Geophysical Journal International

Geophysical Journal International Geophys. J. Int. (2014) 198, 1279 1292 GJI Seismology doi: 10.1093/gji/ggu209 Efficient waveform inversion for average earthquake rupture in three-dimensional structures Ming-Che Hsieh, 1 Li Zhao 2 and Kuo-Fong Ma 1 1 Institute of Geophysics, National Central University, Jhongli, Taoyuan, Taiwan 2 Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan. E-mail: zhaol@earth.sinica.edu.tw Accepted 2014 May 29. Received 2014 May 29; in original form 2014 February 20 1 INTRODUCTION Realistic earthquake sources are often described as distributions of slip in a finite spatial temporal volume. In seismology, a simple yet effective representation of an earthquake source is a point source, which can be considered as the integrated slip concentrated at the centroid location of the spatial temporal volume of the actual slip. The centroid moment tensor (CMT) is a standard description of the point-source model of an earthquake, and its inversion (Dziewonski et al. 1981) is now routinely conducted for moderate and large earthquakes all over the world (http://www.globalcmt.org). In the past few decades, several other methods have been proposed to determine earthquake focal mechanisms by fitting regional and/or teleseismic waveforms (e.g. Patton & Zandt 1991; Dreger & Helmberger 1993; Romanowicz et al. 1993; Zhao & Helmberger 1994; Zhu & Helmberger 1996; Kanamori & Rivera2008; Herrmann et al. 2011; Zhu & Ben-Zion 2013). At present, solving for point-source focal mechanisms is a well-established operation, and reliable SUMMARY Advances in observational, theoretical and computational technologies have made it possible for automatic, real-time solutions of the focal mechanisms of earthquake point sources. However, for earthquakes of moderate and greater magnitudes, the complexity of the source kinematic processes often requires additional characteristics on the source rupture in order to make seismotectonic inferences and to explain the observed directivity effects of the radiation of seismic energy. We develop an efficient and effective approach to determining the average finite-rupture models of moderate earthquakes by fitting synthetic and recorded broadband waveforms. A Green s tensor database is established using 3-D structural model with surface topography to enable rapid evaluations of accurate synthetic seismograms needed for source parameter inversions without the need for high-performance computing. We take a two-step strategy: In the first step, a point-source model is determined by a grid search for the best fault-plane solution. Then, taking the two nodal planes in the point-source model as candidates of the actual fault plane, a second grid search is carried out over a suite of simplified finite-rupture models to determine the optimal direction and speed of the integrated rupture of the finite source. We applied our method to four moderate events (M W 6) in southeastern Taiwan. Results show that our technique provides an effective choice in semi-automatic, near real-time determinations of finite-source parameters for earthquake hazard assessment and mitigation purposes. Key words: Earthquake source observations; Computational seismology; Early warning; Asia. solutions can be automatically obtained within a few minutes for earthquakes as small as M 3. For earthquakes of moderate size or larger, the increased complexity in both space and time of the faulting process demands that more detailed features of the source be considered to characterize its spatial temporal variations in order to capture the directivity effects on the radiation of seismic waves (e.g. Silver & Jordan 1983; Bukchin 1995). A logical extension of earthquake source description from a point CMT model is the so-called second moments of the stress glut tensor (e.g. Backus 1977; Dahlen & Tromp 1998), also known as the finite moment tensor (FMT) of earthquakes (e.g. Bukchin 1995; McGuire et al. 2001; McGuire 2004; Chen et al. 2005). The advantage of the FMT is that it captures the average or integrated properties of the spatial temporal variation of a finite rupture that lead to significant directivity effect. These average properties include the characteristic length, duration and speed of the source rupture. Given its spatial finiteness, the FMT model provides a discriminant for the identification of the actual fault plane between C The Authors 2014. Published by Oxford University Press on behalf of The Royal Astronomical Society. 1279

1280 Ming-Che Hsieh, Li Zhao and Kuo-Fong Ma the two orthogonal auxiliary planes in a point-source model. Chen et al. (2005) used the FMT approach and Green s functions in 1-D structure for Southern California to investigate the 2002 Yorba Linda earthquake, and demonstrated that the FMT parameters can be obtained using synthetics in a local velocity model instead of empirical Green s function. The effect of propagation can be effectively taken into account by a reliable structural model. The FMT approach enables the capability of resolving the issue of fault plane ambiguity without having to invert for a relatively large number of free parameters as required in determining the source s full slip distribution (FSD). Therefore, the FMT model provides a natural and convenient description for moderate-sized earthquakes (M 4 6) which have less source complexity. As an extension to the CMT model which provides the centroid location and moment tensor, the FMT solution has fewer unknown parameters to be determined including only the characteristic dimension of the fault plane and the average rupture speed vector. As a practical matter, it is also important to determine the FMT solution immediately after the occurrence of a moderate-sized event for hazard prevention purpose, since it provides more detailed properties of the earthquake source that are crucial to the prediction of realistic strong ground motion and to the assessment of the likelihood of larger earthquakes to come. Our goal in this study is to develop an effective and efficient approach to obtain the FMT solutions of moderate earthquakes for the purpose of seismotectonic analyses and hazard assessments. Our approach is based on fitting the observed and synthetic seismograms at local distances with the consideration of 3-D velocity structures and surface topography. Similar to the cut-and-paste (CAP) method for CMT inversions (Zhao & Helmberger 1994; Zhu & Helmberger 1996), we use the P and surface waves separately in waveform fitting. We perform a two-step grid search in determining the FMT solutions. A fault-plane solution of the earthquake point source is determined first by a grid search for the optimal depth, magnitude, and strike, dip and rake angles. Then, the two auxiliary planes in the point-source fault-plane solution are taken as candidate rupture planes and a second grid search is carried out to determine the optimal FMT solution of the earthquake. The resulting average rupture direction and speed characterize the integrated behaviour of a moderate-sized event, which can be obtained automatically and in near real-time for hazard assessment and mitigation purposes. 2 METHODOLOGY In this study, our objective is to develop a practical and effective approach to quickly obtain finite rupture parameters of moderate earthquakes. We illustrate our methodology by determining the source rupture parameters for a number of earthquakes in Taiwan using broadband waveforms records from local stations in the Taiwan region. 2.1 Seismic data and moderate earthquakes Taiwan has a high level of seismicity due to the collision between the Eurasia and the Philippine Sea plates. There are on average more than 100 felt earthquakes each year, with occasional moderate ones of M 6.0, providing plenty of observational opportunities. The Broadband Array in Taiwan for Seismology (BATS) was established since 1996 for earthquake monitoring and research (Kao et al. 1998). BATS currently consists of 24 stations and routinely determines the focal mechanisms in the Taiwan region (http://bats.earth.sinica.edu.tw). Fig. 1 showsthebatsstationdistribution and the focal mechanism solutions of four moderate earthquakes in southeastern Taiwan selected as the target earthquakes in this study together with the background seismicity. Details of the target events are also provided in Table 1. The focal mechanisms have been obtained using the BATS waveform records by Figure 1. (a) Four moderate earthquakes (yellow stars, depths and local magnitudes from the catalogue of Taiwan Central Weather Bureau) in southeastern Taiwan used in this study. Seismicity is shown by the dots for epicentres of earthquakes of M 2.0 from 2000 to 2012 (Wu et al. 2008). Beachballs depict the focal mechanisms of the moderate events (red: gcap; blue: GCMT; green: this study). Red lines are active faults in the area. (b) Regional tectonic settings around Taiwan. Black triangles show locations of the BATS stations used in this study. Also indicated are the geological provinces on the island of Taiwan including: Coastal Plain (CP); Western Foothills (WF); Hsueshan Range (HR); Backbone Range (BR); Coastal Range (CR) and Hengchun Peninsula (HP). The rectangle indicates the area shown in (a).

Waveform inversion for average source rupture 1281 Table 1. Information on the four target events used in this study. The gcap solutions are based on the method of Zhu & Ben-Zion (2013). Event Method Time (UTC) Longitude ( ) Latitude ( ) Depth (km) M W M rr M θθ M ϕϕ M rθ M rϕ M θϕ M 0 (N m) Jiahsian100304 (2010 March 4) Wutai120226 (2012 February 26) Taitung060401 (2006 April 1) Taitung060415 (2006 April 15) gcap 00:18.52.10 120.71 22.97 24.0 6.2 0.92 0.12 0.42 0.12 0.50 0.46 2.28e+25 GCMT 00:18:55.40 120.56 22.86 29.1 6.3 1.730 0.373 2.110 0.335 2.070 1.020 3.02e+25 gcap 02:35:00.04 120.75 22.75 32.0 5.9 0.44 0.07 0.62 0.27 0.80 0.08 8.55e+24 GCMT 02:35:01.60 120.70 22.69 25.0 6.0 0.347 0.062 0.408 0.066 0.899 0.231 1.00e+25 gcap 10:02:19.50 121.08 22.88 12.0 6.1 0.09 0.51 0.01 0.26 0.30 0.84 1.49e+25 GCMT 10:02:22.40 121.10 22.89 15.1 6.2 0.111 0.979 0.868 0.772 0.687 1.450 1.99e+25 gcap 22:40:55.40 121.30 22.86 18.0 5.8 0.71 0.04 0.95 0.43 0.10 0.32 4.78e+24 GCMT 22:40:57.40 121.40 22.87 21.7 5.9 0.775 0.057 0.832 0.224 0.068 0.251 8.74e+24 the generalized Cut and Paste (gcap) method (Zhu & Ben-Zion 2013). These moderate events not only generated wide public attention when they occurred, they also excited much interests among seismologists to characterize the sources of these events (Wu et al. 2007; Chenet al. 2013; Leeet al. 2013; Mozziconacci et al. 2013a, b) and to investigate their seismotectonic implications (Kuochen et al. 2007; Wuet al. 2007, 2008; Kuo-Chen et al. 2012). We will compare our results with these existing studies. 2.1 Strain Green s tensor (SGT) database for rapid synthetic calculation One of the new features of this study is the consideration of 3-D seismic structure and surface topography in the modelling of wave propagation. For this purpose we use the finite-difference algorithm of Zhang & Chen (2006), which is based on the non-staggered MacCormak scheme (MacCormack 1969; Gottlieb & Turkel 1976) with optimized dispersion relation preserving (DRP/opt) formulation(tam& Webb1993; Hixon 1997) and a perfectly matched layer (PML) for the absorbing boundary condition (Berenger 1994; Marcinkovich & Olsen 2003). In order to achieve the computational efficiency for calculating synthetic seismograms required by source inversions, we invoke the SGT database approach of Zhao et al. (2006). For a point source at r S with a moment tensor M,then-component displacement at a receiver location r R can be expressed as (e.g. Aki & Richards 2002): 3 3 u n (r R,t; r S )= M ij S j G ni(r R,t; r S ), (1) i=1 j=1 where G(r R, t; r S ) is the Green s tensor from the source at r S to the receiver at r R, and the superscript S in eq. (1) indicates that the differentiation operator acts on the source coordinates. Zhao et al. (2006) introduced the third-order SGT: H ijn (r, t; r S ) = 1 2 [ i G jn (r, t; r S ) + j G in (r, t; r S )]. (2) Taking advantage of the reciprocity property of the Green s tensor G (e.g. Dahlen & Tromp 1998; Aki & Richards 2002), the displacement in eq. (1) can be expressed in terms of the SGT: 3 3 u n (r R,t; r S ) = M ji H ijn (r S,t;r R )oru(r R,t;r S ) i=1 j=1 = M : H(r S,t; r R ). (3) Using eq. (3), the displacement at receiver r R due to an earthquake at r S can be obtained by the SGT from the receiver to the source. Therefore, in studying the earthquakes in a region using a given set of receivers, we can establish a database for the SGT H from all receiver locations to a set of gridpoints in the region enclosing the potential source locations. Then all synthetic seismograms for any source within the region covered by the SGT database can be obtained by simply retrieving the necessary SGT from the database. This makes the use of 3-D structural models in source inversions practical because the SGT database can be pre-established for the 3-D model, which can drastically reduce the CPU time needed in computing synthetic seismograms. The SGT approach has been used for moment tensor inversions using 3-D structural models (Zhao et al. 2006; Leeet al. 2011). In this study for moderate earthquakes in Taiwan, we use a recent tomography model for the Taiwan region (Kuo-Chen et al. 2012) and ETOPO1 (Amante & Eakins 2009) for the surface topography. The main island of Taiwan is sampled by a uniform grid of 2282 points on the surface with a horizontal spacing of 4 km (Fig. 2a). These surface gridpoints serve as virtual receiver locations. Finite-difference simulations are conducted to calculate the SGTs from these virtual receivers to another set of 3-D gridpoints serving as virtual source locations (Fig. 2b). These virtual source gridpoints have a uniform horizontal spacing of 2.4 km and a variable vertical spacing from 0.857 km near the surface to 3.913 km at 40-km depth. The source time function used to compute the SGTs is a Gaussian function with a characteristic width of 2 s, which yields waveforms with frequency content up to 0.8 Hz. A total of 6846 finite-difference simulations have been carried out to obtained the SGTs from all 2282 virtual receivers to all the virtual sources with full 3-D structure and surface topography effects. Using the database of these SGTs, it only takes a fraction of a second to compute the 3-D synthetic seismogram at any location on the surface from an earthquake anywhere in the Taiwan region (down to 60-km depth). This SGT database thus enables near real-time inversions of earthquake source parameters and the prediction of strong ground motions. Our earthquake source inversion using the 3-D SGTs is based on a grid search for the source parameters that provide the best fit between synthetic and recorded seismograms. We separate the process into two steps for point- and finite-source parameters. In the first step, we obtain the optimal solution for a point-source model including the source centroid depth as well as the strike, dip and rake angles of the fault planes. Then, based on the two orthogonal auxiliary planes, we search for the best average finite-rupture model to represent the average rupture behaviour of a finite source.

1282 Ming-Che Hsieh, Li Zhao and Kuo-Fong Ma Figure 2. Locations of virtual source and receiver nodes for the strain Green s tensor (SGT) database. (a) Virtual source nodes for a given depth with a uniform spacing of 2.4 km in both EW and NS directions. The same nodes are used for 26 depths from surface down to 52-km depth with uneven vertical spacing. (b) Virtual receiver nodes (2282 in total) on topography surface with a uniform horizontal spacing of 4 km in both EW and NS directions. 2.2 Grid search inversion for point-source parameters Point-source focal mechanisms can be obtained from the polarizations of first motions, the spectral amplitudes of P and S waves, and the waveforms. Waveforms contain the most complete information of the source and therefore robust source parameter estimations can be obtained with relatively poor station coverage, sometimes even with records from a single station (e.g. Dreger & Helmberger 1993). However, most of the waveform-based methods for focal mechanism determinations have relied on 1-D structure because of the impractical amount of computation involved in computing synthetics in 3-D structure. Here, with the help of the 3-D SGT database approach, we are able to conduct a grid search for point-source parameters in 3-D structure. The ability of using 3-D structure enables us to include waveforms of shorter period in the waveform fitting than using 1-D models, it can also better constrain the source depth. In the CAP method (Zhao & Helmberger 1994; Zhu & Helmberger 1996; Zhu & Ben-Zion 2013), the three-component record at each station from an earthquake is cut into five windows: two P wave and two Rayleigh-wave windows in the vertical and radial components and one Love-wave window in the transverse component. Waveforms in P- and surface waves are fit in different frequency bands. We adopt a similar but slightly different approach: we do not rotate the waveforms and use P- and surface-wave windows in all three components. At first the instrument responses are deconvolved from the records. The resulting ground velocity are then bandpass filtered by a fourth-order Butterworth filter. The pass band of the filter is 0.05 0.3 Hz or 0.04 0.2 Hz for P waves and 0.02 0.1 Hz or 0.02 0.08 Hz for surface waves, depending on the epicentral distance. Then time windows are cut by different criteria depending on first-arrivals or surface waves. We use a cross-correlation operation to align synthetics with respect to observations in order to make allowances for the imperfectness of the velocity model and other possible biases in the arrival times. For the moderate earthquakes studied here (M W 6), limits of 2 and 4 s are set for the shifts in first arrivals and surface waves, respectively. Considering the contribution to waveform shifts from the finite source duration, these limits may need to be increased for larger events. The windowed parts of each record-synthetic pair are used in waveform fitting. In waveform fitting we define the objective function based on the weighted L 2 -norm of the difference between windowed records and their corresponding synthetics calculated using the 3-D SGTs for given set of point-source parameters, including the source centroid depth h, and the strike φ,dipδ and rake λ of the fault-plane solution, and two types of estimated seismic moments M pq 0 = max( r pq (t) ) max( g pq (t) ), and M p 0 = 1 6 6 q=1 M pq 0, (4) where r pq (t) are the windowed and filtered records, and g pq (t) are the corresponding synthetic waveforms calculated with unit seismic moment, that is M 0 = 1. Here the subscripts p and q identify the station and the waveform window, respectively. Thus p takes the values of 1, 2, 3,..., and N if there are N stations, and q varies between1and6withq = 1, 2 and 3 for the P-wave windows and q = 4, 5 and 6 for surface-wave windows. Before taking the difference between a record and its synthetic, the two waveforms

are aligned with the aid of their cross-correlation: g pq (t) r pq (t) = { ˆM :[f pq (t) H pq (t)] } [ f pq (t) r pq (t)], where r pq (t) is the unfiltered record (and instrument response removed) for the pq-window, and H pq (t) and f pq (t) arethesgt and filter, respectively, for the same window. The symbols and stand for convolution and cross-correlation, respectively, and ˆM is the normalized moment tensor for the set of (φ, δ, λ) values.the frequency-domain version of eq. (5) can be written as g pq (ω) r pq (ω) = [ ˆM : H pq (ω) f pq (ω) ][ f pq (ω)r pq (ω) ] = ˆM : H pq (ω) f pq 2 (ω)r pq (ω), (6) where the superscript indicates complex conjugation, and f pq 2 (t) represents the two-pass filter for the pq window. Thus the waveform cross-correlation can be driven by g pq (t) r pq (t) = [ ˆM : H pq (t) ] [ f pq 2 (t) r pq ( t) ], (7) where r pq ( t) is the time-reversed, windowed but unfiltered record. As shown in eq. (7), the cross-correlation between windowed and filtered records and synthetics can be obtained by applying a two-pass filter on time-reversed records. This effectively transfers the filtering of windowed synthetics in eq. (5) to that of the records. Since in determining the source parameters we often need to filter the synthetics calculated for tens of thousands of source models, eq. (7) allow us to drastically improve the computational efficiency. After obtaining the cross-correlations, the record and its synthetic are allowed for a shift in time by an amount based on the peak location in their cross-correlation. This shift accounts for the delays caused by possible imperfection of the structural model and earthquake location errors. We then calculate two types of L 2 -norm objective functions: e pq = r pq (t) s pq (t) 2 = r pq (t) M pq 0 g pq (t) 2 = 1 [ r pq (t) M pq T pq W pq 0 g pq (t)] 2 dt, (8) ẽ pq = r pq (t) M p 0 g pq (t) 2 = 1 [ r pq (t) M p T pq W pq 0 g pq (t)] 2 dt, where T pq is the length of the waveform window W pq.weuse15 and 50 s for the lengths of P- and surface-wave windows, respectively. A P-wave window starts at 1 s before the model-predicted P-wave arrival time, whereas a surface-wave window centres around the time for the peak amplitude. However, because of the time shift allowed between a record and its corresponding synthetic, the exact location of a time window W pq in which the objective functions in eqs (8) and (9) are evaluated is adjusted accordingly. Waveforms at different stations or on different components have different amplitudes, and the L 2 -norm objective functions defined in eqs (8) and (9) are frequently dominated by waveform traces with large amplitudes. Various strategies have been designed to alleviate this effect (e.g. Zhu & Helmberger 1996). In this study, we introduce a weighting factor based on the L 2 -norms of the records w pq = max(r pq )/R pq, (10) where R pq = [r pq (t)] 2 dt. (11) W pq (5) (9) Waveform inversion for average source rupture 1283 Considering the fact that P waves usually have smaller amplitudes and signal-to-noise ratios (SNR) than surface waves, we should also reduce the P-wave contributions to the objective function relative to those of surface waves. Thus we further introduce a weighting factor w pq which takes on a higher value for surface-wave windows than for P waves. With all the above considerations, our final objective function is defined as E = 1 N 6 E pq = 1 N 6 w pq w pq (e pq + ẽ pq ). (12) K K p=1 q=1 p=1 q=1 We tested different choices for w pq and values of 1 and 2 are found to be effective for P and surface waves, respectively. In the algorithm, we can further generalize the function of the factor w pq to assumes the value of zero if the pq window is not used in the grid search. In the end, the value of the quantity K in eq. (12) is the total number of non-zero w pq. The final optimal set of point-source parameters, including the seismic moment M 0, the source centroid depth h, and the strike φ, dip δ and rake λ of the fault-plane solution, are obtained through an iterative grid search process for the minimum of E defined in eq. (12). At each iteration, we examine the waveform residuals E pq of individual traces corresponding to the best solution and discard a number of traces with the largest residual values. The remaining traces are used in the next grid search iteration. This process continues until one of the exit criteria is met. The exit criteria include the level of reduction in the total residual E and the minimum number of traces and stations remaining. An example for the Jiahsian earthquake is given in Fig. 3 to illustrate the grid search for point-source parameters. A total of 12 BATS stations are used in the grid search, and after 11 iterations 30 P and surface waves are kept in determining the final solution. The final optimal solution indicates that the Jiahsian event has a centroid depth of 21.2 km and a moment magnitude of 6.3. These values as well as the nodal planes are all very similar to the gcap solutions. For local events of this magnitude, the CMT solutions obtained from 1-D and 3-D models are usually within the margin of errors. 2.3 Grid search inversion for finite source parameters For an earthquake, the result from the iterative grid search process for its point-source parameters provides two orthogonal auxiliary fault planes. A point-source model describes the integrated strength of the earthquake as well as the orientation of the average stress field in the source region. It also provides the radiation pattern of seismic waves responsible for the observed spatial variation of earthquake-induced strong ground motion. These basic understandings of earthquake sources, although important and necessary, are often not sufficient in making realistic predictions of strong ground motions caused by moderate and large earthquakes. It is also incapable of making the distinction between the two orthogonal auxiliary planes since they are completely equivalent in predicting the seismic wavefields. Thus it is impossible to use a point-source model to identify the actual rupture or fault plane. After the occurrence of moderate earthquakes, it is vitally important to identify the actual rupture plane for seismotectonic analysis in order to make meaningful assessment of the possible impending earthquake disaster. For this purpose, a finite-fault model must be adopted. A complete description of a finite-fault model is the specification of the distribution of the time-dependent slip vector on the fault plane, that is a FSD of the source. However, this specification requires tens or even hundreds of parameters that are often difficult

1284 Ming-Che Hsieh, Li Zhao and Kuo-Fong Ma Figure 3. Grid-search inversion for the point-source fault-plane solution of the Jiahsian earthquake. (a) Residual distribution of the top 20 fault-plane solutions. The red beachball having the smallest residual is taken as the best solution. Red dashed line indicates the value of the objective function, E min = 1.86 10 7, for the optimal solution, whereas the black dashed line shows the value, E 20 = 2.62 10 7, for the 20th best solution. (b) Residual distribution of the best solutions for different source depths. The red beachball shows the best solution. (c) Waveform fitting for the best fault-plane solution. See Fig. 1 for the locations of stations. The horizontal axes are time in sec. Epicentral distances and azimuths are given under the station names. Black and red curves are for recorded and synthetic seismograms, respectively. Time shifts and pass bands are shown in lower-left and upper-right corners of each trace, respectively. to resolve for all but very large events (e.g. M W 7.0). A natural link between the point-source model and the FSD has been the finite-moment tensor (FMT), which is formally defined as the second moments of the stress glut of an earthquake source (e.g. Bukchin 1995; McGuire et al. 2001; McGuire 2004; Chen et al. 2005). Based on a constant CMT, the FMT provides the average or integrated properties of a finite source that are largely responsible for the observed directivity effect, namely the length, duration, direction and speed of the fault rupture. The FMT can be an effective means to discriminate between the two point-source auxiliary planes (Chen et al. 2005). In this study, we describe a finite-source as a simplified FMT model. The source mechanism is give by a constant CMT solution obtained from the point-source search described in the previous section. However, instead of solving for the average length of the rupture, we specify the spatial extent of the fault plane based on

Figure 4. Schematic diagram showing the simplified finite-rupture models. These models are considered on both orthogonal auxiliary planes of a pointsource fault-plane solution. Three fault geometries are considered: circular, unilateral square and bilateral square, with fault dimension determined by regional scaling law. Arrows show rupture directions, and the length of the arrow indicates one of three trial rupture speeds given by the fraction of the shear-wave speed β. A total of 126 simplified finite-rupture models are considered. These include: (a) six circular rupture models, with three possible rupture speeds on either of the two auxiliary planes; (b) 48 unilateral rupture models, including 24 different models on either of the two auxiliary planes formed by eight possible rupture directions with three possible rupture speeds for each direction and (c) 72 bilateral rupture models, including 36 different models on either of the two auxiliary planes formed by four possible rupture directions with nine possible rupture speeds (three possible speeds either way of the bilateral rupture). the scaling law between magnitude and fault dimension. The fault plane thus defined is divided into a number of point sources. These point sources have equal seismic moment but different origin times to simulate rupture propagation. By assuming a constant (or average) rupture propagation speed and direction, we can determine the origin times of the individual point sources, and calculate the synthetic seismogram from the finite-rupture source by summing up the contributions from all point sources. Our determination of finite-rupture model is also based on a grid search for the best set of source parameters that yields the minimum value of the objective function defined in eq. (12). The source parameters include four categories: (1) fault plane selection, that is either Plane A or Plane B of the two orthogonal auxiliary planes in the point-source solution; (2) rupture geometry, where we choose among circular (C), unilateral (U), or bilateral (B) rupture; (3) rupture speed, where we assume a rupture speed of 55, 75 or 95 per cent of the shear-wave speed at the source and (4) rupture direction, where we vary the rupture direction in the fault plane in a 45 interval from vertical to oblique, to horizontal, and so on. The different choices in these four categories form a total of 126 trial finite-rupture models as shown in Fig. 4. These simplified descriptions of finite sources can be considered as stripped-down versions of the FMT model. They involve only a limited number of variables specifying the integrated or average rupture behaviour of the actual source. As a result they can be determined relatively quickly after earthquakes with fewer observations. Yet they can capture a large part of the directivity effect of finite ruptures to Waveform inversion for average source rupture 1285 enable the identification of the fault planes and the prediction of more realistic strong ground motion. We apply the same efficient algorithm for estimating the objective functions based on the L 2 -norm of the difference between windowed records and their corresponding synthetics as described in the previous section to search through the 126 trial models for the optimal simplified finite-fault model. Fig. 5 shows the finite-rupture search for the Jiahsian earthquake. The point-source CMT solution in Fig. 3 provides the two orthogonal auxiliary planes as well as the seismic moment used to define the fault-plane dimension based on the regional scaling law for Taiwan (Yen & Ma 2011). Result suggests that the Jiahsian earthquake is a horizontal bilateral rupture on Plane B, the northeast-dipping nodal plane. The rupture speeds are 0.95β (3.2 km s 1 ) in the strike direction and 0.55β (1.9 km s 1 )in the opposite direction. The distribution of the values of the objective function among the 126 trial finite-rupture models is exhibited in Fig. 6. Most of the rupture models with smaller residuals are bilateral ruptures on Plane B with different rupture speeds, suggesting a better constrained result on the rupture plane, rupture type and direction than on rupture speed. Therefore, for the given size of the earthquake and the data availability, our finite rupture determination provides effective knowledge on the rupture plane, rupture type and rupture direction within a few minutes. 3 APPLICATION TO MODERATE EARTHQUAKES IN TAIWAN In this study, we use four events in southeastern Taiwan, including the Jiahsian earthquake already discussed in Section 2, to illustrate our grid search process in determining the averaged finite rupture models, and to assess the effectiveness of our approach. The events are chosen mainly for two reasons: (1) they are large enough to generate significant finite-rupture effect to be discerned in the waveforms recorded by the BATS stations in Taiwan and (2) there are available detailed studies on the finite sources of these earthquakes which we can use to compare with the averaged finite rupture source models obtained with our approach. The locations and focal mechanisms of these earthquakes are displayed in Fig. 1 and listed in Table 1. 3.1 The 2010 March 4 (M L = 6.4) Jiahsian earthquake The first example is the Jiahsian earthquake which we have presented in Section 2. Various groups have reported the focal mechanism solution of this earthquake. Given in Fig. 1 and Table 1 are the GCMT and gcap solutions. The earthquake excited much interest because it was the strongest on-land earthquake occurred in Taiwan after the disastrous 1999 M W = 7.6 Chi-Chi earthquake. Moreover, seismicity showed that the hypocentre appeared to be located in a low seismicity zone (Chan & Wu 2012; Leeet al. 2013). A FSD model for this event was proposed showing a unilateral rupture in the rake direction on the northeast-dipping plane (Lee et al. 2013). Waveform records from 12 BATS stations are used in the gridsearch inversion for point-source focal mechanism, as illustrated in Fig. 3. The optimal fault-plane solution is similar to both GCMT and gcap results, suggesting a largely thrust event with a nearly vertical plane striking roughly north south and another plane dipping northeast at a shallow angle. The centroid depth of 21.2 km is slightly shallower than both the gcap (24 km) and GCMT (29.1 km) solutions; while the moment magnitude of 6.3 is the largest among the three solutions.

1286 Ming-Che Hsieh, Li Zhao and Kuo-Fong Ma Figure 5. Grid-search inversion for the average finite-rupture model of Jiahsian earthquake. (a) Residual distribution of the top 20 simplified finite-rupture models. Symbols of red and blue colours represent ruptures on the nodal planes of the same colours shown in the beachball in (b). The best rupture model is the shown by the larger square. Red dashed line indicates the value of the objective function, E min = 1.72 10 7, for the optimal solution, whereas the black dashed line shows the value, E 20 = 2.21 10 7, for the 20th best solution. (b) Location (yellow star), point-source focal mechanism (beachball), and finite-rupture model (yellow square) of the Jahsian100304 event. The thicker nodal plane in the beachball denotes the actual fault plane. The top-right plot shows the total moment rate function; (c) Waveform fitting for the best fault-plane solution. The horizontal axes are time in sec. Epicentral distances and azimuths are given under the station names. Black and red curves are for recorded and synthetic seismograms, respectively. Time shifts and pass bands are shown in lower-left and upper-right corners of each trace, respectively.

Waveform inversion for average source rupture 1287 Figure 6. Residuals of the 126 finite-rupture models in the grid-search inversion for the Jiahsian earthquake. Stronger colours represent smaller residuals indicating better solutions. The beachball shows the two nodal planes A and B used as possible fault planes. The thicker nodal plan (blue) indicates the fault plane in the optimal finite-rupture model. The optimal solution is model BB331, a rupture on nodal plane B (B), bilaterally (B) in a horizontal direction (3) with speeds of 0.95β in the strike direction and 0.55β in the opposite direction. The moment magnitude and orthogonal auxiliary planes derived in the point-source solution provide the basis for estimating the spatial dimension as well as the orientation of the trial fault planes in the following grid search for the finite-rupture model. The moment magnitude of 6.3 leads to a fault dimension of 15 15 km 2 based on the scaling law of Yen & Ma (2011), and we divide each of the two orthogonal auxiliary planes into nine equal-area patches, whose centres are the locations of point sources. The same 30 traces retained in the point-source grid search are used in evaluating the objective functions of the 126 trial finite-rupture models, as shown in Fig. 5. The optimal model shows a horizontal bilateral rupture on the northeast-dipping plane with a higher rupture speed (3.2 km s 1 ) to the northwest (strike direction) than that to the southeast (1.9 km s 1 ). In Fig. 5(a), the finite-rupture model with the second lowest value of objective function (BB222, also see Fig. 6) shows similar bilateral rupture but with the same rupture speeds in opposite directions. In presenting the results for the Jiahsian earthquake, we have shown the waveform fits provided by both the point-source and finite-rupture models. For the rest of the events, we will focus on the finite-rupture results and only present waveform fits by the finite-rupture models. 3.2 The 2012 February 26 (M L = 6.4) Wutai earthquake The Wutai earthquake occurred on 2012 February 26, about two years after the Jiahsian event and roughly 50 km to the south. The hypocentral depth of this earthquake was 28.4 km (Fig. 1), slightly greater than that of the Jiahsian earthquake. Chen et al. (2013) used a regional 3-D model (Wu et al. 2007) and a group of densely distributed stations involving several networks to relocate its location and reported a focal depth of 32.3 km. Our grid search (Fig.7b) for the point-source parameters of this earthquakes results in a centroid depth of 23.5 km, similar to the Jiahsian earthquake, and a moment magnitude of 5.9. A total of 29 waveform traces are used in determining the final point-source mechanism, and the result is largely consistent with the GCMT solution. The grid search for the finite-rupture model of the Wutai earthquake using the same 29 traces in determining the point-source reveals that the rupture was an oblique bilateral one on the northeastdipping nodal plane with a slightly lower rupture speed (2.6 km s 1 ) in the direction roughly parallel to the rake direction than in the opposite direction (3.3 km s 1 ). However, the finite-rupture model with the second lowest value of objective function (BB222 in Fig. 7a) shows similar bilateral rupture but with different rupture speeds in opposite directions. Here we have only one station LYUB in the southeast direction, so the constraint on the rupture speed may not be sufficiently robust. 3.3 The 2006 April 1 and 15 (M L = 6.2 and 6.0) Taitung earthquakes The other two events analysed in this study occurred in the Taitung region in southeastern Taiwan, which is the current location of the collision zone between the Philippine Sea and Eurasian plates with a very active seismicity. These two moderate earthquakes happened in a span of about a half month, and are located 20 km apart (see Fig. 1). The earlier one, Taitung060401, has a shallower depth and a nearly strike-slip mechanism; whereas the later event, Taitung060415, has a slightly greater depth and a nearly thrust mechanism.

1288 Ming-Che Hsieh, Li Zhao and Kuo-Fong Ma Figure 7. Grid-search inversion for the average finite-rupture model of Wutai earthquake plotted in the same manner as Fig. 5. The larger square in (a) indicates the optimal solution. The values of the objective functions for the optimal and 20th best solutions are E min = 2.00 10 9 and E 20 = 2.21 10 9, respectively. The thicker nodal plane in the beachball in (b) denotes the actual fault plane. Fewer waveform records are available for the two events in 2006 in comparison to the Jiahsian and Wutai earthquakes. However, we still managed to obtain reasonably robust solutions for both point-source and finite-rupture models for the two 2006 events. The finite-rupture solution for Taitung060401 is shown in Fig. 8. Our point-source solution (Fig. 8b) for this event shows a strike-slip mechanism, a moment magnitude of 6.2 and a shallow centroid depth of 9.9 km, consistent with the previous study of this event using local data (Wu et al. 2006). The finite-source model suggests a horizontal bilateral rupture on the south-dipping plane with the same propagation speed of 2.7 km s 1 in the opposite directions (Fig. 8). The other finite-rupture model with similarly low value of objective function (AB222 in Fig. 8a) shows a similar rupture on the same plane. Finally for the Taitung060415 event, our point-source search leads to a thrust focal mechanism with a moment magnitude of

Waveform inversion for average source rupture 1289 Figure 8. Grid-search inversion for the average finite-rupture model of the Taitung060401 earthquake plotted in the same manner as Fig. 5. The larger circle in (a) indicates the optimal solution. The values of the objective functions for the optimal and 20th best solutions are E min = 4.56 10 8 and E 20 = 5.41 10 8, respectively. The thicker nodal plane in the beachball in (b) denotes the actual fault plane. 5.9 and a centroid depth of 16.6 km. The finite-fault result (Fig. 9b) indicates that this event is a circular rupture on the east-dipping nodal plane with a rupture speed of 2.6 km s 1. 4 DISCUSSION The four moderate earthquakes analysed in this study are all of moment magnitude of about 6. Local strong-motion observations in Taiwan clearly indicate directivity effects due to the finite rupture processes. However, resolving the detailed slip distributions for events of such moderate sizes requires the use of short-period (<10 s) waveform records, which in turn demands for more careful consideration of the effects of 3-D velocity structure. Our approach in this study attempts to resolve the average or integrated behaviour of a finite rupture, thus loosens the above demands. Here, we compare our finite-rupture models for the four moderate earthquakes with previous studies. The results are plotted together in Fig. 10 with both the 3-D structure and the distributions of aftershocks. For the Jiahsian earthquake, the slip distribution inversion of Lee et al. (2013) using both teleseismic and local records as well as GPS observations shows two asperities on the northeast-dipping fault plane. The asperity with the maximum slip is located at 20-km depth, close to the CWB hypocentre of 22.6-km depth, with a dimension of 15 15 km 2, similar to what we estimated by the scalinglawofyen&ma(2011). However, the centre of our finiterupture model is at 21 km depth while at 17 km in Lee et al. (2013). The slip distribution of Lee et al. (2013) shows an oblique unilateral rupture to the northwest with a variable speed towards surface. This is largely consistent with our result, which suggests a horizontal bilateral rupture with a higher speed to the northwest. Fig. 10(b) shows the 7-day aftershock distribution of the Jiahsian earthquake. The aftershocks are relocated in 3-D model (Wu et al. 2007, 2008). A linear distribution of aftershocks dipping to the northeast at 15 25 km depth can be identified, suggesting the orientation of the actual fault plane of the Jiahsian earthquake. The lack of seismicity shallower than 15-km depth suggests that

1290 Ming-Che Hsieh, Li Zhao and Kuo-Fong Ma Figure 9. Grid-search inversion for the average finite-rupture model of the Taitung060415 earthquake plotted in the same manner as Fig. 5. The larger square in (a) indicates the optimal solution. The values of the objective functions for the optimal and 20th best solutions are E min = 2.95 10 9 and E 20 = 3.92 10 9, respectively. The thicker nodal plane in the beachball in (b) denotes the actual fault plane. the Jiahsian earthquake was likely happened on a blind fault which is not well defined by surface rupture observations. For the Wutai earthquake, the depth obtained in our grid search puts its location nicely in the vicinity of the cluster of aftershocks. However, the relocated depth of this event by Chen et al. (2013) seems to be too large, separating it from all the aftershocks (Fig. 10c). The two events in Taitung are both located in the vicinity of or even coincide with the rupture zone of another larger M W = 6.8 earthquake occurred in 2003 December in Chengkung (Mozziconacci et al. 2013b). Mozziconacci et al. (2013a) determined the rupture process of Taitung060401 by using a combination of local strong-motion and teleseismic broadband records, as well as local GPS observations. Their result suggests that the faulting process is complex with ruptures on orthogonal auxiliary fault planes consisting of three segments. The maximum slip occurred on the intersection of two fault planes: the rupture initially nucleated on a north south-striking plane, then propagated onto an east weststriking plane. The cross-cutting structures and the complex rupture process indeed enhance the waveform fitting, and in particular the GPS observations. Using coseismic displacements from GPS observations, Wu et al. (2006) also suggested a west-dipping fault plane. However, their rupture model systematically underestimated the vertical and east west components of the coseismic displacements. Figs 10(d) and (e) show the aftershock distribution of the Taitung060401 earthquake. In Profiles C and D the distribution of aftershocks does not suggest a clear rupture plane. Our result is not inconsistent with that of Mozziconacci et al. (2013a) in the sense that our rupture model provides an integrated picture of the seemingly complex rupture process for this earthquake. Comparing our result for event Taitung060415 with that of Mozziconacci et al. (2013b), we have the same indication on the geometry and orientation of the fault plane, as well as the rupture propagation. The rupture plane can be identified by a seismicity cluster in Profile E in Fig. 10(f). The aftershock distribution also shows a clear linear trend suggesting an east-dipping plane.

Waveform inversion for average source rupture 1291 Figure 10. (a) Summary plot of the point-source and finite-rupture models of the four moderate earthquakes in southeastern Taiwan. Seismicity and tectonic provinces are the same as in Fig. 1. Yellow stars are epicentres of the events. In each beachball the thick blue nodal plane is the actual fault plane in the finite-rupture model. Solid blue lines show the profiles A, B, C, D and E passing through the epicentres depicted in (b), (c), (d), (e) and (f), respectively. (b f) Profiles A E with the shear-speed structure of Kuo-Chen et al. (2012) shown by the background colour and contours at 0.25 km s 1 interval in dashed lines. Circles depict the relocated locations (Wu et al. 2008) of each main shock and its aftershocks with the colours and sizes of the circles indicating the M L magnitudes (see colour scheme in Profile A). The yellow stars show the main shock locations we use with CWB epicentres and depths determined in our grid search in 3-D structure. The thick blue nodal plane in each beachball (plotted in a side-way projection) indicates the actual fault plane. 5 CONCLUSION In this study, we adopt the SGT database approach to achieve the efficiency in calculating synthetic seismograms. With the established SGT database, any synthetics can be evaluated within a fraction of a second on a single-core desktop or laptop computer. In our implementation for earthquakes in Taiwan, we have established a SGT database using 3-D velocity structures and realistic surface topography, which provides more accurate synthetics for earthquake source inversions. The efficient algorithm for synthetic evaluations enable us to perform a two-step waveform inversion procedure to determine the point-source focal mechanisms and average finite-rupture models of earthquakes. Application of this procedure to four moderate earthquakes in southeastern Taiwan indicates that the rupture models are largely consistent with the detailed slip distributions for those events obtained previously in more elaborate studies. These comparisons demonstrate that our two-step approach, with 126 representative rupture scenarios in 3-D velocity structure with realistic topography, yield effective average rupture models, which effectively resolve the issue of fault-plane ambiguity intrinsic to point-source models. In addition, the average finite-rupture models improve the prediction of strong ground motion over point sources by taking the integrated directivity effect into account. As a result, our two-step grid search procedure based on SGT database provides an effective means for automatic, near real-time (within a few minutes) finite-source inversions without the need for high-performance computations, which has a high potential in practical earthquake early warning and hazard mitigation applications. ACKNOWLEDGEMENTS We thank Dr Mozziconacci for insightful discussions on the Taitung earthquakes. This work has been supported by the National Science Council of Taiwan under grants NSC101-2116-M-001-035 and NSC102-2116-M-001-013 (M-CH and LZ). All figures have been produced by the Generic Mapping Tools (GMT) of Wessel & Smith (1998). REFERENCES Aki, K. & Richards, P.G., 2002. Quantitative Seismology, 2nd edn, 700 pp., University Science Books. Amante, C. & Eakins, B.W., 2009. ETOPO1 1 arc-minute global relief model: procedures, data sources and analysis, NOAA Technical Memorandum NESDIS NGDC-24, 19 pp. Backus, G.E., 1977. Seismic sources with observable glut moments of spatial degree two, Geophys. J. R. astr. Soc., 51, 27 45. Berenger, J.A., 1994. Perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Geophys., 114, 185 200. Bukchin, B.G., 1995. Determination of stress glut moments of total degree 2 from teleseismic surface wave amplitude spectra, Tectonophysics, 248, 185 191.