A comparison of t
cand t
p maxfor magnitude estimation in earthquake
early warning
Jang-Tian Shieh,1 Yih-Min Wu,1and Richard M. Allen2
Received 11 August 2008; revised 9 September 2008; accepted 15 September 2008; published 16 October 2008.
[1] We determined the tcandtpmax parameters from the
K-NET strong motion records of 16 earthquakes in Japan with moment magnitude (Mw) ranging from 6.0 to 8.3. A
0.075 Hz high-pass Butterworth filter was applied for determination of tc based on our previous studies. It was
found that different pole selections of the Butterworth filter lead to different uncertainty in magnitude determination. Our results show that using two poles in the filters results in the best magnitude estimates, i.e., minimized the standard deviation in magnitude determination in comparison to Mw
usingtc. Thetpmaxparameters (Allen and Kanamori, 2003)
were also determined with the same dataset using the Wurman et al. (2007) procedure. It was found that tpmax
values obtained from this dataset, and using the Wurman procedure, had a larger uncertainty. However, when a 0.075 Hz high-pass Butterworth filter with five poles was added, the uncertainty intpmax-derived magnitude estimates
decreased minimizing the standard deviation in magnitude determination usingtpmax. This difference in the behavior of
tcandtpmaxcan be used to further reduce the uncertainty in rapid magnitude determination for earthquake early warning. When the magnitude estimations from tc and tpmaxof each event are averaged to provide a new magnitude
estimate, the standard deviation in magnitude estimates is reduced further to 0.27 magnitude units. Citation: Shieh, J.-T., Y.-M. Wu, and R. M. Allen (2008), A comparison oftc
and tpmax for magnitude estimation in earthquake early
warning, Geophys. Res. Lett., 35, L20301, doi:10.1029/ 2008GL035611.
1. Introduction
[2] A central component of earthquake early-warning
(EEW) systems is the determination of the magnitude and location of an earthquake as soon as possible and before destructive energy arrives. Nakamura [1988] first intro-duced the concept of using the frequency content of the initial few seconds of P-wave arrivals. He observed that larger events cause initial ground motion with longer periods than smaller events. Average ground motion period tc and dominant ground motion period tpmax are two
important parameters frequently used to estimate the magnitude in EEW [e.g., Allen and Kanamori, 2003; Kanamori, 2005; Olson and Allen, 2005; Wu and Kanamori, 2005a, 2008a, 2008b; Wu et al., 2007; Wurman et al., 2007; Olivieri et al., 2008]. One measure of P-wave
frequency content is tc which uses the first 3 seconds of
P-wave data. The results of Wu and Kanamori [2005a, 2005b, 2008a, 2008b] and Wu et al. [2006, 2007] show a good relationship between tc and Mw determined from
data collected from Japan, Taiwan and southern California. This suggests that it is possible to estimate the magnitude 3 seconds after the P-wave arrival with the tc method.
[3] Building of the results of Allen and Kanamori [2003]
in southern California, Olson and Allen [2005] also found a good scaling relationship betweentpmaxand Mwfor a global
earthquake dataset. While they used up to 4 sec of P-wave data, tpmax values for most of the records occurred within 2 seconds of the P-wave arrival. This relationship between tpmaxand Mwalso allows estimation of magnitude from the
first few seconds of P-wave data. The fact that their observations were made prior to the termination of the earthquake rupture was also interpreted as suggesting that earthquake rupture is deterministic. This interpretation remains controversial. Rydelek and Horiuchi [2006] used a dataset of earthquakes with M > 6.0 from Japan to investigate the proposed scaling relation and argued that there was no obvious scaling relation betweentpmaxvalues and magnitude.
[4] Here we focus on the applicability of both thetcand
tpmaxparameters for EEW. We use a dataset that is similar to
that of Rydelek and Horiuchi [2006], and compute tc and tpmaxvalues from the vertical acceleration component of the
K-NET strong motion records collected in Japan from 1997 to 2008. There are more than 1000 K-NET stations across Japan, and 16 events were selected in this study (Figure 1). We use the same dataset to determine bothtcandtpmaxand
compare the performance of these parameters as magnitude estimators. We also experiment with the frequency band within whichtcandtp
max
are determined and find that this plays an important role in the robustness of magnitude estimates.
2. Data
[5] The purpose of EEW is to issue a warning before
strong ground motion of a destructive earthquake comes. Thus, sixteen larger earthquakes with Mw 6 (Table S1 in
the auxiliary material) were chosen for analysis in this study.1The criteria for selecting events was: (1) events of 6 Mw < 7 with focal depth less than 30 km and at least
six records within an epicentral distance of 70 km, and (2) events of Mw 7 with focal depth less than 70 km and
at least six records within an epicentral distance less than 200 km. Earthquakes with less than 6 records are not included in this analysis. In this study we use 3 seconds
1
Auxiliary materials are available in the HTML. doi:10.1029/ 2008GL035611.
Here
for
Full Article
1Department of Geosciences, National Taiwan University, Taipei,
Taiwan.
2
Seismological Laboratory, Earth and Planetary Science, University of California, Berkeley, California, USA.
Copyright 2008 by the American Geophysical Union. 0094-8276/08/2008GL035611$05.00
of data in our determination oftcandtpmax. Given that it is
not possible to determine periods greater than12 seconds, i.e., our 3 second data window constitutes 1=
4 of the
wavelength, we apply a 0.075 Hz high-pass filter, and also discard any observations greater than 10 seconds period. Considering the purpose of EEW in this study, for each event, we use the averaged value from the six waveform records with validtcortpmaxnearest to the epicenter.
3. The tc Method
[6] tc is a measure of the average period of ground
motion within some specified time window. It was first
introduced by Kanamori [2005] and is a modified version of the method originally developed by Nakamura [1988]. The period parameter tc is calculated from the first several
seconds of P-wave data as follows:
tc¼ 2p= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ t0 0 _u2ð Þdtt Z t0 0 u2ð Þdtt s ð1Þ
where u is the high-pass filtered displacement of the vertical component ground motion and _u is the velocity differ-entiated from u. Following a series of studies [Wu and Kananmori, 2005a, 2005b, 2008a, 2008b; Wu et al., 2006, 2007], the waveforms have a 0.075 Hz high-pass Butter-worth filter applied to the velocity component during the procedure oftcdetermination (see Text S1, section S1a). A
3 seconds time window starting from first P-wave arrival is set to determine thetcin this study, i.e.,t0in equation (1) is
set as 3 seconds after the P-wave arrival.
[7] In order to study the effect of different numbers of
poles in the 0.075 Hz high-pass Butterworth filter, we tested filters with 1 through 6 poles (Figure S1 shows amplitude response curves) and examined the relationship betweentc
and MW. We average the tc values from the six closest
waveform records to each event and determined the linear relation between Mwand the averagedtcvalues using
least-squares. Figure 2 shows the results of applying filters with 2 and 5 poles. Generally, thetcvalues with a small number of
poles have larger slope versus MW, which is good for
magnitude estimation, but they also have a larger scatter (Figure 2a). A larger number of poles results in a smaller slope versus MW, but with a smaller scatter (Figure 2b).
4. The tp max
Method
[8] tpmax was introduced by Allen and Kanamori [2003]
(in which it is called Tp) and was applied to a global seismic
Figure 1. Epicenter distribution of events (grey stars) used in this study. Small squares show the locations of K-NET stations.
Figure 2. Thetcestimated with (a) two poles and (b) five poles. A 0.075 Hz high-pass Butterworth filter was applied.
Open diamonds represent thetcof each record, and solid circles represent the averagedtcvalues from records of the same
dataset by Olson and Allen [2005]. It is nearly identical to the original concept proposed by Nakamura [1988]. While the purpose of tpmax is the same as for tc in that it is a
measure of frequency content, the approach is quite different. tc is determined by selecting a specific time
window, 3 seconds in this case, and measuring the frequency content of the entire selected window. tp is a
timeseries determined recursively and continuously from the seismic waveform. As such tp at any given time
contains information about the frequency content of the entire waveform up to the given point in time, though the contribution of a given waveform segment decreases with time. This makes tpmax a dominant period parameter of
ground motion while tc is an average period parameter.
tpmaxis also a ratio of the velocity and acceleration signals,
whiletcis the ratio of displacement and velocity signals.
[9] The parametertpis computed by
tpi¼ 2p ffiffiffiffiffi Xi Di r where Xi= aXi1+ xi2 and Di¼ aDi1þ dx dt 2 i ð2Þ
xi is the velocity signal to which both high- and
low-pass filters have been applied [Wurman et al., 2007] (see Text S1, section S1b) anda is a smoothing constant which is set as 0.99 in this study. It is a that determines how quickly the contribution of a given segment of the time series to tp decreases with time. tpis computed at every
time step and the maximum value,tpmax, within some time
window is chosen to be the parameter used to estimate magnitude for EEW. In this study the time window used
was 3 seconds for similarity withtc.tpmax is therefore the
maximum value of tp within 3 seconds of the P-wave
arrival. tpmax is selected from the time window starting at
0.05s rather than from 0.00s because of the recursive nature of the tp calculation as discussed by Olson and Allen
[2005]. As with the tc vs. Mw relations in this study, the
linear relation is shown by the least-squares fit between Mw
and averaged values oftpmax from the same six records for
each earthquake.
[10] Figure 3a showstpmaxvalues for the 16 earthquakes
in this study. Whiletpmaxincreases with Mw, there is a large
scatter in individual station observations for several of the smallest events resulting in the larger averagedtpmaxvalues
than for the larger events. This scatter is likely attributed to processing problems for smaller signal-to-noise ratio wave-forms. Using the appropriate filter reduces the scatter. As withtc, we tried to apply a high-pass Butterworth filter at
0.075 Hz in the tpmax calculation. Figure 3b shows tpmax
when five poles are used. This has the effect of narrowing the frequency band included in the tpmax calculation. The
standard deviation of least-squares fitting of tpmax versus
MWdecreases from 0.48 to 0.22 with the application of this
filter.
5. Discussion and Conclusions
[11] The present study demonstrates that the filter
appli-cation plays an important role in the calculation of tc and
tpmax. In order to determine the best pole setting for the
0.075 Hz high-pass filter, relationships of tc and tp max
versus Mw were analyzed by least-squares fitting for pole
values from 1 to 6. In the EEW application, we use the equation of least-squares fitting of tc or tpmax to estimate
magnitude of an event. Since the purpose of these methods is to estimate the magnitude, standard deviations of estimated magnitude were used as the index to compare Figure 3. The tpmaxestimated (a) by the original method of Allen and Kanamori [2003] and (b) by adding a five pole
high-pass Butterworth filter at 0.075 Hz. Open diamonds represent the tp max
values of each record, and solid circles represent the averagetpmaxvalues from records of the same events. Solid line shows the least-squares fit and the two dashed
results of different pole values. As shown in Figure 4 the best magnitude estimates are obtained from tc when the
number of poles equals 2, which results in a standard deviation of 0.36 in the magnitude estimation. For tpmax,
5 poles had the best result in magnitude estimation resulting in a standard deviation of 0.56. Without the 0.075 Hz high-pass filter the standard deviation in the magnitude estimate fromtpmaxis 2.48.
[12] Based on this result,tcapproach seems more robust
thantpmax. However, these two parameters are based on the
same concept from Nakamura [1988]. For the tc
calcula-tion, a three seconds window after P arrival is used, while the tpcalculation is recursive. Thus, the tp value may be
influenced by signals before P arrival. To abate this influence, we calculated tpby setting waveform values to
zero prior to 0.05 seconds after the P-wave arrival.tpmaxwas
then determined from thetptimeseries up to 3 seconds after
P-wave arrival. A 0.075Hz high-pass Butterworth filter with 5 poles was also applied. Figure 5 shows the tpmaxresults.
The uncertainty in magnitude estimation is decreased resulting in a standard deviation in the magnitude estimate of 0.40. This uncertainty is essentially the same as the uncertainty from thetcmethod.
[13] These tests have shown the importance of filter
application in the calculation of tcandtpmax. We find that
adding a 0.075 Hz high-pass Butterworth filter with a sharp cutoff in frequency (5 poles) is optimal for tpmax analysis
enhancing the relationship between tpmax and magnitude.
Fortc, 2 poles have a best result in magnitude estimation.
The different filter applications totpmaxanalysis results in a
diversity of measurements that may be the cause of the controversy introduced by Rydelek and Horiuchi [2006]. While there is difference in the behavior of tc and tpmax,
when the appropriate specific procedure is applied, both methods have good linear trends with Mw. This suggests
that it may be useful to include both tc and tpmax in the
estimation of magnitude in earthquake early warning
systems. The magnitude estimates of tc with two poles
andtpmaxwith five pole calculated from 0.05 seconds after P
arrival could be averaged to provide a more robust magnitude estimate. This average magnitude estimation has a lower uncertainty than either tc or tpmax alone. The
Figure 4. Standard deviations in magnitude estimation usingtcandtpmaxfor different numbers of poles in the 0.075 Hz
high-pass Butterworth filter.
Figure 5. The tp max
estimated in the same way as Figure 3b, i.e., applying a five pole high-pass Butterworth filter at 0.075 Hz, but with the signals before 0.05s after the P-wave arrival set to zero. Hollow diamonds represent the tp
max
values of each record, and solid circles represent the average tpmax values from records of the same events.
Solid line shows the least-squares fit and the two dashed lines show the range of one standard deviation.
standard deviation of this average magnitude estimate is 0.27 magnitude units.
[14] Acknowledgments. We would like to thank M. Olivieri and one anonymous reviewer for their valuable comments. This research was supported by the Central Weather Bureau and the National Science Council of the Republic of China (NSC96-2625-Z-002-025 and NSC95-2119-M-002-043-MY3). Support was also provided by the USGS NEHRP program (06HQAG0147). Figure 1 was made using Generic Mapping Tool [Wessel and Smith, 1991].
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