Journal of Geophysical Research : Solid Earth Detecting the Onset of Strain Localization Using Two-Dimensional Wavelet Analysis on Sandstone Deformed at Different Effective Pressures

Laboratory brittle deformation experiments have shown that a rapid transition exists in the behavior of porous materials under stress: At a certain point, early formed and spatially distributed tensile cracks interact and coalesce forming a shear plane. In this work, we present and apply a novel image processing tool that is able to quantify this transition between distributed (“stable”) damage accumulation and localized (“unstable”) deformation, in terms of the size, density, and orientation of cracks at the point of strain localization. Our technique, based on a two-dimensional Morlet wavelet analysis, can recognize, extract, and visually separate the multiscale changes occurring in the crack network during the deformation process. We first performed a series of triaxial experiments (σ1 >σ2 = σ3) on core plugs of Hopeman Sandstone (Scotland, UK) at different effective pressures (from 5 to 30 MPa). We then processed high-resolution backscattered electron microscope images of thin sections of these core plugs and found differences in the strain localization process as effective pressure was increased. The critical length of tensile cracks required before the onset of strain localization was reduced from 0.24 to 0.15 mm as the effective pressure was increased to 30 MPa, resulting in a narrow fracture zone. Critically, by comparing patterns of fractures in these deformed sandstone samples, we can quantitatively explore the relationship between the observed geometry of the cracks and the inferred mechanical processes. This will eventually help us to better understand the physics underlying the initiation of catastrophic events, such as earthquakes, landslides, and volcanic eruptions.


Introduction
Understanding the rupture process of initially intact rocks is a basic aim of experimental rock deformation in the brittle field. The propagation of a shear plane (i.e., a fault) by a mechanism of en-echelon tensile crack coalescence has been proposed from field evidence (e.g., Jourde et al., 2002;McGrath & Davison, 1995), from experimental evidence (e.g., Lockner et al., 1991;Petit & Barquins, 1988;Stanchits et al., 2006), and from theoretical and numerical modeling (e.g., Healy et al., 2006aHealy et al., , 2006bReches & Lockner, 1994). It is generally accepted that shearing becomes the dominant mechanism when sufficient tensile damage has been generated. The location of acoustic emission (AE) events, related to microcrack generation throughout the loading cycle (Lockner et al., 1992;Lockner, 1993;Moore & Lockner, 1995), enables the mapping of microcrack formation prior to bulk-that is, whole sample-failure, showing that a fault initiates by the interaction of cracks even at relatively low average crack density. However, in the field of rock mechanics, a quantitative mapping of the proportion of cracks whereat tensile fracturing gives way to a throughgoing shear fracture has rarely been performed. It has been recognized (e.g., Reches & Lockner, 1994) that the interaction starts by dilatation of one or more cracks that induces dilatation of certain neighboring cracks that were themselves on the verge of opening, in an overall process that involves coalescence of tensile cracks, grain-boundary cracks, and porosity (Dunn et al., 1973). However, this effect of self-organized "contamination," which ultimately leads the rock to bulk shear failure, requires the existence of a minimum distance among contiguous en-echelon tensile cracks-that is, a critical density, an interflaw length (Vasseur et al., 2017)-together with a certain minimum 1. Is it now possible to quantitatively determine the point of transition in the length and/or density of microcracking during brittle deformation, to assess and constrain the magnitude of "sufficient tensile damage" in order for shear failure to initiate? 2. How much influence does the applied stress field (i.e., the effective pressure) have on the point of transition between tensile damage and shear failure?
Being able to characterize in a quantitative manner the abrupt changes in the interaction of tensile cracks and incipient faults is crucial to fully comprehend the mechanics of fault initiation, which has direct implications for earthquake nucleation and seismology, and rock-mass stability.
Image analysis techniques succeeded to identify petrographic and textural features-for example, mineral and grain clustering-on rock thin-sections (Reedy, 2006). More recently a number of automatic and semiautomatic algorithms have been developed for identifying cracked regions in composite artificial (e.g., concrete and steel; Yeum & Dyke, 2015) or natural materials (e.g., Arena et al., 2014;Voorn et al., 2013). All these methods employ segmentation procedures based on various object detection techniques such as Radon Transformation (Yeum & Dyke, 2015) and Hessian Filtering (Voorn et al., 2013). In a recent work Rizzo, Healy, Farrell, and Heap (2017) demonstrated the applicability of the Morlet wavelet analysis for detecting fracture size and orientation transitions during the deformation history of a rock sample. Here we extend the method to investigate and characterize the brittle deformation process of a low-medium porosity (11% on average) sandstone, deformed in a conventional triaxial (axisymmetric) deformation apparatus under different effective pressures. The analysis is based on postfailure fracture maps of high-resolution montages of backscattered electron (BSE) microscope images. These montages allow total coverage of the fractured area of the specimens, enabling us to map the crack and incipient-fault interactions for a suite of three samples deformed at progressively higher effective pressures. First, we present the results of the wavelet analysis for each sample. Second, from the derived optimum wavelet coefficient maps, we focus on the geometrical and kinematic attributes of crack/fault relationships to infer the critical lengths, which mark the transition between diffuse tensile cracking and localized shear failure. We demonstrate the ability of wavelet analysis to investigate the evolution through time of crack sizes and their spatial densities using postfailure static images. By comparing the different samples, the role of effective pressure on crack initiation and propagation is also assessed, and the relevance of these findings for interpreting the physical processes at the base of large scale rock failure is also considered.
The wavelet analysis technique presented in this work is part of the FracPaQ toolbox , which can be freely downloaded from https://github.com/DaveHealy-Aberdeen/FracPaQ. With the introduction of this new tool, FracPaQ now allows the user to perform a comprehensive analysis of fracture networks, in both their key "static"-for example, length distribution, spatial density and connectivity-and "dynamic" features-for example, variation in the hierarchy of cracks.

Sample Description and Setup for Laboratory Deformation Tests
We mapped microcracks from images of three centimeter-scale thin sections made from core plugs of the Hopeman Sandstone Formation, a Permian quartz arenite (Edwards et al., 1993;Farrell et al., 2014), taken to failure under triaxial conditions in the laboratory. Intact blocks were taken from the Clashach quarry on the coast of Moray (Scotland, UK), and cylindrical core plugs 20 mm in diameter and 40 mm long were prepared. This quartz-cemented arenite is texturally mature (>92% quartz grains and <8% K-feldspar and subordinate lithics) and well-sorted (optical microscopy showed that grains have a fine to medium grain size and mean grain diameter of 0.25-0.4 mm; . Samples were chosen based on homogeneity: without visible laminations and with uniform grain size, very similar porosity, and cement distribution. Three core plugs of Hopeman Sandstone were deformed in a conventional triaxial compressional apparatus ( 1 > 2 = The bottom image shows the wavelet represented as level curves; here the black ellipse is the envelope (i.e., the wavelet "footprint") that contains most of the energy of the wavelet. Representations of the location parameter (b) and of the orientation parameter ( ) are also viewed in the Cartesian coordinates (x, y). pressure law where = 1) was never exceeded. The confining and pore pressures were kept constant using servo-controlled pumps. The samples were left overnight under the target pressures to ensure microstructural equilibration. The samples were then loaded axially to failure at a constant strain rate of 10 −5 s −1 . Based on the high permeability of Hopeman sandstone (∼ 5 × 10 −13 m 2 ; Farrell et al., 2014), we expect our samples to be drained at a strain rate of 10 −5 s −1 (see Heap & Wadsworth, 2016). During deformation we monitored axial force (using a load cell) and axial displacement (using a linear variable differential transducer located outside the pressure vessel), pore volume change (via the servo-controlled pore pressure pumps), and the output of AE energy. Axial force, axial displacement, and pore volume change were converted to axial stress, axial strain, and porosity change using the sample dimensions. AEs-high-frequency elastic wave packets generated by the rapid release of strain energy such as during microcracking-were recorded using a piezo-electric crystal attached to the top of the upper piston (see Farquharson et al., 2017, for a schematic diagram) and a single-channel AE recording system (Physical Acoustics AE Node) set at a threshold of 40 dB. AE energy is taken here as the area under the received waveform (and is therefore in arbitrary units).
Following deformation, the samples were carefully unloaded at a constant strain rate of 10 −5 s −1 to avoid additional microcrack damage during unloading. The confining and pore pressures were then decreased slowly (0.05 MPa/s), such that the maximum effective pressure was never exceeded. Samples were Epoxy filled under a vacuum then thin sections were cut in the XZ plane of each core plug, that is, orthogonal to the strike of the throughgoing shear plane. The thin sections were hand ground thus minimizing the chance of grain plucking or fracturing induced during machine grinding. Montages of high-resolution (225 pixels per millimeter) BSE images taken at a magnification of ×100 from each thin section were taken on a FEI Quanta 200F environmental SEM at the University of Glasgow (UK). BSE images are very practical for microscale crack and pore mapping, because different minerals are represented with different grey levels and black represents void space, that is, pores and fractures (Farrell et al., 2014;Griffiths et al., 2017;Wibberley et al., 2000). Fracture maps were produced by manually tracing all the visible cracks over the of the BSE mosaic using a vector graphic editor (Adobe Illustrator TM ). A number of automatic and semiautomatic algorithms for the segmentation of crack patterns have been recently developed (e.g., Arena et al., 2014;Delle Piane et al., 2015;Griffiths et al., 2017) and successfully used to isolate crack patterns in crystalline rocks with low primary porosity (e.g., granite and marble). However, these softwares become less accurate when analyzing rocks with more complex microstructures (i.e., sandstones; Griffiths et al., 2017). Therefore, although we recognize that subjectivity issues exist, we still prefer manually retracing the cracks: In this way the user is able to discern and identify specific features, discriminating cracks from pores, grain margins, and artifacts. Once produced, the fracture maps were converted into binary images; the resulting images were embedded in a white square matrix to avoid edge effects during the wavelet analysis (Ouillon et al., 1995).

Two-dimensional Continuous Wavelet Analysis for the Quantification of Multiscale Fracture Patterns
The spatial distribution of crack lengths and densities depend on, and vary with, the scale of analysis: They are multiscale properties (Gaillot et al., 1999;Ouillon et al., 1995Ouillon et al., , 1996. Consequently, for studying fracture trace maps, we need a tool that is able to efficiently perform a multiscale analysis of two-dimensional binary images. Wavelet analysis has emerged as a very powerful tool for signals (i.e., images) in which different scales are combined (Antoine et al., 1993). The method is based on projecting (convolving) the analyzed signal onto a family of elementary functions (filters), obtained by translating and dilating a single basic function, the analyzing wavelet (Ψ) (Figure 1). We based our study on a combination of the fully anisotropic directional Morlet wavelet algorithm of Neupauer and Powell (2005) and the wavelet coefficient method developed by Darrozes et al. (1997). This combination enables sharp unequivocal detection and quantification of any organized feature in a binary image, regardless of its scale, orientation, and location. (2) the position vector (b) defines the location of the wavelet on the image and hence permits the filter to be moved over the whole field of the analyzed image; (3) the anisotropy parameter ( ) allows one to elongate the envelope of the wavelet (Figure 1) in the x-direction producing an elliptical "footprint" with a long axis in the x-direction, setting to unity we recover the isotropic wavelet; and (4) the orientation parameter ( ) permits the rotation of the wavelet for detecting any anisotropic feature (e.g., a fracture trace) in any direction ( Figure 2). The Morlet wavelet is intrinsically directional, with peaks and troughs all aligned in one direction ( Figure 1); when this wavelet is rotated through an angle the peaks and troughs are aligned along the direction of . In addition to being directional, the Morlet wavelet is also intrinsically anisotropic, meaning that it is elongated in the direction of (Neupauer & Powell, 2005).
The convolution, or wavelet transform (WT), of the image with each analyzing wavelet, obtained when we dilate/contract, elongate, rotate, and translate a mother wavelet (Ψ), produces a set of wavelet coefficients (WC), W Ψ (a, b, , ), which are local indicators of the quality of the match between the filter and the local content of the image (Gaillot et al., 2002, Figure 3). This set is called a wavelet image and is represented by a coefficient map (Figure 3). A constant signal would produce a null coefficient. Conversely, if the image shows "irregularities" (i.e., nonnull pixels), the wavelet reacts by producing nonnull coefficients. Small coefficients reflect a poor match between the given analyzing wavelet and the image; and maximum coefficients indicate optimal correlations between the filter and the content of the image under analysis. In practise, a complete and objective detection of all organized structures in a binary image requires the application of a range of filters each having a unique combination of the parameter triplets (a, , ) values . Then the calculated coefficients W Ψ (a, , ) at each point of the image are normalized with respect to the theoretical maximum value obtained for a perfect match between the wavelet and the analyzed structure. This normalization enhances the quality of the coefficient map-that is, produces an optimum map (Ouillon et al., 1995, Figure 3). In Figure 3 we show the basic procedure of wavelet transformation: For simplicity a synthetic fracture network is analyzed employing only two triplets (a, , ) of the wavelet coefficients, two sizes, two orientations, and two anisotropy parameters. The different magnitudes of the coefficients for each resolution allow one to recognize the different level of organization of the image. The anisotropic Morlet wavelet   successfully quantifies the geometry of all features at a given scale a, whatever their location, shape anisotropy, and orientation.
Each optimum map is associated with optimum wavelets with known azimuths ( ); these azimuths correspond to the orientations of the detected structures in a known position b, at a given resolution (a). In our case, the detected structures in the images correspond to cracks formed during the deformation of the sandstone samples. From these azimuthal values, we can draw rose diagrams, which provide an overview of the variations in the cracks orientations with the scale of analysis. It follows that, if a sharp transition in the orientations of the crack traces is observed at a given scale (a), then this scale contains important information about the deformation process encapsulated in the crack pattern (Ouillon et al., 1995). When such a scale is detected, we perform a new analysis to determine its value with better accuracy.
The validity of the described method has been tested by Rizzo, Healy, Farrell, and Heap (2017) through the analysis of synthetic and natural fracture networks. This study found that the anisotropic Morlet wavelet performed better than its more commonly used counterpart, the Mexican hat wavelet, in the detection of oriented linear features, such as fracture traces, because of its inherent anisotropy. In addition, Rizzo, Healy, Farrell, and Heap (2017) described a complementary outcome that arose from the WC maps; from these maps, these authors directly retrieved information on the spatial density distribution of the analyzed features for a specific scale. Therefore, this type of analysis can provide insight into the processes of shear fracture growth from the interaction and coalescence of constituent tensile microcracks (Moore & Lockner, 1995). The analysis of Rizzo, Healy, Farrell, and Heap (2017) was however restricted to a single experiment. Here we use the wavelet analysis method to glean differences between the process of strain localization in sandstone samples deformed at different effective pressures.

Mechanical Tests
The stress-strain curves for the three triaxial experiments on samples of Hopeman Sandstone are shown in Figure 3a. All three curves exhibit the classical stress-strain features of brittle rock in compression. Stress is first a nonlinearly increasing function of strain (Figure 4a), typically attributed to the closure of compliant pores/cracks (as evidenced by the porosity decrease shown in Figure 4b). This stage is followed by a quasi-linear elastic portion (Figure 4a). Following elastic deformation, the stress is a nonlinearly decreasing function (strain hardening) of strain prior to a peak stress ( p ), typically attributed to the formation and growth   of microcracks (as evidenced by the increase in porosity shown in Figure 4b and the appearance of AE activity shown in Figure 4c). The samples enter a strain-softening phase (a stress drop) following the peak stress ( Figure 4a). The development of a shear fracture is often considered to occur during the strain-softening phase, which is characterized by large increases in porosity ( Figure 4b) and significant AE activity (Figure 4c) (due to the growth and coalescence of microcracks). Visual inspection of our postmortem samples revealed the presence of shear fractures. The peak stress of Hopeman sandstone is higher at higher effective pressures: 192.8, 290.8, and 349.7 MPa at effective pressures of 5, 15, and 30 MPa, respectively (Figure 4a). These mechanical data are similar to those previously published on porous sandstones in the brittle regime (e.g., Baud et al., 2000;Klein et al., 2001;Menendez et al., 1996;Wong et al., 1997).

Morlet Wavelet Analysis
We applied wavelet processing to the set of fracture networks obtained from Hopeman Sandstone samples triaxially deformed at three different effective pressures (Figure 4). The inputs were digital, binary, fracture network maps, which mathematically can be described as a function I(x) equal to 1 on pixels corresponding to fracture traces and equal to 0 otherwise. The convolution (or wavelet transform, WT) of I(x) with the analyzing wavelets produces the optimum wavelet coefficients computed in the resulting wavelet maps. Because every resolution (a) and every anisotropy ( ) is explored, the preferred orientations of individual cracks and clusters of cracks can be extracted from the wavelet coefficient maps. Note that in the following descriptions, a high resolution (lower values of a) refers to short length scales, and low resolution (higher values of a) refers to longer length scales.

Low Effective Pressure-Sample 1
This sample was deformed at an effective pressure (P eff ) of 5 MPa (P c = 15 MPa and P f = 10 MPa) and failure occurred through the development of a conjugate set of shear planes (Figure 5a). The first wavelet coefficient to produce results is for a resolution a = 4 ( Figure 6a); at this stage, we detect the presence of diffuse short cracks over about two thirds of the sample area (see Figures 5 and 6). These cracks are oriented parallel or subparallel to the applied maximum stress ( 1 ). On incrementing the wavelet resolution to a = 8 (Figure 6b), while we observed that most cracks were still aligned parallel to the 1 (i.e., vertical), we noticed the appearance of a second prominent crack orientation oriented at an angle of 330 ∘ -that is, assuming that 0 ∘ is parallel to the z-axis. Thus, we identified a first oblique cluster of cracks, whose orientation will become predominant at a resolution of a = 16. This feature corresponded to the left inclined shear plane seen at failure in Sample 1. Finally, at the lowest resolution (a = 32) we recognized a second, now dominant, orientation at 30 ∘ , corresponding to the second shear plane on the sample. However, we also noted that a cluster of cracks with the same orientation was already detected for a = 8 (and consequently a = 16), although relatively minor.  Since we observed the onset of at least two orientation transitions-that is, between a = 4 to 8, and between a = 16 to 32-we further investigated the sample to identify the precise wavelet coefficients at which these transitions are detectable (Figure 7). In particular, we found that the first transition occurred between a = 6 and 7, corresponding to the clustering of initial tensile vertical cracks. The second abrupt change in orientation occurred between a = 20 and 22 corresponding to the propagation of the second and prominent shear plane.

Intermediate Effective Pressure-Sample 2
This sample was deformed at P eff = 15 MPa (P c = 25 MPa and P f = 10 MPa; Figure 8). At the lowest resolution (a = 4), similar to the previous sample, the wavelet method detected the presence of diffuse and short subvertical cracks, aligned parallel to the direction of the maximum compressive stress. When lowering the resolution, for a = 8, we observed a shift in the crack orientation towards an inclined position (∼26-28 ∘ ). This transition marked the change in the fracture orientation to a macroscopic feature, corresponding to the time when the fault started to establish its position at an angle to the applied maximum compressive stress (vertical). At this scale, a minor set of conjugate fractures was also detected. Gradually (Figures 9c and 9d), the conjugate pattern became less resolvable (a = 16) and only the single major shear fracture appeared as a well-defined entity, with an average orientation of 26 ∘ (a = 24). As opposed to the previous sample (Sample 1), the minor conjugate cluster of cracks did not develop into a fully grown shear plane.

High Effective Pressure-Sample 3
This sample was deformed at P eff of 30 MPa (P c = 40 MPa and P f = 10 MPa) and failure occurred along a single well-defined shear plane ( Figure 10). First of all, to detect the initial emergence of small subvertical cracks we had to increase the resolution to a = 2 ( Figure 10). These initial cracks were not diffused through the sample area but were all preferentially located along a relatively narrow zone (Figure 11). For a = 4 the wavelet already detected a single large cluster of cracks oblique (30 ∘ ) to the applied maximum compressive stress. For increasing wavelet resolutions, we did not observe any further orientation transitions for either clusters of cracks nor for the final shear plane (Figure 11).

Discussion
The formation of a fault-a composite shear fracture-is a highly complex process involving the growth and the coalescence of small tensile cracks that are initially subparallel to the 1 -axis (Dunn et al., 1973;Healy et al., 2006a;Lockner et al., 1991;Mandl, 2005;Reches & Lockner, 1994). High-resolution BSE images allowed us to study fault initiation over a significant area within samples deformed under a range of effective pressures (from 5 to 30 MPa; Figure 4).  Conceptually, wavelet coefficient maps for progressively lower resolutions (1∕a) can be compared to a temporal sequence representing the deformation history of the sample, from the early tensile cracking, to crack interaction and eventual shear failure ( Figure 12). The underlying assumption behind this "space for time" substitution in the analysis is that an intact rock transitions to bulk shear failure through the progressive nucleation, growth, interaction, and coalescence of tensile microcracks (Lockner et al., 1992). The fundamental difference seen via the wavelet analysis is that the evolution of crack sizes and orientation transitions between samples can be directly associated with the effect that effective pressure exerts on crack nucleation and propagation. It has been found (Mandl, 2005) that the growth of fractures is very sensitive to compressive stress normal to the crack planes. We therefore infer that shear fracturing becomes the dominant mechanism in two distinct ways in response to the loading conditions ( Figure 12).

Low and Intermediate Effective Pressures
Based on our inferred temporal mapping of the wavelet analysis we observe that at low (5 MPa) and intermediate (15 MPa) effective pressures, the first features to form (a = 4) were pervasive short tensile cracks randomly located throughout the samples (Figure 6a). As time progresses (a = 8), the deformation reached a subsequent stage marked by the first orientation transition (Figure 5b). A previous analysis of the micromechanics of brittle failure in sandstones (Hazzard et al., 2000) showed that "secondary" cracking occurs along grain boundaries by shear rupturing of the cement at the contacts caused by rotation and slip of the grains under an applied load. The same mechanisms were seen by Dunn et al. (1973) as the crucial part of the process, because extension cracks unlock the ends of ready-to-fail grain-boundary cracks. For samples deformed at low and intermediate effective pressure, the application of a differential stress produced comminution and grain rotation until each grain is fixed by its neighbors. This mechanism determined the interlinking of the Deformation starts with pervasive tensile cracks parallel to the applied 1 ; it progresses with the growth of clusters of cracks, which, once reached a critical length and density, start interacting and coalescing to form the final shear planes. In the bottom images (modified after Reches & Lockner, 1994), at relatively high effective pressures (30 MPa), the deformation involves a rapid localization of strain, initiating with the interaction of short en-echelon tensile cracks, which then coalesce and give way to the propagation of the shear plane.
widespread axially (vertically) oriented small tensile cracks. When cracks became interlinked, we observed the formation of scattered clusters of cracks characterized by a different orientation-that is, different relative to the early tensile cracks-as apparent from wavelet coefficient maps and the number of petals in the corresponding rose diagrams (Figures 6 and 8). The distribution and the orientation of these clusters will influence the location of the final shear plane. On average, in both samples, the tensile fracture length is 0.24 mm (Figures 5d and 8d), which is comparable with the mean grain diameter of Hopeman Sandstone (Rizzo, Healy, Farrell, & Heap, 2017). Therefore, we infer that cracks tended to reach the edges of the grains before achieving sufficient interaction and coalescence (Katz & Reches, 2004). Further deformation in the samples caused clusters to expand and join, if favorably oriented, in a shear plane (a = 16, in Figures 6 and 9). Shear fractures then grew accumulating displacement during tensile cracking associated with deformation localizing on individual tensile crack and crack clusters ( Figure 13).
A major effect of increasing the effective pressure from 5 to 15 MPa on the failure process is exemplified at the latest stage of deformation. For Sample 1, deformed at the lowest effective pressure (5 MPa), we observed the formation of two shear fractures with very irregular final profiles (Figures 6c and 6d), as shown by a wide range of associated . On the other hand, for Sample 2, deformed at P eff = 15 MPa, failure occurred along a more localized single plane, even though deformation initially produced similar effects in both samples-that is, randomly located diffuse tensile cracking.
Continuous macroscopic shear planes are therefore formed by the interlinking of axially oriented small-scale tensile cracks with oblique sliding cracks (along quartz grain-boundaries) with the same sense of right-lateral or left-lateral displacement ( Figure 11). The described evolution takes place in a crack nucleation-rather than crack growth-dominated system culminating in a distributed deformation where the load is distributed through the numerous fractures. This model (Figure 12) also assumes that macroscopic shear planes formed due to interaction among many damage points (i.e., cluster of fractures), which formed prior to faulting (Katz & Reches, 2004;Renard et al., 2017Renard et al., , 2018.

High Effective Pressure
Similar considerations regarding our wavelet analysis as time lapse imaging of fault initiation can be applied to Sample 3 deformed at the highest effective pressure (30 MPa). The higher pressure caused the "early" development of the shear plane. This rapid localization did not allow tensile cracks to reach significant lengths; in fact, cracks are shorter on average (0.15 mm, Figure 10d). The cracks are therefore shorter than the average grain size in Hopeman sandstone. Furthermore, at half the wavelet scale (between a = 2 and a = 4, Figure 11), we observed that cracks had already achieved a density sufficient to promote interaction and coalescence compared to the previous specimens deformed at lower effective pressures. Overall, this process resulted in a very localized deformation, meaning that cracks formed simultaneously along a narrow-inclined zone (28 ∘ to the applied 1 ) where the shear plane eventually formed. Cracks were initially aligned parallel to 1 (a = 2), as seen for lower P eff ; we observed a single bulk transition in crack orientation (a = 4), after which no further changes occurred. Reches and Lockner (1994) proposed that, for P c ≥ 50 MPa, faults grow in the wake of a process zone that is a finite size region of high stress concentration at fault tip in which the intact rock is disintegrated. In this model, tensile microfractures would be restricted to the fault vicinity and decay away from the fault. The failure model developed by Reches and Lockner (1994) predicts that the region of maximum stress induced by an opening tensile crack is inclined at an angle of at least 30 ∘ to the crack itself (and to the applied maximum compressive stress, 1 ). This angle exerts a fundamental control in the geometry of the final shear failure plane by promoting the en-echelon interaction of neighboring cracks (Healy et al., 2006b). High P eff allows mechanical interaction predominantly between neighboring cracks which are in a favorable position, determining a specific cluster of en-echelon cracks to coalesce into a macrofracture only on one side of the crack-tip region; otherwise, the en-echelon ensemble would be stopped ( Figure 12). This chain reaction causes the sample to fail by propagation of self-organized en-echelon cracks (Mandl, 2005).

Overall Implications and Applications of the Method
Our wavelet analysis proved useful for sharply distinguishing specific orientations at specific scales of observation, which would otherwise be lost in the complexity of a standard rose diagram containing all fracture trace orientations plotted for all scales simultaneously (e.g., compare Figures 4c and 5). The results obtained from our three initial samples suggest that, for relatively low effective pressures, cracks tend to form in large and diffuse clusters rather than uniformly along the fault zone, as seen for the highest P eff . Furthermore, at relatively high effective pressure, cracks do not need to grow to the same lengths as for lower P eff -that is, they do not scale with the sandstone grain size-to achieve interaction and coalescence (as summarized in Rizzo, Healy, Farrell, & Heap, 2017). This, in turn, results in the two different shear initiation modes observed through our wavelet analysis. Morlet wavelet analysis has detected that the inclination of the final shear plane, in all three samples, is in a range between 26 ∘ and 30 ∘ with respect to the maximum compression axis (Figures 5d,  8d, and 9d). Similar angular values were found both via experimental procedures (20-30 ∘ , Engelder, 1974;Reches & Lockner, 1994;Renard et al., 2018) and numerical models (26 ∘ , Healy et al., 2006a). We emphasize that our results are based on the assumption that all the mapped cracks were produced solely during the compression tests. More experiments and analyses on sandstones with different grain sizes are now required to further explore these findings. Further, we are yet to apply our technique to the process of compactant strain localization (i.e., the growth of compaction bands) in sandstones in the ductile regime (e.g., Baud et al., 2004;Heap et al., 2015).
Our initial application has been to a single rock type in an effort to reduce variables. However, the application of the described image processing technique to rock types with significant mineralogical or textural 10.1029/2018JB015898 heterogeneities might result in improved models for fracture coalescence. The methodology presented here can have important applications for recognizing larger length-scale mechanical heterogeneities controlling fault-scale rupture (e.g., Kim et al., 2004;Peacock, 2002) and fracture propagation during volcanic unrest (e.g., Gudmundsson, 2006;Kilburn & Voight, 1998;Kilburn, 2003). Just as a fault on the "small" scale of a laboratory sample localizes through extension and coalescence of microcracks, larger faults also grow by extension and coalescence of multiple segments (e.g., Cartwright et al., 1995). Rock failure is thus perceived as a multiscale process for which microscopic cracks can ultimately lead to macroscopic faulting. In addition, a variety of phenomena have been suggested as parameters that may be used for the prediction of faulting and the deformation and failure of a volcanic flank or dome, including temporal changes in strain, tilt, and creep. All these phenomena can be associated directly or indirectly with changes in the microcrack geometry.

Conclusions
An image processing method, namely, Morlet wavelet analysis, has been successful in exposing and quantifying the initiation and development of shear fracture formation in laboratory rock deformation tests. The analytical technique presented here has allowed us to detect the variations and abrupt transitions in orientation, length, and density occurring during the deformation, from early tensile cracking to the onset of a final shear fault. The ability of the Morlet wavelet to untangle the geometrical complexities proved straightforward due, for the most part, to its intrinsically anisotropic nature. Based on the derived wavelet coefficients maps obtained for three samples deformed at different effective pressures (5, 15, and 30 MPa), it was possible to derive conceptual models and infer possible mechanisms coupling the role of effective pressure to the localization of the shear plane. We show that, in the case of low and intermediate effective pressures (5 and 15 MPa, respectively), the wavelet analysis detects early tensile damage randomly distributed throughout the samples. An abrupt change in crack orientation corresponds to the start of crack interaction to form clusters, with the detected critical crack length comparable with the average grain size of the samples (0.24 mm). Further transitions in crack orientation occur as the result of a localization of the shear plane, which eventually causes the sample to fail. In contrast, for our highest effective pressure (30 MPa), wavelet analysis detects early vertical tensile cracking occurring along a narrow-inclined zone. Cracks do not need to reach the grain edges to achieve interaction due to their relatively high density, but subsequent deformation suddenly localizes, as identified by the single bulk transition of crack orientations. Thus, the results of this study show how variations in the applied stress field (i.e., effective pressure) on the same material play a significant role in controlling the mechanism of initiation and development of shear fracture.