Journal of Geophysical Research: Solid Earth

Distribution of seismicity across strike-slip faults in California

Authors


Abstract

[1] The distribution of seismicity about strike-slip faults provides measurements of fault roughness and damage zone width. In California, seismicity decays with distance from strike-slip faults according to a power law ∼(1 + x2/d2)γ/2. This scaling relation holds out to a fault-normal distance x of 3–6 km and is compatible with a “rough fault loading” model in which the inner scale d measures the half width of a volumetric damage zone and the roll-off rate γ is governed by stress variations due to fault roughness. According to Dieterich and Smith's 2-D simulations, γ approximates the fractal dimension of along-strike roughness. Near-fault seismicity is more localized on faults in northern California (NoCal, d = 60 ± 20 m, γ = 1.65 ± .05) than in southern California (SoCal, d = 220 ± 40 m, γ = 1.16 ± .05). The Parkfield region has a damage zone half width (d = 120 ± 30 m) consistent with the SAFOD drilling estimate; its high roll-off rate (γ = 2.30 ± .25) indicates a relatively flat roughness spectrum: ∼k−1 versus k−2 for NoCal, k−3 for SoCal. Our damage zone widths (the first direct estimates averaged over the seismogenic layer) can be interpreted in terms of an across-strike “fault core multiplicity” that is ∼1 in NoCal, ∼2 at Parkfield, and ∼3 in SoCal. The localization of seismicity near individual faults correlates with cumulative offset, seismic productivity, and aseismic slip, consistent with a model in which faults originate as branched networks with broad, multicore damage zones and evolve toward more localized, lineated features with low fault core multiplicity, thinner damage zones, and less seismic coupling. Our results suggest how earthquake triggering statistics might be modified by the presence of faults.

1. Introduction

[2] California, with its dense, well-mapped network of faults and high-quality earthquake catalogs, is an excellent setting to investigate seismicity variations in space and time. Earthquake catalogs constructed using improved hypocenter relocation techniques [Ellsworth et al., 2000; Hauksson and Shearer, 2005; Shearer et al., 2005; Thurber et al., 2006] are revealing new details about the 3-D geometry of fault networks [Yule and Sieh, 2003; Carena et al., 2004] and ruptures of large earthquakes [Liu et al., 2003], properties of nascent faults [Bawden et al., 1999], earthquake streaks observed on creeping sections [Rubin et al., 1999; Waldhauser et al., 1999, 2004; Shearer et al., 2005; Thurber et al., 2006], and the space-time behavior of earthquake swarms [Vidale and Shearer, 2006]. These studies, as well as extensive research on the fractal character of fault systems [Tchalenko, 1970; King, 1983; Okubo and Aki, 1987; Hirata, 1989; Robertson et al., 1995; Ouillon et al., 1996; Kagan, 2007], have raised a number of interesting issues regarding the relationship of small earthquakes to major faults. For instance: Are physical properties of a fault zone such as the distribution of secondary faults and fractures, damage zone width, and fault roughness captured in near-fault earthquake rates? Does the rate of small earthquakes in the vicinity of a major fault zone reflect the long-term fault slip rate or cumulative offset of the fault? Do other factors such as heat flow and lithology modulate earthquake rate? How might spatial models of near-fault seismicity improve subsurface fault maps or models of earthquake triggering?

[3] To address these issues, we analyze the variation of seismicity rate normal to near-vertical strike-slip faults in California and examine its relation to stress heterogeneity, damage zones, and degree of seismic coupling. Strike-slip faults were chosen because their locations are constrained by mapped surface traces and their approximate bilateral symmetry of seismicity makes their earthquake distributions simpler to interpret than those of normal and reverse faults. To reveal systematic scaling relationships, we aggregated data from fault segments in a common class, as defined by geographic region, fault length, and aftershock activity. We restricted our analysis to small earthquakes (Mw < 5), which we treated as point sources. Examined this way, the near-fault seismicity shows a power law decay away from the fault surface [Powers and Jordan, 2005, 2007a; Hauksson, 2010].

[4] We examined regional variations in this fault-normal distribution of seismicity by comparing the results from selected fault segments in northern California, between Parkfield and the San Francisco Bay (Figure 1), and a more extensive distribution of faults in southern California (Figure 2). In the northern California region, the fault segments are on or subparallel to the San Andreas master fault, a larger percentage of the faults are creeping [Irwin, 1990], and little seismicity extends below 10 km [Hill et al., 1990] (Figure 3). In contrast, the southern California segments have deeper seismicity [Hauksson, 2000], show little or no creep [Bodin et al., 1994; Lyons and Sandwell, 2003; Shearer et al., 2005], and often intersect at high angles. In southern California, we also investigated the spatial distribution of seismicity near smaller faults (e.g., splays of large faults and unmapped secondary faults) to see how fault length and along-fault variations affect the scaling relations. We declustered each data set and incorporated fault segments that ruptured during large (Mw > 6) earthquakes to constrain how the scaling relations are modified by aftershock activity. To assess the bias and variance in the scaling parameters caused by event mislocation, we supplemented the errors estimated by the hypocenter location algorithms with constraints from intercatalog comparisons and statistical simulations.

Figure 1.

Map of northern California showing locations of seismicity samples (gray boxes with reference numbers; see Table 3). Black lines delineate large faults, and the heavy black line marks the San Andreas fault. Dark gray dots mark the locations of 1.5 < M ≤ 2.5 earthquakes (1984–2002). Reference label background colors reflect the fault-normal intercatalog location uncertainty of each segment: σTNx for the Parkfield segments (51 and 52); σUNx for all others.

Figure 2.

Map of southern California showing locations of seismicity samples (with reference numbers; see Table 3). Black lines delineate large faults, and the heavy black line marks the San Andreas fault. Grey boxes with reference numbers in circles (1–15) indicate the limits of seismicity samples about large strike-slip faults; reference numbers in squares indicate the locations of samples about small (16–29) and aftershock-dominated (30–41) fault segments. Dark gray dots mark the locations of 1.5 < M ≤ 2.5 earthquakes (1984–2002). Reference label background colors reflect the fault-normal intercatalog location uncertainty, σPSx, of each segment.

Figure 3.

Depth distributions of relocated earthquakes in the fault segment catalogs. Events from (a) catalog H and (b) catalog P in southern California and (c) catalogs U and T in northern California. The darker shaded bars mark the medial 90% of all events in the catalogs. Note that seismicity is generally shallower in northern California (Figure 3c). The downward bias in catalog H (Figure 3a) is likely an artifact of the 3-D velocity model used for earthquake relocation.

[5] Our results provide measures of shear localization on faults and indirect evidence that fault damage zones extend through the seismogenic crust. We present a model of fault behavior that incorporates slip on a fractal fault, and discuss the implications of the model on fault evolution using data on fault width, on-fault earthquake density, cumulative offset, and aseismic slip rate. Through comparisons with studies of exhumed faults and drilling results, we relate the structure of a fault damage zone at depth, as defined by seismicity, to near-surface observations.

2. Fault-Referenced Seismicity Catalogs

[6] Hypocenters from six earthquake catalogs (three in southern, two in northern California, and one for the Parkfield region (Table 1)) were used to constrain near-fault seismicity distributions and their uncertainties. The Southern California Seismic Network (SCSN) is the southern part of the California Integrated Seismic Network (CISN), a region within the Advanced National Seismic System (ANSS). The SCSN catalog (here abbreviated as “S”; available at http://www.data.scec.org) is the standard catalog for southern California and contains events reported by all networks in the region. It includes some estimates of hypocentral errors, but all events have a “quality” designation indicative of maximum horizontal and vertical location uncertainties.

Table 1. Earthquake Catalog Sources
Sources by RegionIDaType
  • a

    Used to reference catalog in equations and text.

Southern California
   SCSNSstandard
   Shearer et al. [2005]Prelocated
   Hauksson and Shearer [2005]Hrelocated
 
Northern California
   NCSNNstandard
   Ellsworth et al. [2000]Urelocated
   Thurber et al. [2006], ParkfieldTrelocated

[7] Hauksson and Shearer [2005] relocated events (catalog “H”; available at http://www.data.scec.org) from the SCSN using the double-difference algorithm of Waldhauser and Ellsworth [2000]. They cross-correlated waveforms to measure traveltime differences and relocated seismicity in a three-dimensional (3-D) velocity model. For events lacking sufficient data for double differencing (<10%), they determined hypocenters using Hauksson's [2000] relocation method. They also evaluated errors using this method, because the double-difference code does not compute hypocentral location errors for large data sets.

[8] Shearer et al. [2005] relocated events (catalog “P”; available at http://www.data.scec.org) using a source-specific station term algorithm [Richards-Dinger and Shearer, 2000] that employs a layered (1-D) velocity model. Using event-similarity data from a waveform cross-correlation analysis, they further refined the locations of spatially related events (∼60%) via a cluster analysis. Hypocentral errors reported by Shearer et al. [2005] correspond to the relative locations of events in each cluster.

[9] The catalog of the Northern California Seismic Network (NCSN, abbreviated “N”; available at http://www.ncedc.org/ncsn) is the standard catalog for the northern part of the CISN. Ellsworth et al. [2000] relocated a subset of the NCSN events in the San Francisco Bay area (catalog “U”; available at http://pubs.usgs.gov/of/2004/1083) using the double-difference algorithm of Waldhauser and Ellsworth [2000]. The catalog does not include hypocentral location errors, but Ellsworth et al. [2000] report average horizontal and vertical location uncertainties of 0.1 and 0.5 km, respectively.

[10] On the creeping section of the San Andreas fault in the vicinity of Parkfield, relocated events (denoted catalog “T”; available at http://www.seismosoc.org/publications/BSSA_html/bssa_96-4b/05825-esupp/) were taken from Thurber et al. [2006]. They constructed an improved 3-D wave speed model to first determine station corrections and then relocated events via double difference using a combination of event cross-correlation differential times and traveltime differences from NCSN phase picks. The catalog does not include any hypocentral error information.

[11] For each of the six catalogs, we limited our analysis to events that occurred from the beginning of 1984 until the end of 2002. To allow for intercatalog comparisons, the events were required to have hypocenters in both northern California catalogs (N and U or T and U) or in all three southern California catalogs (S, P, and H). The northern California catalogs, T and U, do not overlap geographically and may only be compared independently to catalog N, whereas the southern California catalogs span the entire lower half of the state and may be compared collectively. Events identified as quarry blasts or ones with intercatalog separations greater than 10 km were discarded.

[12] We constructed catalogs of earthquakes for five classes of near-vertical strike-slip faults (Tables 2 and 3 and Figures 1 and 2): large faults of northern California, the San Andreas fault at Parkfield, large faults of southern California, small faults of southern California, and fault segments with abundant aftershocks of major southern California earthquakes. For large faults, we chose relatively straight segments of named, throughgoing faults (e.g., Figure 4), eliminated fault segments with earthquake density of less than one event per km, and avoided fault junctions and zones of structural complexity. In southern California the SCEC Community Fault Model (CFM, available at http://structure.harvard.edu/cfm) [Plesch et al., 2007] was our guide for fault selection; in northern California, where faults are well defined by near-vertical seismicity distributions, we used surface traces. Where the surface trace or seismicity along a particular fault indicates significant changes in strike or fault-strand overlap, we restricted our selection to smaller fault segments; e.g., the Hayward fault (Figure 1; segments 42–43) and Garlock fault (Figure 2; segments 1–3). Large fault lengths, Lk in Table 3, average 21 km in northern California and 47 km in southern.

Figure 4.

Example of intercatalog earthquake location variation. (a) Map of the Elsinore fault (segment 10, Figure 2) showing relocated seismicity of catalog P and historic (heavy black lines), Holocene (thin black lines), and late Quaternary (thin gray lines) faults. (b) Depth section across the map showing locations of events in the standard catalog S relative to an initial, 3-D fault model based estimate of the fault trace (heavy dashed line); earthquakes are the same magnitude ranges as on the map. (c) Fault-normal distribution of events. (d) Depth section across the map for relocated catalog P. (e) Fault-normal seismicity distribution of relocated events. Note the difference in horizontal bias (black arrow in Figures 4c and 4e) of peak seismicity between the standard and relocated catalog.

Table 2. Earthquake Catalog Statisitics
 NTa (events)bMc
  • a

    All events common to regional and relocated catalogs in Table 1.

Southern California (SoCal)
SCSN catalog291,5410.92.0
Fault classes   
   Large11,0950.91.9
   Small8,9660.81.5
   Aftershock-dominated19,8490.91.8
 
Northern California (NoCal)
NCSN catalog47,7110.81.2
Fault classes   
   Large16,8810.71.2
   Parkfield3,8970.91.2
Table 3. Fault Segment Catalogsa
SegmentSegment NameSize Nk (events)Lk (km)Wkb (km)
  • a

    Down-weighted level N0, Fitting distance xmax.

  • b

    From relocated catalogs P, U, and T as described in text.

SoCal Large (N0 = 1000, xmax = 6 km)
1Garlock (East)56556.313.0
2Garlock (Central)56627.38.0
3Garlock (West)53445.18.7
4Lenwood-Lockhart9231.910.0
5San Andreas (Mojave)74296.910.9
6Santa Cruz–Catalina Ridge7062.515.1
7Palos Verdes11551.614.6
8Newport Inglewood (North)22634.714.4
9Newport Inglewood (South)18279.017.1
10Elsinore-Temecula55454.013.8
11San Jacinto (Anza)5,02737.717.4
12Elsinore–Coyote Mt.41523.612.0
13Cerro Prieto57239.416.6
14Imperial1,17921.214.1
15San Andreas (Coachella)25640.78.3
 Total (NT)11,095702.0 
 Length-weighted average  13.2
 
SoCal Small (N0 = 700, xmax = 2.5 km)
16Scodie Lineament1,27415.09.6
17San Jacinto (Anza)1,16611.011.3
18San Jacinto (Anza)1,0205.814.8
19San Jacinto (Anza)9784.614.0
20San Jacinto (Coyote Creek)4847.012.5
21San Jacinto (Anza)3265.010.9
22San Jacinto (Coyote Creek)8536.811.6
23San Jacinto (Anza)3596.49.6
24San Jacinto (Borrego)4437.59.9
25Superstition Mt.25812.812.8
26Elmore Ranch31819.911.2
27Elmore Ranch (western ext.)1776.88.9
28Elmore Ranch (western ext.)3829.510.7
29Elsinore9287.911.2
 Total (NT)8,966126.0 
 Length-weighted average  9.2
 
SoCal Aftershock-Dominated (N0 = 1500, xmax = 3 km)
30Joshua Tree1,0686.68.0
31Joshua Tree2,3387.58.0
32Joshua Tree1,8154.98.4
33Joshua Tree8616.29.1
34Landers8735.511.0
35Landers66910.38.7
36Landers1,85510.79.5
37Landers2,5397.07.0
38Landers1,3345.77.9
39Hector Mine98411.28.4
40Hector Mine2,87214.38.8
41Hector Mine2,64111.28.6
 Total (NT)19,849101.1 
 Length-weighted average  8.2
 
NoCal Large (N0 = 1500, xmax = 3 km)
42Hayward (North)33644.310.7
43Hayward (South)67445.910.8
44Calaveras (North)2,95813.17.1
45Calaveras (Central)5088.25.9
46Calaveras (South)1,20017.97.1
47Sargent1,08310.97.7
48San Andreas Creeping (North)1,22912.27.6
49San Andreas Creeping (Central)3,01214.98.0
50San Andreas Creeping (South)5,88121.88.3
 Total (NT)16,881189.4 
 Length-weighted average  7.9
 
NoCal Parkfield (No Down Weight, xmax = 3 km)
51San Andreas Parkfield (North)3,41732.48.3
52San Andreas Parkfield (South)48031.910.1
 Total (NT)3,89764.2 
 Length-weighted average  8.4

[13] We identified small and aftershock-dominated faults by sets of earthquakes that define linear structural features of shorter length (average ∼9 km). The small-fault class comprises splays off larger faults and unmapped secondary faults. Aftershock-dominated segments were selected from faults activated by the 1992 Joshua Tree (Mw 6.1), 1992 Landers (Mw 7.3), or 1999 Hector Mine (Mw 7.1) earthquakes. Although the seismicity in the aftershock-dominated class spans the entire 19 year length of the source catalogs, it is dominated by aftershocks from these large events. The seismicity of the small-fault class is more uniformly distributed in time.

[14] We used the most recently updated magnitude data from catalogs S and N; these were generally reported as local magnitude, although there are a few events for which magnitudes were computed using other means. The maximum likelihood Gutenberg-Richter b values [Aki, 1965] for the northern and southern California catalogs are 0.8 and 0.9, respectively (Table 2). Northern California has a lower magnitude of completeness (Mc = 1.2) than southern California (Mc = 2.0), reflecting its smaller area and higher station density.

[15] We filtered the hypocentral depths in each fault catalog to focus on the central part of the seismogenic crust. Averaged across all fault segments, 90% of seismicity falls between 2 and 10 km for catalogs U and T and 2.5 and 17 km for catalog P (Figure 3). The upper 5% of events tend to occur within 2 km of the free surface, and so we set 2 km as an upper truncation depth. The lower cutoff shows significant variation reflecting regional differences in seismogenic thickness [Hauksson, 2000; Magistrale, 2002]. We therefore set the lower truncation depth to exclude the deepest 5% of hypocenters in each fault catalog. The differences between the lower and upper truncation depths for the kth fault segment determined the segment width Wk; values for each fault segment catalog are listed with the fault lengths Lk in Table 3.

[16] For each fault segment, we established a fault-oriented coordinate system by fitting a plane to the seismicity. An initial estimate of the fault plane was derived from the CFM or, in the absence of a fault model, from a vertical plane that approximated the mapped surface or epicenter trace. All parameters of the plane were perturbed to obtain a least squares fit to relocated hypocenters from catalogs P, U or T within 2 km of this initial fault plane. For all fault segments considered, a 4 km wide swath captures most near-fault earthquakes while ignoring off-fault clusters that would contribute to misalignment of the coordinate system with the fault plane. Narrower swaths fail to include near-fault events for fault segments where seismicity exhibits a significant horizontal shift from the initial estimate, as in the case of the Elsinore fault (Figure 4). Rotations permitted by the fitting process further localize events on final, seismicity-based fault planes, as is observed on the Hayward fault (Figure 5).

Figure 5.

Example of how a fault-seismicity-based coordinate system localizes events on a fault and minimizes artificial fault-normal dispersion. (a) Map of the southern Hayward fault (segment 43, Figure 1) showing relocated seismicity of catalog U; fault age representations are the same as in Figure 4. The fault strand cutting across the lower right corner of the map is the northern Calaveras fault. (b) Fault-normal depth section across the fault trace prior to aligning coordinate system to relocated seismicity; earthquakes are the same magnitude ranges as in the map, and the heavy dashed line marks the depth projection of the fault surface trace. (c) Fault-normal distribution of events. (d) Fault-normal depth section across the fault trace after aligning coordinate system to a best fit plane to relocated seismicity. (e) Realigned fault-normal distribution. Note that the fault-seismicity based coordinate system yields a narrower event distribution in Figure 5e.

[17] The hypocenters from the relevant relocated catalog were transformed into a local Cartesian system defined by an origin at one end of the surface trace of the best fit fault plane, a near-vertical z axis, a y axis along the fault strike, and an x axis perpendicular to the fault plane. A final, fault-referenced catalog was then constructed by eliminating events with relocated x coordinates greater than ±15 km, relocated y coordinates beyond the ends of the fault segment, and relocated z coordinates outside the depth limits described above. The fault segment catalogs are summarized in Table 3.

3. Intercatalog Analysis

[18] A proper description of near-fault seismicity distributions requires careful attention to mislocation errors. Information about such errors can be determined from comparisons of hypocenters determined by different methods [e.g., Shearer et al., 2005]. In the present study, we have quantified the intercatalog comparisons on a fault segment basis. For the kth fault segment with Nk events common to the catalog pair A and B, we define the intercatalog fault-normal bias by

equation image

and the fault-normal variance by

equation image

Here, xAi is the fault-normal coordinate of the ith event, and the summation implied by ik is over all Nk events associated with the kth fault segment. Similar expressions can be written for the fault-parallel and near-vertical directions. The intercatalog bias and variance computed for each catalog pair (UN and TN in northern California and HS, PS, and HP in southern California) are listed by individual fault segment in Tables S1–S3 in the auxiliary material.

[19] For each coordinate of the fault-oriented reference frame, the mean intercatalog bias for a fault class was computed by taking the absolute values of the segment biases, weighting them by the number of events for each segment, and averaging over all segments. Likewise, the mean standard deviation was calculated as the square root of the event-weighted segment variances. These averages are given in Table 4. For all classes, the intercatalog biases and standard deviations are largest for the z coordinate, reflecting the uncertainty in estimating hypocentral depth. No systematic differences are observed between the two horizontal coordinate statistics, x and y. In southern California, the z coordinate statistics are especially large, in part because shallow events in the early part of Catalog S were often assigned a default depth of 6 km, and also because the velocity model used to relocate events in Catalog H tends to bias events downward (Figure 3). Because we are interested in the fault-normal distribution of seismicity, we focus our discussion on bABx and σABx.

Table 4. Intercatalog Error Estimates
Catalog Pair (AB)Bias (km)aStandard Deviation (km)
bxABbyABbzABσxABσxABbσyABσzAB
  • a

    See the auxiliary material for individual fault values.

  • a

    Event-weighted mean absolute values.

  • b

    The σxAB estimated by uniform reduction.

SoCal Large
HS0.740.611.361.140.751.233.09
PS0.570.331.361.080.721.163.12
HP0.590.351.060.740.250.772.15
 
SoCal Small
HS0.470.231.690.800.600.822.62
PS0.430.240.960.750.580.772.41
HP0.240.370.980.400.150.451.52
 
SoCal Aftershock-Dominated
HS0.180.222.060.620.420.702.25
PS0.240.271.090.550.390.631.97
HP0.170.140.970.410.190.441.49
 
NoCal
UN (Large)0.100.030.150.300.150.240.52
TN (Parkfield)0.470.130.600.680.420.320.82

[20] The intercatalog statistics for well-instrumented Parkfield region are higher than those of the northern California fault class, particularly the fault-normal values (e.g., σTNx = 0.62 km versus σUNx = 0.31 km). These differences are primarily due to more stations and a greater number of earthquakes being used in the relocation procedure, as well as a strong velocity contrast across the San Andreas fault near Parkfield, which is modeled in the Thurber et al. [2006] relocations but not in the standard catalog.

[21] The intercatalog statistics for the large faults in southern California (e.g., σPSx = 1.04 km) are also substantially higher than for the northern California fault class, which we ascribe to several factors. The southern region has a more heterogeneous crustal structure than the northern region, such as larger and deeper sedimentary basins, increasing the location errors. Moreover, the faults sampled in northern California were restricted to well-instrumented regions of the San Andreas system near the center of the NCSN. In southern California, a number of the large faults are peripheral to the SCSN, and they invariably show bigger intercatalog variations (Figure 2). For instance, σPSx for the coastal Newport-Inglewood fault (segments 8–9) is 2.1 km, and it reaches 2.7 km for the Cerro Prieto fault (segment 13), which is located in Mexico outside the SCSN. In contrast, the values for the more centrally located San Jacinto fault (segments 17–24) are less than 1 km.

[22] In southern California, the large-fault class has a higher intercatalog standard deviation than either the small-fault or aftershock classes (e.g., σPSx = 1.04, 0.75, 0.55 km, respectively). The fault-normal biases show a similar ordering (e.g., bPSx = 0.54 km, 0.43 km, 0.23 km). The network geometry again plays a role, because the latter two classes comprise segments that tend to be more centrally located within the SCSN. In addition, the estimator given by equation (1) accounts only for a constant translational bias; for long segments, other parameters, such as a rotational bias, may be needed to represent the catalog differences, especially for faults on the periphery of the network. The inadequacy of the bias model acts to increase the apparent intercatalog variance.

[23] The standard deviations between the two relocated southern California catalogs (HP) are consistently lower than those involving the standard catalog (HS and PS), satisfying the expectation that relocation reduces the hypocentral variance (Table 4). However, histograms of the fault-normal differences for all three catalog combinations show heavy-tailed distributions with outliers that dominate the variance estimates. Most of these outliers can be explained by the way the different location algorithms respond to anomalous travel times (e.g., picking blunders, large path anomalies). To account for outliers, we applied a method of uniform reduction [Jeffreys, 1932; Buland, 1986] in which we modeled the differences as the superposition of a Gaussian distribution and a nearly uniform distribution. The standard deviations of the best fit Gaussians are listed in Table 4. The largest reductions, more than 80% in variance, are obtained for the HP intercatalog differences. The reduced standard deviations were used to characterize the event mislocations in our subsequent analysis of the errors in the fault-normal scaling parameters.

[24] If the event location errors from the three southern California catalogs are assumed to be statistically independent (possibly a poor assumption) then we can determine catalog-specific biases, bAkx, and standard deviations, σAkx, by solving the three equations for intercatalog bias:

equation image

and the three for intercatalog variance:

equation image

where AB = {PS, HS, HP}. Equations (3) are not linearly independent, and we therefore included the additional constraint that the biases of the individual catalogs should sum to zero, which minimizes the overall bias. Similar sets of equations can be solved for the fault-parallel and depth directions. We averaged the event-weighted absolute values of the fault segment biases to obtain the values in Table 5 (see the auxiliary material for individual fault segment data).

Table 5. Catalog-Specific Error Estimatesa
Catalog (A)Bias (km)b,cStandard Deviationb (km)Catalog Error (km)
bxAbyAbzAσxAσxAdσyAσzAHorizontalVertical
  • a

    See the auxiliary material for individual fault values. NR, not reported.

  • b

    Computed assuming statistical independence.

  • c

    Event-weighted mean absolute values.

  • d

    The σxA estimated by uniform reduction.

  • e

    Catalog averages as reported by Ellsworth et al. [2000].

SoCal Large
S (standard)0.370.290.760.980.711.072.712.354.49
P (relocated)0.280.130.690.450.090.451.550.040.21
H (relocated)0.400.320.720.580.230.621.500.150.25
 
SoCal Small
S (standard)0.280.120.880.720.580.732.281.963.52
P (relocated)0.180.190.300.190.070.240.780.030.13
H (relocated)0.220.180.820.350.140.381.300.100.17
 
SoCal Aftershock-Dominated
S (standard)0.130.161.050.500.380.591.831.763.26
P (relocated)0.130.120.160.210.080.220.710.040.25
H (relocated)0.100.061.010.360.170.381.300.190.33
 
NoCal Large
N (standard)0.300.62
U (relocated)e0.100.50
 
NoCal Parkfield
N (standard)0.560.95
T (relocated)NRNR

[25] The σ values for the relocated catalogs are substantially smaller than those for the standard SCSN catalog, as expected from the intercatalog comparisons, and the σ values for the Shearer et al. [2005] catalog are in all cases smaller than those for the Hauksson and Shearer [2005] catalog. In particular, the values of σPx obtained from the reduced intercatalog standard deviations are only about half the size of σHx, which is consistent with the qualitative observation that the cluster analysis relocation method used to develop the P catalog provides significantly better localization of hypocenters into fault-like structures [Shearer et al., 2005]. For this preferred southern California catalog, the fault-normal standard deviations are less than 0.1 km for all three fault classes. As noted above, our linear removal of bias did not consider possible intercatalog rotations, which could skew the intercatalog variance to higher values, whereas possible correlations in the hypocenter errors between different catalogs would skew them to lower values. On the balance, σPx ≈ 0.1 km appears to be a good estimate.

[26] In Table 5, we compare the results of the intercatalog error analysis with formal location errors listed in the individual catalogs. The standard deviations in depth from the latter sources are always larger than the corresponding mean horizontal standard deviation, in rough agreement with the intercatalog analysis, but the magnitudes are rather different. The mislocation errors included in the standard network catalogs are substantially larger than our computed values. The reverse is generally true for the relocated catalogs, though the agreement is much better. The confidence region of fault-normal hypocentral error for the relocated catalogs is 0.05–0.2 km with the high end of the range represented by a few fault segments at the periphery of the southern California network. For the most part, location error is <0.1 km, consistent with the intercatalog analysis. Further checks on the mislocation errors from seismicity modeling, described below, support the intercatalog analysis.

4. Fault-Normal Seismicity Distributions

[27] Because strike-slip faults in California are nearly vertical, we developed our scaling relations using the fault-normal distance ∣x∣ as the independent variable, ignoring bilateral asymmetry in seismicity. We stacked the seismicity data in each fault group and computed earthquake density as a function of distance ∣x∣ using a nearest-neighbor method [Silverman, 1986] in which the bins are adjusted to contain q neighboring events. For each data set we experimented with a range of q values and selected one that yielded an adequate point density for deriving fault-normal scaling relations (10 ≤ q ≤ 50).

[28] Logarithmic plots of earthquake density versus ∣x∣ for each regional catalog indicate fault-normal distributions that have flat peaks within a few hundred meters of the fault, roll-off as an inverse power law for about an order of magnitude in distance, and merge with irregular backgrounds at distances less than 10 km (Figures 69). Near the fault, the observed distributions can be described by the functional form:

equation image

In expression (5), d is an inner scale that removes the power law singularity on the fault, γ is the asymptotic roll-off of seismicity away from the fault, and the exponent m controls the shape of the distribution for xd, i.e., the sharpness of the corner on a logarithmic plot. By varying the latter parameter, we found that the maximum likelihood fits to the observed fault-normal distributions were obtained for m ≈ 2. We also experimented with exponential and Gaussian distributions but found they were a poor fit to the data.

Figure 6.

Fault-normal earthquake density distributions for large faults in southern California. Distributions for (a) relocated catalog P, (b) relocated catalog H, and (c) standard catalog S using a nearest-neighbor bin interval of q = 50 events. The heavy dashed line marks the limit to which we fit data, xmax; beyond this limit, background seismicity dominates. The black line is a maximum likelihood fit of an inverse power law, with asymptotic slope equation image, to observations within that limit. The inner scale of the distribution is described by equation image.

Figure 7.

Fault-normal earthquake density distributions for (a) small faults and (b) aftershock-dominated fault segments in southern California. Both Figures 7a and 7b use events from relocated catalog P with a bin interval of q = 50 events. Features are the same as in Figure 6.

Figure 8.

Fault-normal earthquake density distributions for large faults in northern California. Distributions for (a) relocated catalog U and (b) standard catalog N using a nearest-neighbor bin interval of q = 20 events. The heavy dashed line marks the limit to which we fit data, xmax; beyond this limit, background seismicity dominates. As in Figures 6 and 7, the black line is a maximum likelihood fit of an inverse power law to observations within that limit.

Figure 9.

Fault-normal earthquake density distributions for the Parkfield fault segments. Distributions for (a) relocated catalog T and (b) standard catalog N using a nearest-neighbor bin interval of q = 10 events. Features are the same as in Figure 8.

[29] Assuming m = 2, we obtained a maximum likelihood fit of equation (5) to the binned data for each fault group out to a maximum fault-normal distance xmax, chosen such that the relative contributions from background seismicity were small; the values of xmax for each fault class are listed in Table 3. The expected number of events in the jth bin of width Δx is the integral:

equation image

We assume the observed value, nj, in each bin is Poisson distributed, which yields the log likelihood function [e.g., Boettcher and Jordan, 2004]:

equation image

Maximizing (7) using a linear approximation to ν(x) over each binning interval (adequate for the small intervals used here) yields the estimates equation image0, equation image, and equation image. These estimates have correlated errors. However, we note that the maximum likelihood estimator for equation image0 is Nmax/∫0xmax(1 + x2/d2)γ/2dx, where Nmax is the cumulative number of events out to xmax. If Nmax is large, its relative error is small (∼Nmax−1/2, the standard population error) and uncorrelated with the errors in equation image and equation image. The latter are positively correlated, as shown in Figure 10, which plots the maximum likelihood estimates and confidence intervals for the various fault groups in the equation imageequation image plane.

Figure 10.

Maximum likelihood solutions and errors for equation image and equation image. Values for (a) southern California large faults, (b) northern California large faults, and (c) Parkfield showing the positive correlation between scaling parameters. Light gray and black ovals mark the 68% and 95% confidence bounds, respectively.

[30] We checked the error estimates from the maximum likelihood procedure with those derived from jackknife resampling [Efron, 1979]. Generally speaking, the two were in agreement, but where they differed, we used the larger estimate. We experimented with the lower magnitude cutoff and depth ranges and found the results to be robust. We also tested a range of bin widths, q, and found little variation in our results.

[31] An important issue is the weighting of individual fault segments in the seismicity stacking. Owing to the variability in seismicity rates, the number of earthquakes per fault segment ranges from a hundred to several thousand (Table 3), and our results will depend on how each is weighted. In our stacking procedure, we applied a positive weighting factor wk to each event in the kth fault segment catalog, which we computed by

equation image

where N0 is a “down-weighted level” that was held constant for each fault group. We varied N0 from Nmin, the minimum of all catalog sizes Nk in each fault group, to Nmax, the maximum in each group. The latter bound corresponds to “one-event-one-vote” (wk = 1), whereas the former corresponds to “one-catalog-one-vote” (wk ∼ 1/Nk). For intermediate values, events from catalogs larger than N0 were down weighted by the ratio N0/Nk, while those from smaller catalogs received unit weight. We experimented with a range of down weight levels for each structural group and found that the maximum likelihood estimates for most of the parameters were stable across a wide range between Nmin and Nmax (Figure 11). Table 3 lists the actual values used in deriving the parameter values discussed below.

Figure 11.

Down-weighted analysis results for small faults in southern California: (a) equation image and (b) equation image vary with down weighted value across the different catalogs. The dashed gray line marks our selected value of N0 = 700. Note that parameter estimates are largely stable within error for most down-weighted values. Only at low values of N0 do parameters start to vary as more box catalogs, including those with few events, are weighted equally.

[32] For all structural groups, there is a well-defined scaling region of at least an order of magnitude in fault-normal distance where the earthquake density shows a power law roll-off before it merges with the background seismicity (Figures 69). In Table 6, we list by fault group the maximum likelihood estimates of equation image and equation image. To assess the effects of earthquake clustering, we declustered the fault segment catalogs using the algorithm of Reasenberg [1985] with default parameters (rfact = 10, xk = 0.5, xmeff = 1.5, τ0 = 2 days, τmax = 10 days, p1 = 0.99) and obtained the distribution parameters for the declustered catalogs and the event clusters.

Table 6. Apparent Fault-Normal Scaling Parameters
CatalogWhole CatalogDeclusteredClusters
equation image (km)equation imageequation image (km)equation imageequation image (km)equation image
SoCal Large
S (standard)0.88 ± 0.091.19 ± 0.040.86 ± 0.091.12 ± 0.040.95 ± 0.191.51 ± 0.12
P (relocated)0.23 ± 0.030.98 ± 0.040.24 ± 0.030.93 ± 0.040.24 ± 0.041.22 ± 0.05
H (relocated)0.27 ± 0.030.98 ± 0.030.28 ± 0.050.94 ± 0.030.27 ± 0.051.18 ± 0.05
 
SoCal Small
S (standard)0.79 ± 0.111.90 ± 0.300.86 ± 0.181.86 ± 0.440.73 ± 0.172.17 ± 0.54
P (relocated)0.19 ± 0.021.37 ± 0.080.19 ± 0.031.27 ± 0.080.22 ± 0.031.62 ± 0.17
H (relocated)0.23 ± 0.021.44 ± 0.090.22 ± 0.031.35 ± 0.100.27 ± 0.041.71 ± 0.19
 
SoCal Aftershock-Dominated
S (standard)0.55 ± 0.041.63 ± 0.11
P (relocated)0.33 ± 0.021.50 ± 0.08
H (relocated)0.31 ± 0.021.44 ± 0.07
 
NoCal Large
N (standard)0.16 ± 0.021.68 ± 0.060.18 ± 0.031.73 ± 0.100.15 ± 0.021.66 ± 0.07
U (relocated)0.08 ± 0.011.60 ± 0.040.10 ± 0.011.62 ± 0.070.07 ± 0.011.61 ± 0.05
 
NoCal Parkfield
N (standard)0.89 ± 0.093.86 ± 0.600.93 ± 0.124.64 ± 0.770.85 ± 0.195.57 ± 1.60
T (relocated)0.13 ± 0.022.52 ± 0.220.14 ± 0.022.50 ± 0.240.10 ± 0.032.70 ± 0.78

[33] Table 6 shows interesting variations across the fault groups and catalog types. Comparing the relocated catalogs P and U, seismicity decays away from the large faults in southern California at a significantly lower rate (equation image = 0.98 ± 0.04) than it decays in northern California (1.60 ± 0.04) or for the small faults in southern California (1.37 ± 0.08). In southern California, clustered events decay more rapidly than independent events, in agreement with the higher decay rate for aftershocks of large southern California earthquakes (equation image = 1.50 ± 0.08).

[34] For the larger faults, the apparent inner scale equation image for relocated catalogs is smaller in northern California (0.08 ± 0.01 km) than in southern California (0.23 ± 0.03 km). The former is comparable to the relocation uncertainty. Significantly higher values are obtained for the standard catalogs (0.6–0.8 km), consistent with more dispersion due to mislocation.

5. Analysis of Bias

[35] The results in Table 6 are biased by two factors: contributions from background seismicity and hypocentral error. The former is apparent when we relax the assumption of bilateral symmetry. We first identified which side of each fault segment had more events for ∣x∣ ≤ xmax and then restacked the data for each of the five fault classes, preserving this asymmetry in seismic abundance (Figure 12 and the auxiliary material). The estimates of equation image obtained for the less abundant side were consistently higher than those on the more abundant side. The values of equation image on the less abundant side of each fault class also increased slightly over those of the symmeterized distributions owing to the positive correlation between equation image and equation image. We did find that, on the side with fewer events, the scaling region extended to greater distances from the fault, in some cases by an order of magnitude, supporting our choice of power law model. Figure 12a shows that exponential and Gaussian distributions are a poor fit to the data. Fault maps show that the truncation of the scaling region on the abundant side can generally be explained by the seismicity increase from another fault branch or splay, proximate to the target fault segment, a common feature of the San Andreas system (e.g., Figure 5).

Figure 12.

Comparison of fault-normal earthquake density for (a) low- and (b) high-productivity sides of a fault for catalog P about large, southern California faults using a nearest-neighbor bin interval of q = 20 events. Features are the same as in Figures 69. Seismicity decays more rapidly on the low productivity side of a fault and spans almost 2 orders of magnitude. In Figure 12a, maximum likelihood fits of exponential (dashed) and Gaussian (dotted) distributions are shown for comparison. In Figure 12b, we additionally fit the data using the parameters from the low-productivity side (Figure 12a) and include a uniform background rate, νbg. See the auxiliary material for distributions of other fault classes.

[36] Uniform background seismicity associated with proximate faults significantly alters the shape of a density distribution close to xmax. We show this by modeling the distribution in each fault class as the superposition of a fault-normal decay of earthquake rate and a uniform background rate (Figures 12b and 13). Parkfield was excluded from the modeling because the background signal is weak and the scaling parameters therefore unbiased. We use the parameters of the low-abundance side of the fault in each class because the contribution from background is significantly lower and the scaling only minimally biased. By varying the ratio of background to decaying events, we recovered the results reported in Table 6, as well as the distributions on the event-heavy side of the fault for each fault class, suggesting that the fault-normal decay of seismicity is similar across strike-slip faults. Values for background-corrected scaling parameters are reported in Table 7.

Figure 13.

Comparison of fault-normal earthquake density for (a) large faults in southern California (catalog P) with (b) that of a synthetic distribution using bin intervals of q = 50 events. Features are the same as in Figures 69. The synthetic distribution is one realization of a Monte Carlo simulation in which fault-normal earthquake density is modeled as the superposition of an unbiased distribution (γ = 1.3, d = 0.3 km; compare Figure 12a) that decays beyond xmax (dashed gray line) and a uniform background. The maximum likelihood fit to the synthetic distribution out to xmax recovers the scaling parameters determined in the initial analysis of symmeterized distributions.

Table 7. Bias-Corrected Scaling Parameters
Fault ClassCatalogBackground CorrectionHypocentral Error Correction
Theoreticalad (km)Simulateda
d (km)γd (km)γ
  • a

    Bias corrections computed assuming σxA = 0.1 km.

SoCal largeP0.26 ± 0.041.23 ± 0.050.230.22 ± 0.041.16 ± 0.05
SoCal smallP0.21 ± 0.031.44 ± 0.070.190.18 ± 0.031.37 ± 0.10
SoCal aftershock-dominatedP0.39 ± 0.041.70 ± 0.060.380.37 ± 0.031.62 ± 0.10
NoCal largeU0.09 ± 0.021.82 ± 0.050.060.06 ± 0.021.65 ± 0.05
NoCal ParkfieldT0.13 ± 0.022.52 ± 0.220.110.12 ± 0.032.30 ± 0.25

[37] We investigated how the results are biased by hypocenter errors in two ways, theoretically and using Monte Carlo simulations. We assumed that the true fault-normal seismicity is governed by the distribution ν(x) in equation (5), and that the catalogs have independent, identically distributed mislocation errors approximated by a zero-mean Gaussian probability density function (pdf),

equation image

where σAx is the standard error for catalog A. The pdf for the observed seismicity can then be computed as the convolution of the two (normalized) distributions:

equation image

A little analysis shows that, if σAx/d is not too large (less than 5 or so), pA(x) can be approximated by equation (5) with an asymptotic slope equation image = γ and an inner scale equation image computed as the intersection of the small-x probability density with the large-x asymptote,

equation image

The bias correction dequation image derived by solving (11) is only weakly dependent on the shape parameter m, so we fixed it at its best fit value (m = 2). We experimented with σAx values spanning the confidence region of 0.05–0.2 km recorded in the relocated catalogs. Because low values have only a minimal effect and the large values are only applicable to a few fault segments in southern California, the conservative estimate of σAx = 0.1 km determined from our intercatalog analysis is appropriate for A = P, U, and T. Figure 14 plots equation image versus d for σAx = 0.1 km and we see that the correction is small for d > σAx and decreases with γ.

Figure 14.

Theoretical relationship between an observed inner scale, equation image, and the true value of d plotted for various γ. Theory assumes that an observed fault seismicity scaling distribution is the product of the true distribution convolved with a Gaussian noise function with standard deviation of 0.1 km. Only at d < 0.2 km do the observed and true values diverge significantly.

[38] The bias-corrected estimates of d obtained from (11) are listed in Table 7. The largest correction, for large-fault seismicity in northern California, changes the estimated inner scale from 0.09 km to 0.06 km, a difference of only 30 m. Note that the magnitude of bias in this worst case, 0.03 km, is smaller than the quadratic estimator equation imaged = 0.05 km. We verified the small size of the bias correction using a Monte Carlo method in which we generated synthetic catalogs that satisfied (12), perturbed them with Gaussian noise (0.05 ≤ σAx ≤ 0.2 km), and calculated a likelihood score of their fit to the distribution curves obtained from the actual data. The values of d and γ that maximized the likelihood for many (∼50) catalog realizations for σAx = 0.1 km are listed in Table 7; an example of a single realization is presented in Figure 15. The estimates of d are nearly identical to the theoretically corrected values. Moreover, the simulations provided bias corrections for γ, which are not zero (as the asymptotic theory predicts) owing to the positive correlation between the estimators of d and γ arising from a finite range of x (see Figure 10). However, the corrections to γ are also small, 10% (for catalog T) or less.

Figure 15.

Sample result from simulation analysis of parameter bias. In each simulation, synthetic distributions were perturbed with Gaussian noise (σAx = 0.1 km) until a combination of γ and d was found that maximized the likelihood score of the fit to the original distribution. The simulation result pictured is for large faults of southern California and illustrates a good correlation between a perturbed synthetic distribution (dots) and our observed distribution (dashed line) for catalog P.

6. Rough Fault Loading Model

[39] Our multicatalog analysis of earthquake hypocenters in California reveals that the seismicity in the vicinity of strike-slip faults can be represented by a three-parameter distribution:

equation image

The constant ν0 describes the fault-normal seismic intensity (in events/km) on the fault surface. Using the data in Table 3, the intensity ν(x) can be normalized by the total fault length Σ Lk, as plotted in Figures 69, or by the total fault area Σ LkWk, which yields a spatial seismic density for the catalog interval T = 19 years. The inner scale d measures the half width of a near-fault region where the seismic intensity is flat (∼ν0), and the exponent γ specifies the power law roll-off of seismic intensity in the scaling region d < xxmax.

[40] Figure 16 presents a conceptual “rough faulting loading” (RFL) model that we will use to explain the seismicity behavior. Our starting point is the observation that fault surfaces can be described by a self-affine (fractal) complexity over a large range of spatial scales [Power and Tullis, 1995; Lee and Bruhn, 1996; Renard et al., 2006] and evolve in time toward surfaces that are less complex in the direction of slip [Wesnousky, 1988; Stirling et al., 1996; Sagy et al., 2007; Finzi et al., 2009]. Here “complexity” refers to the fractal branching of faults into multiple surfaces [e.g., King, 1983; Hirata, 1989] as well as the fractal roughness of individual fault surfaces [e.g., Lee and Bruhn, 1996; Renard et al., 2006; Sagy et al., 2007]. To build a simple model, we begin by considering a single fault surface whose deviations from the planar approximation x = 0 define a fault-normal topography [Saucier et al., 1992; Chester and Chester, 2000; Dieterich and Smith, 2009]. We represent an along-strike (constant z) profile of this topography as the realization of a stationary stochastic process X(y) that has zero expectation, 〈X(y)〉 = 0, and a variogram

equation image

the surface is self-affine, then ξ ∼ ΔyH, where 0 ≤ H ≤ 1 is the Hausdorff measure (sometimes referred to as the Hurst exponent) of the along-strike profile [Feder, 1988; Turcotte, 1997]. Assume the self-affine scaling of fault roughness breaks down above some outer scale Δyouter related to the characteristic segmentation length 〈Lk〉. At profile separations larger than Δyouter, the variogram (13) levels off to 2ξ, where ξ ≡ 〈X2(y)〉1/2 is the root-mean-square (RMS) topographic fluctuation. The power spectrum PX(ky) is then the Fourier transform of the autocovariance function CXy) = ξ2ξ2y). PX(ky) plateaus at a value ∼ξ2 below the characteristic wave number 1/Δyouter and rolls off as ky−2H−1 above this characteristic wave number (Figure 17).

Figure 16.

Schematic representation of the RFL (rough fault loading) model that explains observations of near-fault seismicity distribution. (a) Tectonic loading of a self-affine fault generates a heterogeneous stress field that yields a power law decay of seismicity over a scaling region via stress relaxation [Dieterich and Smith, 2009, Figure 3]. (b) Toward the fault core, small-scale stress heterogeneities of the rough fault are attenuated by low fracture strength across a damage zone of width 2d km. (c) Illustration of how observed seismicity rates vary with distance from a fault (solid black line). The scaling region likely extends beyond xmax, as indicated by the dashed black line, but is masked by interference from proximal fault branches.

Figure 17.

Schematic power spectrum, PX (ky), for the RFL model as a function of along-strike wave number. The spectrum includes upper and lower cutoffs associated with outer, Δyouter−1, and inner, Δyinner−1, scales, as well as a scaling region with slope −(2H + 1), where H is the Hausdorff measure. The inner and outer scales are beyond the resolution of our analysis, which only captures the scaling region (gray box). We find that regional variation in seismicity rates correlates with different levels of fault roughness, as measured by H (inset).

[41] The widths of the seismicity scaling regions, xmax = 4–10 km, are much smaller than the average segmentation lengths in Table 3, so we do not observe a seismicity cutoff associated with Δyouter. Instead, the scaling regions are limited by the background seismicity from proximate faults; that is, the values of xmax are related to fault branching rather than fault roughness.

[42] In the scaling region Δy < Δyouter, the tectonic loading of a self-affine fault will generate near-fault stress heterogeneity characterized by stress lobes with a power law size distribution, as illustrated in Figure 16. Dieterich and Smith [2009] have used two-dimensional numerical simulations based on Dieterich's [1994] rate- and state-dependent model of seismic nucleation to calculate the fault-normal seismic intensity ν(x) from this type of stress loading. They obtain a power law decay in seismicity that satisfies ν ∼ ∣xD, where D = 2 − H is the fractal dimension of the along-strike profile.

[43] If this 2-D approximation applies to strike-slip faults in California, then γD, and the data from Table 7 yield a low fractal dimension for large-fault roughness in southern California: D is close to 1, consistent with self-similar scaling (H = 1). For the small-fault and aftershock-dominated classes in southern California and the large northern California faults, we find D ≈ 3/2, which corresponds to a process with an exponential correlation function (brown noise). Fault traces and profiles across exposed and laboratory-generated fault surfaces typically range between these fractal dimensions [Power and Tullis, 1995; Lee and Bruhn, 1996; Renard et al., 2006], as do profiles across other types of geologic surfaces [Brown and Scholz, 1985; Goff and Jordan, 1988; Brown, 1995]. The seismicity roll-off rate for Parkfield is significantly higher, γ = 2.3 ± 0.25, consistent with H ≈ 0, D ≈ 2. For this type of fault roughness, the power spectrum decays as 1/ky (pink noise). Pink spectra have been observed on a few normal-fault scarps in the direction of slip [Sagy et al., 2007].

[44] Of course, the validity of the fractal quantification can be questioned owing to the simplicity of the calculations (e.g., the Dieterich-Smith model does not account for aftershock diffusion) and the likely role of 3-D effects, including fault branching [e.g., King, 1983; Hirata, 1989] and roughness anisotropy [e.g., Lee and Bruhn, 1996; Renard et al., 2006]. Nevertheless, our data do suggest that the wave number spectra of large southern California faults are “redder” than those of northern California; this inference agrees with observations that many of the large faults in southern California are macroscopically complex [Okubo and Aki, 1987; Wesnousky, 1988, 1990]. By the same token, if we assume the roughness amplitude at short scales (say, Δy ∼ centimeters) is similar for all strike-slip faults in California, as depicted in Figure 17, then Parkfield should have the lowest along-strike roughness amplitude at long scales among the fault classes in Table 7. The Parkfield segments of the San Andreas fault are indeed quite straight. Some evolutionary aspects of the RFL model related to fault complexity are discussed in section 7.

[45] Equally interesting is the interpretation of d. For the fault classes in Table 7, this inner scale of the seismicity distribution varies from 50 m to around 300 m. We could introduce an inner scale of fault roughness, Δyinner, to the RFL model, as depicted in Figure 18, below which ξy) → 0. Such a cutoff would cause the stress-loading amplitudes to level off in a near-fault region of width d ∼ Δyinner. However, the available data suggest that the self-affine scaling of fault roughness continues to much smaller dimensions than d, perhaps even to microscopic scales [Power and Tullis, 1995; Sagy et al., 2007].

Figure 18.

Relationship between fault core multiplicity and damage zone width. Fault cores (heavy black lines) are shown embedded in a damage zone that is surrounded by largely undamaged host rock. Damage intensity is indicated [after Chester et al., 2004]. (a) The damage zone width about a single fault core is narrow, comparable to northern California faults. (b) The width of fault damage zones with a multiplicity of two (paired) is consistent with our observations of the San Andreas fault at Parkfield and field studies of the exhumed Punchbowl fault [Chester et al., 2004]. (c) Fault damage zones with a fault multiplicity greater than 2 are comparable to southern California faults and are manifested as multiple braided or anastamosing fault cores (modified from Faulkner et al. [2003] with permission from Elsevier).

[46] Rather than relating d to a cutoff in surface roughness, we identify it as the half width of a volumetric “damage zone,” where small-scale stress heterogeneity is attenuated by low rock strength (Figure 16). Damage zones with dimensions of tens to hundreds of meters are widely recognized features of exhumed strike-slip faults [Chester and Logan, 1986; Chester et al., 1993; Ben-Zion and Sammis, 2003; Chester et al., 2005; Rockwell and Ben-Zion, 2007], and they have been used to explain vertical low-velocity zones of comparable dimensions inferred from fault zone guided waves [Li et al., 1990]; these low-velocity zones extend at least to several kilometers [Ben-Zion et al., 2003; Peng et al., 2003; Lewis et al., 2005] and perhaps deeper [Li et al., 2004; Wu et al., 2008]. Drill samples across the Nojima fault, which ruptured in the 1995 Kobe earthquake, indicate that shear strength is significantly reduced and the permeability increased within a damage zone surrounding the fault core [Lockner et al., 1999]. A reduction in fracture strength by increased fluid pressures [Unsworth et al., 1997] and the formation of talc and other low-strength minerals within the damage zone [Morrow et al., 2000; Moore and Rymer, 2007] is our preferred explanation of the near-fault stress homogenization implied by the inner scale d.

[47] The best data on damage zone dimensions at seismogenic depths in California come from recent borehole measurements near Parkfield by the San Andreas Fault Observatory at Depth (SAFOD) project. In 2007, SAFOD drilling encountered two principal slip surfaces at measured depths of 3194 m and 3301 m, which were embedded in a zone of variably damaged rock approximately 250 m in fault-normal width [Chester et al., 2007; Zoback et al., 2008]. This value, obtained near the top of the seismogenic zone, is consistent with local studies of fault zone guided waves, which sample somewhat deeper [Korneev et al., 2003; Li et al., 2004], and it agrees with our Parkfield value of 2d = 240 ± 60 m, which averages over the entire seismogenic zone.

[48] For large faults in northern California, the damage zone width inferred from Table 7 is 120 ± 40 m, in line with geologic estimates from large exhumed strike-slip faults [Chester et al., 2004; Frost et al., 2009], faults exposed in mines [Wallace and Morris, 1986], and studies of fault zone trapped waves elsewhere [Ben-Zion et al., 2003; Lewis et al., 2005]. The narrow damage zone is consistent with a simple fault geometry comprising a single fault core bounded on one or both sides by a variably fractured material (Figure 18a), similar to exposures of the San Gabriel, San Andreas, and San Jacinto faults [Chester et al., 2004; Dor et al., 2006]. Such a single-core fault is said to have a “fault core multiplicity of one.” In this terminology, the double-core Punchbowl fault [Chester et al., 2004] and the San Andreas fault at SAFOD have a fault core multiplicity of two, which approximately doubles the damage zone width (Figure 18b).

[49] The average damage zone width obtained for the small-fault class in southern California is significantly larger than at Parkfield: 2d = 360 ± 60 m, suggesting a fault core multiplicity greater than two (i.e., wider, more complex fault zones comprising multiply braided fault cores (Figure 18c). We are not aware of observations from southern California that independently confirm this hypothesis, but some strike-slip faults exposed elsewhere have anastomosing, multicore damage zones at least several hundred meters in fault-normal width [Wallace and Morris, 1986; Faulkner et al., 2003].

[50] The same line of reasoning would attribute an even higher multiplicity to the large southern California faults, for which 2d = 440 ± 80 m. In the case of long faults, however, we must account for an increase in the apparent damage zone width caused by along-strike variability of the fault core surfaces. In a self-similar model (H = 1), the along-strike RMS topography of the large faults, ξyouter) ≈ ξ, should be about 5 times that of the small faults (which are about one fifth as long; see Table 3). Assuming the fault-normal topography is approximately Gaussian, we can estimate its contribution to d by replacing σAx with ξy) in equation (11). A calculation shows that the entire difference between the large and small faults can be explained if ξ ≈ 180 m. Large faults in southern California show at least this much variability (e.g., Figure 4). We conclude that a fault core multiplicity of order 3 can explain the observed d values for both the large-fault and small-fault classes in southern California.

[51] For aftershock-dominated faults, 2d = 740 ± 60 m, the highest of all fault classes. Because these fault segments are quite short (∼8 km), the RMS topography of the fault surfaces should not contribute significantly to the apparent damage zone width. Mislocation errors might be somewhat higher for the aftershock-dominated faults (e.g., owing to saturation of network-processing capabilities during times of high seismicity), but we can discount enhanced mislocation bias as an explanation of higher apparent width, because the subcatalogs of clustered seismicity extracted from the other fault classes do not show an increase in equation image relative to unclustered seismicity (Table 6). More likely, the immature faults of the Eastern California Shear Zone that were activated by the Joshua Tree–Landers–Hector Mine sequence are just more complex than typical southern California faults, as suggested by the broad (∼2 km) compliant zones of induced, and in two cases retrograde, deformation observed following the Hector Mine main shock [Fialko et al., 2002].

[52] A speculative possibility is that the effective width of the damage zone increases in response to strong shaking during large earthquakes and subsequently decreases by logarithmic healing. This mechanism is consistent with the studies of fault zone guided waves following the Landers and Hector mine earthquakes, which indicate significant healing on a decadal time scale [Li et al., 1998, 2003]. We note, however, that the fault zone waveguides do not appear to be anomalously wide in the Landers region [Li et al., 2000].

7. Evolutionary Aspects of Near-Fault Seismicity

[53] The short-term response of fault zones to shaking during large earthquakes and the well-documented long-term evolution of strike-slip faults in California [Wesnousky, 1988; Stirling et al., 1996; Sagy et al., 2007] and elsewhere [Frost et al., 2009] indicate the RFL model may have implications for fault evolution. To augment the geographic variation observed in our results for the aggregated fault classes in Table 7, we provide estimates of the seismicity parameters for 10 individual faults with adequate earthquake rates, taken from the large-fault classes for northern and southern California (Table 8). In three cases, the seismicity catalogs for individual faults correspond to fault segments listed in Table 3 and mapped in Figures 1 and 2: segments 11 (San Jacinto), 14 (Imperial), and 15 (Coachella segment of the San Andreas). In the other seven, we aggregated two or three segments. For instance, the Hayward fault spans two segments (42 and 43) with similar geologic histories and slip rates, and the creeping section of the San Andreas spans three (48–50).

Table 8. Data on Individual Faults
FaultIDSegmentNT (events)equation image (km)equation imageL (km)Wa (km)ν0/LW (events/km3)Cumulative Offset (km)Cumulative Offset ReferencesAseismicity FactorAseismicity Factor References
  • a

    Length-weighted average.

GarlockGA1, 2, 316650.291.07128.712.41.012–64Powell [1993]0.0
          Stirling et al. [1996]  
Newport-InglewoodNI8, 94080.200.84113.718.30.165 ± 5Stirling et al. [1996]0.0
ElsinoreEL10, 129690.530.9377.715.20.4612 ± 3Stirling et al. [1996]0.0
San JacintoSJ1150270.250.9637.719.46.728Powell [1993]0.0
ImperialIM1411790.322.5321.27.020.0? to 85Powell [1993]0.2Genrich et al. [1997]
            Shearer [2002]
San Andreas (Coachella)SAco152560.772.9540.710.30.62160–185Powell [1993]0.2Lyons and Sandwell [2003]
HaywardHA42, 4310100.211.3790.312.81.8100 ± 5Graymer et al. [2002]0.61 ± 0.19WGCEP and NSHMP [2007]
            Lienkaemper et al. [2001]
CalaverasCA44, 45, 4646660.061.7339.28.9130.0160 ± 5Graymer et al. [2002]0.77 ± 0.24WGCEP and NSHMP [2007]
            Galehouse and Lienkaemper [2003]
             
San Andreas (Creeping)SAcr48, 49, 50101220.091.6348.910.1120.0315 ± 10Matti and Morton [1993]0.62 ± 0.18WGCEP and NSHMP [2007]
San Andreas (Parkfield)SApa51, 5238970.142.5264.211.235.0315 ± 10Matti and Morton [1993]0.77 ± 0.09WGCEP and NSHMP [2007]

[54] The same maximum likelihood procedure was employed in fitting equation (5) to the individual fault catalogs. Because these catalogs comprise fewer events, the estimation uncertainties are larger than in Table 7, and we did not apply any background or mislocation bias corrections, which are small enough to be ignored in the following comparisons. The seismicity parameters exhibit interesting internal correlations. In particular, the seismicity is more localized (γ is larger and d is smaller) on faults with higher seismic productivity ν0/LW (Figures 19a and 19b), and it is less localized where the seismogenic zone, as measured by W, is thicker (Figures 19c and 19d).

Figure 19.

Correlations between scaling parameters (a and c) equation image and (b and d) equation image and on-fault earthquake density ν0/LW and fault width W for related subsets of faults. Fault names (see Table 8) are indicated as follows: CA, Calaveras; EL, Elsinore; GA, Garlock; HA, Hayward; NI, Newport-Inglewood; PA, Parkfield; SAcr, San Andreas–Creeping; SAco, San Andreas–Coachella Valley; IM, Imperial; and SJ, San Jacinto. Note that smaller and more productive faults exhibit greater localization of seismicity (small equation image, large equation image). Solid lines are least square fits to the data with correlation coefficients R. Dashed lines are least square fits with outliers removed; in Figure 19c, SAco and SApa were excluded; in Figure 19d, SAco and IM were excluded.

[55] Table 8 also lists cumulative offsets on the individual faults and their aseismicity factors. We aggregated cumulative offset values for each fault from the literature, reporting uncertainties and range estimates where available. Following Wisely et al. [2007], we computed the aseismicity factor (AF) as the ratio of the aseismic slip rate to the long-term slip rate over the full thickness of the seismogenic zone; that is, AF = 0 corresponds to a locked fault, and AF = 1 to stable sliding. The AFs in Table 8, which are weighted by Nk, were primarily derived from the slip data compiled by Wisely et al. [2007] for the 2007 Working Group on California Earthquake Probabilities [Working Group on California Earthquake Probabilities and the USGS National Seismic Hazard Mapping Program (WGCEP and NSHMP), 2007], supplemented with a few additional studies. We did not include faults that exhibit steady or transient surface creep thought to be a dynamic (short-term) or static (long-term) response to some large, regional event (e.g., Superstition Hills/Elmore Ranch [McGill et al., 1989]; Landers [Bilham and Behr, 1992; Bodin et al., 1994]; Loma Prieta [Lienkaemper et al., 1997; Rymer, 2000]; Hector Mine [Rymer et al., 2002]). However, we did assign nonzero AFs to the Coachella Valley [Lyons et al., 2002] and Imperial Valley segments of the San Andreas fault, because they have exhibited transient creep over the full width of the seismogenic crust. In the case of the Imperial Valley fault, we infer aseismic slip over the seismogenic crust by reevaluating Bilham and Behr's [1992] fault model in light of new estimates of seismogenic thickness [Shearer, 2002].

[56] Although the scatter is high, we see that the seismicity tends to localize with increasing cumulative offset (Figures 20a and 20b), conforming to the notion that faults evolve toward more linear, focused structures. Wesnousky [1988] observed greater localization of surface traces with increasing cumulative offset on fault length scales of hundreds of kilometers. Here we observe a similar localization in the seismicity at fault length scales of tens of kilometers. Seismic localization also correlates with AF (Figures 20c and 20d).

Figure 20.

Correlations between scaling parameters (a and c) equation image and (b and d) equation image and cumulative offset and aseismicity factor (a measure of aseismic slip on a fault) for related subsets of faults. Fault names (see Table 8) are the same as in Figure 19. High aseismicity factors tend to be associated with more localized faults. Localization also increases with cumulative offset reflecting fault evolution toward more linear, focused structures. Solid lines are least square fits to the data with correlation coefficients R. Dashed lines are least square fits with outliers removed; in Figures 20b and 20d, SAco was excluded; in Figure 20c, both SAco and IM were excluded.

[57] Our data and the RFL model are consistent with a simple narrative for fault evolution. Faults initiate as multiple overlapping strands and coalesce into throughgoing structures over time. Younger, rougher faults are characterized by multiple fault cores (high multiplicity) and correspondingly wide damage zones. As offset progresses, a fault zone localizes and its roughness spectrum is whitened by a steady decrease in its low wave number components, which localizes the near-fault seismicity by increasing the roll-off rate, γ (decreasing H). This localization also reduces the fault core multiplicity, eventually to a single or paired core, thus reducing the damage zone width, 2d. Fault localization and smoothing promotes aseismic slip. Consequently, stressing rates increase markedly at the boundaries between steadily slipping and locked patches of a fault, driving up the seismic productivity per unit fault area, ν0/LW.

[58] In the real world, various combinations of additional factors may ultimately govern the behavior of a fault. Even if the geology of a fault at depth were known, the interrelated factors of pore pressure, frictional strength, and heat flow will produce widely varying conditions for small earthquake nucleation. Such variability is likely responsible for outliers such as the Coachella Valley segment of the San Andreas fault (SAco in Figures 19 and 20). For example, the Coachella Valley segment is comparable to Parkfield in terms of cumulative offset and fault width, but the Parkfield segment is known to contain serpentinite [Irwin and Barnes, 1975; Zoback et al., 2008] and metamorphic fluids [Unsworth et al., 1997; Bedrosian et al., 2004] that likely encourage aseismic slip along a narrow zone. The Coachella Valley segment, on the other hand, may be dry and therefore have a larger d, lower seismicity, and less aseismic slip.

8. Discussion

[59] The fault-normal seismicity distribution for strike-slip faults in California can be described by an evolutionary rough fault loading (RFL) model that is consistent with constraints on fault structure and evolution derived from other observations. The RFL model predicts no asymmetry in γ, yet we observe asymmetry when we partition each fault class in to groups by low and high seismic abundance. Once we correct for background bias though, the asymmetry disappears and our results are consistent with the RFL model. Asymmetry in d is also of interest as it may correlate with across-fault material contrasts [e.g., Weertman, 1980; Cochard and Rice, 2000; Shi and Ben-Zion, 2006] or asymmetric fault damage [e.g., Dor et al., 2006], properties may be important in controlling rupture directivity. Likewise, material contrasts are known to cause asymmetric fault-parallel distributions of aftershocks [Rubin and Gillard, 2000; Rubin and Ampuero, 2007] and may therefore be reflected in fault-normal earthquake rates. Unfortunately, when we initially selected our fault seismicity catalogs, we allowed the seismicity to guide the choice of coordinate system and effectively symmeterized d values from the start.

[60] We are also considering in more detail how the fault-normal distribution of seismicity correlates with its depth distribution. In this paper, we restricted our data to the central part of the seismogenic crust and verified that there was little variability in our results when the truncation depths were varied. The distribution parameters d and γ were found to correlate with fault width W (Figures 19c and 19d), indicating that the fault-normal seismicity distributions depend on the vertical structure of the fault zones, particularly the geothermal gradient. We have also observed that the near-fault seismicity of some segments is highly localized in depth, more often than not toward the base of the seismogenic zone [Boutwell et al., 2008]. We are investigating what implications depth localization of seismicity might have for the RFL model.

[61] Our results have major implications for earthquake forecasting and prediction, because they suggest how earthquake triggering statistics might be modified by the presence of faults. Epidemic-Type Aftershock Sequence (ETAS) models of triggered seismicity [e.g., Ogata, 1988] are good at predicting the short-term earthquake rates on a regional scale in California, which are dominated by small aftershocks, but they are less effective in forecasting the larger, less frequent earthquakes (M > 6) [Helmstetter and Sornette, 2002, 2003; Gerstenberger et al., 2005; Helmstetter et al., 2006]. The spatial kernels in ETAS models are usually prescribed as an isotropic (radial) power law or exponential decay away from an event [Ogata, 1998; Zhuang et al., 2004; Felzer and Brodsky, 2006], so the forecasts incorporate fault structure only through smoothed representations of the background seismicity. According to such models, a main shock at, say, 2 km from a fault is equally likely to trigger an aftershock at 4 km from the fault as on the fault itself. An isotropic kernel is not consistent with the observed sequences of small aftershocks in California, which show a near-fault bias [e.g., Hauksson et al., 1993], and it is likely to be an even poorer model for the triggering of larger magnitude events. We have observed that aftershock sequences from our fault-referenced catalogs can be described by an ETAS spatial kernel modified to include a fault-normal bias and an strike-parallel elongation, both proportional to ν(x) in equation (5) [Powers and Jordan, 2007b]. This anisotropic, heterogeneous spatial kernel and its consequences for earthquake forecasting will be discussed in a future paper.

Acknowledgments

[62] This research was supported by the Southern California Earthquake Center. SCEC is funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agreement 02HQAG0008. This is SCEC contribution 1247. We are grateful to Bill Ellsworth, Egill Hauksson, Peter Shearer, Cliff Thurber, and their respective coauthors for making high-quality relocated earthquake catalogs available. Efforts on the part of the SCEC Unified Structural Representation group are also appreciated, and we thank Jim Dieterich and Deborah Smith for their insight and opinion.