Citation: Mule, S., D. Suetsugu, T. Hara (2022), Determination of moment tensor solution in the Fiji region
using the waveform inversion technique, Synopsis of IISEE-GRIPS Master’s Thesis.

DETERMINATION OF MOMENT TENSORSOLUTION IN THE FIJI REGION USING THE
WAVEFORM INVERSION TECHNIQUE
Saula Mule1
MEE21704

Supervisors: Daisuke SUETSUGU2, Tatsuhiko HARA3

ABSTRACT
We conducted moment tensor determination for 27 events that occurred in the North Fiji Basin from2015 – 2022 with a magnitude of Mw > 4.0 and depths less than 350 km. We used seismograms fromseismograph stations in Fiji and the neighboring countries. We used a regional velocity model with a
10-km thick crust by Xu and Wiens (1997) to compute Green’s functions. Focal mechanism solutions
were then validated in reference to the Global Centroid Moment Tensor (GCMT) solutions. We foundthat the solutions obtained by the present study are generally consistent with the GCMT solutions interms of focal mechanism and moment magnitude. Stress axes of the focal mechanisms are consistentwith tectonic process reported by previous studies. Focal depths were shallower than the GCMT centroid
depths by 1.4 km in average. We also evaluated an effect of velocity model on focal mechanism solutions
by comparing waveform fitting and similarity to GCMT. We performed the moment tensor inversion
for events with Mw > 5.0 using the original regional model, its modified model with a 20-km thick crust,and a global standard model. For most of the events used for comparison, the waveform fitting is bestfor the original regional model with a 10-km thick crust than the other two velocity models. We
compared focal mechanisms determined from data including and excluding stations from theneighboring countries. The comparison shows that the mechanisms are closer to the GCMT by including
data from neighboring countries, suggesting the importance of international data exchange.
Keywords: Moment Tensor, Fiji, velocity model.
1. INTRODUCTION
Fiji is in the southwest Pacific, an archipelago consisting of 332 islands, of which only 106 are inhabited;
the total land size is 18300km2. Fiji sits on the converging zone of the Pacific Plate and Indo-AustralianPlate. It is a remnant arc in a broad, complex, highly deformed area between the two opposing
subduction zones. Although most of the Fiji Islands are not in the vicinity of a trench of subduction
zones, it is still tectonically active due to the movement of the major plate boundaries. It is subject to an
impact from intraplate seismicity and subduction zone seismicity. The present study highlights a criticalarea needed for the Fiji seismology unit in terms of disaster risk reduction: The determination of moment
tensors in and around the Fiji region. The Mineral Resources department of Fiji established its seismicnetwork in 1979, but the earthquake focal mechanisms were not evaluated routinely. The main objectiveof this study is to determine the focal mechanisms around the Fiji region and to determine the lower
limit of an earthquake whose moment tensors can be determined reliably. The present study should
contribute to resolving this vital earthquake parameter using the Fiji seismic stations on a routine basis
in the future.
2. DATA
1

Mineral Resources Department, Fiji.
Japan Agency for Marine-Earth Science and Technology (JAMSTEC).
3International Institute of Seismology and Earthquake Engineering, Building Research Institute.
21

We analyzed seismograms recorded at seismic networks of Fiji, New Caledonia, Geoscope,and Global Seismograph Network (GSN) in the studied region. The data were retrieved from the Data
Center of the Oceania Regional Seismic Network (ORSNET). The target earthquakes are ones in theFiji region with a magnitude less than 6 with a depth ranging from 0 to 350 km. Broadband sensors areused for moment tensor inversion. Moment tensor determination was performed on 27 earthquakes (M< 6) from 2015 to 2019 and in 2022 (from May to June) as illustrated in Figure 1a, which are in theNorth Fiji Basin. We used hypocenter parameters determined by United States Geological Services(USGS) and ORSNET.
a)b)

FijiNew Caledonia

Figure 1. Seismicity(a) and station distribution(b) used in this study. Theyellow triangles are the Fiji network stations and the red triangles are from the
Global Seismic Network (GSN) and New Caledonia Seismic Network.

Most of the earthquakes in the North Fiji Basin analyzed in the present study are shallow
in depth. Regarding disaster mitigation, shallow events are more hazardous than deeper earthquakes
because of the risk of tsunami and strong ground motion.
3. METHODOLOGY
3.1. Theory of moment tensor
The simplified representation of a seismic source with moment tensors, either the spatial or thetemporal point source, is shown by the formula below.
𝑈𝑛 (𝑥, 𝑡) = 𝑀𝑖𝑗 . 𝐺𝑛𝑖,𝑗 (𝑥, 𝑧, 𝑡),(1)where Un is the observed n-th component of displacement, Gni,j is the nth component of Green’s function
for specific force-couple orientations and Mij is the ij-th component of a seismic moment tensor, whereindices i and j are the geographical directions. The above equation is solved using the linear least squaremethod for a specified depth, where moment tensors are the only parameters to be determined. The
moment tensors are decomposed into double couple (DC) and compensated linear vector dipole (CLVD)
components. The shear faulting has only DC component, which will be used as one of indices ofsolution’s reliability.
Expressing Eq. (1) in vector and matrix form, we have2

𝑈 = 𝐺𝑀,(2)

where U is the matrix of the observed ground motions, G is the matrix of Green’s functions and M is thevector of the moment tensor component. The Green’s function is computed using the wavenumber
integration method (Saikia, 1994) and depends on the seismic structure model (P and S velocities,density, Qp and Qs), focal depth, azimuth, and distance between hypocenter and station. We used avelocity model for the North Fiji Basin (Figure 4), which was determined from surface wave analysis
by Xu and Wiens (1997), which has a 10-km thick crust. The density and quality factors are taken fromthe PREM model (Dziewonski & Anderson, 1981). We also used two other velocity structures, one isthe iasp91 model (Engdahl et al., 1991) and the other is a modified Xu and Wiens model with 20-km
thick crust. These two velocity models are used to know the effects of a velocity model on a momenttensor solution. The moment tensor is obtained by applying a linear least square method
𝑀 = (𝐺 𝑇 𝐺)−1 𝐺 𝑇 𝑈,

(3)

Regarding the focal depth determination, we assume that the greater variance reduction isachieved for the best reliable depth, where a variance reduction (VR) is defined by the followingformula.

𝑉𝑅 = 1 − Σ𝑖[√(𝑑𝑎𝑡𝑎𝑖 − 𝑠𝑦𝑛𝑡ℎ𝑖 )2√𝑑𝑎𝑡𝑎𝑖2∗ 100,(4)]where synth and data represent synthetic and observed seismograms, respectively. The synthetic
seismogram is computed by a multiplication of the moment tensor and Green’s function. This
computation is made for all stations. The greater VR means the better waveform fitting, which will beused as an index of solution’s reliability.
3.2. Computer codes
We used the code TDMT_INVC (time-domain moment tensor inversion code) developed by Dreger(2003). The code is implemented on SEISAN (earthquake analysis software) (Havskov and Ottemoller,2020), which is widely used in the seismological community. This program can be used to compute
several earthquake parameters. This tool also contains different codes to enable us to analyze
earthquakes for research purposes.
4. RESULTS AND DISCUSSION
We obtained moment tensor solutions for 27 earthquakes in the North Fiji Basin. The magnitude range
is from 4.0 to 5.8 and the depth range is from 0 to 350km. We examined the quality of the obtained
focal mechanisms based on the similarity to the corresponding solutions in the Global Centroid
Moment Tensor (GCMT) catalog (http://www.globalcmt.org/; Ekström et al., 2012) in case solutions
are available in the catalog. The similarities were represented by the rotation angle calculated using
the method developed by Kagan (2007), which is also used as an index of solution’s reliability. We
choose 55 ◦ for the threshold of 'similarity of focal mechanisms.’ Among the 27 events (Table 1), thesolutions for 12 events are available in the GCMT catalog. The rotation angles are less than 55◦ for allof the 12 solutions (Figure 2a), which is regarded as reliable (Group A).
For the other 15 solutions, we classified ‘reliable’ for focal mechanism solutions with DC
components greater than 70%, VR greater than 35%. Among the 15 events with GCMT unavailable,
9 events are classified as reliable (Group B) (Figure 2b) (Table 1). In summary, 21 of a total of 27

3

solutions were reliably determined, which is encouraging for reliable determination of focal
mechanisms and moment magnitude in the future, since GCMT solutions are not always available.
The rest of the solutions are regarded as unreliable and classified as Group C (Table 1).
Table 1. Categorization of results in magnitude ranges see definition in text for Group A, B, and C.
GroupABCTotala)

2.9 ≤ Mw ≤ 4.5136104.6 ≤ Mw ≤ 4.93407Mw ≥ 5.082010b)

Figure 2. Distribution of Group A (4.5 ≤ Mw ≤ 5. 9) focal mechanism andGroup B (4.1 ≤ Mw ≤ 5. 2) solutions. (a) Group A Focal mechanisms solution,(b) Group B solutions. The color of the mechanism represents the focal depth.
a)b)Figure3. a) Comparison of Mw determined by this study and those from the GCMT catalog. b)
Comparison of the focal depth determined in this study and those from the GCMT catalog.
We compared moment magnitude (Mw) and the Mw of GCMT as shown in Figure 3a.
Comparing the moment magnitude determined in this study and the Mw determined by GCMT, we
obtained a root mean square of the difference between the two Mw to be 0.26, which indicates that theMw in our solutions is consistent with that of GCMT. We also compared the focal depth of our solutions(Figure 3b) to GCMT focal depth. The depths obtained in this study are shallower compared to GCMT,

4

with an average difference of 1.4km. The difference could be due to the difference in velocity structureused in this study, which is more specific to the region, and the stations used for moment tensor inversion.
The minor differences from GCMT are encouraging from the viewpoint of disaster mitigation since
the precise determination of moment magnitude and the focal depth is directly related to tsunami
potential.
Figure 4 illustrates tensional axes of 19 reliable events with shallower than 50 km (GroupsA and B). The red rectangle in Figure 4 is the strike-slip zone due to the movement along the Fiji
fracture zone. The tensional stress in the green
rectangle depicts normal fault earthquakes withWNW-ESE tension axes, which are associatedwith spreading motion along the NNE-SSWridge. The North Fiji Basin has spreading
ridges, which could be clearly identified by thedirection of the tensional stress.
We examined the effect of thevelocity model on moment tensor solutions
using three different velocity models, asmentioned in section 3.1. We selected six wellconstrained solutions with relatively large VR
and small Kagan angles (Figure 5). The VR for
the Xu and Wiens model is larger than thoseobtained from the other models for five events(Figure5). For event 4, the variance reduction isFigure 4. Map of tensional axes. Events areshallower than 50 km. The white rectangles comparable. We also compared the rotation
depict the direction of the stress on the focal angle and found that the original Xu and Wiensmodel with a 10-km thick crust is better than themechanism determined by this study.
other models in representing a velocity structureof the Fiji region. The other models, particularly the iasp91 model, give different focal mechanisms
from GCMT. Since Green’s function strongly depends on the velocity model, it is vital to use theappropriate velocity model for reliable determination of focal mechanism and moment magnitude.
a)b)

Figure. 5. Effects of velocity model. Comparison of VR(a) and Rotation angle(b)for three velocity
models of Xu and Wiens (1997) with a 10km-thick crust (green circle), Xu and Wiens (1997) withcrustal thickness modified to 20 km (red triangle), and the iasp91 model (blue square) are shown.
We examined the results of the moment tensor inversion concerning the station coverageof the inversion. We show the effects of station coverage by comparing moment magnitude and rotation
angle determined from stations including neighboring countries (New Caledonia and Futuna). The resultshows that the moment tensor inversion using stations including neighboring countries provides Mw
and the focal mechanism consistent with GCMT. It may be because neighboring stations improved thenumber of stations and azimuthal coverage from the events. It is essential to make international dataexchange for reliable determination of moment tensors.

55. CONCLUSIONSIn this study, we determined moment tensor solutions for 27 events in the Fiji region using the seismic
networks in Fiji and neighboring countries. For 12 events for which GCMT solutions are available, wecompared our solutions with the corresponding GCMT solutions. Mw of 4.6 is a lower-limit ofmagnitude whose moment tensor solutions could be reliably determined from stations in Fiji andnonboring countries. Moment magnitudes are determined within the difference of 0.3 from the moment
magnitude of GCMT. The focal depths of the present study are shallower than those of GCMT by 1.4
km on average. Among 15 events for which no GCMT solutions were available, we regarded 9 eventsas reliable from a degree of waveform fitting and consistency of focal mechanism with GCMT for events
nearby. For 21 of the 27 events, focal mechanism and moment magnitude are considered reliable. Eventsnear the Fiji fracture zone have a left-lateral strike-slip mechanism. Normal fault mechanisms wereobtained in the west of Fiji islands, which suggests active spreading ridges in the region.
We examined effect of velocity models on resultant moment tensors. Different velocity
model resulted in substantially different moment tensors. We conclude that the Xu and Wiens (1997)
model with a 10-km thick crust provides better solutions of moment tensors as compared with athickened crust model and a global standard model in terms of waveform fitting and similarity to GCMT.
We also examined the effects of station coverage on moment tensor determination and show
the importance of stations in neighboring countries along with Fiji stations for reliable determination ofmoment tensors.
The present study is encouraging for reliable determination of focal mechanism and moment
magnitude using local and regional data. Understanding earthquake mechanisms is a fundamental stepneeded for earthquake disaster mitigation. The compilation of moment tensor solutions for the Fijiregion will be the first step in creating a seismotectonic model for the Fiji region.
ACKNOWLEDGEMENTS
This research was conducted during the individual study period of the training course “Seismology,Earthquake Engineering and Tsunami Disaster Mitigation” by the Building Research Institute, JICA,and GRIPS. I express my sincere gratitude to my supervisors, Dr. Daisuke Suetsugu and Dr. TatsuhikoHara, for their valuable support in successfully completing my research. I would also like to thank Dr.
Saeko Kita for her comments on this study. I would also like to acknowledge Mr. Angga Wijaya for hishelp in preparing the response file. We used ORSNET data. We used CMT solutions from the Global
CMT project. This study used SEISAN, TDMT_INV, and Generic Mapping Tools (Wessel and Smith,2013). The topography and bathymetry are from GEBCO Compilation Group (2022).
REFERENCES
Dreger, D.S., 2003, 85.11 In International Geophysics (Vol. 81, Issue PART B), 1627.
Dziewonski, A.M., and Anderson, D.L., 1981. Physics of the Earth and Planetary Interiors, 25, 297–356.
Ekström, G., Nettles, M., and Dziewoński, A.M, 2012, Physics of the Earth and Planetary Interiors,
200–201, 1-9.
GEBCO Compilation Group (2022) GEBCO 2022 Grid (doi:10.5285/e0f0bb80-ab44-2739-e0536c86abc0289c).
Havskov, J., Voss, P.H., and Ottemöller, L, 2020, Seismological Research Letters, 91, 1846-1852.
Kagan, Y.Y., 2007, Geophysical Journal International, 171, 411-418.
Kennett, B.L.N., and Engdahl, E.R., 1991, Geophysical Journal International, 105, 429-465.
Wessel, P., Smith, W. H. F., Scharroo, R., Luis, J., and Wobbe, F., 2013, EOS Trans. AGU, 94, 409–410.
Xu, Y., and Wiens, D. A., 1997, Journal of Geophysical Research, 102, 27439–27451.

6