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Corresponding author: F. Renard, ISTerre, University of Grenoble I and CNRS, BP 53, 38041 Grenoble, France. (francois.renard@ujf-grenoble.fr)

Abstract

[1] Many faults and fractures in various natural and man-made materials share a remarkable common fractal property in their morphology. We report on the roughness of faults in rocks by analyzing the out-of-plane fluctuations of slip surfaces. They display a statistical power-law relationship with a nearly constant fractal exponent from millimeter scale micro-fractures in fault zones to coastlines measuring thousands of kilometers that have recorded continental breakup. A possible origin of this striking fractal relationship over 11 orders of magnitude of length scales is that all faulting processes in rocks share common characteristics that play a crucial role in the shaping of fault surfaces, such as the effects of elastic long-range stress interactions and stress screening by mechanical heterogeneities during quasi-static fracture growth.

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[2] In the present study, we propose that the effect of faulting mechanics is expressed in a similar way from micro- to macro-scales. At the scale of the whole Earth's crust, the corrugated shapes of several continents, such as the West coast of Africa and the East coast of South America, have recorded fracturing processes at work during continental breakup since the late Triassic, when the Pangaea supercontinent started to breakup. Other plate boundaries, along major strike-slip faults or in opening rifts, also display corrugated morphologies, as do surface ruptures of major earthquakes or outcropping fault planes at smaller scales [Power et al., 1987]. The question arises of the link between this roughness observed at various scales and the fracturing processes in rocks. This represents an important and still debated challenge because rocks are heterogeneous materials in which short- and long-range stress interactions between defects they contain partly control crack nucleation and propagation into developed fractures and faults [Bonamy and Bouchaud, 2011]. The scale invariance property of the stress interactions between multi-scale defects in rocks (dislocations, fluid inclusions, grains, lithological heterogeneities, pre-existing fractures) suggests that a unique fracturing process could be at work from the micro- to macro-scales. To better characterize the geomorphic manifestation of fracturing processes, we study one of the fingerprints left during the development of faults: their rough topography [Power et al., 1987; Renard et al., 2006; Sagy et al., 2007; Candela et al., 2012].

2 Fault and Fracture Roughness Data Analysis

[3] We use three sets of fault morphology data, where the out-of-plane fluctuations of fault planes (i.e., roughness) are measured in one or two dimensions (see Auxiliary Material). First, at the thousand-kilometer scale, the shorelines of continental passive margins (Africa, Red Sea, India, South America) represent one-dimensional markers of faulting and fracturing processes that induced continental breakup along rifts. The large-scale amplitude of the continent shorelines is considered here as a footprint of the large-scale faulting processes during the initial rifting process, which broke continents apart. The data come from Wessel and Smith [1996]; see the inset in Figure S1a. We do not consider here continental active margins or mid-oceanic ridges, whose morphology was affected by other processes than fracturing, such as subduction or magmatic activity [Abelson and Agnon, 1997]. Secondly, at the pluri-kilometer scale, the traces of several faults and surface ruptures of several earthquakes were analyzed; these data spanning various modes of faulting and rock composition, we consider them as representative of a wide range of faults. The surface trace of the North Anatolian Fault in Turkey was extracted from geological maps [Herece and Akay, 2003]. The trace of seven major segments of this fault was digitized from the maps and then interpolated with a resolution of 2 km. In addition, the surface ruptures of 15 major continental earthquakes were used as one-dimensional markers of the fracturing paths during fault growth and propagation (Figure 1a). These data were either digitized manually from high-resolution geological maps and aerial photographs [Klinger, 2010; Candela et al., 2012] or measured directly on the field in the weeks following the earthquake, using portable GPS [Emre et al., 2003; Pucci et al., 2006]. The trace of a creeping fault was measured using InSAR images [Jolivet et al., 2012]. Thirdly, at the millimeter to meter scale, the two-dimensional morphology of several slip surfaces observed on fresh outcrops of fault zones was acquired using field and laboratory distance meters (Figures 1b and 1c) capable of resolution down to one tenth of a millimeter. Representative one-dimensional topographic profiles along rough faults are displayed in Figure 2, with profile lengths spanning a range of scales from millimeters for outcrop samples to thousands of kilometers for continent shorelines.

[4] The three data sets were analyzed to characterize, if any, scaling relationships, also called fractals by Mandelbrot [1967]. Fractal analyses have been applied in many geophysical systems, including topography of the continents or the sea floor [see Turcotte, 1992, and references therein]. This can be achieved by using various statistical methods able to reveal spatial correlations in a signal [Schmittbuhl et al., 1995a; Candela et al., 2009]. One of these methods involves the Fourier transform of the rough profiles of all one-dimensional data (shorelines, earthquake ruptures, fault traces) and the average of the Fourier spectra of several one-dimensional profiles extracted perpendicularly to the direction of slip from each two-dimensional fault surface. For a signal with fractal properties, the modulus of the Fourier transform shows a linear trend in a bi-logarithmic plot, indicative of a power-law relationship between the modulus of the Fourier spectrum to the square, P(k)^{2}, and the wave number, k, (the inverse of a length scale), expressed by P(k)^{2} ∝ Ck^{− 1 − 2H} [Schmittbuhl et al., 1995b]. The so-called Hurst exponent, H, can be extracted from the slope of the bi-logarithmic plot P(k)^{2} versus k. It describes how the roughness changes with scale while the prefactor C characterizes the amplitude of roughness at a given scale. For a self-affine scale invariant signal, H lies between zero and one. It is related to the fractal dimension D used in several geophysical studies through D = 2 − H for a 1-D signal [Turcotte, 1992].

[5] All one-dimensional data were analyzed using the same procedure. Each data item was considered as a 1-D rough signal for which scaling was analyzed. The signals were first positioned into the same coordinate system, by projecting longitude-latitude coordinates onto a sphere with the same radius as the Earth, to obtain kilometric coordinates. Then, each profile was rotated to adjust its greatest length in the direction of the x axis, the out-of-plane fluctuations (i.e., roughness) being directed along the y axis (Figure 2). Each signal was tapered at both ends using a cosine function, so that 3% of the length of the signal converged smoothly to zero at each extremity. Then the direct Fourier transform was calculated, and its modulus was elevated to the square. The spectra obtained were normalized by the number of points in each signal. All this procedure has been used and discussed in previous studies [Power et al., 1987, 1988; Schmittbuhl et al., 1995a], including the effects of noise in the acquisition device or the presence of missing data [Candela et al., 2009, 2012]. In the case of the two-dimensional data, we also used a 1-D FFT approach, for the sake of consistency. Several hundreds to thousands profiles were extracted in the direction perpendicular to slip, and their Fourier spectra were calculated in the same way as for the one-dimensional data. The only difference is that all the spectra relative to a given surface were averaged to obtain a representative mean Fourier spectrum. Finally, using standard linear regression in bi-logarithmic coordinates, the slope of the Fourier spectra and the Hurst exponent were calculated by dividing each spectrum into 10 bins of lengths following a geometric progression that covers 2–4 decades of length scales and neglecting the highest frequencies when they were altered by the acquisition noise.

3 Results

[6] All individual fault surfaces or fault trace spectra exhibit a power-law behavior over 2–3 decades of length scales (Figure S1), indicating, for each of them, correlations at all scales. Interestingly, all data set spectra align on the same trend with H = 0.77 ± 0.23 (Figure 3), indicating that fault roughness in rocks exhibits a common self-affine fractal statistical property over eleven decades of length scales, from 0.1 mm to 10,000 km.

[7] Since the power spectrum is directly related to the second-order moment of the height distribution of fault surfaces, our analysis also reveals the scale dependence of the standard deviation of fault roughness amplitude. Overall, these out-of-plane fluctuations correspond to ~0.3 to 3 km over a length of 50 km, and 0.02 to 0.2 m over 5 m. The fractal relationship is observed for different modes of faulting (normal, strike slip, thrust), different kinds of rocks (limestone, rhyolite, cherts), and various cumulative displacements ranging from nearly 10 m to tens of kilometers. The value of H does not vary significantly with these parameters (see inset in Figure 3).

4 Discussion and Conclusion

[8] In the case of the continent shorelines and fault traces, the value of H is significantly different from what is observed with natural landscapes, where H is in the range 0.4 to 0.5 [Rodriguez-Iturbe and Rinaldo, 2001]. Indeed, continent topography is the result of not only fracturing processes at work during mountain building or rifting but also other processes such as river incision, erosion, folding, flexure, or gravitational collapse shape the Earth's surface.

[9] Two main characteristics of the plots in Figure 3 can be discussed. First, a common morphological statistical property of roughness is present in systems as different as micro-cracks in fault zones, meter-scale slip surfaces, earthquake rupture traces, continental fault morphology, and continent shorelines. All these systems share a common history: they have formed by breaking rocks apart. Secondly, Figure 3 also shows that for a given length scale, the amplitude of roughness can vary, as the positions of spectra show a vertical spread. Both the value of the Hurst exponent and the value of the roughness at a given scale are discussed below, based on statistical physics studies and laboratory measurements [Måløy et al., 1992; Schmittbuhl et al., 1995b; Hansen and Schmittbuhl, 2003; Santucci et al., 2010].

[10] Interestingly, Mandelbrot [1967], following observations by Richardson [1961], defined the notion of self-affinity based on the fractal property of the length of the coastline of England. The presence of a power-law relationship, covering 2–4 decades of length scales, for fracture roughness in metals was also first recognized by Mandelbrot et al. [1984]; see also the Auxiliary Material for a discussion on the value of the fractal exponents in fault roughness. In the present study, we discuss why the value H = 0.8 observed in faults and in several shorelines along continental passive margins may have a unique origin, the quasi-static fracturing process of rocks [Bouchaud et al., 1990]. The physics of fracture and fault propagation has been studied by adopting an analytical approach in the framework of linear elastic fracture mechanics [Lawn, 1993] and by considering the crack front as an elastic line progressing through a disordered medium [Bonamy and Bouchaud, 2011]. Linear elastic fracture mechanics theory states that stress intensity factors—which depend on the geometry and on the intensity of loading—control both the crack path and the crack velocity. Many attempts to model fault roughness include basic mechanics in the description of fracture of heterogeneous media: a crack acts as a stress concentrator, with steep gradients around it [Hansen and Schmittbuhl, 2003; Bonamy and Bouchaud, 2011, and references therein].

[11] Several processes may modify the stress intensity factors and lead to roughness development with a Hurst exponent H = 0.8, which has been widely observed [Bonamy and Bouchaud, 2011]. Although the disorder pre-existing in the material, such as pores or cracks, modifies out-of-plane fluctuations of the main fracture path [Renard et al., 2009], it is the formation of damage, that is, of cavities, cracks, or other defects ahead of the crack tip during fracture which has been conjectured to be at the origin of such a roughness [Bonamy and Bouchaud, 2011]. This mechanism is actually quite general. It was proposed to explain quasi-brittle fracture in heterogeneous materials such as wood or concrete, where micro-cracks initiate, grow, and coalesce ahead of the crack tip, forming a damage zone of macroscopic size. Similarly, a fault may extend by connecting smaller cracks that coalesce through a hierarchical fault growth process that gives the surface a rough morphology, as observed in laboratory experiments of limestone samples under compression [Otsuki and Dilov, 2005].

[12] A similar process was proposed in quasi-static ductile fracture of metallic materials, where fracture roughness development was numerically simulated by the interplay between the crack front and damage voids created ahead of it at a local scale [Needelmann et al., 2012]. As a consequence, the crack paths zigzag through their interaction with voids, in a regime dominated by inter-cavity elastic and plastic correlations.

[13]Hansen and Schmittbuhl [2003] proposed that correlated stress gradients in the damage zone are responsible for strong elastic interactions between the crack tip and its neighborhood, which are at the origin of the observed fracture roughness. They calculated H = 4/5, close to the result H = 0. 77 ± 0.23 found in the present study. However, this model is still debated. Interestingly, Santucci et al. [2010] observed experimentally that the amplitude of the roughness at a given scale depends on the size of the initial disorder in the fracturing solid. This result is in qualitative agreement with previous observations showing that this amplitude is approximately 1000 times smaller for silicate glass than for metallic materials [Bouchaud, 1997]. Finally, it has to be noticed that once a fault has formed, its roughness evolves because of wear during sliding, healing processes during the inter-seismic period, and damage development in the fault zone during and between earthquakes [Power et al., 1988].

[14] In summary, a picture of the origin of the scaling property of fault roughness emerges. We interpret that the scaling relationship, with a Hurst exponent close to 0.8, could be related to elastic interactions between the fault and the disorder, which appears around it during fracture and screens out long-range interactions over a distance which depends on its nature, size, and density. We also acknowledge that other interpretations of the data may be possible, if one considers, for example, roughness development due to the dynamic propagation of a fracture [Sharon and Fineberg, 1996]. For a given length scale, the amplitude of roughness might be related to the characteristic size of the pre-existing disorder, which explains the variability of this quantity. Finally, once a fault has formed, Brodsky et al. [2011] have observed a slight decrease of the roughness with increasing slip.

[15] The fact that fault surfaces in rocks exhibit a universal self-affine fractal morphology at scales ranging from millimeters to thousands of kilometers remains to be modeled in a comprehensive manner. Such a model would also help in understanding how the presence of morphological heterogeneities on fault surfaces may affect the nucleation and propagation of earthquakes [Ohnaka and Shen, 1999], and the associated frictional processes responsible for stress heterogeneities at all scales.