Bulletin of the Seismological Society of America, Vol. 95, No. 2, pp. 725–730, April 2005, doi: 10.1785/0120040083
E
Earthquake Magnitude Measurements for Puerto Rico
by Dariush Motazedian and Gail M. Atkinson
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 MD (a local magnitude based on
duration) and mb (body-wave magnitude). There is considerable uncertainty over the
interpretation of MD. To reduce this uncertainty, we evaluate moment magnitude (M)
and M1 (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 M1, which closely tracks moment magnitude for small
to moderate events (Chen and Atkinson, 2002). M1 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, M1
can be determined from short-period seismograms. Our values of M and M1 are in
close agreement with each other for small to moderate earthquakes. There is a systematic difference between M1 or M and catalog magnitudes mb or MD, with the
catalog magnitude exceeding moment magnitude by about 0.4 units on average. It
is recommended that M1 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 MD (magnitude based on duration) and mb (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) is MD. MD for Puerto Rico is defined
as MD ⳱ 0.819 log (S-P) Ⳮ 0.639 log (dist) Ⳮ 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
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Short Notes
radiation for an earthquake at a distance R may be modeled
as a point source with an x2 shape (Aki, 1967; Brune, 1970;
Boore, 1983):
A( f ) ⳱ C M0 [(2pf )2/
[1 Ⳮ ( f/f0)2]] [exp(ⳮpfj) exp(ⳮpfR/Qb)/R] S( f ) ,
(1)
where A(f) is the observed acceleration spectrum (horizontal
component of shear waves). M0,f0, 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 ⳱
Rh␾ F V/(4pqb3), where Rh␾ ⳱ 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(ⳮpfj) 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, f0, is given by
1/3
f0 ⳱ 4.9EⳭ6
冢Dr
M冣
,
(2)
0
where Dr is stress drop in bars, M0 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, A0(f). The
Brune model can then be used to relate the source spectrum
to seismic moment, since the acceleration and displacement
spectra, respectively, are:
A0( f ) ⳱ C M0(2pf )2/ [1 Ⳮ ( f/f0)2]
(3)
D0( f ) ⳱ C M0 / [1 Ⳮ ( f/f0)2]
(4)
At low frequencies (f K f0), the displacement spectrum becomes:
D0( f ) ⳱ C M0
(5)
Using equation (5), we can then determine M0 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 f0, can be recovered.
Unfortunately, short-period network data do not generally
extend to sufficiently low frequencies to satisfy the f K f0
constraint, except for very small earthquakes. To get around
this difficulty, Chen and Atkinson (2002) proposed an alternative magnitude measure, M1, which closely tracks moment
magnitude for small to moderate events. M1 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:
A0(1) ⳱ C M0 4p2/ [1 Ⳮ (1/f0)2] .
(6)
If we assume a constant value of 100 bars for Dr, and
use equation (2) to relate Dr to f0, then equation (5) can be
solved numerically by trial and error to find M0, which we
will denote here as M0(1) (to indicate that it is an estimate
only, obtained at a frequency of 1 Hz). The estimate M0(1)
will only equal the actual M0 for events with spectra that
follow the assumed underlying Brune model spectra. For
small earthquakes, the value of M0(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)fK f0 ⳱ C M04p2f 2). M1 is only
sensitive to stress drop for earthquakes of M ⬎ 6. The estimated moment magnitude is calculated using M1 (2/3) log
M0(1) ⳮ 10.7, where the notation M1 is used to indicate that
this magnitude definition is based on the estimate M0(1).
Chen and Atkinson (2002) applied this approach to more
than 3000 earthquakes worldwide.
Derivation of M1 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 M1. The required regional
parameters, to calculate A0(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 A0(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 0.1unit 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 A0(1) for
the trial moment magnitude. The iteration procedure finds
the value of moment magnitude (and consequently M1) for
which the area under the filtered acceleration spectrum, ac-
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Short Notes
cording to the Brune model, most closely matches the area
under the filtered observed acceleration spectrum. The average of the calculated M1 values over all stations that recorded the event defines M1 for that event. As an example,
Figure 1 shows the filtered acceleration source spectrum of
an earthquake of M1 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 M1. The magnitude M1 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 M1 for more than 300 earthquakes in
Puerto Rico. Figure 2 shows M1 versus the catalog magnitudes mb and MD. The reported catalog magnitude for most
of the earthquakes with magnitude ⬍4.0 is MD (duration
magnitude), whereas for most of the earthquakes with magnitude ⱖ4.0 it is mb (body-wave magnitude from the USGS
catalog). There is a systematic difference between M1 and
catalog magnitude. The relationship between M1 and mb is
given by
M1 ⳱ 0.71 mb Ⳮ 0.92 ,
M1 ⳱ 0.76 MD Ⳮ 0.43 ,
(8)
with a standard error of 0.19. mb and MD appear to be compatible estimates of M1, although there is little overlap between the scales and no statistical basis for a relationship
between mb and MD. If we regress M1 against Mcat (assuming
Mcat ⳱ mb ⳱ MD), we obtain:
M1 ⳱ 0.91 Mcat ⳮ 0.08
(9)
with a standard error of 0.22. Table 1 ( E available online at
the SSA Web site) lists the calculated M1 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 M1, 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.
(7)
with a standard error of 0.24. The relationship between M1
and MD is
Figure 2. Relationship between catalog magnitudes mb and MD, and magnitude M1. A systematic
difference exists between M1 and catalog magnitudes.
Table 2
Figure 1.
Illustration of procedure to determine
M1. 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 M1 is
4.7. Note that M1 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.
The Reported CMT Moment Magnitude in Catalog (Mw) and the
Estimated M (this article) for Earthquakes in Puerto Rico
Date
(yyyy/mm/dd)
Distance
(km)
Depth
(km)
Mw
(catalog)
M
(this study)
1996/05/11
1998/08/10
1999/01/18
2001/10/17
186
470
144
191
35
58
33
33
5.1
5.2
5.0
6.0
5.2
4.6
4.8
5.9
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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, D0. 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 D0, a Butterworth filter with the
order of 8 was applied to the source spectra to cut amplitudes
above the high-cut limit of fh ⳱ f0 /3. A matching low-cut
filter was used to cut frequencies ⬍ fh /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 mb or MD, consistent with the relationship observed between M1 and the catalog magnitudes. Because the moment magnitudes are
calculated from a single station, they have significant uncertainty. The determined relations between M and mb
(M ⳱ 0.80mb Ⳮ 0.56) and M and MD (M ⳱ 0.96MD ⳮ
0.28) are thus considered less reliable than those established
between M1 and mb or MD. We note that the determined
relation between M and Mcat would be: M ⳱ 1.02Mcat ⳮ
0.47. Figure 5 compares M1 with M. We conclude that M1
⳱ M for M ⱕ 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 mb and MD.
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 M1. The relationship M ⳱ M1 is followed for earthquakes of M ⬍ 4.5.
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Short Notes
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 M1 has been designed
such that M1 ⳱ 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 M1 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 M1 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. M1 was obtained for all the simulated
acceleration time series as described in the previous section.
Figure 6 shows the relationship obtained between M1 and
M. This figure shows that M1 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 M1 from M becomes
large due to finite fault effects: for M ⬎ 5 the Brune pointsource model that underpins the M1 scale is not generally
applicable. Based on these results, we expect that M1 ⳱ 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 M1 and M based
on stochastic finite-fault simulations for earthquakes
in Puerto Rico. M1 tracks M closely for M ⱕ 5. At
larger magnitudes, finite-fault effects cause M1 to underestimate M.
Conclusions
M1, 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 ⳱ M1 for small to moderate earthquakes (M ⬍ 4.5). The values of both M and M1
are smaller than reported catalog magnitudes MD or mb. The
relationship between M1 and mb is given by M1 ⳱ 0.71 mb
Ⳮ 0.92 with a standard error of 0.24 and the relationship
between M1 and MD is M1 ⳱ 0.76 MD Ⳮ 0.43 with a standard error of 0.19. We recommend that M1 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).
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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.
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Department of Earth Science
Carleton University
Ottawa, Ontario, K1S5B6
[email protected]
[email protected]
Manuscript received 22 April 2004.