Clément Perrin
email: clement.perrin@univ-nantes.fr

Felix Waldhauser

Christopher H Scholz

The Shear Deformation Zone and the Smoothing of Faults with Displacement

Keywords: Fault maturity, shear deformation zone, large earthquakes, aftershock distributions, scaling laws. Plain

Keypoints

Introduction

A fault zone is a complex brittle-frictional system that wears as slip occurs on it. It is formed of three main features, that evolve with fault growth (Fig. 1): (i) the cataclastic core contains the cataclastic detritus of wear of the slipping surfaces of the fault. Its width (W C in Fig. 1) increases linearly with fault displacement at a rate that depends on the strength of the wall rock [START_REF] Scholz | Wear and gouge formation in brittle faulting[END_REF](Scholz, , 2019, pp 132), pp 132). For displacements greater than a few hundred meters, growth of the fault core levels off at a thickness of a few tens of meters (Scholz, 2019, pp 132); (ii) Beyond the fault core lies a region of pervasive tensile fracturing which defines the "dilatant damage zone" (W D , Fig. 1; e.g. [START_REF] Faulkner | Scaling of fault damage zones with displacement and the implications for fault growth processes[END_REF][START_REF] Savage | Collateral damage: Evolution with displacement of fracture distribution and secondary fault strands in fault damage zones[END_REF][START_REF] Vermilye | The process zone: A microstructural view of fault growth[END_REF]. The fracture density in this zone dies off as a power law with distance from the fault (e.g., [START_REF] Ostermeijer | Damage zone heterogeneity on seismogenic faults in crystalline rock; a field study of the Borrego Fault, Baja California[END_REF] and references therein). The dilatant damage zone width increases linearly with fault displacement, and typically levels out at several hundred meters for fault displacements exceeding several hundred meters [START_REF] Savage | Collateral damage: Evolution with displacement of fracture distribution and secondary fault strands in fault damage zones[END_REF]; (iii) Including and extending beyond the dilatant damage zone is what we call the "shear deformation zone" (W S ; Fig. 1) which is defined by a region of enhanced seismicity, first pointed out by Powers & Jordan (2010). This zone shows a region of high seismic activity near the fault with a power law fall-off beyond a corner at W S1 to a full half-width of W S2 (Fig. 1).

The above definitions allow us to distinguish two types of damage zones: the "dilatant damage zone" dominated by volumetric strains, and the "shear deformation zone" dominated by shear strains. The tensile (Mode I) cracks in the dilatant damage zone are dilatant cracks that align parallel to the maximum compression direction and perpendicular to the minimum principal stress. Hence the orientation of cracks provides evidence for the several different mechanisms responsible for their formation [START_REF] Wilson | Microfracture analysis of fault growth and wear processes, Punchbowl Fault, San Andreas system, California[END_REF]. The shear deformation zone is characterized by secondary faults (Mode II and III cracks) and hence are oriented parallel to the maximum Coulomb stress. For example, in the case of a strike-slip fault, this zone is defined by a conjugate set of secondary faults [START_REF] Little | Brittle deformation adjacent to the Awatere strike-slip fault in New Zealand: Faulting patterns, scaling relationships, and displacement partitioning[END_REF], in which one set is parallel to the primary fault.

The evolution of these three zones in figure 1 defines what is called fault maturity. The three zones can be viewed as regions controlled by wear processes, and the fault structural maturity can hence be measured by its degree of wear, which depends primarily on the net fault displacement, assuming a constant slip-vector throughout the fault's history. However, previous studies have shown that, in the absence of data on net fault displacement, several other fault parameters such as the fault initiation age and the geological slip rate can be also used as a proxy of net displacement in evaluating the overall maturity of the fault (e.g. [START_REF] Choy | An overview of the global variability in radiated energy and apparent stress[END_REF][START_REF] Choy | Apparent stress, fault maturity and seismic hazard for normal-fault earthquakes at subduction zones[END_REF][START_REF] Dolan | How well do surface slip measurements track slip at depth in large strike-slip earthquakes? The importance of fault structural maturity in controlling on-fault slip versus off-fault surface deformation[END_REF][START_REF] Hecker | Normal-Faulting Slip Maxima and Stress-Drop Variability: A Geological Perspective[END_REF][START_REF] Ikari | On the relation between fault strength and frictional stability[END_REF][START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF][START_REF] Niemeijer | Fabric induced weakness of tectonic faults[END_REF][START_REF] Perrin | Location of largest earthquake slip and fast rupture controlled by along-strike change in fault structural maturity due to fault growth[END_REF][START_REF] Stirling | Fault trace complexity, cumulative slip, and the shape of the magnitude-frequency distribution for strikeslip faults: a global survey[END_REF][START_REF] Wesnousky | Seismological and structural evolution of strike-slip faults[END_REF]. As these parameters increase, the fault grows and becomes more "mature". Prior studies have suggested that the structural maturity may have a strong impact on earthquake behavior, such as magnitude, stress drop, distribution of slip, rupture velocity, ground motion amplitude, and number of ruptured segments (e.g., [START_REF] Cao | Seismicity simulation with a rate-and state-dependent friction law[END_REF][START_REF] Dolan | How well do surface slip measurements track slip at depth in large strike-slip earthquakes? The importance of fault structural maturity in controlling on-fault slip versus off-fault surface deformation[END_REF][START_REF] Hecker | Normal-Faulting Slip Maxima and Stress-Drop Variability: A Geological Perspective[END_REF][START_REF] Malagnini | Energy radiation from intermediate-to large-magnitude earthquakes: Implications for dynamic fault weakening[END_REF][START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF][START_REF] Perrin | Location of largest earthquake slip and fast rupture controlled by along-strike change in fault structural maturity due to fault growth[END_REF][START_REF] Radiguet | Dependency of Near-Field Ground Motions on the Structural Maturity of the Ruptured Faults[END_REF][START_REF] Stirling | Fault trace complexity, cumulative slip, and the shape of the magnitude-frequency distribution for strikeslip faults: a global survey[END_REF][START_REF] Wesnousky | Seismological and structural evolution of strike-slip faults[END_REF].

The widths of the fault core and dilatant damage zones saturate at fault lengths comparable to the seismogenic thickness, due to the switch from crack like to pulse like earthquake propagation [START_REF] Ampuero | Upper limit on damage zone thickness controlled by seismogenic depth[END_REF]. For larger faults, the evolution of fault maturity involves only changes in the shear deformation zone. In this paper we are concerned with the scaling of large faults (i.e., which breach the brittle seismogenic width) and their associated large earthquakes, so we are only concerned with the shear deformation zone. There is evidence that indicates that large faults become smoother with net displacement [START_REF] Stirling | Fault trace complexity, cumulative slip, and the shape of the magnitude-frequency distribution for strikeslip faults: a global survey[END_REF][START_REF] Wesnousky | Seismological and structural evolution of strike-slip faults[END_REF]. This smoothing is probably the prime attribute of fault maturity. Here we show that the width of the shear deformation zone of large faults decreases with fault displacement, as a consequence of this smoothing.

Precise earthquake locations can be used to image the internal structure of fault and the zone of brittle deformation, often at a resolution similar to field observations (e.g. Powers & Jordan, 2010;[START_REF] Hauksson | Spatial Separation of Large Earthquakes, Aftershocks, and Background Seismicity: Analysis of Interseismic and Coseismic Seismicity Patterns in Southern California[END_REF][START_REF] Valoroso | Earthquakes and fault zone structure[END_REF]. Powers & Jordan (2010) studied the association of small earthquakes with large faults in California. They found that the frequency of small earthquakes is highest in a narrow region surrounding faults and then falls off as a power law at greater distances. They modeled this behavior with a fault model with rough (fractal) topography [START_REF] Dieterich | Nonplanar faults: Mechanics of slip and off-fault damage[END_REF], showing that such a rough fault model would produce high stresses near the fault that could account for the seismicity. The earthquakes they used was from the interseismic period of the faults. Although stacked profiles from many fault sections show the association of seismicity with the faults, individual sections and map views, both in Powers & Jordan (2010) and [START_REF] Hauksson | Spatial Separation of Large Earthquakes, Aftershocks, and Background Seismicity: Analysis of Interseismic and Coseismic Seismicity Patterns in Southern California[END_REF] show that such a tight correlation with the fault is not typical for individual fault segments, which often display a wide variability in distributions. As [START_REF] Hauksson | Spatial Separation of Large Earthquakes, Aftershocks, and Background Seismicity: Analysis of Interseismic and Coseismic Seismicity Patterns in Southern California[END_REF] observed, this variability is likely due to many factors, such as heterogeneity in lithology, the effect of nearby faults both mapped and unmapped, and fault offsets and bends.

As [START_REF] Hauksson | Spatial Separation of Large Earthquakes, Aftershocks, and Background Seismicity: Analysis of Interseismic and Coseismic Seismicity Patterns in Southern California[END_REF] observed further, aftershocks of large earthquakes, in contrast, mostly show a tight clustering around the main fault. At depth, they are mainly distributed at the edges of regions experiencing high coseismic slip and in surrounding areas of low coseismic slip (e.g., [START_REF] Mendoza | Aftershock patterns and main shock faulting[END_REF]. Also, earthquakes that occur during interseismic periods often delineate the edges of stuck patches sometimes surrounded by creeping areas, and potential rupture surfaces at depth (e.g., Perrin et al., 2019;[START_REF] Schaff | High-resolution image of Calaveras Fault seismicity[END_REF][START_REF] Waldhauser | Streaks, multiplets, and holes: High-resolution spatio-temporal behavior of Parkfield seismicity[END_REF]. In both cases, the majority of aftershocks of large earthquakes sample a portion of the main fault zone structure at depth at the edge of the rupture surface, and are often used to delineate it. In an earthquake model with a smooth rupture surface, stress-drop increases to a maximum near the fault surface, thus precluding aftershocks in the near-fault region [START_REF] Kostrov | Evaluation of stress and displacement fields due to an elliptical plane shear crack[END_REF]. Near-fault aftershocks therefore are an indication of a rough fault, as nearfault stresses are generated by dynamic slip on rough topography. They are greatest right after the mainshock, after which they are relaxed by aftershocks and other relaxation mechanisms. In this study, in order to estimate the roughness of active faults, and how they evolve with displacement, we use high-precision aftershock locations of large earthquakes on faults with different net displacements.

Data analysis

Width of the shear deformation zone from aftershock locations

We use high-resolution earthquake catalogs available in the literature to analyze the aftershock distribution of seven large (Mw≥6) continental strike slip earthquakes that occurred on faults with a wide range of net displacement: the 1984 Morgan Hill, 2004 Parkfield, and 2014 South Napa earthquakes in northern and central California [START_REF] Waldhauser | Near-Real-Time Double-Difference Event Location Using Long-Term Seismic Archives, with Application to Northern California[END_REF]Waldhauser & Schaff, 2008), and the 1987Superstition Hills, 1992Landers, 1999Hector Mine, and 2010 El Mayor Cucapah earthquakes in Southern California/Mexico (Hauksson et al., 2012). These earthquakes were selected because high-precision aftershock catalogues, relocated by the double difference method (Waldhauser & Ellsworth, 2000) and with relative lateral location errors derived from bootstrap analysis that are typically less than ≤ 0.03 km on average (Supp. Fig. S1). In order to perform a homogeneous analysis of all sequences, we selected all aftershocks with M L ≥1 that occurred within 2 months of each mainshock.

For each aftershock sequence we determine the three-dimensional fault geometry by applying a principal component analysis (PCA) to all events within boxes that are between 3 and 10 km long along strike and between 3 and 20 km wide across strike (i.e. centered on the surface fault trace), stepping at 1 km intervals along the fault trace (Perrin et al., 2019). For each box, we obtain a plane that best fits the locations of aftershocks. For simplicity, we assume a constant dip of the calculated planes in each box (see also Perrin et al., 2019). In general, the calculated fault plane parameters are in good agreement with existing fault parameters derived from source inversion models (Mai & Thingbaijam, 2014 and references therein; see details in Supp. Table 1 and Supp. Fig. S2).

We then compute the orthogonal distance between each hypocenter and the calculated fault plane segment, count the number of aftershocks as a function of distance from the fault plane within bins of 0.03 km, and normalize the results by the total number of aftershocks in each box (gray lines in Fig. 2b,d,e, f, g and Supp. Fig. S3). Finally, we compute the mean of these normalized counts for each distance bin (black line in Fig. 2 and Supp. Fig. S3) to describe the smoothed distribution of aftershocks as a function of distance from the fault plane for each of the seven earthquake sequences.

We use a power law function to fit the mean normalized aftershock distribution (red curve in Fig. 3 and Supp. Fig. S4) to find the two parameters that describe the width of the shear deformation zone. W S1 is the distance from the fault plane where the fall-off in number of events exhibits the maximum curvature (i.e., maximum in 2 nd derivative). W S2 is the distance from the fault plane where the number of events departs from the power law fit and either flatten out (see Fig. 3a) or drop towards zero (see Fig. 3e). Both parameters represent significant changes in event density, with W S2 indicating the transition from the zone of active faulting to a largely undeformed crustal host rock (see section 4.3 for a comparison with Powers and Jordan, 2010). This transition generally occurs when less than 20% of all boxes used to count aftershocks include events (grey circles in figure 3 and Supp. Fig. S4). We use this criterion to define W S2 for all aftershock distributions considered here.

As examples of our approach, analyses of the 2004 Parkfield and 1999 Hector Mine aftershocks are presented in figure 2 and 3 (see Supp. Fig. S3 and S4 for all earthquake cases). The power law fit to the near-fault aftershock data at Parkfield (red curve, Fig. 3a) has a maximum curvature at about 0.06 km away from the fault plane (W S1 ). Beyond W S2 = 0.56 km, the distribution departs from the red fit, implying that no significant additional aftershock activity (M L ≥1) related to the active fault zone is detected. In cases with multiple ruptures on sub-parallel or sub-perpendicular faults such as Hector Mine, Landers, Superstition Hills and El Mayor Cucapah, we estimated W S1 and W S2 for the individual fault segments that broke during the earthquake (Fig. 2c,d,e,f,g,3b,c,d,e and Supp. Fig. S3 and S4) and then averaged the values so that a single value could be assigned to each aftershock sequence. The variance in the parameters obtained for individual segments is typically small. Table 1 lists W S1 and W S2 we obtained for the seven earthquakes. Also listed in Table 1 are data from the immature Awatere fault. This right-lateral strikeslip fault is a first order splay of the Alpine fault of New Zealand. It crosses the coast at a sea-cliff that offers an almost complete exposure of the entire fault zone from the fault core through the shear deformation zone. This was mapped by [START_REF] Little | Brittle deformation adjacent to the Awatere strike-slip fault in New Zealand: Faulting patterns, scaling relationships, and displacement partitioning[END_REF], who showed that the shear deformation zone consists of a set of right-lateral and conjugate left-lateral secondary faults that decreased in frequency with distance from the primary fault in the same manner as the aftershocks do in our study. The right-lateral set of the secondary faults is nearly sub-parallel to the main fault and the left-lateral set about 60° from that. Values of W S1 and W S2 obtained from that data are indicated in Table 1. An exposure at mid-crustal depths of a strike-slip fault in Austria shows just the same orientation of secondary faults [START_REF] Frost | Progressive strain localization in a major strike-slip fault exhumed from midseismogenic depths: Structural observations from the Salzach-Ennstal-Mariazell-Puchberg fault system, Austria[END_REF]. These observations provide the 'ground truth' for the structures upon which the near-fault aftershocks occur.

Fault parameters

We collect key parameters describing the degree of evolution of the faults that broke during the selected earthquakes. Since faults propagate laterally through time, their structural maturity varies also along strike (Perrin, Manighetti, & Gaudemer, 2016;[START_REF] Perrin | Location of largest earthquake slip and fast rupture controlled by along-strike change in fault structural maturity due to fault growth[END_REF]. Thus, it is necessary to use fault parameters that describe the local fault maturity where the rupture occurred. This is particularly true for long faults, such as the San Andreas Fault, for which fault initiation age varies greatly along the fault, from ancient fault sections in Central California (24 to 29 Ma at Parkfield; e.g. [START_REF] Atwater | Pacific North America plate tectonics of the Neogene southwestern United States: An update[END_REF][START_REF] Liu | Inception of the eastern California shear zone and evolution of the Pacific-North American plate boundary: From kinematics to geodynamics[END_REF] to younger fault sections in Southern California (<12Ma, e.g. [START_REF] Powell | Evolution of the San Andreas Fault[END_REF][START_REF] Sims | Chronology of displacement on the San Andreas fault in central California : Evidence from reversed positions of exotic rock bodies near Parkfield, California[END_REF], and which therefore have different local net displacements. Table 1 presents the fault parameters used in this study. The seven fault sections span a wide range of structural maturity, with various initiation ages (1.1 to 29 Ma), cumulative displacements (1 to 315 km) and geological slip rates (0.2 to 26 mm/yr). Some fault parameters are not fully constrained, especially for the fault associated with the South Napa earthquake. The West Napa fault corresponds to the northern extension of the Calaveras fault, which transfers its slip northwards via the Contra Costa shear zone [START_REF] Brossy | Digital Compilation of Data for the Contra CostaShear Zone for the Northern California Quaternary Fault Map Database: Collaborative Research with William Lettis & Associates, Inc. and the US Geological Survey[END_REF][START_REF] Catchings | Continuity of the West Napa-Franklin Fault Zone Inferred from Guided Waves Generated by Earthquakes Following the 24 August 2014 M w 6.0 South Napa Earthquake[END_REF][START_REF] Unruh | Critical Evalutation of the Northern Termination of the Calaveras Fault, Eastern San Francisco Bay Area, California[END_REF]. We estimated its cumulative slip using a linear fault tip model [START_REF] Scholz | Slip tapers at the tips of faults and earthquake ruptures[END_REF]. The West Napa Fault terminates about 35 km north of the South Napa earthquake, and the tip taper for interacting faults is 0.1-0.8 [START_REF] Scholz | Slip tapers at the tips of faults and earthquake ruptures[END_REF]. Combining these values give a total slip range of 3.5-28 km (Table 1). As the West Napa Fault is genetically linked to the Calaveras fault and situated at its northern tip, we assume that it formed after the formation of the main Calaveras fault (about 12 [START_REF] Ma | Dynamic modeling of the 2004 Mw 6.0 Parkfield, California, earthquake[END_REF][START_REF] Wakabayashi | Distribution of displacement on and evolution of a young system[END_REF] and before the younger surrounded Greenville and Green Valley faults situated to the east (about 6 Ma ago; [START_REF] Wakabayashi | Distribution of displacement on and evolution of a young system[END_REF]. The slip rate on the West Napa fault is estimated to exceed 1 mm/yr, based on geomorphic evidence [START_REF] Wesling | Mapping of the West Napa Fault Zone for Input into the Northern California Quaternary Fault Database[END_REF] and to be 4±3 mm/yr as derived from geodetic data (d'Alessio, 2005;[START_REF] Field | Long-Term Time-Dependent Probabilities for the Third Uniform California Earthquake Rupture Forecast (UCERF3)[END_REF]. For clarity, the estimated geological parameters of the West Napa fault are represented as open symbols in figure 5.

Results

We investigate the relationship between the aftershock distributions, as characterized by W S1 and W S2 , and the various, independently derived fault parameters (Table 1). Our first observation is that W S1 (i.e., the half distance from the calculated fault plane where the aftershock rate saturates) and W S2 (i.e., the full half-width of the shear deformation zone) correlate with each other (Fig. 4), indicating that these two parameters are not independent and that the shape of the shear deformation zone grows in a self-similar way as

W S1 =(0.2±0.1)*W S2
In figure 5 we plot W S1 and W S2 as a function of various fault parameters. When plotted against cumulative fault displacement (Fig. 5a and5b), they both indicate that the width of the shear deformation zone (as characterized by W S1 and W S2 ) decreases with net displacement as a power law with an exponent of -0.54 and -0.39 for W S1 and W S2 , respectively. We note that the field measurements across the Awatere fault (red symbol in Fig. 5) are in good agreement with the trends highlighted from seismological data. This supports the interpretation that the shear deformation zone defined by the width of the aftershock zone corresponds to the width of the zone of active secondary faulting. Furthermore, we observe a decrease of W S1 and W S2 with increasing fault initiation age (fig. 5c and5d) and geological slip rate (fig. 5e and5f), showing a wider shear deformation zone for younger fault sections and slower slip rates. This is also in good agreement with geological observations across the Awatere fault. We note that the power law exponents in figure 5 should be used with caution given the occasionally large uncertainties in the fault parameters.

Figure 6a shows the near-fault aftershock distribution as a function of fault perpendicular distance for each of the seven earthquake sequences (see also black curves in Fig. 2 and Supp. Fig. S3). These curves show a clear pattern: intermediate to mature fault sections (warm colors in fig. 6) are characterized by aftershocks concentrated close to the fault plane. The rapid fall-off in activity away from the fault plane describes a narrow deformation zone at the scale of hundreds of meters. In contrast, intermediate to immature faults (cold colors in fig. 6) exhibit a wider deformation zone at the kilometer scale where events are more widely distributed within the surrounding medium (fig. 6a andb). The distributions remain proportional to one another, indicating that the shape of the shear deformation zone remains constant.

Field measurements of the cumulative number of faults across the Awatere fault (black squares, fig. 6a; [START_REF] Little | Brittle deformation adjacent to the Awatere strike-slip fault in New Zealand: Faulting patterns, scaling relationships, and displacement partitioning[END_REF] show a similar trend and corroborate our seismological observations.

Discussion

The nature of the shear deformation zone and the smoothing of faults

In this study we show that, for fault displacements greater than 1 km, the width of the shear deformation zone (defined by W S1 and W S2 ) decreases with fault displacement. So, what determines the width of the zone of near-fault aftershocks that define the shear deformation zone? If the fault is smooth, the mainshock would produce a stress shadow centered at the fault and there would be no near-fault aftershocks [START_REF] Kostrov | Evaluation of stress and displacement fields due to an elliptical plane shear crack[END_REF]. Slip on a rough fault, on the other hand, would result in high stresses in a narrow region adjacent to the fault, within which earthquakes could be generated. This was the explanation used by Powers & Jordan (2010) to explain near-fault seismicity during the interseismic period. There they used the static rough fault model of [START_REF] Dieterich | Nonplanar faults: Mechanics of slip and off-fault damage[END_REF].

Here we use instead the model of [START_REF] Aslam | Effect of Fault Roughness on Aftershock Distribution: Elastic Off-Fault Material Properties[END_REF] which calculates the stresses resulting from dynamic ruptures propagating on a rough surface. They characterized the fault as a self-affine fractal with Hurst exponent H and fault roughness measured by the RMS of the ratios between the amplitude and the wavelength of the fault surface topography (. This is a fairly realistic rendition of the observed topography of faults (e.g. [START_REF] Candela | Roughness of fault surfaces over nine decades of length scales[END_REF]. Their model predicts large changes in the Coulomb Failure Function (CFF= ; of more than 20 MPa) within a well-defined narrow region, W nf close to the fault. The modeled receiver faults they assumed are parallel to the primary fault, which is consistent with the orientation of most of the secondary faults observed by [START_REF] Little | Brittle deformation adjacent to the Awatere strike-slip fault in New Zealand: Faulting patterns, scaling relationships, and displacement partitioning[END_REF] and [START_REF] Frost | Progressive strain localization in a major strike-slip fault exhumed from midseismogenic depths: Structural observations from the Salzach-Ennstal-Mariazell-Puchberg fault system, Austria[END_REF]. They find that the width of this near-fault region of high stresses is insensitive to H but decreases with . Assuming that our observed shear deformation zone represents the near-fault region of high stresses, this suggests that the observed decrease of W S2 with fault displacement is the result of smoothing of the fault with slip, similarly to observations made during laboratory experiments [START_REF] Goebel | What allows seismic events to grow big?: Insights from b-value and fault roughness analysis in laboratory stick-slip experiments[END_REF] Aslam & Daub (2018) found that the half width of the near fault zone W nf is ~ 2.7 km and ~ 0.9 km for profiles with fault roughness RMS values  of 0.01 and 0.001, respectively.

Comparing these results with the range of W S2 in figure 5 indicates that more than an order of magnitude of roughness change will be required to explain these observations. This indicates a wear rate far greater than that observed for the roughness of individual fault segments measured in the outcrop scale range for fault displacements in the range 10 -1 -10 3 m [START_REF] Brodsky | Faults smooth gradually as a function of slip[END_REF]. Similar measurements made on fault segments in the high porosity and friable Navajo and Entrada sandstones of Utah [START_REF] Dascher-Cousineau | Smoothing of Fault Slip Surfaces by Scale-Invariant Wear[END_REF]) indicate greater smoothing rates but predict fault roughness at our scales orders of magnitude smoother than indicated by our results as interpreted with [START_REF] Aslam | Effect of Fault Roughness on Aftershock Distribution: Elastic Off-Fault Material Properties[END_REF]. However, faults are composed of many segments or sub-faults at many scales [START_REF] Ferrill | Normal fault corrugation: Implications for growth and seismicity of active normal faults[END_REF][START_REF] Klinger | Relation between continental strike-slip earthquake segmentation and thickness of the crust[END_REF][START_REF] Manighetti | Generic along-strike segmentation of Afar normal faults, East Africa: Implications on fault growth and stress heterogeneity on seismogenic fault planes[END_REF][START_REF] Scholz | A further note on earthquake size distributions[END_REF][START_REF] Wallace | Segmentation in fault zones[END_REF] and of nested sub-faults offset from one another [START_REF] Ben-Zion | Characterization of Fault Zones[END_REF][START_REF] De Joussineau | Segmentation along strike-slip faults revisited[END_REF][START_REF] Segall | Mechanics of discontinuous faults[END_REF]. The length distribution of sub-faults follows a power law [START_REF] Scholz | A further note on earthquake size distributions[END_REF] and they are offset by jogs that are self-similar (de Joussineau & Aydin, 2009). The combination of the latter relations would produce a fractal topography at a hierarchy higher than that of the roughness of the individual subfault. A fault smoothing relationship based on roughness at this supra-segment level, measured as fault step spacing, shows an approximately inverse linear reduction of fault roughness with fault displacement [START_REF] Stirling | Fault trace complexity, cumulative slip, and the shape of the magnitude-frequency distribution for strikeslip faults: a global survey[END_REF]. This is a much higher rate of smoothing than that of individual fault segment surfaces, and is comparable to that we infer from the narrowing of the shear deformation zone. Converting the roughness parameter step frequency of [START_REF] Stirling | Fault trace complexity, cumulative slip, and the shape of the magnitude-frequency distribution for strikeslip faults: a global survey[END_REF] to  using the typical step width 1-2 km [START_REF] Wesnousky | Seismological and structural evolution of strike-slip faults[END_REF] and their smoothing curve allows us to recast the two fiducial points of Aslam & Daub, 2018 from (W nf ,to (W nf , D) coordinates (i.e. values of 0.001 and 0.01 corresponding to fault displacements "D" of 100 and 10 km, respectively).

These are plotted in Figure 7 (blue circles) where they are compared with our W S2 measurements from figure 5b (squares). The good agreement supports the hypothesis that narrowing of the shear deformation zone with fault displacement is the consequence of the smoothing of the fault with wear (see also discussion in section 4.3).

The effect of fault maturity on earthquake properties

Once a fault reaches the seismogenic thickness, with a length of ~ 10 km and slip ~ 100 m, its core and dilatant damage zones are fully formed and further maturation occurs by the fault becoming smoother through the processes of frictional wear. It follows that the characteristics of earthquakes may also evolve with the smoothing of the fault. It has been suggested, for example, that earthquake stress drop, apparent stress, and radiation efficiency vary with fault maturity [START_REF] Choy | An overview of the global variability in radiated energy and apparent stress[END_REF][START_REF] Hecker | Normal-Faulting Slip Maxima and Stress-Drop Variability: A Geological Perspective[END_REF][START_REF] Ross | Dissipative Intraplate Faulting During the 2016 Mw 6.2 Tottori, Japan Earthquake[END_REF]. These are not entirely independent parameters. The radiation efficiency, 𝜂 𝑅 , is given by:

𝜂 𝑅 = 𝐸 𝑅 𝐸 𝑅 +𝐸 𝐺 = 2𝜇𝐸 𝑅 Δ𝜎𝑀 𝑜 = 2𝜎 𝑎 Δ𝜎 ,
where 𝐸 𝑅 is radiated energy, 𝐸 𝐺 is energy dissipated in damage, Δ𝜎 is (static) stress drop, 𝜎 𝑎 is apparent stress, 𝑀 𝑜 is seismic moment, and 𝜇 is shear modulus. Here we focus on stress drop, which is the most commonly reported of these properties.

Five of the earthquakes in our study have three or more estimates of stress drop (Δ𝜎).

Following the procedure of Hardebeck (2020), we assume that log 10 (Δ𝜎) has a normal distribution [START_REF] Baltay | Variability in earthquake stress drop and apparent stress[END_REF] and calculate its mean and the standard error of the mean, as reported in Table 2. The corresponding values of mean Δ𝜎 are plotted vs. the cumulative fault slip in figure 8. The correlation Δ𝜎  D -0.23 , compared to our earlier finding that 𝑊 𝑆2 ∝ 𝐷 -0.39 implies that stress drop scales with 𝑊 𝑆2 and hence, from figure 7, that stress drop scales with fault roughness. [START_REF] Hecker | Normal-Faulting Slip Maxima and Stress-Drop Variability: A Geological Perspective[END_REF] measured the maximum slip to length ratio of prehistoric earthquake scarps on intraplate dip-slip faults in the western U.S. They found that this ratio, an indicator of stress drop, tends to decrease with net fault displacement, consistent with our finding.

However, net fault slip usually correlates with slip rate (see Fig. 5), so we also find a similar correlation between stress drop and that parameter, as shown in figure 9a (the relation between stress drop and fault initiation age is shown in Supp. Fig. S5). This is shown also in log-linear coordinates in figure 9b to compare it with the form expected from fault healing. The correlation of stress drop with the log of slip rate or its inverse, the recurrence time, is well known [START_REF] Kanamori | Earthquake repeat time and average stress drop, in Earthquake Source Mechanics[END_REF][START_REF] Scholz | Scaling differences between large interplate and intraplate earthquakes[END_REF] as is its correlation with strain rate [START_REF] Hauksson | Average Stress Drops of Southern California Earthquakes in the Context of Crustal Geophysics: Implications for Fault Zone Healing[END_REF]. The size of the effect seen in figure 9b is about -2 MPa/decade in slip rate, in rough agreement with the other studies. [START_REF] Scholz | Scaling differences between large interplate and intraplate earthquakes[END_REF] pointed out that this behavior could either reflect a rate effect due to fault healing as predicted by the rate-and state-dependent friction laws [START_REF] Dieterich | Time-dependent friction in rocks[END_REF] or a roughness effect in which friction increases with surface roughness, as indicated by several experimental rock friction studies [START_REF] Biegel | Micromechanics of rock friction 1. Effects of surface roughness on initial friction and slip hardening in westerly granite[END_REF][START_REF] Byerlee | Theory of friction based on brittle fracture[END_REF], or some combination of the two.

Observations of repeating earthquakes in which seismic moment increased with recurrence time promised a separation of these effects [START_REF] Marone | Fault healing inferred from time dependent variations in source properties of repeating earthquakes[END_REF][START_REF] Vidale | Variations in Rupture Process With Recurrence Interval in a Repeated Small Earthquake[END_REF]. Simulations with one dimensional spring-slider models with rate-state friction indicated that these data could be explained with fault healing at a rate of 1-3 MPa/decade. However, subsequent studies using a 3D model of a repeating earthquake occurring on a velocity-weakening patch imbedded within a creeping velocitystrengthening fault showed that the recurrence time -moment scaling could occur without any increase in stress drop [START_REF] Chen | Scaling of small repeating earthquakes explained by interaction of seismic and aseismic slip in a rate and state fault model[END_REF]. Observations at Parkfield, California, showed both positive and negative moment-recurrence time scaling of repeating earthquakes [START_REF] Chen | Postseismic variations in seismic moment and recurrence interval of repeating earthquakes[END_REF]. Both types of scaling can be predicted with the [START_REF] Chen | Scaling of small repeating earthquakes explained by interaction of seismic and aseismic slip in a rate and state fault model[END_REF] model by varying the ratio of nucleation to patch size.

In the [START_REF] Chen | Scaling of small repeating earthquakes explained by interaction of seismic and aseismic slip in a rate and state fault model[END_REF] model, aseismic slip propagates from the external creeping region into the velocity-weakening patch such that only a fraction of its slip is seismic. The ratio of seismic to aseismic slip in the velocity-weakening patch is sensitive to parameters such as the radius of the velocity-weakening patch and its ratio to the nucleation radius. As a result, it behaves very differently from a simple velocityweakening block slider model in which the slip is entirely seismic. The recurrence timemoment scaling can be reproduced by varying the patch radius without varying stress drop. In our problem, however, we are considering the fault to be fully coupled seismically so that its slip is entirely seismic. In that case, the simple spring slider model should be more applicable. If so, the analysis of [START_REF] Beeler | Earthquake stress drop and laboratory-inferred interseismic strength recovery[END_REF] finds that stress drop should increase logarithmically with recurrence time at about 0.5 to 1.5 MPa per decade, assuming typical laboratory values for the friction parameters a and b. This is in reasonable agreement with the slope in Fig. 9b of -2 MPa/decade in slip rate, although, as we said earlier, the effect of fault roughness on stress drop cannot be ruled out.

Strengths, limitations and comparison with other studies.

In our study, we use aftershock distributions to study how the early postseismic events are reactivating the long-term damage zone around active faults. We define W S2 as the distance from the fault plane where the mean aftershock distributions start to flatten out or drop abruptly (Fig. 3 and Supp. Fig. S4). Either the flattening or the abrupt termination of the number of events with distance away from the fault would indicate that there are no faults or fractures big enough to accommodate a large number of events. Powers and Jordan (2010) observed a flattening of the interseismic event distribution away from the fault which they attributed to background seismicity. Since we are only using 2 months long aftershock sequences, with significantly larger number of events compared to the interseismic period, we don't systematically observe the curve flattening and cannot reliably estimate the off-fault background seismicity level as Powers & Jordan (2010) did for interseismic periods. If significant off-fault seismicity exists during our observational period, then it may have occurred shortly after the mainshock, and went undetected due to the low signal to noise ratio in the coda of the mainshocks. However, a few of our aftershock distributions show curve flattening (e.g. Parkfield, South Napa, Morgan Hill). These sequences occurred on intermediate to mature faults where creep has been observed (e.g. [START_REF] Harris | Detection of a Locked Zone at Depth on the Parkfield, California, Segment of the San Andreas Fault[END_REF][START_REF] Schaff | High-resolution image of Calaveras Fault seismicity[END_REF][START_REF] Xu | Interseismic Ground Deformation and Fault Slip Rates in the Greater San Francisco Bay Area From Two Decades of Space Geodetic Data[END_REF]. Our hypothesis is that these aftershock distributions have fewer absolute number of events because they are related to smaller mainshocks (M~6), and also lower stress drops (see Fig. 8). Moreover, the presence of creeping sections might increase the regional background seismicity level.

In this study, we selected aftershocks of M l >1, thus including events with magnitudes that are below magnitudes of completeness, M c , levels for most aftershock sequences, especially in southern California. Assuming that small events are randomly distributed in the stress field, recovering and including all small events in our analysis would increase the number of events in all boxes. This would result in shifts along the y axis in figure 6a, but would not change the shape of the curves. It might lead to slight increases in the W S2 values, which therefore should be considered here as lower bounds of our shear deformation zone widths. This is in good agreement with figure 7, where some measurements corresponding to southern California sequences seem to be underestimated compared to the expected theoretical models.

Our estimates of the shear deformation zone width might also be affected by the aftershock productivity, which depends on the magnitude of the mainshock. The number of aftershocks does increase with magnitude, but also linearly with fault area [START_REF] Wetzler | Regional and stress drop effects on aftershock productivity of large megathrust earthquakes[END_REF][START_REF] Yamanaka | Scaling relationship between the number of aftershocks and the size of the main shock[END_REF], and therefore the density of aftershocks per unit area is constant. This is what we show in supplementary figure S6, where the mean number of events in boxes per km 2 are quite similar and do not show any magnitude dependency (Parkfield and Morgan Hill being slightly off because of the creeping areas that bound the rupture zones, which produce abundant interseismic earthquakes on the fault, contaminating the data at small distances). The apparent correlation between mainshock magnitude and W S2 in our study results rather from the happenstance that the larger earthquakes happen to have occurred on low maturity intraplate faults whereas the smaller ones occurred on mature faults.

As mentioned earlier, Powers & Jordan (2010) estimated the normal distance of the earthquakes away from a single vertical fault plane inferred from the surface trace of the fault. They defined the zone of shear deformation from the near fault seismicity during the interseismic period and averaged over the entire fault length, possibly neglecting local variations in fault plane orientation and dip, leading to wider zones compared to our results. In addition, geological heterogeneity over long distances along the fault and other effects (such as triggering or regional stress release caused by other earthquakes) may play a role, as pointed out by [START_REF] Hauksson | Spatial Separation of Large Earthquakes, Aftershocks, and Background Seismicity: Analysis of Interseismic and Coseismic Seismicity Patterns in Southern California[END_REF]. Finally, our approach of fitting a single fault plane to the local aftershock distribution as we step along the fault assumes that the rupture occurred mainly on one fault strand and that aftershocks are homogeneously distributed in the surrounded medium. While individual secondary faults are distinguishable around mature fault sections (see for instance oblique linear faults around Morgan Hill in Supp. Fig. S3), it is more difficult to identify them around immature fault sections, within the broader zone of deformation, especially near the rupture tips (see the northern part of the El Mayor Cucapah earthquake for instance; Supp. Fig. S3).

Previous studies have estimated damage zone width or low velocity zones across several faults in California using various seismic and/or geodetic methods (see Yang, 2015 for a review). We compare some measurements with our results in Supp. Table S2. First, we note that, different methods result in different estimates of "damage zone" widths.

Secondly, all studies using seismic fault zone trapped waves (FZTW) techniques find a similar low velocity zone that is 100-200 m wide, regardless the fault structural maturity. The FZTW derived width correspond to the constant width of dilatant damage zones around large faults, already described by geological studies (e.g., [START_REF] Faulkner | Scaling of fault damage zones with displacement and the implications for fault growth processes[END_REF][START_REF] Savage | Collateral damage: Evolution with displacement of fracture distribution and secondary fault strands in fault damage zones[END_REF]; also called the inner damage zone in [START_REF] Perrin | Location of largest earthquake slip and fast rupture controlled by along-strike change in fault structural maturity due to fault growth[END_REF]. However, other seismic and geodetic studies (sometimes used together; see Cochran et al., 2009), find greater damage zone widths, that are in general agreement with our results. This discrepancy between results from different methods was also pointed out by Spudich & Olsen (2001). We tested the relations found in Figure 5a and 5b to deduce the size of the shear damage zone of the San Jacinto fault, for which both a narrow damage zone using FZTW methods (e.g., Lewis et al., 2005) and wider damage zone using tomography (e.g., Allam et al., 2014) have been described. Assuming a cumulative slip of 24 km [START_REF] Sharp | San Jacinto Fault Zone in the Peninsular Ranges of Southern California[END_REF], we obtain a half width of the shear deformation zone ranging from 0.2 (i.e. W S1 ) to 1.5 km (i.e. W S2 ), which is in better agreement with tomographic studies (half width of 1.5 to 2.5 km; Allam et al., 2014).

Thus, we believe that the studies using FZTW are mainly sensitive to the most damaged inner part of the fault zone, also called the dilatant damage zone, which we don't seem to be able to resolve with our aftershock data. The other seismic and geodetic studies found damage zone widths that are between 2(W S1 ) and 2(W S2 ), which we define as the inner and outer bounds of the shear deformation zone.

A recent geodetic study from [START_REF] Scott | The M 7 2016 Kumamoto, Japan, Earthquake: 3-D Deformation Along the Fault and Within the Damage Zone Constrained From Differential Lidar Topography[END_REF] has shown that the dilatant damage zone is equivalent to slightly wider than the shear deformation zone. These results come from surface measurements (i.e., LiDAR) along the active fault zone responsible for the 2016 Kumamoto earthquake. It is possible that extensional strains are greater at the surface than at depth. As cracks close down rapidly under pressure, dilatant strains in the dilatant damage zone will decrease more rapidly with depth [START_REF] Yoshioka | Elastic properties of contracting surfaces under normal and shear loads 1. Theory[END_REF] than the shear strains. This may explain why their results from surface measurements differ from ours measured at depth.

In our study we average the near-fault aftershock distributions assuming that the degree of fault maturity is locally constant along each broken fault sections we investigate.

However, it has been shown that the fault maturity can vary along strike and this can affect the heterogeneity of fracture density around the fault core [START_REF] Ostermeijer | Damage zone heterogeneity on seismogenic faults in crystalline rock; a field study of the Borrego Fault, Baja California[END_REF], the distribution of secondary faults at greater distances away from the fault (Perrin, Manighetti, & Gaudemer, 2016) and finally the behavior of earthquakes (e.g., [START_REF] Cappa | Off-fault long-term damage: A condition to account for generic, triangular earthquake slip profiles[END_REF][START_REF] Huang | Earthquake Rupture in Fault Zones With Along-Strike Material Heterogeneity[END_REF][START_REF] Perrin | Location of largest earthquake slip and fast rupture controlled by along-strike change in fault structural maturity due to fault growth[END_REF].

Consequently, for long or multiple-fault earthquake ruptures, the fault normal distribution of aftershocks could change along strike, and present a shear deformation zone that widens towards the most immature parts of the rupture. Such a behavior would be in good agreement with mapped faults at the surface showing that the off-fault damage zone widens in the direction of long-term fault propagation (e.g., [START_REF] Manighetti | Slip accumulation and lateral propagation of active normal faults in Afar[END_REF]Perrin, Manighetti, & Gaudemer, 2016). Future work is needed to relate such observations to the occurrence of seismicity.

Conclusion

Our study presents strong correlations between independent datasets that describe the near-fault distribution of aftershocks following large earthquakes and geological fault parameters such as cumulative displacement, initiation age and slip rate. We find that for large faults, defined as those that have ruptured the entire brittle crust, the zone of active shear deformation narrows as a power law with cumulative fault displacement, hence with fault maturity. This result is predicted by a dynamic rough fault model combined with an empirical roughness/displacement relation, indicating that the narrowing of the shear deformation zone is the result of fault smoothing with cumulative displacement. Earthquake stress-drop also decreases with fault displacement and hence fault smoothness or slip rate. 1. Blue circles are for step widths of 1 and 2 km, which are typical step width range [START_REF] Wesnousky | Seismological and structural evolution of strike-slip faults[END_REF]. [START_REF] Ancheta | NGA-West2 database[END_REF][START_REF] Baltay | Earthquake Stress Drop and Arias Intensity[END_REF][START_REF] Hanks | M-logA Observations for Recent Large Earthquakes[END_REF][START_REF] Hauksson | The 1999 Mw 7.1 Hector Mine, California, earthquake sequence: Complex conjugate strike-slip faulting[END_REF]Kaverina et al., 2002

Figures and figure captions

Earthquake name

Number of measurements

Mean

  Figures and figure captions:

Figure 1 :Figure 2 :

 12 Figure 1: Simplified view of the architecture of a fault zone and the density of

Figure 4 :

 4 Figure 4: Relations between W S1 and W S2 for the seven earthquakes analyzed in

Figure 5 :Figure 6 :

 56 Figure 5: Relations between W S1 (circles; a, c, e), W S2 (squares; b, d, f) of seven

Fig

  Fig.S3). Each colored curve represents one earthquake sequence. Curves with the same

Figure 7 :

 7 Figure 7: Comparison between the outer bound of the shear deformation zone (W S2 ;

Figure 8 :

 8 Figure 8: Earthquake stress drop (Δ𝜎) as a function of cumulative displacement of the

Figure 9 :

 9 Figure 9: Earthquake stress drop (Δ𝜎) as a function of geological slip rate (SR) for

  Figure 1.

  Figure 2.

  Figure 4.

log 10 ()

				Mean	Mean	References
				standard	stress drop
				error	(in
					MPa)
	Hector Mine,	8	0.672	0.011	4.7 ± 1	Allmann & Shearer, 2009;
	1999

Table 2 : Mean stress drop values available from the literature for five of the seven earthquakes analyzed in this study.

 2

	; Price &
	Burgmann, 2002

ACKNOWLEDMENTS

This work is supported by the Brinson Foundation (CP) and the National Science Foundation (NSF-EAR #1520680). Earthquake catalog data are available at http://ddrt.ldeo.columbia.edu and https://scedc.caltech.edu/researchtools/altcatalogs.html. We thank the editor R. Abercrombie, associate editor M. Cooke and reviewers E. Nissen, N. Abolfathian and K. Dascher-Cousineau for their fruitful comments that greatly improved the manuscript. This paper is LDEO contribution 8477.

 S4for all cases). Blue circles represent the mean hypocenter distances from each fault section within 0.03 km wide bins (see black curve in Fig. 2 and Supp. Fig. S3). Grey circles represent the mean distances of events that are only included in less than 20% of all boxes. The red curve is the best (power law) fit to the blue circles, taking into account the power law singularity on the fault (see Supp. Fig. S4 for details). The vertical gray dashed lines labeled W S1 and W S2 point out the locations where the power law fit shows the maximum curvature and where the data (blue [START_REF] Dibblee | Evidence of strike-slip movement on northwest-trending faults in the western Mojave Desert, California[END_REF][START_REF] Dokka | Geology Displacements on late Cenozoic strike-slip faults of the central Mojave Desert, California Displacements on late Cenozoic strike-slip faults of the central Mojave Desert, California[END_REF][START_REF] Dokka | Late Cenozoic strike-slip faulting in the Mojave Desert, California[END_REF][START_REF] Garfunkel | Model for the Late Cenozoic Tectonic History of the Mojave Desert , California , and for Its Relation to Adjacent Regions[END_REF][START_REF] Jachens | Relationship of the 1999 Hector Mine and 1992 Landers fault ruptures to offsets on neogene faults and distribution of late Cenozoic basins in the eastern California shear zone[END_REF][START_REF] Oskin | Slip rate of the Calico fault: Implications for geologic versus geodetic rate discrepancy in the Eastern California Shear Zone[END_REF]USA Landers, 1992, M [START_REF] Dibblee | Evidence of strike-slip movement on northwest-trending faults in the western Mojave Desert, California[END_REF][START_REF] Dokka | Geology Displacements on late Cenozoic strike-slip faults of the central Mojave Desert, California Displacements on late Cenozoic strike-slip faults of the central Mojave Desert, California[END_REF][START_REF] Dokka | Late Cenozoic strike-slip faulting in the Mojave Desert, California[END_REF][START_REF] Garfunkel | Model for the Late Cenozoic Tectonic History of the Mojave Desert , California , and for Its Relation to Adjacent Regions[END_REF][START_REF] Jachens | Relationship of the 1999 Hector Mine and 1992 Landers fault ruptures to offsets on neogene faults and distribution of late Cenozoic basins in the eastern California shear zone[END_REF][START_REF] Rockwell | Paleoseismology of the Johnson Valley, Kickapoo, and Homestead Valley faults: Clustering of earthquakes in the Eastern California shear zone[END_REF] Atwater [START_REF] Atwater | Pacific North America plate tectonics of the Neogene southwestern United States: An update[END_REF][START_REF] Critelli | Provenance and stratigraphy of the Eocene Tejon Formation, Western Tehachapi Mountains, San Emigdio Mountains, and Southern San Joaquin Basin, California[END_REF][START_REF] Crowell | The San Andreas fault system through time[END_REF][START_REF] Graham | Oligocene and Miocene paleogeography of central California and displacement along the San Andreas fault[END_REF][START_REF] Liu | Inception of the eastern California shear zone and evolution of the Pacific-North American plate boundary: From kinematics to geodynamics[END_REF][START_REF] Matthews | Correlation of Pinnacles and Neenach volcanic formations and their bearing on San Andreas fault problem[END_REF][START_REF] Revenaugh | Cumulative offset of the San Andreas fault in central California: A seismic approach[END_REF][START_REF] Toké | Late Holocene slip rate of the San Andreas fault and its accommodation by creep and moderate-magnitude earthquakes at Parkfield, California[END_REF]