Moment tensor inversion of near source seismograms
Yuji Yagi and Naoki Nishimura
ABSTRACT
We constructed a program set for estimating moment tensor solutions using near source seismograms. 	 
We take the effect of the source time function into account, which has been neglected in predominant
program sets of the moment tensor analysis with near source seismograms. In our program set, weapproximate the horizontal location of the centroid to be epicenter, and then estimate a seismic source
duration and depth of centroid in the moment tensor inversion procedure. To evaluate our program set,
we applied it to three earthquakes: an aftershock (MJMA 6.4) of the 2003 Tokachi-oki great earthquake,the 2004 Nemuro-oki earthquake (MJMA 7.1) and its aftershock (MJMA 4.2). The results show that theobtained moment tensor solutions for large earthquakes are consistent with those in other studies. We
also found that the neglect of the source time duration will result in an underestimation of the seismic
moment.
INTRODUCTION
The moment tensor solution is one of the most important information of earthquakes. From themoment tensor solutions, we can obtain seismic moment (M0), moment magnitude (Mw) and fault
type of earthquakes. The moment tensor solution is also necessary information for investigation of adetailed seismic source process.
The moment tensor solution has been estimated by inversion analysis of seismic waveforms
observed by local and global seismic networks (e.g. Dziewonski et al., 1981; Kikuchi and Kanamori,1991; Kawakatsu, 1995). For example, U.S. Geological Survey (USGS) and Global Centroid Moment
Tensor Project have estimated the moment tensor solutions for earthquakes M > 5.5 using global
seismic network, and created a catalog of earthquakes. The moment tensor information can be
downloaded from their web sites: USGS web-site (http://earthquake.usgs.gov/earthquakes) and GCMT
web-site (http://www.globalcmt.org). Using local seismic networks, many institutes estimated momenttensor solutions for earthquakes M > 3 and a created catalogs of earthquakes (e.g. Dreger andHelmberger, 1993; Pasyanos et al. 1996). In Japan, the National Research Institute for Earth Scienceand Disaster Prevention (NIED) has estimated the moment tensor solutions using a broadband seismicobservation network (F-net) since 1997 (Fukuyama et al., 1998).
Since the occurrence rate of middle-size earthquakes is higher than that of large-size earthquakes, it
is important to estimate moment tensor solutions of middle-size earthquakes for understanding stressfield and faulting systems in local regions. So far, broadband seismic networks have been establishedin developing countries, we can estimate moment tenser solutions for earthquakes M > 3.5 in thedeveloping countries and construct a catalog of moment tensor solutions. The program set of moment
tensor inversion and its lecture become important for seismologists in the developing countries.
In general, the effect of source time function has been neglected in predominant program sets of themoment tensor inversion with near source seismograms, for simplicity (e.g. Dreger and Helmberger,
1993; Ito et al., 2006). The neglect of the effect of seismic source duration should result in anunderestimation of the seismic moment. In this study, we constructed a formulation to estimate
moment tensor solutions based on a point source model and a simple source time function, and thenwe applied it to local seismic data for evaluating our program set. Through application of new
program set, we confirmed that the seismic moment is underestimated if we neglect the effect of thesource duration. Our program set is introduced in the lecture provided by the International Institute ofSeismology and Earthquake Engineering, the Building Research Institute.
University of Tsukuba

METHODOLOGY
In general, observed seismic waveform of c component at a station j due to seismic moment releasein a source volume V is given by6ucj ( t ) = ∑ ∫∫∫ G cjq ( t, ξ ) ∗ M q ( t, ξ ) dξ + ecj′ ( t ) ,v

q =1(1)

whereis the Green’s function of basis moment tensor (Kikuchi and Kanamori, 1991),
is a
spatio-temporal moment density function, and e’ is an observed error. Since the volume change during
earthquakes is too small to detect, we can neglect volume change component. To estimate stable
moment tensor solutions, we should assume a simple source model such as the point source model, inwhich we assume the seismic waveform to be radiated from one point. Following the point source
model, we rewrite eq. (1) as
5

ucj ( t ) = ∑ G cjq ( t, ξ c ) ∗ M q ( t ) + ecj ( t ) ,(2)M q ( t ) = ∫∫∫ M q ( t, ξ ) dξ ,(3)

q =1

with

V

where
is the location of the centroid, and e contains observed and modeling errors. We assumed
that the focal mechanism is kept constant during an earthquake, and approximated the shape of thesource time function to be an isosceles triangle with half duration of tr. Based on the assumption of theconstant focal mechanism and the simple source time function, we can rewrite eq. (2) as,(4)withGcjq ( t,t r , ξ c ) = G cjq ( t, ξ c ) ∗ T ( t,t r ) ,(5)where T is a source time function with half duration of tr. In general, we apply low-pass filter toobserved data for mitigating the aliasing effect in re-sampling procedure, the effect of the complicatedseismic source process and the effect of the heterogeneity of the local velocity structure. Therefore,relationship between the observed seismic waveform and data waveform is represented by,(6)

where B is the function of the low-pass filter. Substituting eq. (4) into (6), and then we rewrite in thefollowing simple vector form:,(7)

where d and e are N-dimensional data and error vectors, respectively; m is a 5-dimensional modelparameter vector; A is a N x 5 coefficient matrix. The solution of the above matrix equation is
obtained by least square approach if we assume the duration of the source time functionand thecentroid location. In the study, we assumed that horizontal location of centroid can beapproximated to the epicenter, and estimated optimal depth of the centroid and half duration using thegrid-search method, which minimizes normalized L2-norm as.

APPLICATION
We performed a moment tensor inversion of one large earthquake and two medium earthquakes for
examining the validity of our program set.
Data and the Green’s Function
Figure 1 shows the epicenters of three earthquakes and locations of stations, which are used in thisstudy. Two aftershocks of the 2003 Tokachi-oki great earthquake were selected for the evaluation of
our program set. We retrieved near source data observed by broadband seismograph F-NET stations,NIED. 9 components at 3 stations were used to estimate moment tensor solutions. The observed raw
data were corrected for seismometer responses and converted to ground velocity motions. The data
were filtered in the bandpass to mitigate the effect of the heterogeneity of the local velocity structure,and re-sampled with 1 Hz. Following the NIED moment tensor analysis (Fukuyama et al., 1998), we
selected filtering range 20 – 100 sec for events A and B, and 20 – 50 sec for an event C. We did not
integrate observed waveform to avoid the increase of off-diagram of the data covariance components(Yagi and Fukahata, 2008). Green’s function was calculated by the discrete wave number method
developed by Kohketsu (1985) with the simple J-B structure.

Fig. 1 Location map of F-net broadband stations and earthquakes used in our analysis. The triangles
and the stars represent the location of stations and the epicenters, respectively.
Results
Now, we applied our program to an aftershock (MJMA 7.1; origin time: 29 Sep 2003 11:37 JST) ofthe 2003 Tokachi-oki great earthquake, the 2004 Nemuro-oki earthquake (MJMA 7.1; origin time: 29

Nov 2004 03:32 JST) and its aftershock (MJMA 4.2; origin time: 30 Nov 2004 13:02 JST), which arelabeled event A, B and C, respectively. We approximate the horizontal location of centroid to be theepicenter determined by the Japan Meteorological Agency (JMA). Figure 2 shows the focal
mechanism and the comparison between the observed waveform (black line) and the synthetic
waveform (gray line) for each event. As can be seen from the focal mechanisms, three earthquakes are
typical thrust earthquakes along plate boundary. The estimated focal mechanisms, depths and seismic
moments of three events are consistent with the F-NET MT solutions.

Fig. 2 Results of moment tensor inversion for the event A (a), the event B (b) and the event C (c). For
each case, the focal mechanism (top left-hand graph), the information of source parameters (topright-hand graph), and observed and synthetic waveforms (black lines and gray lines, respectively;bottom graph) are shown. The numbers below the station code indicate the maximum amplitude (inunit of cm/sec).
Table 1 shows the depth of centroid, the seismic moment, the half duration of source time function,and the normalized L2 norm for each event with our formulation and a conventional formulation thatneglects the effect of source duration. As can be seen from the values of the normalized L2 norm intable 1, the waveform fitting is improved by our formulation. It should be noted that the seismic
moment estimated by our formulation is always larger than that estimated by the conventional
formulation. This gap should increase with the seismic moment. This result shows that the neglect ofsource duration is inappropriate assumption for analysis of large earthquakes, because the duration of

the source time function increases with the seismic moment. In fact, moment magnitude (Mw) of the2003 Tokachi-oki earthquake estimated by NIED (Ito et al., 2004) is 7.9, which is meaningfully
smaller than Mw 8.1 estimated by seismic source analysis (e.g. Yagi, 2004). The estimated half
duration of event C is 4.0 sec that is somewhat large duration for M4.2 earthquake. Since thenormalized L2 norm in the low frequency waveform is insensitive to value of the half duration, thevalue of half duration tends to have large estimation errors.
Table 1 Comparison between the source parameters obtained by our formulation and the conventional
formulation.
(a) Event A

Our formulation
Conventional formulation

Seismic
moment (Nm)
7.63 x 1018
5.54 x 1018

Depth (km)Seismic
moment (Nm)
4.05 x 1019
3.18 x 1019

Depth (km)Seismic
moment (Nm)
2.75 x 1015
2.46 x 1015

Depth (km)3535

Half duration
(sec)
6.0
N/A

Normalized L2
norm
0.113
0.140

Half duration
(sec)
6.0
N/A

Normalized L2
norm
0.129
0.139

Half duration
(sec)
4.0
N/A

Normalized L2
norm
0.102
0.104

(b) Event B
Our formulation
Conventional formulation

5050(c) Event C
Our formulation
Conventional formulation

4040

CONCLUSIONS
We constructed a program set for estimating moment tensor solutions using near source
seismograms, which will be introduced in the lecture provided by the International Institute ofSeismology and Earthquake Engineering, the Building Research Institute. We applied our program set
to real data, and confirmed that the obtained moment tensor solutions for large earthquakes are wellconsistent with the F-NET MT solutions. We found that the neglect of the effect of seismic source
duration will result in an underestimation of the seismic moment, since the neglect of seismic source
duration is inappropriate assumption for analysis of large earthquakes
ACKNOWLEDGMENT
The seismic waves used in the present study are from F-NET stations provided by NIED. The
epicenter information used in the present study is from JMA. We thank anonymous reviewer forvaluable comments that improved the manuscript. This work was supported by Science andTechnology Research Partnership for Sustainable Development (SATREPS) from Japan Science andTechnology Agency (JST) and Japan International Cooperation Agency (JICA).
REFERENCE
Dreger, D. S., and D. V. Helmberger, 1993, Determination of source parameters at regional distanceswith single station or sparse network data, J. Geophys. Res., 98, 8107-8125.

Dziewonski, A. M., T. A. Chou and J. H. Woodhouse, 1981. Determination of earthquake source
parameters from waveform data for studies of global and regional seismicity, J. Geophys. Res., 86,2825–2852.
Fukuyama, E., M. Ishida, D. S. Dreger and H. Kawai, 1998, Automated seismic moment tensor
determination by using on-line broadband seismic waveforms, Zisin 2, 51, 149-156.
Ito, Y., H. Matsubayashi, H. Kimura, T. Matsumoto, Y. Asano and S. Sekiguchi, 2004, Spatial
distribution for moment tensor solutions of the 2003 Tokachi-oki earthquake(MJMA=8.0) andaftershocks, Earth Planets Space, 56, 301-306.
Ito, Y., S. Sekiguchi, T. Okada, R. Honda, K. Obara and S. Hori, 2006, Performance of regional
distance centroid moment tensor inversion applied to the 2004 mid-Niigata prefecture earthquake,Japan, Geophys. J. Int., 167, 1317-1331.
Kawakatsu, H., 1995, Automated near‐realtime CMT inversion, Geophys. Res. Lett., 22, 2569–2572,
doi:10.1029/95GL02341.
Kikuchi, M. and H. Kanamori, 1991. Inversion of complex body wave-III, Bull. Seism. Soc. Am., 81,2335-2350.
Kohketsu, K., 1985, The extended reflectivity method for synthetic near-field seis- mograms, J. Phys.
Earth, 33, 121–131.
Pasyanos, M. E., D. S. Dreger, and B. Romanowicz, 1996, Towards real-time determination ofregional moment tensors, Bull. Seism. Soc. Am., 86, 1255-1269.
Yagi, Y., 2004, Source rupture process of the 2003 Tokachi-oki earthquake determined by joint
inversion of teleseismic body wave and strong ground motion data, Earth Planet and Space, 56,311-316.
Yagi, Y. and Y. Fukahata, 2008, Importance of covariance components in inversion analyses ofdensely sampled observed data: an application to waveform data inversion for seismic source
processes, Geophys. J. Int., 175, 215-221.