Short period and broadband teleseismic waveform data and three-component strong motion records were analyzed to obtain the source parameters of the 1991 Sierra Madre earthquake. Close-in strong motion velocity records (analyzed from 5 s to 5 Hz) show two distinct pulses about 0.35 s apart, requiring some rupture complexity. The near-field shear wave displacement pulse from this event has a relatively short duration (about 1 s) for the magnitude of the event, requiring a particularly high average stress drop. To further constrain the rupture process, the data were used in a finite fault source inversion to determine the temporal and spatial distribution of slip. We chose a fault plane orientation striking S 62 degrees W and dipping 50 degress toward the northwest as required by the distribution of aftershocks, the first motion mechanism and the teleseismic bodywave point source inversion. In addition to the aftershock locations, depth constraints are provided by teleseismic short period and broadband recordings which require a centroid depth of 10--11 km. Our inverse modeling results indicate that both the teleseismic and strong motion data sets can be fit with a compact rupture area, about 12 km^2, southwest and up-dip from the hypocenter. The average slip is approximately 50 cm, and the maximum slip is 120 cm. The seismic moment obtained from either of the separate data sets or both sets combined is about 2.8 +- 0.3 x 10^24 dyne-cm and the potency is 0.01 km^3. Using the area of significant slip estimated from the finite fault inversion, the resulting stress drop is on the order of 150--200 bars.
The near-field shear wave displacement pulse from this event has a relatively short duration (about 1 s) for the magnitude of the event, requiring a particularly high overall stress drop [ Kanamori et al., 1991]. Moreover, source parameters determined from regional and local waveforms substantiate the short duration and compact nature of the source [ Dreger and Helmberger, 1991]. Hence, we are interested in quantifying the spatial and temporal distribution of slip in order to better understand the source of damaging ground motions from this thrust fault and other potential blind thrust earthquakes. The recent series of deep strike-slip and blind thrust events within and around the heavily populated Los Angeles metropolis [ Hauksson, 1991; Hauksson and Jones, 1991 b], and the potential for similar and larger events in the future mandates a careful examination of data recorded from the Sierra Madre earthquake.
Fortunately, a large number of acceleration recordings were recovered for this earthquake due to the density of California Division and Mines and Geology (CDMG) and USGS instruments in the epicentral area. The 1987 Whittier Narrows earthquake (ML=5.9) also produced useful local strong motion data which were analyzed by Hartzell and Iida [1990]. They modeled the overall rupture area from the Whittier Narrows earthquake as approximately 70 km^2, with slip concentrated over a smaller region of perhaps 25 km^2. Here we are testing the resolving power of the strong motion data and teleseismic data for a somewhat smaller source dimension than for the Whittier Narrows earthquake. It will be useful to compare the rupture process of these two thrust style earthquakes in the Los Angeles region.
In order to better understand the ground motion behavior and hazards in Los Angeles, it is very helpful to remove the contribution of the effects of source finiteness and rupture heterogeneity from the effects of ground motion amplification due to propagation and site response. When these source effects are distinguished from others, it is possible to scale up the source of moderate-sized events like the 1987 Whittier Narrows and the 1991 Sierra Madre earthquakes and make reasonable estimations of ground motions and attenuation relationships specific to the Los Angeles region from larger anticipated earthquakes (C. K. Saikia, Simulated ground motions in Los Angeles due to a large hypothetical earthquake on the Elysian thrust fault zone, Submitted to Bulletin of the Seismological Society of America, 1991). Here we attempt to characterize the source contribution of the Sierra Madre earthquake by modeling local ground motion and teleseismic waveforms. Some independence from the effects of local path and site contamination is provided by the teleseismic data which are less likely to be seriously contaminated by site effects and, in general, show more stability in overall amplitude than the strong motion data. But, it is also important to consider only those strong motion stations which have coherent waveforms and appear to be relatively free from site effects. The strong motion velocity records we model are dominated by energy with frequencies of about 3--5 Hz. This frequency band is at the cross-over between longer periods which, on average, show amplification of soil sites relative to rock sites, and higher frequencies for which the relationship is reversed [ Aki, 1988]. Therefore, we expect some independence from site amplification from the near-source strong motion data used here due to the band-limited nature of the velocity waveform observations.
Although there is fairly good azimuthal coverage of the source from the strong motion stations shown in Figure 2, there is less adequate coverage for stations within 20 km of the epicenter. In our waveform inversions we initially gave more weight to the close-in stations by considering the vertical, radial and tangential components, whereas for the more distant stations normally we use only the tangential component. This approach is justified, since the close-in stations show very similar larger amplitude waveform features, which are not easily identified in the more distant, smaller amplitude stations (Figure 3). In subsequent modeling we also gave equal weight to the distant stations to analyze their contribution to the source model. Note the similarity of waveforms from the close-in stations. In particular, the tangential records at COG, ETN, GSA, GVR and MTW begin with distinct, double SH arrivals easily traced from record to record (Figure 3). The time separation between the two arrivals is nearly identical at these stations, consistently 0.35 s. The same feature is observed on the radial components but is not as clearly recognized. Further, note the corresponding feature on the tangential components at stations MOR, MTB, and UPL.
Examination of aftershocks at stations MTW and GSA indicate that several of the aftershocks also exhibit double arrivals at the time of the direct SH wave. The two arrivals for the aftershocks were, in general, roughly 0.15 to 0.20 s apart but show some variability. Initially, we attributed the second of the two arrivals to a reflection from a deeper, near-source velocity interface. If so, the variation in time separation was due to varying source depths and hence varying relative locations with respect to the reflecting interface. However, the double arrival nature of the mainshock velocity records cannot be attributed to propagation complexities alone. Since this feature is observed at stations ranging in location from directly above the source area (COG and MTW) as well as along a southwest profile of stations (ETN, GSA, and GVR in Figure 2) at distances up to at least 26 km from the epicenter, a near-source reflection is not a likely candidate. While it is possible to generate a reflection comparable in amplitude to the incident phase with a simple dipping structure and a reasonable impedance contrast, it requires a near-critical incidence and hence will not be sufficient for producing comparable reflection amplitudes over the range of distances at which this feature is observed. Moreover, the relative timing between the phases would vary systematically as a function of distance, and a characteristic phase shift would be apparent. It is also difficult to reproduce the comparable relative amplitudes of the two arrivals from a reflection below the source at stations almost directly above the source. From these considerations, we attribute the double arrival of the mainshock to source complexity. The source of the double arrivals observed on some of the aftershocks remains yet unresolved.
In this study we represent the Sierra Madre rupture area as a 7-km-long plane striking S 62 degrees W and dipping 50 degrees toward the northwest (Figure 1 a). The fault extends from a depth of 9.4 km to 14.0 km (Figures 1 b and 1 c), giving a downdip width of 6 km. We fix the hypocentral depth to the value of 12.0 km determined by Hauksson and Jones [1991 a] using the Southern California Seismic Network (SCSN) short period array and chose the overall dimensions of the fault to enclose the region of major aftershock activity. The hypocentral depth is slightly shallower than the best point source depth as determined from the short period data, suggesting an updip rupture propagation. The strike and dip values of our fault plane were chosen from our point source broadband teleseismic inversion results, discussed in a later section. This geometry is nearly identical to that determined from the first-motion solution determined from the short-period array [ Hauksson and Jones, 1991 a] and fits neatly into the distribution of aftershocks (Figures 1 a,1 b,1 c). We discretized the fault area into 10 subfault elements along strike and 10 elements downdip, giving each subfault a length of 0.7 km and a downdip width of 0.6 km. The subfault elements are shown as a gridded overlay on later figures, which display the modeled slip distribution.
The synthetic ground motion contribution for each subfault is computed using the Green's function summation and interpolation method of Heaton [1982] and is only briefly summarized here. The subfault motions are obtained by summing the responses of a number of point sources distributed over the subfault. We sum nine equally spaced point sources appropriately lagged in time to include the travel time difference due to the varying source positions and to simulate the propagation of the rupture front across each subfault. In all, 900 point sources are summed to construct the teleseismic and strong motion synthetics at each station. We fix the rake at 82 degrees, nearly pure thrusting with the northwest side moving up and to the southwest. A variable rake was tested, but the results indicated very little variation from the constrained value, so we prefer the fixed model since it has fewer free parameters.
The point source responses, or Green's functions, for teleseismic P or SH body wave synthetic seismograms are computed using the generalized ray method [ Langston and Helmberger, 1975]. We include the responses of all rays up to two internal reflections in a layered velocity model, including free surface and internal phase conversions. An attenuation operator [ Futterman, 1962] is applied with the attenuation time constant t* equal to 0.75 and 3.5 s for P and SH waves, respectively.
The point source responses for the strong motions, including near-field terms, are computed for a layered velocity model for frequencies up to 10 Hz using the frequency-wavenumber (FK) methodology. This method allows the computation of complete waveforms and also includes the effects of attenuation. In practice, we calculate a master set of Green's functions for increments in depth from 9 to 15 km and for ranges between 0 and 45 km to allow for the closest and furthest possible subfault-station combinations. Then for each subfault-station pair the required subfault response is derived by the summation of nine point source responses obtained by the linear interpolation of the closest Green's functions available in the master set. The linear interpolation of adjacent Green's functions is performed by aligning the waveforms according to their shear-wave travel times. The amount of slip on each subfault is determined in the waveform inversion.
The velocity model used to compute the FK Green's functions is given in Table 3. \marginboxed{Table 3} The P wave velocities were obtained by Hauksson and Jones [1991a] using a joint inversion of the aftershock data for location and velocity structure. We have also added a thin, slower layer to this model to better approximate the average site velocity just beneath the strong motion stations. S wave velocities were obtained by assuming that the structure is a Poisson solid. The Q structure in Table 3 was estimated based on the velocity structure and was made to be consistent with the determinations for the total path attenuation to TERRAscope station PAS for this earthquake [K. F. Ma, personal communication, 1991]. The velocity model used to compute the teleseismic Green's functions does not include the shallowest 0.3 km thick layer employed for the local strong motion modeling.
The subfault synthetics are convolved with a dislocation time history which we represent by the integral of an isosceles triangle with a duration of 0.2 s. This slip function was chosen based on a comparison of the synthetic velocity pulse width for a single subfault with the shortest duration velocity pulse width observed. As suggested by Heaton [1990], this is likely a maximum duration for the slip function, and in fact, using a 0.1-s triangle matches the data equally well. Slip durations longer than 0.2 s, however, substantially increase the waveform misfit.
The rupture velocity is assumed to be a constant 2.7 km/s, or about 75\% of the shear wave velocity in the source region (Table 3). Some flexibility in the rupture velocity and slip time history is obtained by introducing time windows [ Hartzell and Heaton, 1983]. Each subfault is allowed to slip in any of three identical 0.2-s time windows following the passage of the rupture front thereby allowing for the possibility of a longer slip duration or a locally slower rupture velocity. In this study each time window is separated by 0.1 s, allowing a small overlap of the 0.1-s duration subfault source-time function. With a constant rupture velocity this model implies that, at most, a ribbon having a width of 1.1 km is slipping at any one time. Due to the short duration and high frequency nature of the ground velocities, the slight travel time inaccuracies inevitable in synthetics from a one-dimensional model would result in poor alignment between synthetics and observations and thus detract from the inversion for slip distribution. For this reason, we align the synthetics with the impulsive initial S wave arrival in the data (best seen in the recorded accelerations) and do not try to match the absolute timing.
A constrained, damped, linear, least squares inversion procedure is used to obtain the subfault dislocation values which give the best fit to the strong motion velocity waveforms. The inversion is stabilized by requiring that the slip is everywhere positive and that the difference in dislocation between adjacent subfaults (during each time window) as well as the total moment is minimized. These constraints have been previously discussed by Hartzell and Heaton [1983]. Both the strong motion observations and subfault synthetic seismograms are bandpass filtered from 0.2 to 5.0 Hz with a zero-phase Butterworth filter and are resampled at a uniform time step of 20 samples/s. The teleseismic data were similarly filtered from 0.025 to 2.5 Hz and are resampled at 10 samples/s. We modeled the first 3 s of the strong motion records and 10 s of the teleseismic data.
To be certain that the fault geometry suggested by the aftershock distribution and local first motion mechanism adequately reflects longer period moment release for this earthquake, we first performed a point source inversion of the available P and SH broadband displacement waveforms using the methodology of Kikuchi and Kanamori [1991]. Initially, we constrained the time function to be a simple 0.75-s triangle and determined the best fitting mechanism and the source depth. The observed (top) and synthetic (bottom) waveforms resulting from this inversion are shown in Figure 6. \marginboxed{Figure 6} The best point source was determined to be at a depth of 10.7 km with a moment of 2.8 x 10^24 dyne-cm. The mechanism, as shown, has a strike, dip, and rake determined to be 243 degrees, 49 degrees, and 82 degrees, respectively. Although most of the SH waveforms are not impressive for this event, the nodal nature of this arrival at MAJO and the SH waveform at ALE do play roles in constraining the mechanism (Figure 6). Next, by fixing the mechanism and allowing for multiple point source locations relative to the hypocenter, the data required all the energy release to be updip and southwest of the hypocenter.
The combined strong motion and teleseismic rupture model is similar to that derived from the teleseismic data alone in that slip is concentrated updip and towards the southwest. Again, the region of substantial slip is very limited in size, but more so than from the teleseismic data alone. The moment is 2.8 x 10^24 dyne-cm. There is a trade-off between the fit to the strong motion data and the relatively narrow depth interval required to fit the depth phases of the teleseismic waveforms. Inverting only the strong motion data yields an asperity similar to the combined inversion, although the centroid is forced slightly deeper (12 km) and southwest of the hypocenter rather than southwest and updip.
Examination of the combined data teleseismic and strong motion inversion solution indicates that the double arrival seen at most close-in strong motion stations was modeled by employing the time windows. Figure 11 \marginboxed{Figure 11} shows the slip contributions during time window 1 (top), 2 (middle), and 3 (bottom). The initiation of time window 2 is 0.1 s after time window 1; time window 3 begins 0.2 s later than window 1. The 0.35-s time separation between the two arrivals is obtained in the model by rupturing two adjacent fault regions (Figure 11, top and bottom slip concentrations). The total time separation of 0.35 s is obtained from the combined effects of a 0.15-s delay due to rupture across the additional distance to the region shown in the bottom of Figure 11 relative to the region at the top of the figure, and a 0.2-s time delay between time window 3 and time window 1. This may be considered to be a complex slip function or a local retardation of the rupture front. The alternative to a complicated slip function or locally variable rupture velocity is to allow only one time window and to force two slip concentrations separated in time by 0.35 s. This is not as favorable as the more complex scenario since it failed to adequately predict the observed waveforms.
Considering slip occurs in all three time windows, the total slip duration in the region of the largest slip is 0.4 s. As the peak slip is slightly greater than a meter, this requires a slip velocity of over 2 m/s. Similar slip velocities are demanded by models for other well-studied earthquakes [ Heaton, 1990]. We also investigated the effect of different rupture velocities on the slip distribution. Although rupture velocities of 3.0 and 3.3 km/s, corresponding to roughly 80\% and 90\% of the source region shear-wave velocity, were tried, they did not produce any improvement in the match to the observed waveforms. Thus, slightly faster rupture velocities are not favored, or ruled out. The fact that the earlier time window contributes a large portion of the slip suggests that a slower rupture velocity is not appropriate.
The waveforms most similar to the observations result from the model "SW", which best approximates the solution determined independently of regional waveforms by the inversion of teleseismic and strong motion records. In particular, note the improvement to the SmS arrivals on the tangential and vertical components at station GSC and the similarity to the observed P wave train on the radial and vertical components at all three stations. This exercise supports the finite fault solution obtained above and suggests that the broadband regional waveforms contain sufficient information to extract source information pertaining to rupture directivity and relative slip location in addition to focal mechanism and seismic moment. Note that the tangential component at ISA is close to a radiation node for the Sierra Madre mechanism and is dominated by off-azimuth multi-pathed arrivals; thus, it is not well fit by any of the models.
We noted that the static displacements are not very sensitive to the location and concentration of the slip on the inferred fault plane. The location, amplitude and shape of the uplift pattern are more sensitive to the fault geometry and total slip than its relative location updip or along strike. This suggests that only with a very dense network of geodetic monuments would it be possible to resolve variations of slip on a buried fault plane with dimensions comparable to that of the Sierra Madre earthquake (7 km by 6 km).
The updip, southwestward rupture in our model is also compatible with other observations. First, the hypocentral depth is slightly deeper than the point source depth of the teleseismic data requiring propagation updip. Second, the broadband regional waveform data, which independently provide updip and downdip rupture constraints, are best modeled with updip slip. Finally, the location of the hypocenter with respect to the aftershock distribution and the extent of the distribution itself indicate probable updip extent to the slip (Figure 1).
Figure 8 indicates that many of the aftershocks surround the central asperity found in our dislocation model. Most noteworthy, the two largest aftershocks (both magnitude 4.0) outline the edge of the significant coseismic slip inferred from our waveform modeling (Figure 8). This result is consistent with the observation of Mendoza and Hartzell [1988] that aftershocks for many earthquakes often cluster along the margin of regions of the fault that experienced large coseismic slips. They attribute the aftershock patterns to a secondary redistribution of stress following primary failure on the fault. Note that the overall dimensions of the active aftershock perimeter of the Sierra Madre earthquake define an area substantially larger than the modeled coseismic region.
From teleseismic and local strong motion waveform modeling we find that only a small portion of the fault was responsible for producing significant ground motions, implying a substantial stress drop. It is often difficult to estimate stress drop since one must normally make assumptions concerning the relationship of the rupture duration and the rupture area. The finite fault approach allows us to determine both the amount of slip and the area over which it occurred. In this case, the stress drop expression of Eshelby [1957] for a circular fault is appropriate, Delta sigma = ( 7 pi mu u )/(16a), where mu is the rigidity, u is the average dislocation, and a is the radius. Using mu = 3.9 x 10^11 dyn cm^2, u = 57 cm, and a = 2.0 km, we obtain the stress drop of 150 bars. Since the choice of the area of significant rupture is still subjective, we also evaluate the stress drop for a slightly smaller rupture area. For a = 1.8 km, the stress drop becomes 190 bars. This event consists of a relatively high stress drop region with relatively little additional slip in the surrounding regions.
There are serious ramifications to our observation that damaging ground motion radiation can be attributed to such a compact fault region. It is often considered that fault segmentation limits the maximum size of earthquakes that can occur along a given fault zone. The relatively large localized slip in the Sierra Madre earthquake suggests that thrust faults of even limited dimensions are capable of producing potentially hazardous ground motions. This is substantiated by the high stress drops and substantial ground motions from the 1987 Whittier Narrows (ML=5.9), 1988 Pasadena (ML=4.9), and 1990 (ML=5.5) Upland earthquakes. The localized slip concentrations and the heterogeneous nature of the final dislocation imply that strain was released on only a small area of each fault. Further, the large gradients in slip modeled from the Sierra Madre earthquake (this study) and from the Whittier Narrows earthquake [ Hartzell and Iida, 1990] suggest that regions immediately adjacent to the main asperities of these events likely experienced stress increases during the earthquake and are therefore more likely to fail now than prior to the earthquake. Recall that the area that actually slipped is substantially smaller than the region which experienced aftershocks.
It is worthwhile to compare and contrast the Sierra Madre earthquake and our rupture model with the 1987 Whittier Narrows earthquake (ML=5.9) and seismological studies of that event. Both events were deep thrust earthquakes and both were costly to the residents of metropolitan Los Angeles. The felt area for the Sierra Madre event was approximately 59,000 km^2, considerably less that the 110,000 km^2 felt area for the Whittier Narrows earthquake [ Stover and Reagor, 1991]. For comparison, the 1987 Whittier earthquake had a seismic moment of about 1.0 x 10^25 dyne-cm as estimated from strong motion data [ Hartzell and Iida, 1990] and 1.1 to 1.4 x 10^25 dyne-cm based on teleseismic waveforms [ Bent and Helmberger, 1989], roughly 4--5 times the moment of the Sierra Madre earthquake.
Estimates of the rupture area for the Whittier Narrows earthquake (and thus stress drop) vary considerably. Hauksson and Jones [1989] show that the overall aftershock area was approximately 25--30 km^2, but proposed a rupture area of 13 km^2 based on the temporal and spatial clustering of the aftershocks. Wald et al. [1988], using a semi-empirical forward modeling technique, inferred that a rupture area of about 20 km^2 could adequately simulate the acceleration recordings. Hartzell and Iida estimate that the total rupture area from the Whittier earthquake is approximately 70 km^2, with more concentrated slip isolated over a smaller region of perhaps 25 km^2. Still, the majority of the coseismic slip was contained within the overall aftershock distribution. For the Sierra Madre earthquake, nearly all the slip lies within an area of 13 km^2 with a majority of the moment release within a region on the order of 4 km^2, easily confined to a small portion of the total active aftershock perimeter. For both the Whittier Narrows and the Sierra Madre events, the hypocenter is located in a region of the fault at the edge of an asperity, not within the major slip area.
Note, however, that although the total moment was a factor of 4--5 lower than that of the Whittier earthquake, the peak ground velocity of 25 cm/s recorded at station ETN from the Sierra Madre earthquake is comparable to that recorded during the Whittier Narrows earthquake at stations WTR (26 cm/s) and DOW (28 cm/s). The relatively large ground motions for the Sierra Madre earthquake may reflect the highly localized nature of the source. Of course, damage resulting from the Sierra Madre earthquake was greatly moderated by the relative remoteness of the epicentral region and areas of high ground motion relative to the heavily populated area of the Whittier Narrows earthquake.
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