Heterogeneous Slip Distribution Self-Similarity on a Fault Surface

The earthquake slip distribution self-similarity is investigated in this study. We complied finite fault slip models for earthquakes in the Taiwan orogenic belt and global earthquakes to determine the slip distribution self-similarity. Forty-one earthquakes (19 Taiwan earthquakes and 22 global earthquakes) in the Mw = 4.6 8.9 magnitude range were examined. The fault slip exhibited self-similar scaling between the rupture slip and area. The average area ratio (Rs) and slip ratio (Rd) follows a scaling of R 10 ( ) s a n Rd = . Slip self-similarity implies that a fault rupture exhibits fractal behavior. The scaling exponent can be considered as a measure for the roughness degree of the slip distribution on the fault surface. This study suggests that the slip distribution for large earthquakes (Mw > 7.0) tends to have a more homogeneous slip. Scaling exponents can provide insight into earthquake rupture mechanics and the scaling of heterogeneous slips on the fault surface provides a basis for ground motion simulation for a finite fault for an earthquake scenario, particularly for near-fault motion.


IntroDuctIon
Scale invariance occurs in many natural phenomena (Callan 1970;Krug 1997;Turcotte 1997;Sornette 1998;Ostojic et al. 2006;Caldarelli 2007) that show similar statistical characteristics at various scales and is a precise form of selfsimilarity.The most recognized scaling relationship for representing the self-similarity of earthquakes is that between the number of earthquakes and magnitudes reported by Gutenberg and Richter (1944).Mandelbrot (1983) proposed another concept, fractal geometry, and stated that a fractal dimension can describe the scale invariance of natural phenomena.This concept has been widely applied to describe the spatial distribution and time series of earthquakes (Turcotte 1989(Turcotte , 1997;;Ogata and Abe 1991;Hirabayashi et al. 1992;Papadopoulos and Dedousis 1992;Koyama et al. 1995;Wang 1996;Wang and Lee 1997;Wang et al. 2014) and fault activities (cf.Okubo and Aki 1987;Lee and Schwarcz 1995).The self-similarity of earthquake slip distributions is an essential physical characteristic of earthquake kinematics and was discussed in previous studies (Beresnev and Atkinson 2002;Manighetti et al. 2005Manighetti et al. , 2007;;Wesnousky 2008;Klinger 2010).Wesnousky (2008) analyzed earthquake rupture traces and coseismic surface slip and obtained the scaling relationship between the slip and rupture length for strike-slip earthquakes.In addition, Manighetti et al. (2005) analyzed the scaling relationship between maximum displacement and length as well as width in 76 inverted slip models for global events.The authors showed that the overall shape of slip profiles was basically triangular and strongly asymmetric.However, the aforementioned studies focused mainly on the relationship between the amount of slip and earthquake magnitude and did not consider the scaling of the slip area partitioned in a finite fault.
The slip distribution on a finite fault is a major concern regarding the forward simulation of an earthquake scenario.In several studies the scale invariance property in earthquakes has been used for simulating the rupture process.For instance, Wang and Lee (1997) applied a fractal distribution of the breaking strengths in a spring-block model to simulate earthquakes to study the scaling relation between the earthquake frequencies and rupture length.Ide and Aochi (2005) and Aochi andIde (2009, 2011) proposed a model of the wide-scale growth of dynamic rupture during an earthquake.Ide and Aochi (2013) used a multiscale heterogeneous model for simulating the rupture process of the Tohoku-Oki earthquake and applied numerous circular patches in the model for characterizing the heterogeneous frictional properties of the fault plane.In their analysis the area of the slip partitioned in the finite fault was simulated using several sized circular patches following a power law. Lee et al. (2012) modeled fault rupture using a fiber bundle model which associated fibers with the asperities on a fault.The authors determined that a system at a critical transition from the nucleation to rupture phases produces a self-similarity scaling relationship between the cluster fail fibers area and the cumulative number of clusters.For determining the self-similarity properties of the slip area in a finite fault during rupture, Somerville et al. (1999), examined earthquakes in California and Murotani et al. (2008) examined earthquakes in Japan, showing that the area where slips are 1.5 times larger than the average slip is approximately 20% of the total fault rupture area.This area, defined as the area of asperity (Sa), was considered the area most responsible for generating extreme ground motion acceleration.
Source-scaling relationships provide insight into the underlying rupture process mechanics and also provide a crucial reference for obtaining deterministic parameters for ground-motion prediction for earthquake hazard mitigation.For instance, Aagaard et al. (2010) constructed a kinematic earthquake rupture model for simulating scenarios for the fault system in Southern California.They used three-dimensional ground-motion simulation and employed geophysical constraints and empirical relationships for estimating the shaking of scenario earthquakes.The results indicated that the ground motions were sensitive to the slip distribution.In addition, Frankel (2009) applied a broadband system for constructing earthquake models with M w = 5.5, 6.5, and 7.5 by combining deterministic synthetics for plane-layered models at low frequencies and stochastic synthetics at high frequencies.All of the aforementioned studies suggested that slip heterogeneity is an essential constraint in groundmotion simulations.
In view of the recent developments in scenario earthquake simulations for predicting ground motions toward seismic hazard assessments, the placement of the slip distribution (amount and area) within a finite fault is essential.For comprehensive datasets on earthquakes in Taiwan, Yen and Ma (2011) compiled data from 19 earthquakes in Taiwan (with 22 finite fault slip models) and 7 global events (M w > 7.0) for observing the source scaling relationships of effective fault dimensions and mean slip with the seismic moment in which the source dimensions showed scale invariance.Based on these findings we further explored the relationships among the areas of slips partitioned in a finite fault.Our results suggest the fault slip exhibited self-similar scaling between the rupture slip and area.For comparing the scaling relationship of the slip distribution heterogeneity on the fault surface for Taiwanese events with that from the global dataset we included 15 additional global events listed in Mai and Beroza (2000).A comparable self-similarity feature was evident, particularly for larger events (M w > 7.0).This scaling relationship between the self-similarity of the slip and slip area heterogeneity provides a reference for constructing the slip distribution of a finite-fault for an earthquake scenario toward ground-motion prediction.

Slip Models
The main slip models used in this study are those developed by Yen and Ma (2011) and compiled slip models for global earthquakes developed by Mai and Beroza (2000) from a database of finite-source rupture models (Mai 2007;Mai and Thingbaijam 2014).Yen and Ma (2011) inferred the slip models for Taiwan earthquakes from the regional high-quality data recorded by the Taiwan Strong Motion Instrumentation Program (Shin and Chang 2005) and teleseismic data using the finite fault inversion technique proposed by Hartzell and Heaton (1983).The earthquake distribution and corresponding focal mechanisms are shown in Fig. 1.There are 12 dip-slip earthquakes, including the reverse-, normal-, and oblique-type events, and 7 strike-slip events.Their magnitudes M w are 4.6 -7.7, corresponding to seismic moments of 7.75 × 10 15 -4.7 × 10 20 Nm (estimated using referenced slip models; Table 1).Among the 19 events, 14 events were located in the fold-and-thrust belt of the Taiwan orogenic collision zone.For global events, according to Yen and Ma (2011), 7 global event slip models were used for compensating for the lack of larger events (M w > 7.0) in the Taiwan region.
For comparison with the aforementioned slip models mainly for Taiwanese events, we considered additional slip models for global events.Several slip models of global earthquakes were compiled and can be viewed at: http:// www.seismo.ethz.ch/static/srcmod/Homepage.html.The Web site includes several new source models, updates for existing source model entries, and appended additional information (Mai 2007, database of finite-source rupture models).In addition, the Web site displays finite-source rupture models that describe the spatiotemporal evolution of earthquake rupture through the modeling or inversion of seismic and geodetic data.The Web site contains 152 rupture models for 80 earthquakes (M w = 4.1 -8.9) in various tectonic regimes.Mai and Beroza (2000) listed 31 published slip models for 18 events that are the most accurate estimates of true fault dimensions as noted in their compilation.These slip models are typically derived from the finite fault inversion of low-pass-filtered strong-motion recordings (Archuleta 1984;Beroza and Spudich 1988).There are 10 dip-slip and 8 strike-slip events in the database, with magnitudes M w being 5.9 -8.1, corresponding to seismic moments of 2.2 × 10 17 -1.2× 10 21 Nm (estimated using referenced slip models; Table 2).Since three events overlapped with the previous databases of Yen and Ma (2011), 15 additional global earthquakes, with their corresponding 28 slip models, were included in addition to the slip models of Yen and Ma (2011) for the analysis and comparison.The number of earthquakes in study is 41, including 19 Taiwan earthquakes and 7 global events, from Yen and Ma (2011) and 15 global earthquakes from Mai and Beroza (2000), with magnitudes M w ranging from 4.6 -8.9 with various fault types.

Determination of Source Dimensions and Scaling Parameters
For determining effective source dimensions, the spatial autocorrelation scheme developed by Mai and Beroza (2000) was adopted for defining the effective fault dimensions of length and width of the slip models.In addition, a spatial slip function along the strike direction, where the slips on each subfault along the dip direction are summed, was considered.Similarly, a spatial slip function along the dip was obtained by summing the slip on each subfault along the strike direction.The spatial slip function was defined as f(x) in Eq. ( 1).Furthermore, an autocorrelation method was used for estimating the effective dimensions of two spatial functions in the strike and dip directions.An example of an estimate of the effective dimensions using the slip model proposed by Ma et al. (2001) for the 1999 M w 7.6 Chi-Chi, Taiwan earthquake is shown in Fig. 2. The effective dimensions are defined using the autocorrelation slip function divided by the maximum value of autocorrelation function at zero lag as follows: where W ACF denotes the effective dimension of L e or W e , where L e and W e are the effective length and width along the strike and dip directions, respectively.The mean slip is defined using the relationship between seismic moment and source dimensions as follows: where n is the rigidity of crustal rock, and d m is the average effective slip or the mean slip.The derived effective source parameters, i.e., L e , W e , and d m for all events obtained through this analysis are listed in Table 1.We also consider the fault area as A e , which denotes the effective area of L e × W e .For a scaling analysis of the amount of slip and slip area, the following parameters were defined.The area of asperity, Sa, which is the area where the slips are 1.5 times larger than the mean slip, d m , as definition of Somerville et al. (1999).For determining the area of the slip partitioned in a finite fault, R d and R s , are introduced.R d and R s are the normalized slip and slip area, respectively, of the compiled slip models.R d is defined as the ratio of d to d m , i.e., d/d m , where d is the slip within the finite-fault and d m is the mean slip; R s is defined as the ratio of A to A e , i.e., A/A e , where A is the slipped area with a slip amount larger than R d and A e is the effective ruptured area.Here, R d was considered 0.25 -4.0 for every 0.25 increment, referring to the slip area where slips are > (0.25 -4.0)d m for every 0.25d m .As previously mentioned the slip models were inverted mainly from the strong motion data of individual studies (Tables 1 and 2); therefore, the resolution of each inverted slip distribution might vary.However, using the normalization parameters as R d and R s for each slip model minimizes the influence of model resolution on our scaling analysis.

Scaling relationship of Asperity to Magnitude
To examine the relationship between Sa and M w , the asperity to magnitude scaling relationship is illustrated using the slip models in Fig. 3. Two scaling relationship regressions, Sa = 1.58M w -(8.29 ± 0.33) for earthquakes with M w = 4.5 -7.0 and Sa = 0.83M w -(3.77 ± 0.36) for  Wald (1996) Table 2. Locations and corresponding source parameters of earthquake models used (Mai and Beroza 2000).The symbols are the same as those in Table 1.Fault type events: R, reverse slip; S, strike slip; N, normal slip; O, oblique slip.
earthquakes with M w > 7.0, were performed.The Sa value is approximately 20% of A e , which is similar to those calculated by Somerville et al. (1999) and Murotani et al. (2008) for earthquakes in California and Japan, respectively (Fig. 4).
Figure 5 shows the scaling relationship for asperity to magnitude by considering additional global events from Taiwanese events using a total of 41 events.Two scaling relationship regressions, Sa = 1.07M w -(5.44 ± 0.49) for earthquakes with M w = 4.5 -7.0 and Sa = 0.82M w -(3.73 ± 0.35) for earthquakes with M w > 7.0, were conducted.

Slip Area to Slip Scaling in a Finite Fault
According to the definitions of R d and R s , we obtained the slip area as a function of the slip from the compiled slip models.Figure 6 shows the plot of log(R s ) versus R d for the fault slip models provided by Yen and Ma (2011) (see Table1).The results show a self-similar scaling of slip values, with R 10 ( ) For finite fault models of moderate magnitude (M w = 4.5 -7.0), the relationship between R s and R d shows two scaling exponents (n), 0.46 and 0.32, for R d ranges of 0 < R d ≤ 2.0 and 2.0 < R d ≤ 4.0, respectively (Fig. 6).However, as also shown in Fig. 6, when only the events with M w > 7.0 are considered, the scaling relationship for R s as a function of R d follows only one scaling relation as logR s = -0.69Rd + (0.09 ± 0.16).
For considering additional global events (Mai and Beroza 2000), the relationship between R s and R d for the compiled 43 finite fault models with M w = 4.5 -7.0 and 14 finite fault models with M w > 7.0 is shown in Fig. 7.For M w = 4.5 -7.0, the scaling exponents (n) are 0.66 and 0.23 for R d ranges of 0 < R d ≤ 2.0 and 2.0 < R d ≤ 4.0, respectively.In addition, the scaling relationship between R s and R d for large earthquakes with M w > 7.0 follows a scaling relation of logR s = -0.68Rd + (0.11 ± 0.21), similar to the results of the fault slip models shown in Table 1.
To disaggregate the results among events the value of

DIScuSSIon
Comparing the Sa values for Taiwanese and global earthquakes, the values for Taiwanese events are generally larger for M w = 4.5 -7.0 as the scaling slope for these events in Fig. 3 is larger than that in Fig. 5.However, similar trends are evident for earthquakes with M w > 7.0 (Figs. 3 and 5).The differences in this might due to the differences in stress drops, which were not taken into account in this scaling analysis.As in the studies of Yen and Ma (2011), several moderate events in the Taiwan fold-and-thrust belt region exhibit large stress drops.These events with large stress drops might yield larger Sa values for Taiwanese earthquakes than for global ones.However, for larger earthquakes M w > 7.0, the variation in stress drop become insignificant, which, thus, yields similar scaling in the whole dataset.
The slip area to slip scaling gives the area of slips partitioned in a finite fault.Our study shows that the scaling relationship follows the relation of R 10 ( ) The scaling exponent n represents the degree of slip heterogeneity, where the higher n is the smoother the slip distribution.The earthquakes with moderate magnitudes exhibit more significant slip self-similarity for a slip less than twice the mean slip (d ≤ 2.0d m ) for less variance compared to the slip larger than twice the mean slip (d > 2.0d m ).The difference in n for d ≤ 2.0d m and d > 2.0d m suggests that the region with larger slip is more isolated for the area with larger slips (d > 2.0d m ).These features were found not only in Taiwanese (Fig. 6), but also in the global dataset (Fig. 7) of Mai and Beroza (2000).However, if we considered only the earthquakes with magnitudes greater than 7.0, as shown in Figs. 6 and 7 for Taiwanese and the global dataset, the selfsimilarity becomes invariant throughout the whole range of slip ratios.Although Manighetti et al. (2005) suggested that the maximum displacement on the fault is less than twice the average slip.The slip model we examined did reveal the slips, which are larger than twice the average slip.From the adopted global dataset we note that 10 out of the 31 slip models present a slip with d > 2.0d m , including multiple models from a common event.
Slip self-similarity implies that fault rupture is a fractal behavior with the fractal dimension of n.The fractal character of slip distribution is to describe the variety of fault structures that are irregular with roughness or heterogeneity.The rupture process is generated by the stress field under the heterogeneity of the fault surface.A fractal distribution of fault surface roughness has been shown in the field observations (Aviles et al. 1987;Okubo and Aki 1987) and laboratory experiments (Brown and Scholz 1985;Power et al. 1987).Wang and Lee (1997) suggested that the fractal geometry of a fault might have a strong relationship with the fractal heterogeneous distribution of the breaking strengths over the fault.Our study shows different scaling exponents (n) between larger earthquakes of M w > 7.0 (n ≈ 0.7) and moderate earthquakes of M w = 4.5 -7.0 (n < 0.7), suggesting the slip distribution for larger magnitude earthquakes tend to be more homogenous.The different scaling relationships could be attributed to the difference in rupture dynamics.The more heterogeneous the slip distributions for moderate earthquakes might be related to the variations in breaking strength over the fault plane as suggested by Wang and Lee (1997); while the more homogenous slip distribution for larger earthquakes (M w > 7.0) might suggest more significant influence from dynamic rupture process, e.g., thermal pressurization or melting (Kanamori and Brodsky 2004;Ma et al. 2006).
Discrepancies in source scaling for earthquakes have been observed and discussed previously (Shimazaki 1986;Wang 1997;Wang and Ou 1998;Hanks andBakun 2002, 2008;Manighetti et al. 2007;Shaw and Wesnousky 2008).Wang and Ou (1998) analyzed the scaling relation among seismic moment, average displacement, rupture width and rupture length of earthquake fault models.Their results showed that rupture width is independent of rupture length for large earthquakes.Mai and Beroza (2000) analyzed source dimension scaling and observed that scale invariance breaks down for large strike-slip events.They suggested that the breakdown is caused by rupture width saturation.In addition, Yen and Ma (2011) observed that the bilinear feature of magnitude-area scaling appears at a ruptured area of approximately 1000 km 2 for a seismogenic thickness of 35 km in Taiwan.They suggested that the seismogenic thickness controls the evolution of earthquake scaling in fault geometry.Comparatively, in Gutenberg-Richter scaling, several authors observed a breakdown in scaling for large earthquakes (Rundle 1989;Pacheco et al. 1992;Wu et al. 2013).Rundle (1989) suggested that a transition of the b value (the scaling exponent of magnitude and frequency) from 1 -1.5 is caused by the approximately equidimensional (width ≈ length) for smaller events, whereas larger earthquakes (long, shallow strike-slip earthquakes) have considerably greater lengths than depths.Furthermore, Manighetti et al. (2007) analyzed the relationship between displacements and rupture lengths and discovered a scaling break between moderate and large earthquakes.The authors suggested that the scaling break results from the multipleevent rupture process of large earthquakes (Klinger 2010).The behavior in scaling breaking between M w > 7.0 and M w = 4.5 -7.0 observed in this study is similar to that in those aforementioned studies.However, in our study, we indicated further that the fractal heterogeneous distribution also exhibits the breakdown, which suggests a difference in rupture dynamics between moderate and large earthquakes.Although the earthquake rupture dynamics are still difficult to model specifically, the difference in the fractal heterogeneity could be empirically presented by R 10 ( ) s a n Rd = as shown in this study.

concLuSIonS
This study compiled published finite fault models (19 of Taiwan earthquakes and 22 of global earthquakes) for analyzing the scaling self-similarity of the slip distribution in earthquakes.A scaling relationship between the fault slip and area was examined.Our results suggest a self-similarity in the heterogeneity in slip distribution over the fault.The self-similar scaling exponent indicates the degree of fractal dimension in the fault slip system.The spatial slip distribution for large earthquakes (M w > 7.0) follows a scaling of logR s = -0.68Rd + (0.11 ± 0.21), and tends to have a more homogeneous slip distribution compared to the moderate events.In addition, a breaking in the asperity source scaling between M w > 7.0 and M w = 4.5 -7.0 was found.The difference in scaling behavior might be attributed to the rupture dynamics.
In addition to the physical implication of earthquake dynamics from the self-similarity of heterogeneous slip distribution, the scaling relationship for the slip partition in a finite-fault derived in this study provides an important basis for ground-motion prediction, which is particularly crucial in assessing seismic hazards and simulating earthquake scenarios.Our investigations over the databases for Taiwan and Global earthquake events, individually, give the comparison on the validation of the scaling relations for the Taiwan region and its general features for the global dataset.For the comprehensive seismic dataset of Taiwan, our study, thus, also provides an essential reference for global communities involved in the ground-motion prediction of earthquake scenarios.This ground motion prediction is essential for establishing the next-generation attenuation relationships, especially for the near-fault motions, and, thus, in probabilistic seismic hazard assessment.

Fig. 1 .
Fig. 1.Distribution of earthquakes and their corresponding focal mechanisms used in this study.Star colors represent the earthquake depth.Triangles represent the distribution of strong-motion stations of the Taiwan Strong Motion Instrumentation Program (TSMIP) (light green triangle) and Broadband Array in Taiwan for Seismology (BATS) networks (dark green triangle).(Color online only)

Fig. 2 .
Fig. 2. Illustration of how effectively fault length and width were estimated using the slip model of the 1999 Chi-Chi earthquake derived from Ma et al. (2001).Two slip functions were obtained by summing the slip of each subfault along the down-dip and strike directions and were used for obtaining the effective fault width (W e ) and length (L e ) (Yen and Ma 2011).(Color online only) R s in terms of M w for different R d values (R d = 0.25 -4.0) is presented in Fig. 8.In addition, the variance (Δσ) of R s for different R d values was also calculated.The result indicates that the average R s value decreases with increasing R d .Furthermore, the Δσ of R s is approximately < 0.1 when R d is > 1.0, and the Δσ of R s decreases with R d .The ratio of slipped area (R s ) with lower bound of R d (e.g., R d = 0.25 and 0.5) is lower in Taiwan earthquakes than that in global earthquakes.Except for the lower bound of R d , most of the earthquakes exhibited a similar effective area ratio with a slip larger than R d times the mean slip R s (d > R d × d m ).The similar ratio of the effective area R s (d > R d × d m ) of the slip models is a feature of the scale invariance of the slip distribution.

Fig. 8 .
Fig. 8.The variance of R s with magnitude for different R d values.R d is d/d m , the ratio of the slip, d, to the mean slip (d m ).R s is A/A e , the ratio of the fault area (A), where slip d > R d × d m , to the effective area (A e ).The effective area and slip were determined according to the normalized autocorrelation of slip length with slip width.Dashed lines represent the average R s values.Blue dots denote Taiwan earthquakes, while red dots denote global earthquakes.(Color online only)

Table 1 .
(Yen and Ma 2011)responding source parameters of earthquake models used(Yen and Ma 2011).M 0 , seismic moment from slip models; L e