroy this roughness that causes the heterogeneity of the stress field in the fault zones. 57
Other studies have also shown the importance of non-planar structures in the rupture propagation 58
(e.g. Aochi and Madariaga, 2003) and the close relationship between the rupture geometry and its 59
propagation velocity (Vallée et al., 2008; Bouchon et al., 2010). Numerical models of earthquake 60
rupture and strong motion need accurate 3-D morphological models of fault surfaces to improve 61
simulations of rupture scenarios. 62
As direct observations are not possible at the depths of earthquake nucleation, data of exhumed 63
fault scarps (Power et al., 1987; Renard et al., 2006; Sagy et al., 2007; Candela et al., 2009; Brodsky 64
et al., 2011 and references therein) or earthquake surface ruptures (Wesnousky, 2006, 2008; Klinger, 65
2010) provide the means to characterize fault plane morphology over a wide range of spatial scales. 66
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Pioneering fault roughness measurements (Brown and Scholz, 1985; Power et al., 1987; Power 67
and Tullis, 1991; Schmittbuhl et al., 1993) performed on exhumed scarps found that their roughness 68
cannot be described by a single number such as the standard deviation of the roughness amplitude. 69
Rather, fault surface topography was measured as non-stationary and more particularly self-affine 70
fractal, the amplitude of the topography increasing with the wavelength under consideration. 71
A 2-D rough profile (Figure 1) is self-affine if it remains statistically invariant under the scaling 72
transformation zzxx H , (Feder, 1988; Meakin, 1998), where x is the coordinate 73
along the 2-D profile, z the roughness amplitude and H the Hurt exponent (or roughness 74
exponent). If the power law scaling exponent lies in the range 10 H , different magnification 75
factors will be needed in the directions parallel and perpendicular to the profile for a small portion of 76
the profile to appear statistically similar to the entire profile (Figure 1). As a consequence, in this case, 77
the slope at large scales along a self-affine profile scales as 1
/
H
xxzs , and tends to flatten 78
for long wavelengths, suggesting a significant role of the small spatial scales (Schmittbuhl et al., 79
1995a). Since the exponent H does not contain any information on the amplitude of the signal, and is 80
related only to the progression in space, a second parameter is needed to describe fully the power law 81
and the amplitude of the scaling behavior. In this study, we call this parameter the pre-factor of the 82
power law. 83
Pioneering studies measured 2-D profiles on exhumed fault planes using mechanical 84
profilometers (Brown and Scholz, 1985; Power et al., 1987; Power and Tullis, 1991). Recently, with 85
the development of a new generation of 3-D laser scanners, fast and accurate acquisitions of 86
topographic data are now available, allowing more fault surfaces to be characterized (Renard et al., 87
2006; Sagy et al., 2007; Candela et al., 2009; Brodsky et al., 2011). These first results confirmed the 88
roughness of several faults over a wide range of spatial scales and their morphologically anisotropic 89
property with a Hurst exponent smaller in the slip direction than perpendicular to it. 90
In the present study, we investigate roughness properties of five fault surfaces in different 91
geological contexts (with varying lithology, accumulated displacement, tectonic regime) using three 92
independent scanner devices (a Light Detection And Ranging apparatus – also called LiDAR, a laser 93
5
profilometer, and a white light interferometer), spanning a range of scales from 5.10-5
m to 10 m. In 94
addition, we analyzed the geometry of eight map-scale high resolution rupture traces of large 95
continental earthquakes, giving access to a range of scales from 200 m to 50 kilometers. 96
The availability of LiDAR has greatly facilitated the measurement of topography data from 97
natural fault surfaces. However, the level of noise inherent in the measurements of these instruments, 98
which is estimated by scanning flat reference surfaces (see Section 3), requires caution when 99
interpreting the results of LiDAR topography measurements. In contrast to the noise level of the 100
LiDAR instruments, the noise level inherent in the laboratory laser profilometer and the white light 101
interferometer (WLI) profilometer is much lower, well below the magnitude of the surface 102
topography. For this reason the laser profilometer and WLI data can be considered to be essentially 103
noise free for the measurements presented here. 104
We characterize the self-affine scaling of both data sets (scanned fault surfaces and rupture 105
traces) using the same statistical tool, i.e. the Fourier power spectrum analysis. This is a robust 106
technique well-suited for characterizing self-affine roughness of fault zones (Candela et al., 2009). A 107
self-affine model implies that the Fourier spectrum as a function of either the spatial frequency or 108
wavelength plots as a linear trend in log-log coordinates. Two parameters describe such a self-affine 109
model in the spectral domain: the slope of the power spectrum and its pre-factor (i.e. the intercept at a 110
given length scale) on a log-log plot of the Fourier spectrum. The slope (directly proportional to H ) 111
describes how the roughness changes with scale, while the pre-factor determines the magnitude of the 112
surface roughness at a given scale (Mandelbrot, 1983, p. 350, Power and Tullis, 1991). Both 113
parameters are necessary and sufficient to describe a self-affine geometry. We compute these two 114
parameters for the fault surfaces and rupture traces in order to decipher if a global tendency emerges 115
and/or to characterize the fluctuations. 116
In Section 2, the characteristics of both the scanned fault surfaces and rupture traces are described. 117
Section 3 is devoted to the presentation and the application of the roughness analysis method. In 118
Section 4, we present results that indicate an anisotropic self-affine model with two different Hurst 119
exponents in perpendicular directions and varying pre-factors, could fit each fault scarp data set, 120
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despite different geological contexts. Secondly, we highlight a similar self-affine regime that fits the 121
eight earthquake rupture traces. In Section 5 we discuss the possibility that a unique anisotropic self-122
affine geometrical model is maintained from the micrometric scale to the map-scale earthquake 123
surface rupture. The variability of the pre-factors for each data set (fault surfaces and ruptures traces) 124
is discussed with respect to the different geological contexts sampled and more particularly the 125
accumulated slip. Physical processes at work in the generation of the observed roughness are also 126
debated. 127
128
2. Fault roughness data 129
2.1. Exhumed fault scarps 130
We have analyzed five natural fault surfaces which were selected because of their particularly 131
well preserved slip surfaces, large exposures and few pits or weathering patterns. Existing data sets on 132
the Vuache-Sillingy (Renard et al., 2006; Candela et al., 2009, Angheluta et al., 2011) and Magnola 133
(Candela et al., 2009) faults have been updated and extended with three new faults (Corona Heights, 134
Dixie Valley, and Bolu). A complete list of our fault data, including GPS locations, nature of the rock, 135
direction of motion, and estimated finite geological offset, is given in Table 1. The finite geological 136
offset is always the most difficult of these parameters to estimate. In fact, if the total offset of a fault 137
zone can be estimated from the displacement of geological markers, the total displacement could have 138
occurred on several parallel slip surfaces, making it difficult to estimate the total slip on a given slip 139
surface. For each of our scanned surfaces, we will consider a range of slips bracketed by the extreme 140
values: the total slip of the fault zone and a minimum slip we could estimate from field observations. 141
The Corona Heights and Dixie Valley faults cut through silicate rocks. The Dixie Valley (Basin 142
and Range province in Nevada) fault has a mainly normal slip component and cross-cuts through 143
rhyolite. Chemical changes during faulting at depth have altered the mineralogy and chemical 144
composition of the rock in the fault zone (Power et al., 1987; Power and Tullis, 1989, 1992). The 145
material that forms the fault consists almost entirely of secondary quartz, a mineral which is 146
extremely resistant to weathering and allows extremely good preservation of the slip surfaces. Using 147
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gravity studies combined with reflection seismology studies (Okaya and Thompson, 1985), Power and 148
Tullis (1989) estimate that the total normal slip is probably between 3 and 6 km. Geological and 149
mineralogical constraints indicate that the slickenside surface formed at depths of less than 2 km and 150
temperatures less than Co270 (Power and Tullis, 1989). Major historical earthquakes and 151
microearthquakes occurred on the region of the studied fault scarp (Wallace and Whitney, 1984; 152
Dozer, 1986). Additionally, Power and Tullis (1989) have argued that seismic faulting played a role in 153
the development of the slickenside surfaces, based on textural features they described in the fault 154
surface materials. 155
The Corona Heights strike-slip fault (Figure 2), located in the Castro district of San Francisco, 156
cross-cuts brown Franciscan cherts and was exposed by post-1906 earthquake anthropogenic 157
quarrying. The relatively recent exposure of the fault and the high resistance of cherts to weathering 158
allows for excellent preservation of the slip surface (Figure 2). In most places on the outcrop, an 159
anastomosing set of slip surfaces is present (Figure 2). Individual patches of the fault surfaces may 160
have been activated at different times and different depths. Although the total slip for the fault zone as 161
a whole could be large (> 1000 m), individual surfaces have recorded smaller (~1 m) displacements 162
which are difficult to precisely estimate due to the absence of well-defined structural markers. 163
The other three faults offset limestone rocks (Vuache-Sillingy in the French Alps, Magnola in the 164
Appenines, and Bolu in Turkey). The Vuache-Sillingy fault is an active strike-slip fault system in the 165
western part of the French Alps and has accumulated a total displacement in the kilometer range 166
(Thouvenot, 1998). The fault surface we analyzed (Renard et al., 2006; Candela et al., 2009) lies on a 167
short segment of this fault system, where the accumulated slip was small, in the range of 3010 168
meters, as estimated on aerial photographs. The fault plane was exhumed in the 1990s by quarrying 169
and, as a consequence, the LiDAR measurements were performed on fresh, vegetation free surfaces, 170
where weathering is minimal. The Magnola fault (Candela et al., 2009), in the Fuccineo area, is part 171
of the extensive fault system in central Apennines, Italy. This 15 km long normal fault shows 172
microseismic activity and presents an average vertical displacement larger than 500 meters. The site 173
we study has been recently exhumed (Palumbo et al., 2004; Carcaillet et al., 2008) with less alteration 174
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by weathering than older exhumed portions of the fault. The Bolu fault is part of the North Anatolian 175
strike-slip fault system. The study area (Figure 3) is a part of the section that ruptured during the 1944 176
earthquake (Kondo et al., 2005; Kondo et al., 2010; Barka, 1996). The small vertical component of 177
the motion (~1 m), compared to the dominant horizontal strike-slip motion (~3.5 m), was responsible 178
for the partial exhumation of the fault plane (Figure 3) during the 1944 earthquake (Barka, 1996). 179
More recently, anthropogenic activity (excavation for a garbage dump) also contributed to the 180
exhumation of the outcrop. The total geological offset of the North Anatolian fault can be of about 181
2585 km (Hubert-Ferrari et al., 2002), but it is not easy to define the slip accommodated 182
specifically on each individual sub-parallel slip surface constituting the fault zone of the Bolu 183
segment. Paleo-seismological investigations on the Bolu segment give a lower bound of 184
approximately 20 m (Kondo et al., 2005; Kondo et al., 2010). 185
The five faults studied show slip activity during the Quaternary. The Bolu fault records both the 186
propagation and termination of the 1944 earthquake. The Dixie Valley outcrop lies north of a segment 187
that broke in 1954; it is the same outcrop studied by Power and Tullis (1987). For three of these 188
faults, the slip surfaces were exhumed recently by anthropogenic activity during the 20th century 189
(Vuache-Sillingy, Bolu, and Corona Heights). These three surfaces were therefore exposed to 190
atmospheric alteration for only a short period of time and therefore their roughness reflects only 191
faulting processes. The Dixie Valley fault was exhumed by normal faulting activity combined with 192
local quarrying. In this region, desert weather conditions and the silicate rocks result in slow chemical 193
alteration. For this fault, we have chosen surfaces free of mechanical erosion, where the mirror-like 194
polishing due to the latest slips activity was still present. Finally, the Magnola slip surfaces were those 195
for which the alteration was the most important. For this fault, we selected slip surfaces that were 196
exhumed by the quaternary vertical activity of this normal fault, and for which the erosion was 197
minimal (i.e. surface away from a stream that could have increased the erosion rate); however we 198
cannot discount that weathering has altered the roughness properties of these surfaces. We will show 199
later that the roughness properties of the Magnola fault do not deviate significantly from those of the 200
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other faults and interpret this observation as evidence that weathering processes did not influence our 201
roughness analyses at the spatial scales we considered. 202
Even if it is difficult to accurately determine under which conditions at depth (confining 203
pressure, temperature, strain rate and chemical environment) fault surfaces were built, the five fault 204
surfaces studied here sample a range of different control parameters (total fault zone displacement 205
(10’s of meters to ~10’s of kilometers), lithology (rhyolite, chert and limestone), tectonic regime 206
(strike-slip, oblique and normal) which possibly could have controlled fault surface roughness. 207
208
2.2. Scanner devices and digital elevation models of fault roughness 209
Fault surface topography was scanned in the field using five different 3-D portable LiDAR laser 210
scanners that use the time of flight of a light beam to accurately measure distances. The laser scanner 211
records the topography of each exposed fault surface by collecting a cloud of points whose three 212
dimensional coordinates correspond to points on the fault surface (Renard et al., 2006; Sagy et al., 213
2007; Candela et al., 2009; Resor and Meer, 2009; Zhanyu et al., 2010). 214
The actual point spacing depends on the distance between the target and the scanner and a 215
chosen angular spacing. For each fault outcrops, fresh sub-surfaces were selected and scanned for 216
higher resolution acquisition (Figure 2, 3). 217
Our data sets cover surface scales from 21 m to
2800 m at a spatial length scale resolution x 218
from mm1 to mm30 . This spatial length scale resolution x corresponds to the point spacing after 219
the data processing (see Section 3), and is systematically taken to be twice as large as the average 220
irregular spacing during the acquisitions, that is from mm5.0 to mm15 . The actual precision in the 221
spatial positioning is estimated to be at most half the original average spacing, that is mm25.0 to 222
mm5.7 . The height precision achievable depends on scanning conditions and is closely related to 223
the spatial length scale resolution and to the roughness amplitude of the surface. In Table 1, z 224
represents the estimated amplitude of the instrumental noise. 225
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The scans were combined with digital photographs to distinguish clear slip surfaces from eroded 226
areas. This manual cleaning of the extremely large datasets (several tens of millions of points) was 227
completed using 3-D Reshaper software, a point cloud editor and visualization tool. Once all non-fault 228
features such as trees, grass, or anthropogenic structures were removed, the whole fault scarp or 229
selected smaller patches were analyzed (Figure 2, 3). Typically, in our data sets, less than 5% of 230
spurious points were removed from the raw scanner data. As pointed out by Candela et al. (2009), the 231
estimation of the fault surface properties was not significantly biased by the presence of randomly 232
distributed holes and missing data in the cloud of points (see Appendix A.1 for a quantitative analysis 233
of the bias inherent to data acquisition in the estimation of the geometrical properties of fault 234
surfaces). 235
In the laboratory, we used a home-made laser profilometer (Méheust, 2002), to measure samples 236
of the fault surfaces (between 2200 mm and
21000 mm , Figure 2 and 3), set on a 2-axis moving 237
table. Each surface is scanned by geometric triangulation, measuring the distance between the sample 238
and a laser head (Schmittbuhl et al., 2008; Candela et al., 2009). One difference between the LiDAR 239
and the profilometers is that with the laser profilometer, the data are regularly spaced. The spatial 240
length scale resolution x is equal here to the horizontal step, i.e. 20 microns. The actual precision in 241
the spatial positioning is 1 micron and the vertical resolution ( z ) is better than 1 micron. The 242
reliability and accuracy of the cloud of points obtained with this laser profilometer required that only 243
few spurious points were removed (less than 0.01%). 244
At the millimeter scale, the topography of several slip surfaces (between25.0 mm and
240 mm , 245
Figure 2 and 3) was measured using white light interferometry microphotography (Dysthe et al., 246
2002). This is done with a microscope that uses a broad-band white light source and that is coupled to 247
a Michelson interferometer. A reference arm creates interference fringes with maximum intensity at 248
equal optical path lengths of the imaging beam and reference beam. By vertical movement of the 249
sample and simultaneous image capturing, the interference, intensity envelope, and thereby the 250
relative height of the imaged surface at each pixel is determined with a resolution of nmz 3 . The 251
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horizontal resolution depends on the lens used; the highest magnification it is at the diffraction limit 252
of white light, of about 5.0 micron. In the present study, we have selected horizontal steps ( x ) 253
between 1 and 2 microns (Table 1). The actual precision in the spatial positioning is estimated to be 254
025.0 micron. As for data acquired with the laser profilometer, the cloud of points obtained are 255
regularly spaced and only some spurious points have been manually removed. The whole suite of 256
characteristics of the scanners devices and digital elevation models (spatial precision, resolution x , 257
noise on the data z ) used in this study are shown in Table 1. 258
As examples, two fault surfaces (Corona Heights fault, Figure 2 and Bolu fault, Figure 3) have been 259
selected to illustrate the topographic data acquired at all scales with the three techniques presented. 260
These fault surfaces are composed of many discrete slip surfaces delimiting bumpy lenses elongated 261
in the direction of slip (Figure 2). These multi-scale bumpy lenses give the wavy aspect of the fault 262
surfaces that are overprinted by fine linear polished striations and coarser corrugations generated by 263
abrasions (Figures 2 and 3). 264
265
2.3. Earthquake surface rupture data 266
In contrast to the exhumed fault scarps presented in Section 2.1, where the 3-D roughness is 267
characterized, surface rupture data at larger scales provide only 2-D measurements (Figure 4), since 268
full fault surfaces are not accessible for direct measurement. Fault trace roughness has been 269
previously analyzed (Scholz and Aviles, 1986; Wechsler et al., 2010). In the present study, high 270
resolution geological maps of large continental strike-slip earthquake surface ruptures, in various 271
geological settings, were analyzed using the dataset of Klinger (2010); see for an example Figure 5, in 272
Klinger (2010). For each event, the surface ruptures length, earthquake magnitude, and total 273
geological offset are provided in Table 2. 274
Of these three parameters, we note that the total slip of a fault is the least constrained one 275
because of the difficulty in finding markers of the displacement over long distances on strike-slip 276
continental faults. For Owens Valley fault (Beanland and Clark, 1994), the total offset was estimated 277
in the range 20-30 km. For the Haiyuan fault (Zhang et al., 1987) it is in the range from 10-15 km 278
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(Burchfiel et al. 1991) to 95±15 km (Gaudemer et al. 1995). For the Gobi-Altay fault, the total 279
displacement lies in the range of 2-20 km (Kurushin et al., 1997). For the Superstition Hills fault, the 280
total slip is still debated and, by comparison with the closeby San Jacinto Fault, the total displacement 281
is estimated to be less than 24 km (Sharp et al. 1967). For the Luzon earthquake, the cumulated 282
displacement of the fault is in the range of 50-200 km (Karig, 1983; Mitchell et al. 1986), and 283
Rigenbach et al. (1993) noticed that because of the activity of this fault since the beginning of the 284
Pleistocene, the total offset should be smaller than 200 km. The Hector Mine and Landers earthquakes 285
occurred on faults that are part of the East California Shear Zone that has a total geological offset 286
close to 65 km (Jachens et al., 2002). The individual geological total slip for Landers (3.1 to 4.6 km) 287
and Hector Mine (3.4 km) faults were estimated from the offset of magnetic anomalies (Jachens et al., 288
2002). Finally, Korizan (1979) and Zirkuh (1997) earthquakes occurred on the Abiz fault that 289
accommodates 60 km of right lateral motion along the Sistan suture between Iran and Afghanistan 290
(Berberian et al., 1999). 291
These rupture traces have been acquired using field cartography that allows mapping of the 292
geomorphologic traces of the rupture, combined with slip distributions and high resolution satellite 293
images (Klinger et al., 2005, 2006; Klinger, 2010). The actual point spacing is irregular and its 294
average is between 200 m and 1300 m. However, the precision of each point based on the combined 295
field observations and high resolution satellite images is close to the meter-scale (Klinger et al., 296
2005). This last detail is crucial for interpreting a multi-meter fine description of the roughness of the 297
rupture trace. 298
To avoid any bias due to local wiggles of the digitized rupture trace, the data set is re-sampled to 299
ensure consistent spatial sampling, independent of the length of each rupture (Klinger, 2010). This re-300
sampling procedure does not affect the Fourier transform computation and makes it possible to keep 301
the scaling information of the rupture traces (see Appendix A.2). 302
Geometric discontinuities that are commonly observed on the eight high resolution maps of large 303
continental strike-slip earthquake surface ruptures are fault azimuth changes or bends, and relay 304
zones, which are also referred to as jogs. These discontinuities reflect the multi-scale en-echelon 305
13
pattern of the fault system, and range from a few hundred meters to several kilometers in size. Due to 306
the presence of abrupt steps associated with relay zones, which influence the Fourier transform and 307
therefore bias the roughness analysis, the ruptures traces are divided into individual segments (Figure 308
4). A reconstruction of the entire rupture trace can still be made when removing steps but the 309
information contained in the first order large scale segmentation has been removed. Note that for 310
some earthquake surface traces (Luzon, Superstition Hills, Hector Mine), no abrupt steps were 311
detected and the whole rupture trace can be directly analyzed. 312
313
3. Analysis of scaling properties of roughness data 314
In this section we detail the procedure to characterize the scaling properties of the scanned fault 315
surface topography. The same approach is followed for the digitized earthquake surface ruptures since 316
they are considered as rough profiles extracted along one direction of a fault surface. 317
First the original cloud of points (Figure 3) with irregularly spaced points was rotated, so that the 318
mean rupture surface was horizontal and parallel to major axis (X, Y). The 3-D set of points was 319
transformed in to 2-D ( YX , ) + 1-D ( Z ) data set where Z is the vertical direction, perpendicular to 320
the mean fault plane (Renard et al., 2006; Sagy et al., 2007; Candela et al., 2009; Zhanyu et al., 321
2010). A set of parallel cuts was taken through the cloud of points to obtain a series of thin bands of 322
points striking at an angle from the X axis. Then, each band of points was projected to obtain a 323
series of profiles with irregularly spaced points. The thickness of the bands of points is closely related 324
to the average spacing (X, Y) of the original raw data. Linear interpolation on a regular spacing is 325
performed independently on all profiles to yield a set of heights iXh , function of the coordinate 326
iX along the cut. The regular spacing taken for the linear interpolation has been systematically 327
chosen to be twice as large as the average irregular spacing of the original profiles. 328
To describe the scaling properties of these rough profiles (Figure 1), we search for possible spatial 329
correlations of the height fluctuations. Along each profile, we computed auto-correlation functions. If 330
the auto-correlation function of a rough profile is a power law and scales as 331
14
Hxxxhxh 2)(),( , then the rough profile is self-affine with H the Hurst exponent, if 332
multi-affinity is excluded. One way to estimate the Hurst exponent is to compute the Fourier 333
transform. This method is well-suited and robust for recognizing and characterizing self-affine 334
roughness, as demonstrated by Candela et al. (2009). The Hurst exponent H can be estimated from the 335
Fourier power spectrum, which has a power law form for a 2-D self-affine profile (Barabàsi and 336
Stanley, 1995; Meakin, 1998). The steps in the procedure to compute the Fourier power spectrum of 337
each profile are as follows: 1) First, removing of the residual drift in the signal is performed in order 338
to avoid any ramp artifact for the Fourier analysis (see Schmittbuhl et al., 1995b for a quantitative 339
analysis of this trend artifact). Indeed, adjusting the reference mean fault plane is always hard to 340
determine exactly a priori. In addition, even if corrections are possible a posteriori, trend suppression 341
is non-trivial and complex. All length scales are involved in the faulting process, even very large 342
ones, and therefore suppressing macroscopic information may influence the scale invariance analysis 343
at large scales. For the following analyses, and as suggested by Schmittbuhl et al. (1995a), the trend 344
has been defined simply as the line fit through the first and last point. 2) In order to ensure that there 345
are no step functions at the end of the finite window, we apply a 3% cosine taper. Note that a taper 346
function of 5 and 10% has been tested and the results appear robust. 3) The Fourier power spectrum 347
)(kP , i.e. the square of the modulus of the Fourier transform, is calculated as a function of 348
wavenumber k . 4) The Fourier power spectrum is normalized by dividing the power at each 349
wavenumber by the length of the profile. 5) Each cloud of points is computed as a whole by stacking 350
and averaging all 2-D Fourier transforms to reduce the noise associated with individual profiles. In 351
other words, for each fault patch, power spectral estimates with regularly spaced wavenumbers is 352
obtained by averaging the power spectra of the individual profiles in a geometric progression (Figure 353
5). 354
As an example, the computed Fourier power spectra along the slip direction and perpendicular to 355
it from patches of the Corona Heights fault surface acquired with each device (LiDAR, laser 356
profilometer, WLI) are displayed in Figure 6. The uncertainty in the average spectral values obtained 357
for each fault patch can be estimated following the method of Bendat and Piersol (1986). A one sigma 358
15
confidence interval for the spectral power is given by: 1
)(ˆ)(
1
)(ˆ
s
s
s
s
n
kPnkP
n
kPn with 359
1)( kynkn ys , where )(kP and )(ˆ kP are the actual and calculated spectral power, 360
respectively; yn and sn are the total number of profiles spaced a distance y apart perpendicularly 361
to the profile direction, and the number of independent profiles used to calculate )(ˆ kP , respectively. 362
Note that sn depends on scale. For the largest wavenumbers, there are many more independent 363
estimates of the total spectral power, and hence the error estimate is smaller than at the smallest 364
wavenumbers (see Figure 6). 365
Note that for wavelengths below 50 mm, the fault surfaces we scanned with the LiDAR are so 366
smooth that at this small scale, their spectral power falls within the range of those of the flat plate we 367
use as a planar reference surface (Figure 6). That means that even if our LiDAR surface 368
measurements were acquired at a spatial length scale resolution of mm1 to m10 (cf. Section 2.2), 369
total power estimates of the surfaces are accurate only between 0.05-10 m scales. In contrast, the laser 370
profilometer data and WLI data can be considered to be essentially noise free since their inherent 371
noise level falls well below the magnitude of the fault surface topography (Figure 6). 372
When plotting the average power spectrum as a function of wavenumber in a log-log space, a 373
self-affine function reveals a linear slope, which is itself a function of H through HkCkP 21)( 374
(with C the pre-factor). Taking into account the possible uncertainties in the spectral power as 375
previously described for our entire data set of fault surface patches, the upper limit of the error bar on 376
the Hurst exponent estimated using a least square method is equal to 05.0 and does not vary 377
significantly with the wavenumber. Due to the fact that only one rough profile for each rupture trace 378
is analyzed (Figure 4), the noise in the spectrum is higher compared to the fault surface patch 379
(constituted of an average of a multitude of profiles). For a single power spectral estimate, the 380
standard deviation is equal to the mean (Press et al., 2007). So that the upper limit of the error bar in 381
the estimated Hurst exponent of the eight rupture traces analyzed is equal to 1.0 . 382
16
383
4. Fault roughness results 384
4.1. Two universal Hurst exponents 385
In order to extract the Hurst exponents characterizing the scaling behavior of the faults 386
roughness, we have fit the linear part of each averaged spectrum obtained in the slip direction and 387
perpendicular to it for each individual fault patch (Figure 6). Because we are ultimately interested in 388
the variability of pre-factors, as discussed in the next section, we have worked separately on each fault 389
patch with each type of instrument instead of calculating average spectra for the full surface. We 390
have focused our analysis of fault surface roughness between the largest scale accessible with the 391
LiDAR, i.e. approximately m10 , and to the scale of mm05.0 accessible by the WLI. However, the 392
WLI gives access to scales down to 1 micron. 393
Figure 7 compiles all Hurst exponents derived from each individual fault patch (41 sub-surfaces 394
in total) in the direction of slip and perpendicular to it. The five faults scanned can be characterized by 395
two different global Hurst exponents between mm05.0 and m10 (see Table 1): 07.058.0// H 396
in the direction of slip and 04.081.0 H perpendicular to the slip. We will refer to 6.0// H 397
and 8.0H afterwards. This result highlights the fact that, for one direction (parallel or 398
perpendicular to slip), the relative amplitude of short and large wavelengths remains identical for the 399
whole data set. Fault surfaces with a Hurst exponent smaller or larger than one standard deviation of 400
their distribution ( 07.0 in slip direction and 04.0 normal to slip) exist, but all Hurst exponents 401
fall within two standard deviations (Figure 7). Consequently, our global Hurst exponents appear to 402
appropriately characterize the scaling behavior of the five faults studied when taking into account the 403
fluctuations indicated in Table 1. 404
A systematic bending of the Fourier power spectra (along both the parallel and the perpendicular 405
slip directions) at the length scale mm05.0 is observed on the WLI data (Figure 6, 9). This change of 406
regime occurs at larger length scales relative to the expected WLI resolution, and we suspect that it 407
could be the hallmark of the transition between two physical processes appearing at the grain scale. 408
17
Chen and Spetzler (1993) suggest that a characteristic length scale appearing at the grain scale is due 409
to a change in the dominant mode of deformation from small scale intergranular cracking to 410
intraganular cracking at large scale. The same observation and interpretation was made by Meheust 411
(2002) on tensile cracks. Sagy et al. (2007) also suggested a change in behavior at the sub-centimeter 412
scale for large-slip faults. However, it is beyond the scope of this work to quantitatively characterize 413
this possible characteristic length scale and we limit our scaling analysis to scales larger than 414
mm05.0 . 415
416
4.2. Pre-factor variability 417
In Section 4.1, we focused the analysis on the slope of the power-law, which appears to be 418
constant along slip and perpendicular to it for each fault surface. In addition, for each scanned fault 419
patch, the along slip direction is smoother than perpendicular to it. This is seen by the lower position 420
of power spectrum amplitudes at each scale (Figure 6). More precisely, the power-law relationships 421
between )(kP and k have both lower power-law exponents and pre-factors. However, our results 422
point out that the pre-factor derived from each fault patch is variable along the fault scarps. Here, we 423
illustrate this variability on the data acquired with the LiDAR on the Corona Heights fault, along the 424
slip direction. A significant vertical shift is observed on the Fourier power spectra corresponding to 425
the six different patches of the Corona Heights fault (Table 1, Figure 8). The slopes, however, are 426
similar (Table 1, Figure 7). We emphasize that, even if for some fault patches it is not the case, for 427
other patches this spatial variability on the pre-factor is clearly larger than the size of the error bars in 428
the average spectral values (see Figure 8, for example between Corona-A and Corona-E). 429
As proposed by Power et al. (1987) and Power and Tullis, (1989), who have observed a similar 430
vertical shift in their Fourier power spectra, a tempting explanation of the variability of the pre-factor 431
within a same fault surface is that the finite displacement accumulated by rougher fault patches (or 432
segments with larger pre-factors) is smaller than for the smoother fault patches. In other words, it is 433
possible that the spatial heterogeneity on the pre-factor illustrated in Figure 8 highlights variable 434
accumulated displacement (probably much less important than the total offset recorded by the fault 435
18
zone) on each individual segment (or fault patch) constituting the whole fault zone. However, as 436
emphasized by Power and Tullis (1989), this possibility is difficult to quantify because the total 437
displacement for sub-parallel individual surfaces (which represent a part of the total offset of the fault 438
zone) cannot be observed with certainty in the field. Alternative explanations for the difference in 439
surface characteristics would include formation of the individual fault patches at different conditions 440
(depth, confining pressure, temperature, strain rate) or the presence of different initial heterogeneities 441
in the rock before deformation. It is also possible that the variability of roughness reflects the natural 442
variability of an underlying stationary pre-factor distribution. 443
In order to highlight the specific trend of each fault in the spectral domain, we have averaged the 444
similar spectra (that means spectra with an approximately identical slope but with a slightly different 445
vertical position) obtained from each scanner device. Each curve on Figure 9 therefore represents an 446
average of similar spectra obtained for multiple fault patches. In this way, this technique gives a 447
smoother spectrum that represents the global self-affine character of the entire fault surface at the 448
specific scales accessible by each device, while preserving good wavelength resolution. However, as a 449
direct consequence of the previously described spatial variability of the pre-factor along the fault 450
surface (Figure 8), the global spectra obtained for each device are vertically shifted in Figure 9. 451
Consequently, even if each global spectrum presents a similar slope, in most cases it is difficult to 452
connect them all together with a unique linear trend (Figure 9). It is worthwhile to point out here that 453
when we compute the scaling from the LiDAR measurements, we average all the 2-D power spectra 454
of all individual rough profiles extracted from different fault patches (Figure 5, 8). Each of these 455
profiles has a different pre-factor but the average gives one specific pre-factor characterizing the 456
global 3-D geometrical property of the fault surface at the LiDAR scale. At smaller wavelengths, with 457
another device (i.e. laser profilometer or WLI), we select several sub-regions with one given average 458
pre-factor among the whole population that we explored at larger scale (i.e. with the LiDAR) but not 459
necessarily equal to the average pre-factor of the large scale measurements (see Figure 8, 9). In other 460
words, with the laser profilometer or the WLI, we expect to sample rougher or/and smoother sub-461
regions of the fault (i.e. with a pre-factor magnitude that is, respectively larger or smaller) than the 462
19
average behavior recorded by the LiDAR by combining data over the full surface. As a consequence, 463
it is clear that for the same fault scarp, the specific global spectra calculated for each device by 464
stacking similar spectra obtained for several fault patches, might be vertically shifted in some cases as 465
observed in Figure 9. Finally, this variability in the pre-factor: (i) explains why the Hurst exponents 466
have to be necessarily calculated separately for each device (cf. Section 4.1), and (ii) implies that it is 467
not necessary to invoke a change in slope to connect the vertically shifted global spectra in Figure 9. 468
To summarize, our analysis highlights that the roughness of the five fault scarps studied can be 469
characterized over more than 5 decades of length scales (between mm05.0 and m10 ) by two 470
universal Hurst exponents in the direction of slip and perpendicular to it even if it is difficult to point 471
out a single pre-factor in both directions. This description of the fault surfaces is independent of the 472
geological context, i.e. lithology (rhyolite, chert and limestone) and tectonic regime (strike-slip, 473
oblique and normal). Moreover, no clear relationship is observed between the range of pre-factor 474
magnitude extracted from the global spectra shown in Figure 9 for each fault (see Table 3 for a 475
complete list of the maximum and minimum pre-factors of each faults), and the slip estimated for the 476
fault zone (see Figure 10). 477
478
4.3. Roughness of large continental earthquakes surface ruptures 479
In Figure 11, a compilation of the roughness results is provided for the eight surface ruptures. 480
Results of the Fourier power spectrum analysis are shown for the individual segments (bounded by 481
abrupt steps), and the whole rupture trace. For each surface rupture trace, each individual segment has 482
a roughly identical self-affine exponent (see also Table 2) at large scales (i.e. above the regime 483
controlled by data re-sampling – see Section 2.3). Moreover, the profiles obtained for the entire 484
“reconstructed” rupture traces represent an average of the roughness over all the individual segments 485
and keep an approximately identical self-affine exponent (Figure 11). 486
The variability in the power spectra amplitude (i.e. the pre-factor) of the profiles extracted from 487
the exhumed fault scarps (see Section 4.2) is also observed between each individual segment of the 488
rupture traces. More precisely, along the rupture traces, individual segments display variability in the 489
20
roughness amplitude but keep the same self-affine scaling properties of amplitude versus length scale 490
(Figure 11). Since the 2-D roughness scaling properties of the reconstructed rupture traces correspond 491
to a sum of those of the individual segments, it is to be expected that their spectra fall between the 492
smoothest and the roughest short segments (Figure 11). In this way, the range of pre-factors inferred 493
by the individual segments can be interpreted as a typical fluctuation or error in the estimated pre-494
factor of the “reconstructed” rupture traces. 495
On the same log-log graph (Figure 12a), a stack of all the spectra calculated on the whole rupture 496
traces complements Figure 11 and emphasizes that all of the surface rupture data can be described by 497
a unique self-affine exponent of 1.08.0 . Another interesting result highlighted in Figure 12a is 498
that, even though the eight rupture traces analyzed clearly sample variable geological settings, the 499
same self-affine exponent fits all of the data best. In particular, even though the surface rupture traces 500
are related to fault zones which have accumulated a large range of finite geological offsets (see Table 501
2), no trend is revealed between this parameter and the 2-D roughness scaling of the rupture traces. 502
Indeed, taken into account both (i) the average pre-factor of each whole rupture traces and (ii) their 503
respective typical fluctuations extracted from the corresponding individual segments, no correlation 504
appears with the finite geological offset (Figure 12b), admitting however that the standard deviation 505
in the data is quite large and could hide the physical correlation, if any. 506
507
5. Discussion 508
5.1. Can surface measurements of faults be correlated with earthquake processes at depth? 509
Firstly we discuss how our measurements, performed on surface outcrops, could be relevant 510
to faulting processes that occur at crustal depths up to 15 km. On the five faults we have studied, two 511
of them (Magnola and Dixie Valley) have a mainly normal component and their roughness has 512
therefore recorded processes at work in the first kilometers of the crust. For the three other strike-slip 513
faults, the roughness has recorded only shallow depth faulting processes. Dixie Valley and Bolu 514
outcrops have recorded the surface termination of major earthquakes in the last century. For all faults, 515
we find roughness exponents in the range 0.6 – 0.8. This can be compared to a recent study of the 516
21
roughness of exhumed faults in the Sierra Nevada and Italian Alps (Griffith et al. 2010, Bistacchi et 517
al. 2011). In these studies, the roughness of fault traces was characterized on outcrops for which the 518
deep origin (c.a. 10 km depth) of the faulting process could be identified because of the presence of 519
pseudotachylites and specific mineral assemblages. Griffith et al. (2010) measured Hurst exponents 520
with a quite large variability, in the range 0.4 – 1. Interestingly, Bistacchi et al. (2011) report Hurst 521
exponents in the range 0.6 – 0.8 for spatial scales in the range 0.5 mm – 500 m, similar to those 522
observed in our data and suggesting that the roughness property of faults is also maintained at depth. 523
524
5.2. Implications of the self-affine scaling of the fault roughness on the machinery of 525
earthquakes 526
One of the main contributions of the present study is to provide a robust and realistic fault 527
geometry model, which is still currently missing. Here, we demonstrate that one way to describe fault 528
surfaces with few parameters is to consider them as self-affine. Using our realistic geometry of rough 529
fault surfaces in seismic rupture dynamic models will provide new keys to move forward towards a 530
better understanding of the fundamentals of earthquake behavior. In this sense, based on self-affine 531
geometrical model of fault surfaces as highlighted in our present work, recent numerical and 532
theoretical studies have shown results in agreement with seismological observations. Indeed, 533
quantification of roughness has been used to infer the scaling of the stress and slip variations along 534
the fault plane after an earthquake (Schmittbuhl et al., 2006; Candela et al., 2011a, 2011b). Fault 535
roughness is also useful for calculating the expected seismic radiation (Dunham, 2011). 536
537
5.3. Reconciling the Hurst exponent of the scanned fault surfaces and rupture traces 538
The rupture trace geometry of strike-slip earthquakes should correspond to the extrapolation of 539
the fault surface roughness sampled on fault scarps along the slip direction and the Hurst exponent 540
should be close to 0.6. However, our results highlight a small difference in the Hurst exponents of the 541
eight rupture traces compared to the average trend in slip direction of the five fault surfaces scanned. 542
The spatial correlations of the rupture trace irregularities are characterized by 1.08.0 RH and 543
22
pre-factors magnitudes, close to those in the direction perpendicular to slip, collected on the five fault 544
surfaces that were scanned. We propose that the roughness increase at the rupture trace scale is due to 545
the fact that the slip direction is not strictly sampled as it is the case for fault surfaces scanned. Slip 546
distributions of strike-slip earthquakes show that at some locations a vertical component could be 547
significant (Florensov and Solenko, 1965; Kurushin et al., 1997). Therefore, at the surface rupture 548
scale, a slight vertical slip component along the whole rupture trace of the strike-slip earthquake 549
would explain how we could sample a roughness oblique to the slip direction. 550
This interpretation is investigated in Figure 13. The Hurst exponents and the pre-factors of a 551
synthetic anisotropic self-affine surface (see Appendix A.1), computed using a Fourier based method 552
(Candela et al., 2009), and with a roughness root-mean-square standard deviation that scales as 553
6.0005.0 LRMS in slip direction and 8.0015.0 LRMS normal to it, were calculated on series of 554
2-D profiles extracted at an angle from the slip direction. The same procedure was performed on a 555
scanned patch of the Corona Heights fault (Corona-A in Table 1). A clear similarity is observed in the 556
angular variability of the roughness exponent computed for the synthetic surface and the natural fault 557
patch (considered as representative of our set of fault surfaces). Both surfaces expose the same non-558
linear dependence of the Hurst exponent with the azimuthal direction of profile extraction ( ), as 559
pointed by Renard et al. (2006) and Candela et al. (2009) using different statistical tools. When 560
departing a few degrees from the direction of the smallest exponent, i.e. the slip direction, the Hurst 561
exponent sampled is already very close to the largest exponent, i.e. the direction normal to slip 562
(Figure 13). This effect could explain the slight difference in the roughness exponents observed 563
between the surfaces ruptures and fault surface scanned along slip. In the same way, this non-564
alignment of the ruptures traces with the slip direction, could also explain that their pre-factors 565
magnitudes are close to the maximum sampled for the fault surface scanned. 566
Moreover, the traces of the ruptures show a structural complexity underlined by relay zones 567
(compressive and extensive jogs), and bends, which could locally accommodate a significant vertical 568
component at many locations along the fault (Klinger et al., 2006). In this case, the rupture trace 569
morphology corresponds to a combination of topography along slip direction and normal to it, which 570
23
might partially explain the increase of the measured Hurst exponent and pre-factor compared to those 571
of the along-slip fault surfaces that were scanned. 572
Taking into account the previous arguments, we propose that the self-affine regime observed at the 573
outcrop scale is also present at the map-scale of earthquake rupture traces. In other words, even 574
though there is a gap of data between the scanned fault surfaces and the rupture traces that remains to 575
be investigated and the traces do not directly constrain the slip-parallel geometry, a unique self-affine 576
geometrical model could fit our measurements from the WLI scale to the map-scale of rupture traces. 577
Indeed, considering the variability of the pre-factor observed both on the scanned fault scarps (see 578
Section 4.2) and the rupture traces (Section 4.3), it seems reasonable to directly connect both data sets 579
by a unique line covering 9 decades of length-scales (Figure 14). As already mentioned, analyzing the 580
roughness of fault traces in Italian Alps, Bistacchi et al. (2011) have recently reported Hurst 581
exponents in the range 0.6 – 0.8 for spatial scales in the range 0.5 mm – 500m and therefore bridging 582
our gap of data. 583
584
5.4. Pre-factor variability independent of displacement 585
In Section 4.2, by combining the measurements acquired with various scanner devices and 586
covering complementary scales, we have shown that no clear relationship exists between the range of 587
roughness amplitude (i.e. the range of pre-factor magnitude) of each exhumed fault surface and the 588
slip estimated for the scanned fault surface (Figure 10). Note here that, following Sagy et al., (2007), 589
even if we focus on the smoothest sub-surfaces of each fault (i.e. the lower bound of the range of pre-590
factors), no trend is revealed with the estimated offset accommodated (see Figure 10). Other recent 591
studies have noted changes in fault roughness and damage parameters as a function of slip for faults 592
spanning a range of offsets shifted from the dataset here (Sagy et al., 2007; Mitchell and Faulkner, 593
2009; Savage and Brodsky, 2011; Brodsky et al., 2011). Sagy et al. (2007) noticed a difference for 594
faults that have slip more than 10 m versus those that slip less than 1 m. The fault roughness data 595
presented in the present study only spans offsets that are within the large-slip faults population by 596
these criteria. The lack of a discernible evolution signal is consistent with the suite of data of Brodsky 597
24
et al. (2011) that also shows that smoothing is a very weak process once >10 m of offset is 598
accumulated. 599
Another important detail of our roughness scaling analysis is that no correlation was observed 600
between the finite displacement by the fault zone hosting the rupture trace and its roughness (see 601
Section 4.3 and Figure 12b). This observation is consistent with the study of Klinger (2010), who 602
showed that the correlation between the characteristic fault segment length and the thickness of the 603
seismogenic crust is maintained, independently of the slip accumulated. In addition, we suggest in this 604
study that a consistent spatial organization persists over the entire range of length scales accessible 605
(i.e. from several hundred meters to ~50 km), independent of the total geological offset. Note here 606
that in following the reasoning of Klinger (2010), a specific length scale should appear at 607
approximately 20 km, i.e. the thickness of the seismogenic crust. However, because of the lack of 608
sufficient frequency content between 20 km and 50 km, this probable characteristic length scale is not 609
clearly revealed by our analysis. 610
Smoothing of fault geometry has been suggested by other studies (Wesnousky, 1988; Manighetti 611
et al., 2007), and is thought to result from surface roughness of faults being inversely related to their 612
total displacement. Such smoothing of a fault trace has primarily been studied by examining large 613
scale geometrical asperities, such as step-overs of several kilometers. In our study, given that we 614
remove these first-order geometric discontinuities to perform our Fourier transform analysis, we lose 615
these hallmarks of the fault maturity. 616
The notion of "geometric regularization" or “maturity” refers to the intuitive idea that the fault 617
zone geometry becomes more and more flat during the successive slips by abandoning or smoothing 618
the complexity of the initial structures (segments). Ben-Zion and Sammis (2003) recommended that it 619
is necessary to separate the abandoned structural units from those which actively participate in the 620
accommodation of the slip to reveal a possible regularization of the geometry of the fault zone with 621
slip. It is therefore important to emphasize that both the fault scarps scanned on the field and the 622
rupture traces are markers of the morphology of the active structures of fault zones. Our results thus 623
demonstrate that the active portions of faults from the spatial scale of micrometer up to at least the 624
25
thickness of the seismogenic crust preserve a complex geometry during successive displacements. In 625
other words, the complexity of an active fault surface is spatially organized following a self-affine 626
geometrical model, independently of the lithology and the tectonic regime. Our observations, as those 627
of Brodsky et al. (2011) and Klinger (2010), suggest that a re-roughening mechanism is active in the 628
fault zone to maintain the geometrical complexity during successive slips. For example, as proposed 629
by Klinger (2010), rupture might naturally branch on preexisting secondary faults as a result of the 630
dynamic stress build-up ahead of the rupture (Poliakov et al., 2002; Bhat et al., 2004). This effect 631
could explain the persistence of some level of complexity and prevent the complete smoothing of the 632
fault geometry. 633
634
5.5. Scale-free process at the origin of fault roughness 635
Previous studies on natural fault surfaces have shown that the slope of the Fourier spectrum 636
could change from large to small scales (e.g. Lee and Bruhn, 1996), implying that different processes 637
may be involved during the generation of surface fault textures at different spatial scales. In fact, any 638
inflexion of the spectrum would correspond to a characteristic scale. Such a crossover scale could be 639
interpreted as the transition between physical processes, as is the case for dissolution surfaces such as 640
stylolites (Renard et al., 2004). In the present study, five fault surfaces in different geological settings 641
have been scanned with three different scanner instruments spanning complementary length scales. 642
Considering the variability of the pre-factor (see Section 4.2), an anisotropic mono-affine geometric 643
model characterized by two different global Hurst exponents 6.0// H and 8.0H best fits the 644
entire data set between m10 and mm05.0 . In our data set, because the slopes follow a consistent 645
trend, and considering the level of noise, no variation of the Hurst exponent could be detected 646
between m10 and mm05.0 . This observation could be evidence that a unique scale-free process 647
could be at the origin of the scanned fault scarps. 648
In contrast to mode I fracture roughness for which a lot has been done in physics and mechanics 649
communities, few studies have focused on the processes at work in the generation of the shear surface 650
roughness. Since the 1990s, increasing experimental evidence showed that the roughness exponent for 651
26
fractures surfaces had a universal value of about 0.80 (Bouchaud et al., 1990; Bouchaud, 1997). 652
Simultaneously with these experiments, theoretical and numerical works have been produced at a 653
steady rate with the aim of modeling the fracture propagation in order to identify the origin of the self-654
affine scaling of fracture roughness (Bouchaud, 1997, Alava et al., 2006; Hansen & Schmittbuhl, 655
2003; Bonamy and Bouchaud, 2011). The common denominator of all this work is to model fracture 656
propagation as a network of elastic beams, bonds, or electrical fuses with random failure thresholds 657
and subject to an increasing external load. In other words, in order to reproduce the self-affine 658
roughness exponent experimentally observed, the fracture propagation is assimilated as a damage 659
coalescence process in a heterogeneous material. The damage becomes localized by long-range elastic 660
interaction between multi-scale cracks. In this sense, we propose that elastic interactions related to 661
linkage of many discrete slip surfaces, controlling the generation of the multi-scale bumpy lenses 662
observed on the scanned fault outcrops, could be the scale-free process at the origin of fault 663
roughness. Indeed, even if shear cracks involve significantly higher energy release rates than tensile 664
cracks (Atkinson, 1991), some similarities in the elastic influences on the stress field at the crack tip 665
can be obtained theoretically (Gao and Rice, 1986; Gao et al., 1991; Atkinson, 1991; Schmittbuhl et 666
al., 2002). 667
Even if a unique self-affine regime is maintained up to the map-scale of the rupture traces, it 668
remains difficult to support the fact that their roughness is also controlled by the process at work for 669
the scanned fault scarps. However, it is noteworthy that the rupture traces seem to be also formed by 670
multi-scale segments (see for example the Figure 5 in Klinger, 2010) whose growth and coalescence 671
could be also controlled by long range elastic interactions. 672
673
6. Conclusion 674
The roughness properties of fault surfaces is summarized in Figure 14, where the Fourier power 675
spectra of the Corona Heights fault surfaces perpendicular to the slip direction and the spectra 676
obtained for the continental earthquake surface ruptures are plotted together. First, even if it is 677
difficult to point out a unique power-law fit due to the pre-factor variability, a single anisotropic self-678
27
affine model ( 6.0// H and 8.0H ) best fit our 3-D scanned fault measurements from 679
mm05.0 scale to m10 . Secondly, an identical self-affine behavior is consistent with the map-scale 680
of the 2-D rupture traces. In order to connect both data sets while considering the variability of the 681
pre-factor, we propose that a unique self-affine geometrical model is maintained over nine decade of 682
length scales (between 0.05 mm and at least the thickness of the seismogenic crust, i.e. ~20 km). Even 683
if the lack of data between the scanned fault surfaces and the rupture traces remains to be investigated, 684
recent works of Bistacchi et al. (2011) bridging the length scale gap appear to support our 685
interpretation. 686
In addition, even if in both cases, for the scanned fault surfaces and the ruptures traces, we have 687
focused our analysis on the active portion of the fault zone, it appears that once a small amount of 688
offset has been achieved, the geometric complexity is maintained regardless of the amount of further 689
slip accommodated. Consequently, we propose that processes that create this roughness and processes 690
that destroy it must reach a dynamic equilibrium. 691
692
Acknowledgments: This study was supported by the Agence Nationale pour la Recherche grant ANR-693
JCJC-0011-01, ANR ECCO TRIGGERLAND and ANR RiskNat SUPNAF. Field work was 694
supported in part by NSF Grant EAR-0711575. The authors are very grateful to Michel Bouchon and 695
Fabrice Cotton for encouragements and scientific stimulations. William L. Power is really thanked for 696
his detailed and constructive comments on an early version of this work. The first author gratefully 697
acknowledges Andrew P. Rathbun and Megan Avants for their constructive comments that helped to 698
improve the content and clarity of the manuscript, and James D. Kirkpatrick for fruitful discussions 699
concerning various aspects of this paper. 700
701
702
703
704
28
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37
Figure captions 923
924
Figure 1: Two-dimensional roughness profiles from the Corona Heights fault surface parallel and 925
perpendicular to slip. A: Profiles parallel to the slip direction and, B: perpendicular to the slip 926
direction. For both directions, a magnified portion of the profiles has a statistically similar appearance 927
to the entire profiles when using the scaling transformation zzxxH
,//, . For each scale, 928
profiles have been shifted vertically for clarity. 929
930
Figure 2: Corona Heights fault. Multiple bumpy discrete slip surfaces constituting lenses and 931
striations can be detected at all scales, from the measurement resolution of each scanner device to the 932
size of the entire exposure. A: Whole outcrop view. The inset corresponds to the surface shown on 933
Figure 2C. B: Zoom on the fault showing different segments constituting the surface. C&D: Map of 934
fault surfaces scanned using LiDAR topography. The inset in C corresponds to the patch shown on 935
Figure 2D. E and F: Maps of fault surfaces scanned with the laser profilometer. G and H: Zoom on 936
the Figure 2E-F scanned with the white light interferometer. 937
938
Figure 3: Bolu Fault. A: Photograph of the fault zone in the Bolu limestone. Dashed red contour 939
corresponds to the limits of the cloud of points shown in B. C: Photograph of a well-preserved slip 940
surface constituting the fault zone. D: LiDAR data of surface in C. E and F: Zoom on the fault surface 941
and the corresponding topographic map acquired with the laser profilometer, which still includes 942
anisotropic roughness features in the slip direction. 943
944
Figure 4: Digitized surface rupture trace of the three largest segments of the Landers earthquake. The 945
data correspond to Figure 5G in Klinger (2010). The inset zooms on one of the steps removed for the 946
Fourier transform analysis. Each segment is individualized by different colors and the steps are 947
represented in pink. The reconstructed trace by removing steps is displayed in gray. 948
38
949
Figure 5: Example of the 3-D average power spectrum of one LiDAR fault patch (Corona-B) 950
obtained by averaging in a geometric progression several thousands 2-D power spectra of individual 951
profiles (we have changed the color of the spectra every 20 profiles successively extracted from the 952
surface). 953
954
Figure 6: Typical average power spectra in the slip direction and perpendicular to it for fault patches 955
scanned with the three devices used in our study. Errors bars with 68% confidence interval (one 956
sigma), and power law fits performed in linear portions of each average spectra, are shown. At the 957
LiDAR scale, the black arrow indicates the lower limit used for the fits. This lower limit underlines 958
the length scale at which spectra flatten out when they intersect the noise spectrum calculated by 959
scanning smooth, planar reference surfaces. For the laser profilometer and the WLI, the spectral 960
power levels of our natural fault roughness data fall at a vertical higher position than the noise spectra. 961
However, at the WLI scale, the vertical dashed blue line at the bending of spectra indicates the lower 962
limit of our fits. Even if this systematic change of slope at the length scale mm05.0 seems to be 963
related to intrinsic physical fault roughness properties, we arbitrarily limit our scaling analysis to 964
scales larger than mm05.0 . 965
966
Figure 7: Plot of the Hurst exponents (see Table 1) in slip direction (B) and normal to it (A) of the 41 967
scanned fault patches. The average Hurst exponent is equal to 58.0 along the direction of slip and 968
81.0 perpendicular to it. The shaded area and dashed lines indicate 1 and 2 confidence 969
intervals, respectively. 970
971
Figure 8: Illustration of the pre-factor variability along the Corona Heights fault. Log-log graph 972
gathering the global laser profilometer spectrum with the 6 averaged Fourier power spectra obtained 973
at the LiDAR scale. Note the intercepts range of all 2-D LiDAR spectra, performed from each 974
individual profiles extracted from the six fault patches, highlighted by the two dashed dark power 975
39
laws. The global laser profilometer spectrum falls in this range of intercepts sampled by the whole 976
population of the individual profiles that we explore at the LiDAR scale. 977
978
Figure 9: Global Fourier power spectra from the five faults analyzed along the slip direction (left) 979
and perpendicular to it (right). Each curve at each scale (LiDAR, laser profilometer, white light 980
interferometer) includes together the average spectra of several sub-surfaces (or fault patches in Table 981
1). Power-law fits (thick gray lines) with a roughness exponent of 6.0// H and 8.0H , 982
connecting the field and laboratory data in both directions, are shown on plot for eye guidance. The 983
black arrow and the vertical dashed blue line indicate, respectively: the level of noise of the LiDAR 984
and the lower limit for the fit performed at the WLI scale. Dotted black lines indicate the range of pre-985
factors,
//,21
//,//, )(,H
kCkPC , extracted from the power-law fits at mk 1 (see all the 986
values in Table 3), and used for the Figure 10. One way to more easily interpret this log-log graph and 987
to compare our results with previous studies like Sagy et al. (2007), is to convert the power spectrum 988
module in term of root-mean-square ( RMS ) roughness amplitude using Parseval’s Theorem 989
(Brodsky et al., 2011). Indeed if 10 //, H , for a profile of length L , the integration of 990
//,21
//,)(H
kCkP over the wavelength (with k/1 ) yields that the RMS roughness 991
correlates as
//,
5.0
//,
//,
2H
LH
CRMS . In this sense, red lines represent power-law fits for 992
three self-similar rough surfaces (i.e. H = 1) with various pre-factors, that are: LRMS 1.0 , 993
LRMS 01.0 , LRMS 001.0 . 994
995
Figure 10: Log-log plots of the range of global pre-factor magnitude of the five faults extracted from 996
Figure 9 (see also Table 3) versus the estimated slip (Table 1). 997
998
Figure 11: Compilation of the surface rupture roughness results: Digitized rupture traces (top) and 999
corresponding Fourier power spectrum (bottom). Because of the abrupt steps biasing the Fourier 1000
40
transform computation, we have performed this roughness analysis on each individual segment 1001
between two steps of the whole profile. The same color code is respected between the individual 1002
segments and the corresponding spectra. In addition, the gray rupture profile and the corresponding 1003
gray Fourier power spectrum represent a reconstruction of the entire profile. Power-law fits and the 1004
inferred Hurst exponents on the linear part of each curves at large scale (above the cross-over length 1005
scale indicated by the gray vertical bar and marking the beginning of the regime controlled by data re-1006
sampling) are represented on each graph. Note here that for each rupture trace, the curves 1007
corresponding respectively to the selected segments and to the reconstructed profiles are characterized 1008
by a similar scaling exponent but variable pre-factors. 1009
1010
Figure 12: Variability of the pre-factor and slip accumulated for the eight studied rupture traces. A: 1011
Stack of all the Fourier power spectra of the whole surface ruptures stacked, underlining the global 1012
trend of the self-affine behavior at large scale. Power-law fit giving an average Hurst exponent of 1013
1.08.0 is indicated. B: Log-log plot of the average pre-factors (corresponding to those of the 1014
spectra of the whole rupture traces and indicated in Table 2) with their respective typical fluctuations 1015
(i.e. the range of pre-factors extracted from the spectra of the individual segments constituting the 1016
whole rupture traces) of each rupture traces in function of the finite geological offsets (see Table 2). 1017
1018
Figure 13: Angular dependence of the Hurst exponent (left) and the pre-factor (right) computed on a 1019
synthetic anisotropic self-affine surface with two input exponents in perpendicular directions 1020
( 6.0// H and 8.0H ), and for a patch (Corona-A, see Table 1) scanned with the LiDAR on the 1021
Corona Heights fault surface. The Hurst exponents and the pre-factors were calculated on series of 2-1022
D profiles extracted at an angle between the slip direction ( 0 ) and the perpendicular direction 1023
( 90 ). 1024
1025
Figure 14: Comparison of the roughness of the earthquakes surface ruptures with the Corona Heights 1026
fault topography. The Fourier power spectra normal to the slip direction of the Corona Heights fault 1027
41
surface are plotted on a log-log graph together with those obtained for the eight continental strike-slip 1028
earthquakes surface rupture traces. The Corona Heights fault data are identical to those plotted on 1029
Figure 9, and those of the surface ruptures correspond to that of Figure 12. Both data sets are 1030
connected with a unique power-law fit (grey line) with a Hurst exponent of 0.8. 1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
42
Table 1. Laser scanner characteristics and fault roughness results.
Fault Name Lithology
& Slip* Sense
Fault
Patches Scanner
§
x
Spatial
precision
#
z
)05.0(
//
H
)05.0(
H
Average
//H H
Vuache-
Sillingy
45°57’14.5’’N
6°2’56’’E
Limestone
10-30m
Strike
slip
Surf-1 GS 100
(Trimble)
20
mm 5mm
4.5
mm 0.60 0.78
0.60
0.07
0.81
0.02
Surf-7 GS 100
(Trimble)
20
mm 5mm
4.5
mm 0.68 0.82
Surf-6
LMS
Z420i
(Riegl)
30
mm
7.5mm 10.2
mm 0.50 0.82
Surf-
JPG
S10
(Trimble) 1
mm
0.25mm 0.9
mm 0.63 0.83
Small Lab.
profilo-
meter
20
µm 1µm
<
1µm
0.65 0.81 0.61
0.05
0.80
0.01 Vu-1-G 0.57 0.80
Vu-A-1
WLI
2µm
0.025 µm
3nm
0.58 0.76 0.60
0.04
0.79
0.03
Vu-A-2 2µm 0.55 0.78
Vu-A-7 1µm 0.65 0.84
Vu-A-8 1µm 0.61 0.79
Corona
Heights
37°45’55’’N
122°26’14’’E
Chert
several m
to >1km
Strike
slip
Corona-
A
HDS 3000
Leica
5
mm
1.25mm 2
mm
0.57 0.85
0.65
0.04
0.83
0.03
Corona-
B 0.65 0.81
Corona-
C 0.67 0.87
Corona-
D 0.64 0.81
Corona-
E 0.69 0.79
Corona-
F 0.67 0.85
P3 Lab.
profilo-
meter
20
µm 1µm
<
1µm
0.66 0.86 0.63
0.04
0.85
0.03 Co-
AGU 0.60 0.85
Co-A-4
WLI
2µm 0.025
µm 3nm
0.62 0.82 0.62
0.03
0.83
0.02 Co-A-9 1µm 0.63 0.85
Bolu
40°41’07’’N
31°34’04’’E
Limestone
20m to
85km
Strike
slip
Stack
2345
Ilris-3D
Optech
20
mm 5mm
20
mm
0.48 0.79
0.50
0.07
0.78
0.02
Stack67 0.50 0.78
Stack12 0.44 0.78
W-
detail-2 0.45 0.76
E-detail-
3 0.49 0.80
E-detail-
2 0.46 0.80
E-detail-
1 0.65 0.74
Bolu-1 Lab.
profilo-
meter
20
µm 1µm
<
1µm
0.58 0.79 0.55
0.05
0.76
0.04 Bolu-2 0.51 0.74
1046
43
1047
Fault Name Lithology
& Slip* Sense
Fault
Patches Scanner
§
x
Spatial
precision #z
)05.0(
//
H
)05.0(
H
Average
//H H
Dixie Valley
39°56’48’’N
117°56’43’’E
Rhyolites
several m
to 3-6km
Normal
Dixie-1
HDS 3000
Leica
5
mm
1.25mm 2
mm
0.66 0.79 0.59
0.08
0.82
0.03
Dixie-2 0.63 0.80
Dixie-3 0.61 0.84
Dixie-4 0.47 0.84
Map-1 Lab.
profilo-
meter
20
µm 1µm
<
1µm
0.57 0.82 0.59
0.03
0.81
0.03 Map-2 0.61 0.81
Dixie-D
WLI
2µm
0.025 µm
3nm
0.56 0.88 0.56
0.05
0.88
0.01
Dixie-H 1µm 0.50 0.87
Dixie-E 2µm 0.61 0.89
Dixie-C 2µm 0.59 0.89
Magnola
42°7’N
13°28’31’’E
Limestone
several m
to >500m
Normal
A32 Ilris-3D
Optech
20
mm 5mm 20
mm 0.59 0.77
M2
Lab.
profilo-
meter
20
µm 1µm
<
1µm 0.60 0.83
1048 * Except for the Vuache-Sillingy fault surface, a lower and upper bound of the displacement is given. 1049
Although total geological cumulated slip for the fault zone as a whole can be kilometric (i.e. the upper 1050
bound), scanned individual surfaces within the fault zone may have experienced considerably less slip 1051
(i.e. the lower bound). 1052
§ Spatial length scale resolution ( x ). 1053
# Vertical resolution ( z ). 1054
1055
1056
1057
1058
1059
1060
1061
1062
44
Table 2. Characteristics and roughness results of earthquake rupture maps used in this study.
(references for each earthquake are given in the text)
Name Year Magnitude Rupture
length (km)
Total geological
offset (km) RH Pre-factor
(m3)
Owens Valley
(USA) 1872 Mw 7.5-7.8 81 20-30 0.6 0.1 4 x 10-2
Haiyuan
(China) 1920 Ms 8 to 8.7 200 15 to 95 0.8 0.1 7 x 10-3
Gobi-Altay
(Mongolia) 1957 M 8.3 235 2 to 20 0.7 0.1 2 x 10-2
Superstition
Hills (USA) 1987 M 6.6 18 24 0.7 0.1 2 x 10-3
Luzon
(Philippine) 1990 Mw 7.8 107 50 to 200 0.8 0.1 1 x 10-2
Landers
(USA) 1992 Mw 7.2 65 3.1 to 4.6 0.8 0.1 1 x 10-3
Zirkuh
(Iran) 1997 Mw 7.2 104 60 0.9 0.1 3 x 10-4
Hector mine
(USA) 1999 Mw 7.1 39 3.4 0.7 0.1 7 x 10-3
1063 1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
45
Table 3. Pre-factor (exhumed slip surfaces)
Parallel to slip Perpendicular to slip
Vuache-Sillingy Pre-factor (m3)
min 8 x 10-8 2 x 10-6
max 2 x 10-5 1 x 10-4
Corona Heights Pre-factor (m3)
min 4 x 10-7 1 x 10-5
max 5 x 10-5 2 x 10-4
Bolu Pre-factor (m3)
min 2 x 10-8 3 x 10-7
max 5 x 10-5 2 x 10-4
Dixie Valley Pre-factor (m3)
min 1 x 10-7
2 x 10-5
max 1 x 10-5
Magnola Pre-factor (m3)
min 1 x 10-7 5 x 10-6
max 5 x 10-4 8 x 10-4
1081 1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
46
Figure 1 1098
1099
1100
47
Figure 2 1101
1102
1103
48
Figure 3 1104
1105
49
Figure 4 1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
50
Figure 5 1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
51
Figure 6 1144
1145
1146
1147
52
Figure 7 1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
53
Figure 8 1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
54
Figure 9 1169
1170
1171
1172
1173
1174
1175
1176
55
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
56
Figure 10 1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
57
Figure 11 1203
1204
1205
1206
1207
1208
1209
1210
1211
58
1212
1213
1214
1215
1216
1217
1218
1219
1220
59
Figure 12 1221
1222
1223
1224
1225
1226
1227
1228
1229
60
Figure 13 1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
61
Figure 14 1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
62
Appendix A: Potential bias in roughness data 1265
A.1. Reliability of the roughness results at the LiDAR scale: effect of the noise in the acquisition 1266
system 1267
Before analyzing the main biases inherent to LiDAR data acquisition, we emphasize that raw 1268
scanner data acquired with the home-made laser profilometer and the WLI are considered as quasi-1269
ideal since the level of the intrinsic white noise of these instruments is well below that recovered from 1270
the fault surfaces (see Figure 6). A detailed description of the conditions in which measurements with 1271
this scanner device departs from the reality is given by Méheust (2002). In addition, the slope of the 1272
spectra computed on WLI scans at spatial scales larger than mm05.0 are consistent with those of 1273
laser profilometry for the same range of length scales (see Figure 9). Given that LiDAR data include 1274
the largest bias, we focus our noise analysis on this instrument. Consequently, the results provided by 1275
the following analyses can be considered as end-members of the noise effect estimation for the three 1276
instruments used in our study. 1277
In the spirit of the work of Schmittbuhl et al. (1995) on the reliability of a self-affine 1278
measurement on 2-D rough profile, Candela et al. (2009) have reviewed different statistical 1279
methodologies which allow the assessment and characterization of the anisotropic self-affine behavior 1280
of fault topography. This work was mainly devoted to precisely define the intrinsic error of the 1281
statistical methods (such as the Fourier power spectrum) to estimate the scaling properties of fault 1282
surface roughness. 1283
Here, a new test is performed by taking into account the error encountered in the spatial position 1284
( ZYX ,, ) of each points measured by the 3-D laser scanner. We use a synthetic anisotropic self-1285
affine surface (Candela et al., 2009) of m55 with an original regular point spacing of mm5 1286
(Figure A1), and with two different Hurst exponents in perpendicular direction ( 6.0// H and 1287
8.0H ). In order to simulate the error inherent to LiDAR data acquisition on the spatial position 1288
of each points ( ZYX ,, ), we add Gaussian white noise with a distribution of ]5.2,0[ mm on the 1289
original position of each point, to obtain the perturbed grid ( ZYX ,, 11). Then the height
1Z of each 1290
63
point at these new positions (11,YX ) is computed by interpolating (bilinear interpolation) the four 1291
nearest points of each of these new positions on the original ideal model (Figure A1). In a final step, 1292
Gaussian white noise, with a standard deviation equal to mm5 , is randomly added on the 1293
interpolated heights 1Z , to yield the error of the LiDAR data in the vertical position. 1294
After generating this noisy cloud of points (111 ,, ZYX ), we extract profiles oriented along slip 1295
direction and perpendicular to it as is done for the measurements, and estimate both Hurst exponents. 1296
The results are then averaged over 100 realizations of synthetic surfaces. Due to the noise, the Fourier 1297
power spectra flatten at short length scales (as for examples in the slip direction shown in Figure A1), 1298
which results in a slight underestimation of the Hurst exponents. For both directions, along slip and 1299
perpendicular to it, we find that the median estimates of //H and H of the biased synthetic surfaces 1300
are 06.059.0 and 11.074.0 respectively, compared to the noise-free data where the Hurst 1301
exponents were 07.06.0 and 05.078.0 , respectively. Note that the error bar of the estimated 1302
Hurst exponent of the biased synthetic surfaces is approximately twice larger in the direction 1303
perpendicular to slip relative to that in the slip direction. In both directions, even if the Hurst exponent 1304
is slightly underestimated for the noisy data, its value is still included in the range given by the 1305
standard deviation of the noise-free data. Therefore, the noise in the LiDAR data could be estimated 1306
as well as the reliability of Hurst exponent values. 1307
1308
A.2. Reliability of the roughness results at surface rupture scale: effect of re-sampling 1309
For each earthquake, once the surface rupture map has been digitized, the data set is re-sampled 1310
to a regular spacing to ensure consistent spatial sampling, independent of the length of each rupture. 1311
This re-sampling is performed to avoid bias due to local wiggles of the rupture trace (Klinger, 2010). 1312
We verify here how this re-sampling process affects the spectra of the Fourier transform. The original 1313
digitized rupture trace of the Hector mine earthquake, taken as an example, is re-sampled with various 1314
constant values of x in the range 60-620 m (Figure A2). In the Fourier power spectra of this set of 1315
digitized rupture traces, two regimes can be observed. At small scales, i. e. large wave numbers, 1316
64
(between approximately m120 and m1200 ) the behavior can be attributed to data re-sampling. At 1317
large scales, i. e. small wave numbers (above m1200 ), a power law giving a Hurst exponent 1318
74.0RH represents the best fit. The cross-over length scale between the two regimes corresponds 1319
to the maximum spacing between two points in the original data. Whatever the value of x taken for 1320
re-sampling the data, the cross-over length scale remains at the same position. We propose that the 1321
regime at large scales, characterizing roughness properties of the digitized ruptures traces is therefore 1322
not affected by data re-sampling; both the slope and the pre-factor of each spectrum are identical 1323
(Figure A2). 1324
The same re-sampling procedure has been performed on ideal synthetic self-affine profiles in 1325
order to precisely define if the scaling property could be modified (Figure A3). An original self-affine 1326
profile (6.001.0 LRMS ) with regular spacing of m500 and a total length of km100 (extracted 1327
from a synthetic surface as previously presented), is modified by adding to the X coordinates a 1328
Gaussian white noise perturbation with a distribution of ]250,0[ m (Figure A3). This altered profile is 1329
re-sampled with different values of x in the range [40-350] m. The Fourier spectra (Figure A3) 1330
indicate that for the different values of x , the large scale regime (above m750 ) characterizing the 1331
input self-affine behavior with a Hurst exponent of 6.0 is not biased. This test validates that the re-1332
sampling procedure makes it possible to keep the scaling information of the rupture traces at spatial 1333
scales above 1200 m. 1334
1335
65
Figure A1: Effect of noise inherent to LiDAR data acquisition. A: Ideal synthetic m55 self-affine 1336
surface with an original regular point spacing of mm5 . B: Illustration of the addition of a noise in the 1337
regularly spaced original grid ),,( ZYX . C: Comparison of the Fourier power spectra in direction of 1338
slip obtained for the ideal synthetic surface and the noisy synthetic surface. Both vertical dashed grey 1339
lines indicate limits taken for fitting the Hurst exponent. D: Distribution of measured Hurst exponents, 1340
on 100 simulations, for the direction of slip and perpendicular to it. Black bars show the ideal 1341
simulated fault surface models, and the gray bars correspond to the noisy simulated fault surface 1342
models. The solid lines (black for the ideal case and gray for the natural case) represent the fits of the 1343
measurements to a normal distribution with mean and standard deviation given at the top left 1344
for the noisy natural case and the top right for the noise-free case. 1345
1346
Figure A2: Effect of re-sampling effect on earthquake surface rupture roughness: example of Hector 1347
Mine earthquake. A: Digitized surface rupture traces of the Hector Mine earthquake. The original 1348
rupture map with an irregular point spacing (pink profile) is re-sampled in order to ensure consistent 1349
spatial sampling with a regular spacing )( x . The inset indicates the position of the zoom located on 1350
the right, showing the irregular point spacing on the top and the gradual increase of the re-sampling 1351
(or decreasing of x ) with a regular point spacing of profiles downwards. B: Fourier power spectra of 1352
the digitized rupture traces shown in A. Spectra colors are the same than in A. On the right: the 1353
spectra have been shifted vertically to improve the visibility. The blue vertical bar on both graphs 1354
highlights the cross-over length scale, at approximately m1200 , between both regimes: the lower 1355
regime is attributed to data re-sampling, the upper regime characterizes roughness properties of the 1356
digitized profiles with a Hurst exponent 74.0RH . 1357
1358
Figure A3: Effect of re-sampling on synthetic self-affine profiles. A: Example of synthetic rough 1359
profiles with a standard deviation that scales as 6.001.0 LRMS , analogue of digitized surface 1360
rupture traces. The green profile at the top of the left figure is the original ideal synthetic profile with 1361
66
a regular point spacing x of m500 . A Gaussian white noise perturbation with a distribution 1362
]250,0[ m is added on the original regular spacing to obtain a noisy profile (pink curve) similar to 1363
that of original ruptures maps. This noisy synthetic profile with irregular point spacing is re-sampled 1364
with a regular spacing x . The inset zooms on the synthetic profiles located on the right. B: Fourier 1365
power spectra of the synthetic self-affine profiles shown in A. Colors of each curves correspond to 1366
those of profiles. On the right: the spectra have been shifted vertically to improve the visibility. 1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
67
Figure A1 1389
1390
1391
68
Figure A2 1392
1393
1394
69
Figure A3 1395
1396
1397
1398