Earthquake Magnitude Measurements for Puerto Rico

Bulletin of the Seismological Society of America, Vol. 95, No. 2, pp. 725 730, April 2005, doi: 10.1785/0120040083 Earthquake Magnitude Measurements for Puerto Rico by Dariush Motazedian and Gail M. Atkinson E Abstract Reliable determination of earthquake magnitude is a fundamental building block of seismic hazard assessment. The seismicity catalog for Puerto Rico is dominated by small earthquakes (M 5), mostly M D (a local magnitude based on duration) and m b (body-wave magnitude). There is considerable uncertainty over the interpretation of M D. To reduce this uncertainty, we evaluate moment magnitude (M) and M 1 (1-Hz magnitude) for events within the catalog and develop relationships between these and other magnitude measures. The available seismographic data are mostly short-period records, because broadband instruments in Puerto Rico have been installed only recently. A difficulty with the calculation of moment magnitude is that short-period data do not generally extend to sufficiently low frequencies to reliably obtain the displacement spectrum at low frequencies. Moment magnitudes for small earthquakes in Puerto Rico are thus estimated from a single broadband station and subject to much uncertainty. To get around this difficulty, we used M 1, which closely tracks moment magnitude for small to moderate events (Chen and Atkinson, 2002). M 1 is obtained from the spectral amplitude at 1 Hz and is defined such that it will equal moment magnitude for earthquakes following a Brune point-source model. Unlike moment magnitude, M 1 can be determined from short-period seismograms. Our values of M and M 1 are in close agreement with each other for small to moderate earthquakes. There is a systematic difference between M 1 or M and catalog magnitudes m b or M D, with the catalog magnitude exceeding moment magnitude by about 0.4 units on average. It is recommended that M 1 be used as a regional magnitude scale for earthquakes in Puerto Rico, and as an estimate of M for events of M 5. Online material: List of earthquakes in Puerto Rico from 1993 through 2002. Introduction In Puerto Rico, there has been a long-standing problem with regional magnitude determinations. What is really needed is a moment-magnitude-based catalog. Moment magnitude provides an objective measure of earthquake size and is compatible with earthquake hazard analysis, which calculates ground motions based on moment magnitude and distance. The seismic event catalog for Puerto Rico is dominated by earthquakes with M D (magnitude based on duration) and m b (body-wave magnitude) smaller than 5, for which conventional estimates of moment magnitude (such as the Harvard Centroid Moment Tensor solutions) are not available. The magnitude reported by the Puerto Rico Seismic Network (PRSN) ism D. M D for Puerto Rico is defined as M D 0.819 log (S-P) 0.639 log (dist) C, where S-P is in seconds, dist is epicentral distance in kilometers, and C is a constant that is site dependent (PRSN, 1996). The PRSN, consisting of 13 vertical-component shortperiod seismograph stations, has been digitally recording seismograms since 1991. There is also a strong motion network in Puerto Rico (PRSMN), which has been installed gradually since 1994, and now comprises 32 strong motion stations. A single Incorporated Research Institutions for Seismology/U.S. Geological Survey (IRIS/USGS) station (SJG) has also been operating in Puerto Rico since 1993. During the past few years, the PRSN has been gradually adding more broadband stations to its network. The purpose of this study is to use the available post-1991 digital data to develop a useful magnitude scale, related to moment magnitude, which will improve the earthquake magnitude estimates in the regional seismicity catalog of Puerto Rico. Regional Magnitude Scale for Puerto Rico Ideally, a regional magnitude scale should be closely correlated with moment magnitude. Based on the Brune (1970, 1971) model, the acceleration spectrum of the shear 725

726 Short Notes radiation for an earthquake at a distance R may be modeled as a point source with an x 2 shape (Aki, 1967; Brune, 1970; Boore, 1983): 2 A(f) CM 0 [(2pf) / (1) 2 [1 (f/f 0)]][exp( pfj) exp( pfr/qb)/r] S(f), where A(f) is the observed acceleration spectrum (horizontal component of shear waves). M 0,f 0, and R are seismic moment, corner frequency, and distance (hypocentral) from the observation point, respectively. S(f) represents site amplification ( 1 for a very hard rock site). The constant C R h FV/(4pqb 3 ), where R h radiation pattern (average value of 0.55 for shear waves), F free surface amplification (2.0), V partition onto two horizontal components (0.71), q density, and b is shear-wave velocity (Boore, 1983). The term exp( pfj) is a high-cut filter to model near-surface kappa effects (Anderson and Hough, 1984); this is the commonly observed rapid spectral decay at high frequencies. The quality factor, Q(f) [log(e)pf]/(cb), is inversely proportional to anelastic attenuation, c(f). The implied 1/R attenuation term is applicable for body-wave spreading in a whole space and can be modified based on the geometric spreading behavior of seismic waves. The corner frequency, f 0, is given by 1/3 Dr f 4.9E 6, (2) 0 where Dr is stress drop in bars, M 0 is in dyne cm, and b is shear-wave velocity in kilometers per second (Boore, 1983). If we calculate the Fourier spectrum of a recorded time series, divide out frequency-dependent site amplification, S(f), then play back the attenuation effects (including anelastic and geometric behaviors), all according to equation (1), then we can obtain an estimate of the source spectrum, A 0 (f). The Brune model can then be used to relate the source spectrum to seismic moment, since the acceleration and displacement spectra, respectively, are: M 0 2 2 A 0(f) CM(2pf) 0 /[1 (f/f 0) ] (3) 2 D 0(f) CM/[1 0 (f/f 0) ] (4) At low frequencies (f K f 0 ), the displacement spectrum becomes: D (f) CM (5) 0 0 Using equation (5), we can then determine M 0 and hence moment magnitude M. This approach to calculating moment magnitude can be used for broadband records, from which low-frequency amplitudes, with f K f 0, can be recovered. Unfortunately, short-period network data do not generally extend to sufficiently low frequencies to satisfy the f K f 0 constraint, except for very small earthquakes. To get around this difficulty, Chen and Atkinson (2002) proposed an alternative magnitude measure, M 1, which closely tracks moment magnitude for small to moderate events. M 1 is an intermediate-period magnitude obtained from the spectral amplitude at 1 Hz and is defined such that it will equal moment magnitude for earthquakes following a Brune point-source model (Chen and Atkinson, 2002). Consider equation (3) at f 1 Hz: 2 2 A 0(1) CM0 4p /[1 (1/f 0) ]. (6) If we assume a constant value of 100 bars for Dr, and use equation (2) to relate Dr to f 0, then equation (5) can be solved numerically by trial and error to find M 0, which we will denote here as M 0 (1) (to indicate that it is an estimate only, obtained at a frequency of 1 Hz). The estimate M 0 (1) will only equal the actual M 0 for events with spectra that follow the assumed underlying Brune model spectra. For small earthquakes, the value of M 0 (1) is not sensitive to the assumed stress drop, because source spectral amplitudes are independent of stress drop at frequencies lower than the corner frequency (recall A(f) fkf0 CM 0 4p 2 f 2 ). M 1 is only sensitive to stress drop for earthquakes of M 6. The estimated moment magnitude is calculated using M 1 (2/3) log M 0 (1) 10.7, where the notation M 1 is used to indicate that this magnitude definition is based on the estimate M 0 (1). Chen and Atkinson (2002) applied this approach to more than 3000 earthquakes worldwide. Derivation of M 1 for Puerto Rico We use digital ground-motion data from the PRSN shortperiod stations, PRSMN strong motion stations, and the SJG broadband station to determine M 1. The required regional parameters, to calculate A 0 (f), were determined by empirical analysis of these data as described by Motazedian (2002) and Motazedian and Atkinson (2005a). We calculated the Fourier spectrum of each observed vertical-component waveform and applied a bandpass filter to the acceleration spectrum, centered at 1 Hz (a Butterworth filter with the order of 8 from 0.7 to 1.3 Hz). This defines A 0 (1). There is an implicit assumption that the vertical component record is a reasonable estimate of the horizontal component record before any site amplification. This is consistent with the commonly used horizontal-to-vertical ratio technique as an estimate of site response (Lermo and Chavez-Garcia, 1993). For trial values of moment magnitude from 1 to 8 (in 0.1- unit increments), we use equation (2) to specify corner frequency, and equation (3) to calculate the predicted acceleration source spectrum for that magnitude (arbitrary stress drop of 100 bars assumed) according to the Brune model. The calculated Brune spectrum was filtered in the same way as the observed spectrum to obtain the predicted A 0 (1) for the trial moment magnitude. The iteration procedure finds the value of moment magnitude (and consequently M 1 ) for which the area under the filtered acceleration spectrum, ac-

Short Notes 727 cording to the Brune model, most closely matches the area under the filtered observed acceleration spectrum. The average of the calculated M 1 values over all stations that recorded the event defines M 1 for that event. As an example, Figure 1 shows the filtered acceleration source spectrum of an earthquake of M 1 4.7 after application of these procedures, in comparison with the Brune source model for M 4.7. The total area under both curves is the same for the determined M 1. The magnitude M 1 is well determined by this procedure, as a deviation of 0.1 magnitude units would cause a significant deviation in the area under the curve, as shown on the figure. We determined M 1 for more than 300 earthquakes in Puerto Rico. Figure 2 shows M 1 versus the catalog magnitudes m b and M D. The reported catalog magnitude for most of the earthquakes with magnitude 4.0 is M D (duration magnitude), whereas for most of the earthquakes with magnitude 4.0 it is m b (body-wave magnitude from the USGS catalog). There is a systematic difference between M 1 and catalog magnitude. The relationship between M 1 and m b is given by M 0.76 M 0.43, (8) 1 D with a standard error of 0.19. m b and M D appear to be compatible estimates of M 1, although there is little overlap between the scales and no statistical basis for a relationship between m b and M D. If we regress M 1 against M cat (assuming M cat m b M D ), we obtain: M 0.91 M 0.08 (9) 1 cat with a standard error of 0.22. Table 1 ( E available online at the SSA Web site) lists the calculated M 1 and reported catalog magnitudes for all study events. The number of stations used per event ranges from 1 to 13. The average standard deviation for the measurement of M 1, excluding events with only one observation, is 0.22. Independent moment magnitude estimates for a few of the larger events (the only events for which such values are available) are provided in Table 2. M 0.71 m 0.92, (7) 1 b with a standard error of 0.24. The relationship between M 1 and M D is Figure 2. Relationship between catalog magnitudes m b and M D, and magnitude M 1. A systematic difference exists between M 1 and catalog magnitudes. Figure 1. Illustration of procedure to determine M 1. The area under the 1-Hz filtered observed spectrum most closely matches that under the 1-Hz filtered Brune spectrum for M 4.7; thus, the value of M 1 is 4.7. Note that M 1 is well determined, as an increase or decrease in value by 0.1 units causes a significant difference in the degree of fit of the model to the observed spectrum. Table 2 The Reported CMT Moment Magnitude in Catalog (M w ) and the Estimated M (this article) for Earthquakes in Puerto Rico Date (yyyy/mm/dd) Distance (km) Depth (km) M w (catalog) M (this study) 1996/05/11 186 35 5.1 5.2 1998/08/10 470 58 5.2 4.6 1999/01/18 144 33 5.0 4.8 2001/10/17 191 33 6.0 5.9

728 Short Notes Estimates of Moment Magnitude from Displacement Spectra We can use equation (4) or (5) to calculate seismic moment from the long-period level of the displacement spectrum, D 0. As an example, Figure 3 shows the displacement spectrum of a M 4.8 earthquake and the Brune source displacement spectrum model for that earthquake; the flat portion of the spectrum covers the frequency band from about 0.1 to 0.8 Hz. To obtain the spectral amplitudes from which to determine the average D 0, a Butterworth filter with the order of 8 was applied to the source spectra to cut amplitudes above the high-cut limit of f h f 0 /3. A matching low-cut filter was used to cut frequencies f h /5. For example, the Butterworth filter for M 4.8, with the assumed corner frequency of 1.76, retains the spectral amplitudes for frequencies from 0.12 to 0.59 Hz as shown on Figure 3. The seismic moment is determined from the spectral amplitudes between these filter frequencies. Figure 4 shows M based on the recorded waveforms at SJG versus catalog magnitude. There is a systematic difference between M and catalog magnitudes m b or M D, consistent with the relationship observed between M 1 and the catalog magnitudes. Because the moment magnitudes are calculated from a single station, they have significant uncertainty. The determined relations between M and m b (M 0.80m b 0.56) and M and M D (M 0.96M D 0.28) are thus considered less reliable than those established between M 1 and m b or M D. We note that the determined relation between M and M cat would be: M 1.02M cat 0.47. Figure 5 compares M 1 with M. We conclude that M 1 M for M 5.0, but underestimates M for larger events. In Table 2, we compare our calculated M values with more reliable global estimates based on CMT solutions. The largest catalog event has a moment magnitude (CMT) of 6.0, whereas our estimated M for that earthquake is 5.9. Overall, Figure 4. Moment magnitude (single-station estimate) versus catalog magnitudes m b and M D. Figure 3. Displacement source spectrum for an event of M 4.8, along with the corresponding Brune model spectrum. The filtered spectra used to determine the long-period displacement level for the seismic moment calculation are also shown. Figure 5. Relationship between moment magnitude (single-station estimate) and M 1. The relationship M M 1 is followed for earthquakes of M 4.5.

Short Notes 729 Table 2 indicates that our estimated M values are reasonably close to CMT moment magnitude, especially considering that our estimates are based on a single station. An exception is the large difference between our estimate of M (4.6) and the CMT value (5.2) for the earthquake on 10 August 1998; the large distance (500 km) from SJG may have been a factor in this discrepancy. Our regional magnitude scale M 1 has been designed such that M 1 M for small to moderate earthquakes. In the next section, we test the general applicability of this conclusion for earthquakes in Puerto Rico based on stochastic finite-fault modeling. Expected Relation Between M 1 and M Based on Ground-Motion Modeling We use a stochastic finite-fault modeling exercise to simulate acceleration time series for earthquakes with magnitudes from M 2.0 to M 8.0 and hypocentral distances from 10 to 500 km. Stochastic finite-fault modeling is a wellknown tool for the investigation and modeling of ground motions over a wide range of magnitudes, distances, and tectonic settings (Hartzell, 1978; Irikura, 1983; Joyner and Boore, 1986; Schneider et al., 1993; Tumarkin and Archuleta, 1994; Beresnev and Atkinson, 1997, 1999). The application of the stochastic finite-fault model to Puerto Rico is described by Motazedian (2002) and Motazedian and Atkinson (2005a,b). For small earthquakes, a point-source model is typically assumed. For larger events, finite-fault effects such as fault geometry, directivity, and heterogeneity of slip on the fault plane can profoundly influence the amplitudes, frequency content, and duration of ground motion. For such events, a finite-fault version of the stochastic model is used. An extended fault plane is modeled by a number of subsources, each of which is modeled as a point source. The contributions to the ground motion from all subsources on the fault plane are summed at the observation point. The input parameters to the stochastic finite-fault model used in the calculation of M 1 for the simulated Puerto Rico earthquakes are given in Motazedian (2002) and Motazedian and Atkinson (2005a,b). We simulated 650 vertical-component acceleration time series for magnitudes from M 2.0 to M 8.0 and distances from 10 to 500 km. M 1 was obtained for all the simulated acceleration time series as described in the previous section. Figure 6 shows the relationship obtained between M 1 and M. This figure shows that M 1 is expected to be a good estimate of M for small to moderate earthquakes, up to M 5.0. For larger magnitudes the deviation of M 1 from M becomes large due to finite fault effects: for M 5 the Brune pointsource model that underpins the M 1 scale is not generally applicable. Based on these results, we expect that M 1 M for small to moderate earthquakes as proposed by Chen and Atkinson (2002). These conclusions match what we would expect based on our simulation results (Fig. 5). Figure 6. Relationship between M 1 and M based on stochastic finite-fault simulations for earthquakes in Puerto Rico. M 1 tracks M closely for M 5. At larger magnitudes, finite-fault effects cause M 1 to underestimate M. Conclusions M 1, an earthquake magnitude measure based on the amplitude of the Fourier spectrum at 1 Hz, has been calculated for earthquakes that occurred in Puerto Rico from 1991 to 2003. Moment magnitude, based on the displacement spectrum at lower frequencies, has also been estimated from the recorded waveforms at a broadband station. Our estimates of moment magnitude agree reasonably well with independent global estimates for the few earthquakes that are large enough to have independent moment estimates. Both simulations and actual data show that M M 1 for small to moderate earthquakes (M 4.5). The values of both M and M 1 are smaller than reported catalog magnitudes M D or m b. The relationship between M 1 and m b is given by M 1 0.71 m b 0.92 with a standard error of 0.24 and the relationship between M 1 and M D is M 1 0.76 M D 0.43 with a standard error of 0.19. We recommend that M 1 be used as a regional magnitude scale for earthquakes in Puerto Rico, and as an estimate of moment magnitude for events of M 5. Data Sources The seismographic time series were provided by PRSN and PRSMN. We also benefited from the seismic data from the IRIS station SJG in Puerto Rico, downloaded from www.iris.edu. A soft copy of calculated magnitudes is available to interested parties by writing to the authors (dariush @ccs.carleton.ca).

730 Short Notes Acknowledgments This work was funded by U.S. National Earthquake Hazards Reduction Program Grant 02HQGR0054. We appreciate the comments of Bill McCann, Roland Laforge, David Oppenheimer, Cezar Trifu, and an anonymous reviewer. References Aki, K. (1967). Scaling law of seismic spectrum, J. Geophys. Res. 72, 1217 1231. Anderson, J., and S. Hough (1984). A model for the shape of the Fourier amplitude spectrum of acceleration at high frequencies, Bull. Seism. Soc. Am. 74, 1969 1993. Atkinson, G., and W. Silva (1997). Empirical source spectra for California earthquakes, Bull. Seism. Soc. Am. 87, 97 113. Beresnev, I., and G. Atkinson (1997). Modeling finite fault radiation from the w n spectrum, Bull. Seism. Soc. Am. 87, 67 84. Beresnev, I., and G. Atkinson (1999). Generic finite-fault model for ground motion prediction in eastern North America, Bull. Seism. Soc. Am. 89, 608 625. Boore, D. (1983). Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra, Bull. Seism. Soc. Am. 73, 1865 1894. Brune, J. (1970). Tectonic stress and the spectra of seismic shear waves from earthquakes, J. Geophys. Res. 75, 4997 5009. Brune, J. (1971). Correction to Tectonic stress and the spectra of seismic shear waves from earthquakes, J. Geophys. Res. 76, 5002. Centroid Moment Tensor (CMT) Catalog, www.seismology.harvard.edu/ CMTsearch.html (last accessed February 2003). Chen, S., and G. Atkinson (2002). Global comparisons of earthquakes source spectra, Bull. Seism. Soc. Am. 92, 885 895. Hartzell, S. (1978). Earthquake aftershocks as Green s functions, Geophys. Res. Lett. 5, 1 14. Irikura, K. (1983). Semi-empirical estimation of strong ground motions during large earthquakes, Bull. Disaster Prev. Res. Inst. Kyoto Univ. 33, 63 104. Joyner, W., and D. Boore (1986). On simulating large earthquakes by Green s function addition of smaller earthquakes, in Earthquake Source Mechanics, M. Ewing (Editor), American Geophysical Monograph 37, 269 274. Lermo, J., and F. Chavez-García (1993). Site effect evaluation using spectral ratios with only one station, Bull. Seism. Soc. Am. 83, 1574 1594. Motazedian, D. (2002). Development of ground motion relations for Puerto Rico using the stochastic finite-fault method, Ph.D. Thesis, Carleton University, Ottawa. Motazedian, D., and G. Atkinson (2005a). Ground motion relations for Puerto Rico, Geol. Soc. Am. Bull. GSA Special Paper 385 (in press). Motazedian, D., and G. Atkinson (2005b). Stochastic finite fault modeling based on a dynamic corner frequency, Bull. Seism. Soc. Am. (in press). Puerto Rico Seismic Network (PRSN) (1996). Seismic Bulletin Local, Regional and Teleseismic Earthquakes Recorded and Processed by the Puerto Rico Seismic Network, internal report, 39 pp. Schneider, J., W. Silva, and C. Stark (1993). Ground motion model for the 1989 M 6.9 Loma Prieta earthquake including effects of source, path and site, Earthquake Spectra 9, 251 287. Tumarkin, A., and R. Archuleta (1994). Empirical ground motion prediction, Ann. Geofis. 37, 1691 1720. Department of Earth Science Carleton University Ottawa, Ontario, K1S5B6 gma@ccs.carleton.ca dariush@ccs.carleton.ca Manuscript received 22 April 2004.