Generation of self-affine surfaces
A Self-affinity in 1D and 2D
Surface roughness analysis provides a statistical characterization of a surface which is simpler and easier to use than a complete deterministic description. In geophysics, Brown and Scholz (1985) and Power et al. (1987) demonstrated the self-similar property of natural fault surfaces at field scale. Coming from statistical physics, a more general scaling transformation has been proposed: self-affinity (Mandelbrot, 1985; Mandelbrot, 1986; Voss, 1985) that was successfully used for the quantitative description of fault roughness (Schmittbuhl et al., 1993, Renard et al., 2006).
A self-affine 1D profile remains unchanged under the scaling transformation δx → λ δx, δz → λHδz for 1D profiles (Figure 2.1) extracted from a surface (Meakin, 1998). Here, δx is the coordinate along the profile and δz is the roughness amplitude. For a self-affine profile, the scaling exponent H, also called Hurst exponent, lies in the range 0 ≤ H < 1. Accordingly, self-affinity implies that a profile appears less rough as the scale increases. In other words, if a profile is self-affine, a magnified portion of it will appear statistically identical to the entire profile if different magnifications are used in the x and z-directions (Figure 2.1).
For 2D surfaces, this self-affinity property can be described for sets of 1D parallel profiles extracted from the surface. Moreover, if the surface is striated along some given orientation, anisotropic scaling behavior can emerge if H varies for different directions in the plane of the surface. An anisotropic self-affine surface Zx, y with coordinatesx, y obeys the property: Z1 / H // x,1 / H y Zx, y, where is a positive dilation factor, H // and H are the Hurst exponents, comprised between 0 and 1, in two perpendicular directions of the surface. H // is defined along a direction parallel to the main striations, and H is defined along a direction perpendicular to the striation (Figure 2.1b).
B. Synthetic anisotropic self-affine surfaces
To calculate synthetic fault surfaces (Figure 2.1b), we used a Fourier based method to simulate a matrix scaling random Gaussian field on a 2D grid, where an anisotropy matrix eigenvalues of this matrix correspond to the inverse of the two roughness exponents H // , and H that characterize the self-affine properties of the generated surface in two perpendicular directions (Bierme et al. 2007, Clausel and Vedel, 2008). The code to generate an anisotropic 2D self-affine surface, written in Matlab©, is given in the appendix 2.A and can be run easily on a desktop computer.
In the following sections, we decompose the signal processing analysis of rough surfaces in two stages. Firstly, we present the six signal processing tools used to estimate the self-affine property of an isotropic surface with a single Hurst exponent (Figure 2.1a), as observed for example for fresh mode I brittle fractures in rocks (Power et al., 1987; Schmittbuhl et al., 1995b; Bouchaud, 1997). For this, we have synthesized several isotropic surfaces with an exponent in the range [0.1 – 0.9] and grid sizes in geometrical series: 129 × 129 points, 513 × 513 points, 2049 × 2049 points. Secondly, we analyse synthetic anisotropic surfaces (Figure 2.1b) with H // in the range [0.7 – 0.9] and H in the range [0.4 – 0.9].
Statistical signal processing methods
We have used six different methods that characterize the amplitude of the roughness at various spatial wavelengths. All these methods, presented in the following sub-sections, are based on the analysis of 1D profiles (Figure 2.1c) that are extracted from the 2D Digital Elevation Model (DEM) of 2D surfaces (Figure 2.1a, b). For each surface, a set of 1D parallel profiles in a specific direction are extracted, detrended and then analyzed. Then, the properties are averaged over all the 1D profiles to characterize the 2D surface in the chosen direction. To study the azimuthal dependence of the statistical properties of the surfaces, we have extracted profiles in several directions, following a 360° rotation. Estimating the statistical properties of the surface in various directions (Renard et al., 2006) allows characterizing a morphological anisotropy.
For the application to natural fault surfaces, we also tested how the noise in the data and the presence of missing points could affect the estimation of fault surface. Indeed, the raw scanner data consist of clouds of points, with x, y, and z coordinates, sampled more or less regularly. Sometimes, data are missing (vegetation on the fault plane, low reflectivity of the scanner light beam), and the surface is incomplete. An interpolation is then necessary, which induce a bias in the estimation of scaling exponents that need to be estimated too.
Figure 2.1. Digital Elevation Models (DEM) of 2D synthetic self-affine surfaces (up) and 1D profiles (down) generated using the algorithm of the appendix. (a) Surface with an isotropic self-affine property characterized by a Hurst exponent of 0.8. (b) Anisotropic self-affine surface with two Hurst exponents ( H // 0.6 and H 0.8 ) in perpendicular directions. (c) Representative 1D profiles extracted in two perpendicular directions of surface (b). Inset: magnified portion of a profile along the H // direction (parallel to the striations), which has a statistically similar appearance to the entire profile when using a rescaling δx → λ δx, δz → λHδz.
A. Root-mean-square correlation (RMS) and maximum-minimum height difference (MM) methods
Consider a 1D profile L(x) that is divided into windows of width x and indexed by the position of the first point x0 of the band. The standard deviation x of the height Lx and the height difference h x between the maximum and minimum height are computed for each band, and then averaged over all the possible bands of fixed width x spanning the profile, by varying the origin x0 . We then obtain x and h x Note that this technique is useful when H is not too close to 0 or 1, where a significant error can be measured (see Figures 2.3a, b – 2.4a, b). Usually, levelling off ofx at small x values is due to the noise in the data (see Figure 2.7c, d), and leveling-off at largex isdue to the finite size of the profile.
B. Height-height correlation function (COR) method
For a signal Lx, we consider the height-height correlation which estimates the average height-height difference between two points of the profile separated by a distancex . For a self-affine profile, the correlation function follows a power-law such that C x x H where H is the Hurst exponent.
Standard deviation of the correlation function (RMS-COR) method
For a profile Lx containing N points, the height difference L between each pair of points separated by a distance x is calculated. The window size x is varied between the sampling distance and the size of the system and, for a given x , the standard deviation of the height difference Lx is calculated. For a self-affine surface this measurement follows a power-law such that x x H . This method was successfully applied to characterize the self-affine properties of the Vuache fault plane (Renard et al. 2006).
Fourier power spectrum (FPS) method
The Hurst exponent H can be estimated from the Fourier power spectrum which has a power law form for a 1D self-affine profile (Barabasi and Stanley, 1995; Meakin, 1998). For each parallel profile, the Fourier power spectrum Pk, i. e. the square of the modulus of the Fourier transform, is calculated as a function of the wave-number k . Then the spectrum of the whole surface is calculated by stacking all the 1D Fourier transforms to reduce the noise associated with individual profiles. For each profile of length L containing N increments, the spatial frequencies range between 1/ L and the Nyquist frequency N / 2L (i.e. the reciprocal of the interval between data points). When plotting the power spectrum as a function of k in log-log space, a self-affine function reveals a linear slope, which is itself a function of the Hurst exponent H through Pk k12H .
Average wavelet coefficient power spectrum (WPS) method
The average wavelet coefficient method consists of decomposing the input signal into amplitudes that depend on position and scale. The wavelet transform of each 1D profile Lx wavelet coefficients are averaged over the translation factor b for each length scale a : Wa Wa,b b . If the profile is self-affine, the wavelet transform verifies statistically that, for any positive dilatation factor , Wa,bLx H Wa,b . Accordingly, the averaged wavelet coefficients scale as Wa aH1/ 2 . A wide range of wavelet functions can be used. For a simple and efficient implementation, we chose the Daubechies wavelet of order 12 as suggested in Simonsen et al. (1998).
Quantitative estimation of the accuracy of roughness analysis methods
Synthetic isotropic and anisotropic rough surfaces
Figures 2.1a and 2.1b display the topography of synthetic rough surfaces where the data set includes 2049 × 2049 points regularly spaced on a grid. Figure 2.1a shows an isotropic rough surface, whereas Figure 2.1b shows an anisotropic surface, with corrugations elongated parallel to the direction of smaller Hurst exponent (analogue to the direction of slip on a natural fault surface) and covering a wide range of scales.
The roughness amplitude of a profile parallel to the striation direction (green curve in Figure 2.1c) is smaller than that of a perpendicular profile. The profile extracted along the direction with the smallest exponent (green curve) appears more jagged at small scales compared to a perpendicular profile, showing the different effects of the anisotropy of the surface on the waviness and amplitude of the profiles.
The outputs of the statistical methods described in section 2.2.3 are represented on Figure 2.2 Each curve is calculated by averaging the outputs of all possible parallel 1D profiles extracted from the anisotropic surface of Figure 2.1b. The results are represented in a log-log plot, allowing visualizing the linear portion of the curve that characterizes a power-law distribution (Figure 2.2). This linear portion is binned in a small number of increments, and a power-law fit is performed to extract the Hurst exponent that characterizes the self-affinity of the profile. The best fits are performed for each curve and a value of the « output » self-affine exponent is then calculated for all the six signal processing methods.
Using the RMS correlation function, we have also extracted sets of parallel profiles in several directions, at an angle to the direction of the striations. For each set of profile, we have calculated the correlation function and estimated the value of H . The angular dependence of H could be represented on a polar plot (inset in Figure 2.2d) (Renard et al., 2006). The anisotropy of such plot characterizes the anisotropy of the surface: an isotropic surface is represented as a circle of radius H , whereas an anisotropic one has a more complex elliptical shape.
Isotropic surfaces: effect of size and input exponent on the output estimation of the Hurst exponent
The comparison between the input Hurst exponent used to calculate an isotropic synthetic surface and the output Hurst exponent estimated using the six different methods is represented on Figure 2.3, for different system sizes. The RMS, MM, COR and RMS-COR methods are all mainly sensitive to the value of the input self-affine exponent (the typical trend of the curve is not parallel to the diagonal). Small self-affine exponents are systematically overestimated whereas large exponents are underestimated. In contrast, the error for the WPS method is mainly function of the system size (the response is more or less parallel to the diagonal). The FPS method appears the most accurate technique, with only slight sensitivity to the input self-affine exponent and size effects. This conclusion should however be interpreted cautiously as the algorithm used to generate the synthetic surface is based on a Fourier transform approach. The conclusion of this comparison tests is that the FPS, WPS, and RMS-COR methods are the most reliable because they have a small dependence on the value of the input Hurst exponent and a slight dependence on system size.
Figure 2.2. Outputs of the six signal processing techniques applied on the data of the anisotropic self-affine surface shown in Figure 2.1b. (a) Root-mean-square correlation (RMS), (b) maximum-minimum height difference (MM), (c) correlation function (COR), (d) RMS correlation function (RMS-COR), (e) Fourier power spectrum (FPS), (f) Wavelet power spectrum (WPS). The dashed lines represent fits of the curves to obtain the output . The inset in (d) displays a polar plot of H obtained by the RMS-COR method and allowing then to determine the azimuth dependence of H . The black points correspond to the Hurst exponents for the two profiles shown on this plot.
Figure 2.3. Comparisons between the “input” Hurst exponents introduced in the construction of isotropic fractional Brownian surfaces and the “output” exponent measured on these surfaces using the six independent signal processing techniques. The three outputs illustrate effects of system size. (a) Root-mean-square correlation (RMS), (b) maximum-minimum height difference (MM), (c) correlation function (COR), (d) RMS correlation function (RMS-COR), (e) Fourier power spectrum (FPS), (f) Wavelet power spectrum (WPS). The gray line in (a), (b), and (c) indicate for which input exponent the error in the estimation is closest to zero.
Anisotropic surfaces: Interaction between the two input roughness exponents
For synthetic self-affine anisotropic surfaces, we have calculated the error on the estimation of the two Hurst exponents. For this, we have built surfaces (2049×2049 points, similar to Figure 2.1b) for which the Hurst exponents H and H input in two perpendicular directions were varied in the range [0.4 – 0.9]. We have then used the six signal processing techniques to estimate the values of these same exponents. The absolute error in the estimation of each Hurst exponent (Figure 2.4) depends on the input value of these parameters and also on the amplitude of their difference ( H input// H input ).
This error is particularly large for the RMS (up to 20%), MM (up to 25%), and COR (up to 35%) methods. When the input anisotropy ( H input// H input ) increases, the absolute error on the two output exponents increases accordingly. The absolute error is smaller in the direction of the smallest exponent (analogue to the direction of striation on a natural fault surface) than perpendicular to it. Moreover, it is also noteworthy to mention that these three techniques show significant errors in the estimation for input exponents close to 0.8-0.9 even if the anisotropy is minimal, demonstrating the limited reliability of these methods to detect an exponent close to one. The RMS-COR analysis is also sensitive to the input anisotropy (Figure 2.4d), however such an effect is not strongly pronounced (the absolute errors are smaller, up to 15%). For this method, the error does not depend on the values of the two input Hurst exponents. For example, with a synthetic surface defined by H input// = 0.8 and H input = 0.6, the absolute error in the estimation of each Hurst exponent is almost identical. As shown in Figure 2.4d, when anisotropy is small, the errors do not increase significantly for input values close to 1 unlike the three previous methods.
Table of contents :
Chapter 1 : Introduction
1 1.1. Introduction
3 1.2. Complexité géométrique des zones de failles
4 1.3. De la fracturation au frottement
1.4. Variation de la contrainte pendant le glissement
1.5. Hétérogénéité spatiale du glissement et de la contrainte
1.6. Plan du manuscrit
Chapter 2 : Characterization of fault roughness at various scales: implications of three-dimensional high resolution topography measurements
2.1. Chapter 2 overview (Présentation du Chapitre 2)
2.2. Characterization of fault roughness at various scales: implications of three-dimensional high resolution topography measurements
2.2.2. Generation of self-affine surfaces
2.2.3. Statistical signal processing methods
2.2.4. Quantitative estimation of the accuracy of roughness analysis methods
2.2.5. Quantitative Acquisition of roughness data on natural faults at various scales.56
2.2.6. Roughness results and interpretation
2.2.7. Discussion & Conclusion
2.2.8. Appendix 2.A
Chapter 3 : Roughness of fault surfaces over nine decades of length scales: self- affinity, variability, and absence of characteristic length scale
3.1. Chapter 3 overview (Présentation du Chapitre 3)
3.2. Roughness of fault surfaces over nine decades of length scales: self-affinity, variability, and absence of characteristic length scale
3.2.2. Fault roughness data
3.2.3. Analysis of scaling properties of roughness data
3.2.4. Fault roughness results
3.2.7. Appendix 3.A
Chapter 4 : Fault slip distribution and fault roughness
4.1. Chapter 4 overview (Présentation du Chapitre 4)
4.2. Fault slip distribution and fault roughness
4.2.2. Self-affine correlations of seismological slip fields
4.2.3. Fault surface roughness
4.2.4. From fault geometry to the spatial distribution of slip
4.2.5. Discussion and conclusion
Chapter 5 : Stress drop during earthquakes: effect of fault roughness scaling
5.1. Chapter 5 overview (Présentation du Chapitre 5)
5.2. Stress drop during earthquakes: effect of fault roughness scaling
5.2.2. Roughness of natural fault surfaces
5.2.3. Spatial variability of the stress drop on the fault plane
5.2.4. Evolution of the stress drop with rupture size
Chapter 6 : Effect of surface morphology on the dissipation during shear and slip along a rock-rock interface that contains a visco-elastic core
6.1. Chapter 6 overview (Présentation du Chapitre 6)
6.2. Effect of surface morphology on the dissipation during shear and slip along a rock-rock interface that contains a visco-elastic core
6.2.2. Rough geological interfaces
6.2.3. Visco-elastic shear flow between rough walls
6.2.4. Concluding remarks
Chapter 7 : Perspectives: Origine de la rugosité des failles et développement des surfaces de glissements
7.1. Chapter 7 overview (Présentation du Chapitre 7)
7.3. Architecture des zones de failles
7.4. Propriétés géométriques des lentilles
7.5. Modèle mécanique de la formation des lentilles
Chapter 8 : Conclusion