Measurement of differential rupture durations as constraints on the source finiteness of deep-focus earthquakes

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jb004001, 2006 Measurement of differential rupture durations as constraints on the source finiteness of deep-focus earthquakes Linda M. Warren 1 and Paul G. Silver 1 Received 17 August 2005; revised 20 January 2006; accepted 1 March 2006; published 13 June [1] The physical mechanism of deep earthquakes can be constrained by identifying their fault planes. Resolving the fault plane ambiguity is a classical problem in seismology, and we present a method to distinguish the fault plane using observations of source finiteness. Source finiteness is observable on seismograms at different azimuths and distances, for single-event ruptures, as variations in the apparent rupture duration and, for complex ruptures, as differences in the traveltime delay between subevents. For each earthquake, the rupture duration (or traveltime delay) will be shortest in the direction of rupture propagation and longest in the opposite direction. Rather than measuring the actual rupture duration at each station, we use a cross-correlation technique that includes a stretching factor to measure the differential rupture duration between each pair of stations. These differential measurements then allow us to identify the rupture direction, rupture velocity, and fault plane for each earthquake. First, we test the method on two synthetic earthquakes, which represent earthquakes composed of one and two events. The method works well for both examples, although attenuation can bias the determined rupture direction for the single-event case. Next, we apply this method to P waves from broadband seismograms from four intermediate- and deep-focus earthquakes composed of two subevents: the 23 January 1997 Bolivian earthquake (M W 7.1, 276 km depth), the 27 October 1994 earthquake south of the Fiji Islands (M W 6.7, 549 km depth), the 21 July 1994 Japan Sea earthquake (M W 7.3, 471 km depth), and the 11 November 1998 Fiji Islands earthquake (M W 6.3, 149 km depth). Each focal mechanism contains a subvertical and a subhorizontal nodal plane. For three of the events, our analysis shows that rupture propagated subhorizontally, and we identify the subhorizontal nodal plane as the fault plane. For the smallest event, the rupture azimuth, but not the rupture dip, is well constrained, and we cannot conclusively identify the fault plane. Rupture velocities vary from 0.18 to 0.63 of the local P wave velocity. Citation: Warren, L. M., and P. G. Silver (2006), Measurement of differential rupture durations as constraints on the source finiteness of deep-focus earthquakes, J. Geophys. Res., 111,, doi: /2005jb Introduction [2] Since the discovery of deep earthquakes in the 1920s [Wadati, 1927], their physical mechanism has been debated. A viable mechanism needs to explain why seismological observations of deep and shallow earthquakes are so similar despite physical differences in their source regions. In the top km of the Earth, elastic strain energy accumulates and then is released by sudden brittle failure (an earthquake). Deeper in the Earth, the increased confining pressure and resulting increased normal stress on faults makes brittle failure less likely, so that material is expected to deform in a ductile rather than brittle mode [Leith and 1 Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, D. C., USA. Copyright 2006 by the American Geophysical Union /06/2005JB004001$09.00 Sharpe, 1936]. Deep earthquakes are only located within subducting slabs, so most proposed mechanisms for their initiation and propagation are derived from processes expected to occur in slabs. Scientists have proposed several mechanisms (including dehydration reactions [Raleigh and Paterson, 1965; Meade and Jeanloz, 1991], phase transformations [Kirby, 1987; Green and Burnley, 1989; Kirby et al., 1991], and thermal runaway instabilities [Ogawa, 1987; Hobbs and Ord, 1988; Kanamori et al., 1998]) to explain why these earthquakes occur at depths down to nearly 700 km. For each mechanism, rupture is likely to initiate and propagate in different ways. Some mechanisms appeal to the reactivation of faults formed at shallow depth, while others imply the creation of new faults. Thus, to distinguish between these mechanisms, we need measurements of properties such as the rupture velocity, rupture dimensions, rupture geometry, fault plane orientation, and seismic efficiency for numerous earthquakes from different subduction zones and depth ranges. We begin this 1of17

2 analysis by introducing a semiautomated method to identify the rupture directions and fault planes of deep earthquakes. [3] Since the seismic radiation pattern of an earthquake is symmetric, the focal mechanism provides two possible fault planes. Distinguishing the fault plane from the auxiliary plane of the focal mechanism is a classical problem is seismology. Surface ruptures, aftershock locations [e.g., Wiens et al., 1994], rupture directivity [e.g., Beck et al., 1995; Silver et al., 1995; Chen et al., 1996; Antolik et al., 1999; Tibi et al., 1999, 2002, 2003], and the second-degree moments [e.g., Silver, 1983; McGuire et al., 2001] are commonly used to constrain the fault plane. Since surface ruptures can only be mapped for some shallow earthquakes, and aftershocks, which are scarce in some subduction zones, do not always occur on the same plane as the main rupture [Willemann and Frohlich, 1987], we choose to primarily use the directivity of earthquake ruptures to distinguish their fault planes. The second-degree moments constrain the long-period parameters of an earthquake whereas our methodology takes advantage of the higher-frequency information provided by broadband data. Both unilateral and bilateral ruptures result in differences in the observed rupture duration at stations at different angles from the direction of rupture propagation [e.g., Haskell, 1964]. As a result, the symmetry of the focal mechanism is broken and the fault plane can be determined. [4] Most directivity studies rely on an analyst to pick specific features, such as the beginning and end of moment release and the peaks of individual subevents, on each seismogram and consistently identify them from one station to the next. Each of these features can then be located relative to each other or modeled with synthetic seismograms to determine the direction and speed of rupture propagation. This method works well for large earthquakes that are recorded with high signal-to-noise ratios at many stations. However, as the signal-to-noise ratio decreases it is harder to precisely pick these features. Instead, if these features can be automatically identified based on the similarity of the waveforms, it increases the consistency of the picks and reduces the amount of time the analyst needs to spend on each earthquake, allowing more and smaller events to be studied. [5] In this paper, we describe and apply a new, semiautomated method that uses the similarity of broadband waveforms to analyze the directivity of the rupture. As we demonstrate with several synthetic unilateral earthquakes, we estimate the differential rupture duration between each pair of seismograms by stretching or compressing one time series to maximize its cross-correlation coefficient with the other. After we apply this cross-correlation technique to all pairs of seismograms for an earthquake, we use the measured differences in rupture duration to calculate the rupture direction, and then we determine which nodal plane is most consistent with this direction. Finally, we successfully apply this method to broadband data from four teleseismic earthquakes of different sizes and depths in different subduction zones. 2. Methods 2.1. Earthquake Rupture Models [6] Evidence of source finiteness is apparent when seismograms from a variety of azimuths and distances are compared: the rupture duration and relative timing of subevents will vary among stations. How the apparent rupture durations vary depends on the rupture geometry. Following Bollinger [1968], in unilateral rupture, an earthquake begins at one end of a fault of length L and propagates entirely in one direction at constant velocity v r. The apparent rupture duration t varies with the angle q between the direction of rupture propagation and the takeoff vector to the station: tq ðþ¼ L L cos q v r c ¼ L 1 v r v r c cos q ¼ a 1 v r a cos q ; ð1þ where a is the rupture duration, c is the local seismic velocity, and a is the P wave velocity (substituted for c in the final line since we will be analyzing P waves). If we plot the seismograms as a function of q for a strike-slip earthquake with east-west and north-south nodal planes and horizontal rupture to the north on the north-south plane (Figure 1), a coherent increase in rupture duration with increasing angle is apparent when the seismograms are aligned relative to the actual rupture direction but not for other directions. Thus the directivity of the rupture can be used to distinguish the fault plane from the auxiliary plane of the focal mechanism. [7] If an earthquake rupture propagates bilaterally, it also generates coherent, predictable variations in rupture duration over the focal sphere, but in a different pattern than for a unilateral rupture. For a symmetric bilateral rupture, the rupture duration as a function of the angle from the rupture direction is tq ðþ¼a 1 þ v r a jcos qj ; ð2þ and directivity can be used to identify the fault plane in this case as well. [8] Finally, if an earthquake is well represented by an instantaneous point source (i.e., the rupture duration is short compared to the dominant period of the P wave), there will be little variation in rupture duration over the focal sphere. If either unilateral or bilateral rupture models are fit to such observations, the resulting apparent rupture velocity will be zero, and the fault plane cannot be distinguished. In general, the same effect will bias our estimates of rupture velocity to lower values Measurement of Differential Rupture Duration [9] Before we can use any of these rupture models to determine the rupture direction of an earthquake, we need to map the distribution of differential rupture durations over the focal sphere. We make these measurements of the differential rupture durations of P wave arrivals between each pair of seismograms independent of the rupture model and without any assumptions about the complexity of the source time function. The rupture duration on seismograms at angles q i and q j from the rupture direction is related by a stretching factor s ij : t(q i )=s ij t(q j ). Thus, for a unilateral 2of17

3 Figure 1. (left) Cartoon showing a map view of the angular variations in pulse width for a unilateral rupture propagating toward direction A. (right) The seismograms plotted as a function of the angular separation between candidate rupture directions A and B and the station takeoff vectors. For direction A, the true rupture direction, the pulse width increases with increasing angle. For direction B, there is no coherent angular variation in pulse width. rupture, the stretching factor between seismograms recorded at stations i and j is 1 v r s ij ¼ a cos q i 1 v : ð3þ r a cos q j When s ij >1.0 the rupture duration is longer at station i than at station j whereas s ij < 1.0 indicates a longer rupture duration at station j than at station i. A value of s ij = 1.0 means that the rupture duration is the same at stations i and j. [10] Following the cross-correlation formulation of VanDecar and Crosson [1990], we estimate the stretching factor between the seismograms recorded at stations i and j by cross correlating the two time series at different time offsets t off and with different stretching factors s. For a given time offset and stretching factor, the cross-correlation function is c ij ðt off ; sþ ¼ Dt T XT=Dt k¼1 y i ti P þ t 0 þ k Dt t off =2 y j tj P þ t 0 þ skdt þ t off =2 ; ð4þ where y i is the time series for the ith seismogram, Dt is the sampling rate of the seismogram, T is the time length of the correlation window, t i P is the preliminary arrival time at station i, and t 0 is the time interval between the preliminary arrival time and the beginning of the correlation window. To allow different stretching factors for time series j, we resample y j by interpolating between points. When s < 1.0, we increase the length of the time window to T/s so that the end of the wave arrival is not cut off. We locate the maximum (or minimum, if the seismograms are of opposite polarity) of the cross-correlation function. The maximum cross-correlation coefficient between seismograms from stations i and j is r ij ¼ c ij toff max ij ; s max ij p ffiffiffiffiffiffiffiffiffiffi ; ð5þ c ii c jj where t max off ij and s max ij are the time offset and stretching factor that maximize jc ij j. For highly correlated pairs, we use the corresponding stretching factor in our analysis. We typically choose a minimum cross-correlation coefficient of 0.9. We also confirm that s ij is approximately 1/s ji. They are not identical because we have resampled one of the time series with finite increments in the stretching factor. For an earthquake recorded at n stations, we have m n(n 1) measured stretching factors (after excluding pairs with i = j) Determination of Rupture Direction [11] To find the rupture direction most consistent with the measurements of differential rupture duration, we search over all candidate rupture unit vectors ^v on the focal sphere. For each ^v, specified by azimuth f r and dip g r from horizontal (in 10 increments for each), we compute the RMS misfit for values of v r /a from 0.0 to 0.8 for both unilateral and bilateral rupture models. Using only the selfconsistent, highly correlated measurements, we define the RMS misfit M as M ^v; v r a vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 X n X n;6¼i h i 2 ¼ t s measured ij s predicted ij : ð6þ m i¼1 j¼1 3of17

4 Figure 2. Focal mechanism for the synthetic earthquakes. The solid black diamonds indicate the takeoff vectors to the 60 uniformly distributed stations. The open black square shows the input rupture direction. For a unilateral rupture, the misfit is M ^v; v r ¼ a vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 X n X n;6¼i 1 v 3 r s measured ij a cos q 2 i u 6 4 m i¼1 j¼1 1 v 7 t r a cos q 5 j With three independent parameters and m/2 constraints, this system is highly overdetermined. For each candidate rupture direction in both rupture models, we determine the value of v r /a that minimizes the misfit. Next, we search over the minimum misfits to determine the best fitting rupture direction over the entire focal sphere. The misfit for a point source, given by M(^v, 0) for any direction ^v, is the ð7þ maximum possible misfit. Thus we compare the minimum misfit in each direction to the misfit for a point source and report this ratio. We also record the minimum misfit on each nodal plane of the focal mechanism, since the rupture must be on a fault plane, and identify the fault plane as the nodal plane with the lower misfit. The rupture velocity v r can be estimated by substituting appropriate values for P wave velocity a. [12] We estimate error in the determined rupture direction with bootstrap resampling [Efron, 1982]. We compute the rupture direction 100 times with different combinations of the stretching factors. For each computation, we take a random sample of n stations. The resampling scheme means that some stations are represented multiple times and others are excluded. Since different combinations are represented in each resample, the resampling provides a measure of the uncertainty in the rupture direction based on statistical scatter in the measurements. It does not account for error due to attenuation or deviations from the tested rupture model Two Synthetic Earthquakes [13] In this section, we illustrate our method with two synthetic unilateral earthquakes. To represent most intermediate- and deep-focus earthquakes, we use a focal mechanism with a subvertical and a subhorizontal nodal plane. The input rupture direction is on the subhorizontal plane (Figure 2). For these synthetic earthquakes, we generate seismograms at n = 60 uniformly distributed stations located at epicentral distances of from a 450-km-deep earthquake. In the first example, the earthquake is composed of a single event with directionally varying rupture duration. We generate seismograms with rupture durations determined by t(q) =4 ( cos q) seconds. In the second example, the earthquake is composed of two subevents and the time delay (in seconds) between them is determined by t(q) =3 ( cos q). Both subevents Table 1. Misfits (M) and Rupture Velocities (v r /a) for the Best Fitting Rupture Directions on Each Nodal Plane for the Synthetic Earthquakes Unilateral Rupture Bilateral Rupture Horizontal Horizontal Plane Vertical Plane Plane Vertical Plane Synthetic Description Input v r /a m a M v r /a M v r /a M v r /a M v r /a One event Directivity only Directivity and noise PREM 1-D attenuation only WS00 1-D attenuation only WS02 lateral attenuation only Directivity and PREM 1-D attenuation Directivity and WS00 1-D attenuation Directivity and WS02 lateral attenuation Two subevents Directivity only Directivity and noise PREM 1-D attenuation only WS00 1-D attenuation only WS02 lateral attenuation only Directivity and PREM 1-D attenuation Directivity and WS00 1-D attenuation Directivity and WS02 lateral attenuation a The variable m is number of measurements. 4of17

5 Figure 3. Variations in the cross-correlation coefficient with time offset and stretching factor. For the synthetic earthquake composed of two subevents, we plot synthetic seismograms for stations with takeoff vectors located (a) 134, (b) 30, and (c) 71 from the rupture direction. For each pair of seismograms, we cross correlate the time series at different time offsets and with different stretching factors to maximize the cross-correlation coefficient. The contour plots indicate the cross-correlation coefficients as a function of time offset and stretching factor for the seismogram pairs indicated in the lower left of each subplot (e.g., a-a, a-b, a-c, etc.). Solid contour lines indicate positive correlation coefficients whereas dashed contour lines indicate negative correlation coefficients. The star shows the maximum (or minimum) of the cross-correlation function. have a 2-s duration, but the amplitude of the second subevent is half the amplitude of the first subevent. These parameters result in the merging of the two pulses for propagation directions close to the rupture direction. We have also added realistic noise to the directivity synthetic seismograms by extracting the time series from a noise window preceding the P wave arrival for some of the seismograms we analyze later, and adding it to the synthetics in the measured signal-to-noise ratios. The input parameters and results for these synthetic earthquakes are summarized in Table 1, along with the results for additional experiments discussed in the auxiliary material. 1 1 Auxiliary material is available at ftp://ftp.agu.org/apend/jb/ 2005jb [14] For each pair of stations, we cross correlate the seismograms at different time offsets and with different stretching factors. For example, for the two-subevent example (Figure 3), we select the stations with takeoff vectors farthest from (134, time series a) and closest to (30, time series b) the rupture vector, as well as a station with an intermediate rupture-takeoff angle (71, time series c). When we cross correlate each seismogram with itself (subplots a-a, b-b, and c-c), the maximum of the crosscorrelation function is, of course, equal to 1.0 and located at a time offset of 0.0 and a stretching factor of 1.0. The maximum is more sharply defined when the two pulses are separated (i.e., a-a and c-c) than when they merge together (i.e., b-b). 5of17

6 Figure 4. For the synthetic earthquake composed of two subevents, plots of record sections for seismograms with cross-correlation coefficients 0.90 (or 0.90) for stations at rupture-takeoff angles of (a) 134, (b) 30, and (c) 71. The seismograms are aligned relative to the input rupture direction. As plotted, each seismogram covers 8 in rupture-takeoff angle. The actual rupture-takeoff angle is the midpoint of this range. In each plot, the bold grey seismogram is the original time series. The other seismograms have been stretched and offset according to the maximum of their cross-correlation functions with the original record. [15] When we cross correlate pairs of different seismograms, the stretching factors differ from 1.0, allowing us to determine their relative durations. For example, a visual comparison of seismograms a and c shows that time series c needs to be stretched out to align the second pulse of each record. The contour plots of the cross-correlation function for this pair of seismograms (subplots a-c and c-a) confirm that seismogram c needs to be lengthened by a factor of 1.7. Similarly, subplots b-c and c-b show that seismogram b needs to be lengthened by a factor of 1.4 to align the second pulses. Visually comparing seismograms a and b, we can see that time series b needs to be stretched out to align the second pulse of each record. However, subplots a- b and b-a show that the minimum of the cross-correlation function ( 0.878) instead corresponds to time series a being lengthened by a factor of 1.6. This stretching factor would align the first pulse of seismogram a with the merged pulses of time series b. While this is mathematically the best alignment, it is not physically correct. We can avoid such erroneous cross correlations with appropriate values for the minimum allowable cross-correlation coefficient. When we require a cross-correlation coefficient 0.9 (or 0.9) for these seismograms, we find, as expected, that they tend to be most highly correlated with records of similar angular distance from the rupture direction (Figure 4). Since each time series is stretched out and time-shifted according to the maximum (or minimum) of the cross-correlation function, the apparent duration of the individual subevents changes. For example, in Figure 4a, where we plot the records that are highly correlated with time series a (the station with the rupture-takeoff vector of 134 ), the peaks remain 4.4 s apart while their apparent duration increases from 2 s at 134 to 3 s at 82. [16] With a grid search over the entire focal sphere, we use the measured stretching factors to determine the best fitting rupture direction and rupture velocity for models of unilateral (Figure 5) and bilateral rupture (the auxiliary material). For the unilateral rupture model, the region of best fitting rupture directions (solid pink and red diamonds) is elongated along the input rupture azimuth of 120, but the 6of17

7 Figure 5. Misfit for rupture in each direction in the focal sphere for the synthetic unilateral earthquakes with noise composed of (a and b) one event and (c and d) two subevents. For each solution, the open brown circles mark the best fitting rupture directions on each nodal plane and over the entire focal sphere. The open black diamonds show the rupture directions found from bootstrap resampling. The misfit for unilateral rupture, indicated by the colored diamonds at the bottom of each column, is measured relative to the misfit for a point source. unilateral rupture model is fit to both examples, there is little scatter for the best fitting rupture direction on the horizontal nodal plane (Figures 6a and 6c). Again, the vertical nodal plane can be ruled out as the fault plane in each case because the measured and predicted stretching factors are effectively uncorrelated for the best fitting rupture direction on it (Figures 6b and 6d). Similarly, the measured and predicted stretching factors are uncorrelated for both planes with the bilateral rupture model (auxiliary material). [19] Finally, we confirm that the subhorizontal plane provides a better fit by aligning the seismograms relative to the best fitting rupture direction on each nodal plane (Figures 7 and 8 and auxiliary material). For the best fitting rupture model on the true fault plane, the increase is rupture duration with increasing rupture-takeoff angle is apparent in both examples. In contrast, the record sections for the best fitting rupture directions on the auxiliary planes for the unilateral rupture model and both nodal planes for the bilateral rupture model do not show any systematic changes in rupture duration with rupture-takeoff angle. This is particularly reassuring for the example composed of two subevents because of the very different shape of the seismograms for stations close to and far from the rupture direction. In comparison with the results for synthetics without added noise, the computed rupture direction rupture dip is not as tightly constrained. For both synthetic earthquakes, the observations are better fit by the unilateral rupture model, and the best fitting rupture direction coincides with the input rupture direction. This allows us to correctly identify the subhorizontal nodal plane as the fault plane. The minimum misfit (relative to a point source) is 0.28 for the single-event earthquake and 0.33 for the twosubevent example. In both cases, the vertical nodal plane has high misfit values (0.91 relative to a point source) for all possible rupture directions on it, allowing us to exclude it from being the fault plane. The rupture directions we find from bootstrap resampling are tightly clustered around the true rupture direction, with a range in dip and 10 range in azimuth. [17] The lowest misfit values (0.34 and 0.54, for the oneand two-subevent cases, respectively) for the bilateral rupture model are larger than for the unilateral rupture model, and the corresponding rupture directions do not fall near either nodal plane. The lowest misfit value on either nodal plane is Since this is little improvement over a point source, it shows that the rupture could not have propagated bilaterally. [18] The better fit for unilateral rupture on the horizontal nodal plane is also apparent in scatterplots comparing the measured and predicted stretching factors for the best fitting rupture direction on each nodal plane (Figure 6). When the Figure 6. Comparison of the measured and predicted stretching factors for the best fitting rupture directions on the horizontal and vertical nodal planes for the synthetic earthquakes with noise. For the single-event, unilateral earthquake, the measured stretching factors are compared with the predicted stretching factors for the best fitting rupture direction for models of unilateral rupture on the (a) horizontal and (b) vertical nodal planes. For the twosubevent, unilateral earthquake, the measured stretching factors are compared with the predicted stretching factors for the best fitting rupture direction for models of unilateral rupture on the (c) horizontal and (d) vertical nodal planes. 7of17

8 Figure 7. Record sections for the synthetic, single-event, unilateral earthquake with noise aligned relative to the best fitting rupture direction on each nodal plane ((a) horizontal plane; (b) vertical plane) for a model of unilateral rupture. remains the same, although the misfit increases and the rupture vector error ellipse is larger. [20] While the rupture speed is not our primary parameter of interest, we estimate it as part of the misfit calculation. For the single pulse we find v r /a = 0.27, which is within the bootstrap error. For the two-subevent case, we find v r /a = 0.46 rather than the input v r /a = We underestimate v r /a because of the duration of each subevent and the interaction between the two pulses. However, the underestimation of rupture velocity does not affect the identification of the rupture direction or fault plane Effect of Focal Mechanism Changes [21] For the two-subevent example, we have modeled the effect of a rotation in the focal mechanism between subevents. If the focal mechanism changes, the relative amplitudes and polarities of the two pulses can change. The changes are largest for stations on or near the nodal planes, so the size of this effect is very dependent on the orientations of the nodal planes and their positions relative to the station takeoff vectors. For example, for the two-subevent example discussed above, if both the strike and dip of the focal mechanism are rotated by 15, we can still recover the input rupture direction. However, the misfit for this direction increases and several records are erroneously aligned. To reduce errors from focal mechanism rotations in our analysis of actual earthquakes, we exclude nodal stations when changes in the focal mechanism are visible in the seismograms Effect of Attenuation [22] As seismic waves propagate through the Earth, highfrequency energy is dissipated more quickly than lowfrequency energy, so pulses will be broadened and the apparent rupture duration lengthened. This effect can appear similar to the pulse broadening caused by a unilateral rupture, particularly for single-event earthquakes. To determine how much attenuation could bias our directivity results, we create additional synthetic seismograms for both constant duration and unilaterally rupturing earthquakes in combination with one-dimensional and laterally varying attenuation models. The results of the experiments are summarized in Table 1, and described in more detail in the auxiliary material. [23] We analyze the effects of two different onedimensional attenuation models (PREM [Dziewonski and Anderson, 1981] and Warren and Shearer [2000] (hereafter referred to as WS00)) that have very different range dependencies. In PREM, the path-integrated attenuation, which is quantified with t*= R dt/q, increases by 0.5 s for P waves at 8of17

9 Figure 8. Record sections for the synthetic, two-subevent, unilateral earthquake with noise aligned relative to the best fitting rupture direction on each nodal plane ((a) horizontal plane; (b) vertical plane) for a model of unilateral rupture. epicentral distances from 30 to 90. In WS00, t* increases by 0.1 s over the same distance range. For synthetics with directivity and WS00 attenuation, which is derived from data in the same frequency band as our seismograms and therefore more representative of the actual error due to attenuation, the calculated rupture direction varies by just 10 from the true rupture direction. Since the amount of attenuation increases with increasing distance from the earthquake, the rupture direction appears more vertically up than it actually is. [24] To determine the effect of lateral variations in attenuation, we use the upper mantle attenuation model of Warren and Shearer [2002] (hereafter referred to as WS02) and an earthquake in Tonga. In this model, t* variations span 0.5 s. For the single-event synthetics, the best fitting rupture direction is 10 from the input rupture direction, and the range of best fitting rupture directions is constrained to 10 in azimuth and 40 in dip. Since the lateral variations in attenuation are coherent over only small portions of the focal sphere, they tend to average out and not hinder the determination of the rupture direction. [25] Attenuation has an even smaller effect on ruptures composed of two or more subevents since the time separation between the subevents is not changed even though the duration of each subevent is lengthened. As a result, we concentrate on earthquakes composed of multiple subevents in the following analysis. 3. Analysis of Four Teleseismic Earthquakes 3.1. Data Selection and Processing [26] To illustrate the method described above, we analyze broadband seismograms from four intermediate- and deepfocus earthquakes. We selected these earthquakes, which are Table 2. Earthquake Information for the Four Analyzed Earthquakes a Event Date Time, UT Latitude Longitude Depth, km M W T,s n m 1 23 Jan Oct Jul Nov a T, duration of signal analyzed; n, number of stations; and m, number of measurements. 9of17

10 Figure 9. Locations and focal mechanisms of the four analyzed earthquakes. listed in Table 2 and mapped in Figure 9, because they span different magnitudes (M W ), depths ( km), and subduction zones. While we have selected earthquakes 100 km depth because the P wave arrival precedes the depth phases by >25 s, this method can also be applied to shallower events if the direct arrival can be isolated. For each earthquake, we select seismograms recorded at epicentral distances of We remove the known instrument response and integrate each time series to displacement. Next, we pick the P wave arrival time, discarding records with poor signal-to-noise ratios or contamination from other phases (such as PcP). We use these picks as the preliminary arrival times in the cross-correlation procedure. We also note the maximum rupture duration to determine the window length for cross correlation. Plots for the unilateral rupture model are included in this article, while the corresponding plots for a bilateral rupture model, which generally yields poorer fits to the observations, are provided in the auxiliary material. The best fitting rupture directions for both rupture models are summarized in Table The 1997 Southern Bolivia Earthquake [27] On 23 January 1997, an M W 7.1 earthquake occurred at a depth of 276 km beneath southern Bolivia. The station distribution, as shown in Figure 10a, is more concentrated in the northern portion of the focal sphere. However, this station distribution is sufficient to allow us to identify the horizontal nodal plane as the fault plane and determine that rupture propagation was more unilateral than bilateral. The misfit for each direction on the focal sphere, plotted in Figures 11a 11b for a unilateral rupture model, shows that the rupture propagated toward an azimuth of With a horizontal (±40 ) rupture dip for the unilateral rupture model, this range in rupture directions is most consistent with rupture on the subhorizontal nodal plane. The middle of the cloud of bootstrap-resampled rupture directions falls on the subhorizontal nodal plane, and none Table 3. Best Fitting Rupture Directions and Fault Planes, Assuming Models of Unilateral and Bilateral Rupture Propagation, for the Four Analyzed Earthquakes a Strike, deg Dip, deg Rake, deg f r, deg g r, deg v r /a M Event 1 Unilateral rupture Fault plane Auxiliary plane Focal sphere Bilateral rupture Fault plane Auxiliary plane Focal sphere Event 2 Unilateral rupture Fault plane Auxiliary plane Focal sphere Bilateral rupture Fault plane Auxiliary plane Focal sphere Event 3 Unilateral rupture Fault plane Auxiliary plane Focal sphere Bilateral rupture Fault plane Auxiliary plane Focal sphere Event 4 Unilateral rupture Nodal plane Nodal plane Focal sphere Bilateral rupture Nodal plane Nodal plane Focal sphere a M, misfit; f r, rupture azimuth; g r, rupture dip; and v r /a, rupture velocity. 10 of 17

11 Figure 10. Lower hemisphere focal mechanisms and station takeoff directions for the (a) 23 January 1997, (b) 27 October 1994, (c) 21 July 1994, and (d) 15 November 1998 earthquakes. of the directions overlap with the subvertical nodal plane. If the observed differences in rupture duration are modeled with a bilaterally propagating rupture, the rupture azimuth is similar to the unilateral case but the rupture dip does not intersect either nodal plane, allowing us to exclude bilateral rupture as a viable rupture model for this earthquake. [28] To confirm this fault plane identification, we compare the measured and predicted stretching factors for the best fitting rupture direction on each nodal plane (Figures 12a 12b). The scatter is significantly less for unilateral rupture on the subhorizontal plane than for unilateral rupture on the subvertical plane or bilateral rupture on either plane. [29] The better fit for unilateral rupture on the subhorizontal plane can also be seen by comparing record sections aligned relative to the best fitting rupture direction on each nodal plane (Figure 13). The earthquake is composed of two subevents, with the second peak arriving s after the first peak. The time delay between the two subevents increases more coherently for the rupture direction on the subhorizontal nodal plane than on the subvertical nodal plane. This change in arrival time of the two subevents can Figure 11. Misfit beach balls for unilateral rupture for the 23 January 1997, earthquake in southern Bolivia ((a) horizontal plane; (b) vertical plane), the 27 October 1994, earthquake south of the Fiji Islands ((c) horizontal plane; (d) vertical plane), the 21 July 1994, earthquake in the Japan Sea ((e) horizontal plane; (f) vertical plane), and 15 November 1998, earthquake in the Fiji Island Region ((g) horizontal plane; (h) vertical plane). For each solution, the open brown circles mark the best fitting rupture directions on each nodal plane and over the entire focal sphere. The open black diamonds show the rupture directions found from bootstrap resampling. The misfit, indicated by the colored diamonds at the bottom of each column, is measured relative to the misfit for a point source. 11 of 17

12 Figure 12. Comparison of the measured and predicted stretching factors for the best fitting rupture directions on the horizontal and vertical nodal planes for the analyzed earthquakes. For the 1997 southern Bolivia earthquake, the measured stretching factors are compared with the predicted stretching factors for the best fitting rupture direction for a model of unilateral rupture on the (a) horizontal and (b) vertical nodal planes. For the 1994 earthquake south of the Fiji Islands earthquake, the measured stretching factors are compared with the predicted stretching factors for the best fitting rupture direction for a model of unilateral rupture on the (c) horizontal and (d) vertical nodal planes. For the 1994 Japan Sea earthquake, the measured stretching factors are compared with the predicted stretching factors for the best fitting rupture direction for a model of unilateral rupture on the (e) horizontal and (f) vertical nodal planes. For the 1998 earthquake in the Fiji Islands region, the measured stretching factors are compared with the predicted stretching factors for the best fitting rupture direction for a model of unilateral rupture on the (g) horizontal and (h) vertical nodal planes. be explained by v r = 0.55a, which corresponds to a rupture velocity of 4.8 km/s. [30] This event was previously studied by Tibi et al. [2002], and our rupture direction is consistent with theirs. From studying the directivity of P and SH waves and modeling waveforms, they determined that the second event was located at an azimuth of 331 ±4 from the first event and at the same depth. Even though this direction lies on the subhorizontal nodal plane, the authors could not unequivocally distinguish it as the fault plane because the rupture vector lies near the intersection of the two nodal planes. With our error estimates, however, we are confident that we can identify the subhorizontal plane as the fault plane The 1994 Earthquake South of the Fiji Islands [31] On 27 October 1994, an M W 6.7 earthquake occurred south of the Fiji Islands at a depth of 549 km. As Figure 10b shows, the station coverage is denser to the west than the east. As a result, the lowest misfit values and the bootstrapresampled rupture directions cover a relatively large portion of the focal sphere (Figures 11c 11d). Still, we are able to identify the subhorizontal nodal plane as the fault plane because the misfit for every possible rupture direction on the subvertical nodal plane is equivalent to a point source for both the unilateral and bilateral rupture models. [32] For unilateral rupture, the best fitting rupture direction has rupture propagating subhorizontally toward an azimuth of 300, which is 20 from the closest direction on the subhorizontal plane. As Figures 12c 12d demonstrate, the measured and predicted stretching factors are well correlated for rupture on the subhorizontal nodal plane but not on the subvertical nodal plane. Rupture on the subvertical nodal plane can explain none of the measured differences in stretching factors and requires v r /a = For rupture on the subhorizontal plane, we estimate v r /a = 0.18 (1.8 km/s). [33] The better fit for the horizontal plane is also apparent in the record sections (Figure 14) aligning the seismograms relative to the best fitting rupture direction on each nodal plane. The earthquake is composed of two subevents, with the relative amplitude of the second subevent appearing barely above the noise level on some seismograms and being the same amplitude as the first subevent on others. For stations with takeoff vectors closest to the rupture direction, the two pulses are separated by 3 s, whereas for stations with takeoff vectors farthest from the rupture direction the two pulses are separated by 5 s. The time separation between the two subevents clearly increases with increasing rupture-takeoff angle. The same pattern is not 12 of 17

13 Figure 13. Record sections for the 23 January 1997 earthquake in southern Bolivia aligned relative to the best fitting rupture direction on each nodal plane ((a) horizontal plane; (b) vertical plane). The gray lines on the lower and upper axes indicate the time window analyzed. seen when the seismograms are aligned relative to the best fitting rupture direction on the subvertical plane. [34] When we compare the observed differences in rupture duration with a model of bilateral rupture, most of the possible rupture directions cover 140 in azimuth and 50 in dip. As in the previous example of the 1997 southern Bolivia earthquake, these directions do not overlap with either nodal plane. In addition, the scatterplots for the best fitting rupture direction on both nodal planes (Figures 12g 12h) show that bilateral rupture can explain little of the measured variations in rupture duration. Thus we conclude that this earthquake rupture could not have propagated bilaterally on either nodal plane The 1994 Japan Sea Earthquake [35] On 21 July 1994, an M W 7.3 earthquake occurred at a depth of 471 km beneath the Japan Sea. As shown in Figure 10c, the earthquake is recorded by stations at welldistributed azimuths and epicentral distances. The good station distribution allows us to narrow the rupture azimuth and dip more than in the previous examples. With our directivity analysis for unilateral rupture (Figures 11e 11f), we find that the rupture propagated toward an azimuth of along a subhorizontal vector. The rupture dip is not as well constrained, as we saw with the synthetics. Unfortunately, the rupture azimuth runs through the intersection of the two nodal planes. While the best fitting rupture direction on the subhorizontal nodal plane gives a slightly better fit than the best fitting rupture direction on the subvertical plane (0.68 vs. 0.69), they are too close to use this information to identify the true fault plane. Similarly, scatterplots (Figures 12e 12f) of measured versus predicted stretching factors show little difference. Record sections aligned relative to the best fitting rupture directions on each nodal plane (Figure 15), which vary by only 35, show that the earthquake was composed of two subevents and that, for both candidate rupture directions, moveout of the second event relative to the first is visible. For either fault plane, we find that the rupture velocity is 1/3 of the P wave velocity, or 3.1 km/s. [36] The bilateral rupture analysis for this event excludes bilateral rupture on the subhorizontal plane. While misfit (0.71) for the best fitting rupture direction on the subvertical plane is only slightly larger than for unilateral rupture on either plane, there is only one station located more than 90 from the rupture direction. Thus there is not enough information to determine whether the rupture propagated unilaterally or bilaterally. 13 of 17

14 Figure 14. Record sections for the 27 October 1994 earthquake south of the Fiji Islands aligned relative to the best fitting rupture direction on each nodal plane ((a) horizontal plane; (b) vertical plane). The gray lines on the lower and upper axes indicate the time window analyzed. [37] Previous studies of this event also determined that rupture propagated toward the south-southeast. Antolik et al. [1999] inverted P and SH arrivals for the distribution of slip over the fault plane for rupture on each nodal plane. Since the misfit was significantly lower for slip on the subhorizontal than on the subvertical nodal plane, they were able to identify the fault plane in addition to the rupture vector. Their method allows slip to occur over a plane whereas we restrict rupture to a vector, and this difference probably let them confidently identify the fault plane. Chen et al. [1996] and Tibi et al. [2003] used directivity and subevent relative locations to determine a rupture azimuth close to the intersection of the two nodal planes, but excluded the subvertical plane from being the fault plane because they felt that it was unlikely for horizontal rupture to occur on a vertical plane The 1998 Fiji Islands Earthquake [38] On 11 Novemember 1998, an M W 6.3 earthquake occurred at a depth of 149 km in the Fiji Islands region. Since this event is smaller and shallower than the previous examples, it is not recorded with a high signal-to-noise ratio at as many stations. As shown in Figure 10d, there are no stations in the southeastern portion of the focal sphere. Still, the earthquake is composed of two subevents and our directivity analysis can constrain the rupture direction to an azimuth of Unfortunately, as shown in Figures 11g 11h, the range of possible rupture dips for a unilaterally propagating rupture includes both nodal planes. Since the misfit (0.68) is equivalent on both planes, we cannot distinguish which was the fault plane. The equivalence of the two directions can also be seen in scatterplots comparing the measured and predicted stretching factors (Figures 12g 12h). When the seismograms are aligned relative to each of these rupture directions (Figure 16), the time delay between the two subevents increases with increasing rupture-takeoff angle. [39] In the bilateral rupture analysis, we find that rupture propagated in the same azimuthal direction as the unilateral case. The dip of the rupture vector is more tightly constrained and only compatible with rupture along the subhorizontal nodal plane: the misfit (0.69) is lower on the subhorizontal plane than on the subvertical plane (0.83). For the solution on the subhorizontal plane, rupture propagates toward an azimuth of 81 at a dip of 15 above horizontal and toward an azimuth of 261 at a dip of 15 below horizontal, and the time between the subevents decreases as the rupture-takeoff angle increases. However, there is no 14 of 17

15 Figure 15. Record sections for the 21 July 1994 earthquake beneath the Japan Sea aligned relative to the best fitting rupture direction on each nodal plane ((a) horizontal plane; (b) vertical plane). The gray lines on the lower and upper axes indicate the time window analyzed. evidence that the rupture propagated bilaterally rather than unilaterally: the misfit for bilateral rupture on the subhorizontal plane is equivalent to the misfit for unilateral rupture on either plane, and the rupture-takeoff angles for bilateral rupture on the subhorizontal plane do not cover both halves of the double cosine curve. 4. Summary [40] To determine which nodal plane of the focal mechanism slipped during an earthquake, we analyze the directivity of the rupture from broadband seismograms. In this paper, we describe a novel, semiautomated method to determine the rupture direction of an earthquake and thereby identify its fault plane. This method, which does not depend on an analyst to visually pick individual subevents of the rupture or precisely identify the beginning and end of moment release on each seismogram, automatically measures the differential rupture duration of P waves between all stations recording the earthquake. Like standard multichannel cross-correlation methods, we cross correlate each pair of seismograms with different lag times to maximize the crosscorrelation coefficient. We also search over an additional parameter, the stretching factor, which we define as the ratio of the absolute rupture durations observed at the two stations being compared. The location of the maximum of the crosscorrelation function provides the differential rupture duration between each pair of stations. Using a grid search over all potential rupture directions and rupture velocities on the focal sphere, we compare the measured differences in rupture duration with models of unilateral and bilateral rupture to determine the rupture direction and fault plane. [41] To illustrate the method, we first apply it to two synthetic earthquakes with subhorizontally propagating unilateral ruptures. In these examples, which are for an earthquake composed of a single event with variable rupture duration and an earthquake composed of two subevents with varying time separation between the pulses, we generate synthetic time series at 60 uniformly distributed stations. When the synthetics include only the directivity signal, we can recover the input rupture direction and fault plane in both cases. When we add an attenuation model to the synthetics, the dip of the rupture vector becomes biased by just 10 for realistic attenuation models. This bias does not prevent us from identifying the input rupture direction and fault plane for both cases. [42] We have selected four intermediate- and deep-focus earthquakes, each composed of two subevents with variable time separation between pulses, to demonstrate the validity and usefulness of the method. Multiple subevents are a 15 of 17