# Magnitude determination using duration of high frequency energy radiation and displacement amplitude: application to the 2011 off the Pacific coast of Tohoku Earthquake

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## Abstract

We have applied a technique to determine earthquake magnitudes, using durations of high frequency energy radiation and the maximum displacement amplitudes, to the 2011 off the Pacific coast of Tohoku Earthquake. The estimated duration of high frequency energy radiation and magnitude are 170.5 s, and 8.96, respectively. This agrees well with preliminary analyses for this earthquake. Compared with the December 26, 2004, Sumatra earthquake (*M*_{w} 9.0), this event is characterized by a shorter duration of high frequency energy radiation and a larger displacement amplitude. The measured durations of high frequency energy radiation show azimuthal dependence, which indicates rupture propagation in the southwest direction. This result, together with rupture models obtained by other studies using lower frequency seismic signals or tsunami waveforms, suggests that there were two distinct rupture propagations in this event: one in a southwest direction from which high frequency energies were radiated efficiently, and the other in an east direction from which a very large seismic moment was released. We measured the time differences between *P*-wave arrivals and the times at which the absolute amplitudes of high bandpass (2–4 Hz) filtered *P*-waves became the largest. Most of the measured time differences, normalized by twice the centroid time shift, are in the range between 50 and 80 per cent. This is consistent with the frequency distribution that we obtained previously for a set of 68 large shallow earthquakes.

### Key words

High-frequency energy radiation magnitude rupture propagation## 1. Introduction

Hara (2007a) developed a method to determine earthquake magnitudes using durations of high frequency energy radiation, which we refer to as HFER hereafter, and maximum displacement amplitudes measured from first arriving *P* -waves recorded at a tele-seismic distance range. He showed that this method was applicable to huge earthquakes such as the December 26,2004, Sumatra earthquake (*M*_{w} 9.0 after the Global Centroid Moment Tensor Project, which we refer to as GCMT hereafter).

The mega-thrust earthquake occurred on March 11, 2011 (origin time 14:46:18.1 JST; location 38.10°N, 142.86°E; depth 24 km; *M*_{w} 9.0: Japan Meteorological Agency, JMA) and caused huge tsunamis and devastating damage in the northeast part of Japan. In the present study, we have applied the method of Hara (2007a) to this event. When the earthquake occurred, we were in Tokyo away from our institute located in Tsukuba. Since transportation facilities were down, we could not start data analyses immediately. We obtained a set of estimates of HFER duration and magnitude on the 13th in Tsukuba. The estimates of HFER duration and magnitude shown in this paper are those determined on that day. In April, we compared the results of this event with those of other large earthquakes considered in our previous works and we investigated the azimuthal dependence of durations of HFER, which we also present in this study. We also investigated the time differences between *P* -wave arrivals and the times at which the absolute amplitudes of high bandpass (2–4 Hz) filtered *P* -waves became the largest. Hara (2008) analyzed 68 large shallow earthquakes to show that the frequency of these time differences, normalized by twice centroid time shifts of the corresponding earthquakes, was high around 50% (corresponding to centroid time shifts).

## 2. Durations of HFER

## 3. Magnitude

*M*is the earthquake magnitude,

*A*is the maximum displacement (m) during the estimated duration of HFER from the arrival time of a

*P*-wave, Δ is the epicentral distance (km), and

*t*is the estimated duration (s) of HFER. The estimates of magnitude for the stations shown in Fig. 1 are 8.97, 8.71, and 8.79 for QIZ, CTAO, and MIDW, respectively. The median of magnitudes for all stations is 8.96. This agrees well with

*M*

_{w}9.0 issued by JMA (http://www.jma.go.jp/jma/press/1103/13b/201103131255. html) and from the USGS WPhase moment solution (http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/neic_c0001xgp_wmt.php), and 9.1 from the GCMT.

*M*

_{w}≥ 7.2) shallow (depths ≤ 50 km) earthquakes that occurred from July, 2006 to the 2011 off the Pacific coast of Tohoku Earthquake. The total number of events is 115. For the December 26, 2004, Sumatra earthquake (

*M*

_{w}9.0 from the GCMT), Hara (2007a) determined that the duration of HFER and the magnitude were 396.2 s and 9.03, respectively. The duration of HFER of the 2011 off the Pacific coast of Tohoku Earthquake, 170.5 s, is shorter than that of the Sumatra earthquake (about 43 per cent), while the maximum amplitude with the distance correction is larger (about 169 per cent). Tsunami earthquakes (Kanamori, 1972) are characterized by small amplitudes and long durations (the crosses in Fig. 3). Thus, our result does not infer that the 2011 off the Pacific coast of Tohoku Earthquake was a tsunami earthquake, although it does not exclude the possibility that the rupture process had a tsunami earthquake-like component.

## 4. Temporal Distribution of the Largest High Frequency Signals

*P*-wave arrivals and the times at which the absolute amplitudes of high bandpass (2–4 Hz) filtered

*P*-waves became the largest. Following Hara (2008), we normalized these time differences by twice the centroid time shift, 69.8 s, from the GCMT. Most of the normalized time differences are in the range of 50 to 80 per cent (Fig. 4). The mean is 69.9 per cent (corresponding to 97.6 s). This is consistent with the distribution obtained by Hara (2008), who showed that frequencies of these time differences are high around centroid time shifts (figures 3 and 4 in his paper). Thus, the result for this event again implies that HFER is weak in the vicinity of rupture starting regions.

## 5. Discussion

After the December 26, 2004, Sumatra earthquake occurred, many ideas and techniques have been proposed to determine reliable magnitudes more rapidly for tsunami early warning. For the 2011 off the Pacific coast of Tohoku Earthquake, INGV (Istituto Nazionale di Geofisica e Vulcanologia) obtained in real-time an *M*_{wpd} (Lomax and Michelini, 2009) estimate of 8.5 at 12 minutes after the origin time, and a final *M*_{wpd} 8.8 within 15 minutes after the origin time (Lomax and Michelini, 2011). When *M*_{wpd} with the “moment correction” of Lomax and Michelini (2009) is applied to a large set of off-line seismograms, the *M*_{wpd} estimate is 9.0 (Anthony Lomax, personal communication). INGV obtained real-time estimates of high-frequency rupture duration *T*_{0} of 143 s at 12 minutes after the origin time and a final *T*_{0} of about 170 s within 15 minutes after the origin time (Lomax and Michelini, 2011). USGS obtained an *M*_{w} estimate of 8.9 from Wphase moment tensor (Kanamori and Rivera, 2008) 34 minutes after the origin time (“Tohoku, Japan 2011 *M* 9.0 Educational Slides”, http:// earthquake.usgs.gov/learn/topics/Tohoku2011.ppt). GFZ (Deutsches GeoForschungsZentrum) obtained an *M*_{w} (*m*_{B}) (Bormann and Saul, 2008) estimate of 8.6, 8 minutes after the origin time (Joachim Saul, personal communication). Since *m*_{B} is derived from a single maximum velocity amplitude, this technique is simpler and also slightly faster than methods involving additional measurements of rupture duration. Thus, all these techniques worked well for the 2011 off the Pacific coast of Tohoku Earthquake mainly using tele-seismic data, although *M*_{w} (*m*_{B}) was smaller than others.

The initial *M*_{wp} (Tsuboi et al., 1995, 1999; Whitmore et al., 2002) estimate reported by PTWC (Pacific Tsunami Warning Center) was 7.9 (http://ptwc.weather.gov/ptwc/?region=1 ID=pacific.2011. 03.11.055508), which is much smaller than the value of *M*_{w} 9.0–9.1 issued by JMA, USGS, and GCMT. Since the technique of Hara (2007a) is similar to that of *M*_{wpd}, it is possible to determine magnitudes about 15 minutes after the origin time by this technique, when it is implemented on a real-time based system. Therefore, this technique would have produced a much better magnitude estimate than *M*_{wp}, which is currently in place. It could have helped in getting a better picture of the size of the Tohoku earthquake earlier on. It is, therehore, highly desirable for the techninque to be operational before the next big earthquake.

The azimuthal dependence of the durations of HFER shown in Section 2 indicates the rupture propagation in a southwest direction, while some rupture models placed large slips in an east direction from the epicenter (e.g., Fujii et al., 2011; Hayes, 2011; Shao et al., 2011). These results suggest that there were two distinct rupture propagations in this event: one in a southwest direction from which high frequency energies were radiated efficiently, and the other in an east direction from which a very large seismic moment was released. The latter is likely to have cased the huge tsunamis that struck areas along the Sanriku coast.

## Notes

### Acknowledgments

In this study, we have used SAC (Seismic Analysis Code) (Goldstein et al., 2007) for seismic waveform data analyses. We appreciate the input of Dr. Anthony Lomax and an anonymous reviewer whose comments and suggestions greatly improved the original manuscript.

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